From meyer@imap.theory.csail.mit.edu Mon Nov 28 14:37:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jASJbv5o004382 for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 14:37:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jASJbtDC025710 for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 14:37:55 -0500 (EST) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-FIFTY-TWO.MIT.EDU [18.235.1.152]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jASJbqba027382 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 14:37:53 -0500 (EST) Message-Id: <6.2.1.2.2.20051128140633.03e5fa00@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Mon, 28 Nov 2005 14:38:02 -0500 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 12 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 355 X-IMAPbase: 1126748715 2877 NonJunk $Label4 $Label1 $Label2 $Label3 $Label5 Junk $MDNSent $Forwarded NotJunk X-UID: 1884 X-Status: A X-Keywords: NotJunk In section 4.3 of the Course Notes on Probability, disjointness is mentioned in comparison to independence. I understand that mathematically, when two events are disjoint, Pr{AintB} = 0 as opposed to Pr{AintB} = Pr{A} x Pr{B}, which applies for independence. However, in terms of events, how could this be described? Thanks, Praveen Pamidimukkala From meyer@imap.theory.csail.mit.edu Mon Nov 28 18:12:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jASNCW5o003459 for <6042-probs@theory.lcs.mit.edu>; Mon, 28 Nov 2005 18:12:32 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jASNCV3o018516 for <6042-probs@theory.lcs.mit.edu>; Mon, 28 Nov 2005 18:12:31 -0500 (EST) Received: from hzhou.mit.edu (BURTON-SIX-TWENTY-FOUR.MIT.EDU [18.247.7.113]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jASNCCpc009185 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 28 Nov 2005 18:12:24 -0500 (EST) Message-Id: <5.2.1.1.2.20051128181100.02b043b8@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 28 Nov 2005 18:12:15 -0500 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: David-tp13 reading comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 144 X-UID: 1885 X-Keywords: NonJunk NotJunk Hey, Reading and lecture were clear. I found the ability to get a percentage confidence for sampling data to be interesting. - Steven Zhou From meyer@imap.theory.csail.mit.edu Mon Nov 28 19:19:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT0Jf5o021596 for <6042-probs@theory.csail.MIT.EDU>; Mon, 28 Nov 2005 19:19:41 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT0Je3o003544 for <6042-probs@theory.csail.MIT.EDU>; Mon, 28 Nov 2005 19:19:40 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAT0JbOW024603 for <6042-probs@theory.csail.MIT.EDU>; Mon, 28 Nov 2005 19:19:38 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAT0Jb5R027958; Mon, 28 Nov 2005 19:19:37 -0500 Received: from ab6.draper.com (ab6.draper.com [192.80.95.243]) (User authenticated as icharny@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 28 Nov 2005 19:19:37 -0500 Message-ID: <20051128191937.pl45v83e6tdwgcgk@webmail.mit.edu> Date: Mon, 28 Nov 2005 19:19:37 -0500 From: Isaac Charny To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Week 13 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 121 X-UID: 1886 X-Keywords: NonJunk NotJunk I don't understand binomial distribution (section 3.4, page 11). Please discuss this in the next lecture. ~Isaac Charny From meyer@imap.theory.csail.mit.edu Mon Nov 28 19:51:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT0pe5o001999 for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 19:51:40 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT0pJ3o023281 for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 19:51:20 -0500 (EST) Received: from [18.234.1.61] (DP-SIXTY-ONE.MIT.EDU [18.234.1.61]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAT0pCVM001023 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 19:51:13 -0500 (EST) Message-ID: <438BA605.10505@mit.edu> Date: Mon, 28 Nov 2005 19:51:17 -0500 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] week 13 reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 612 X-UID: 1887 X-Keywords: NonJunk NotJunk The section on polling was very confusing, I haven't quite understood it yet. I would like this to be explained better in class. There are a lot of variables and equations and it's hard to follow. I forgot what the goal of all this was by the fourth paragraph, and had to reread it a couple of times to understand why each equation brought us closer to what we wanted. Also, in the tutor problems, there was a question that included the concept of variance, but this was not mentioned in the notes (or at least I couldn't find it). This made it a lot more difficult to solve the problem. Adriana From meyer@imap.theory.csail.mit.edu Mon Nov 28 22:02:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT32r5o029585 for <6042-probs@theory.csail.MIT.EDU>; Mon, 28 Nov 2005 22:02:53 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT32paG014663 for <6042-probs@theory.csail.MIT.EDU>; Mon, 28 Nov 2005 22:02:52 -0500 (EST) Received: from w20-575-41.mit.edu (W20-575-41.MIT.EDU [18.187.0.60]) (authenticated bits=56) (User authenticated as cwong08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAT32jfL028697 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Mon, 28 Nov 2005 22:02:46 -0500 (EST) Received: (from cwong08@localhost) by w20-575-41.mit.edu (8.12.9) id jAT32jO5021918; Mon, 28 Nov 2005 22:02:45 -0500 Subject: [Sayan] Week 13 Comments From: Christopher Wong To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 28 Nov 2005 22:02:45 -0500 Message-Id: <1133233365.21884.1.camel@w20-575-41.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 74 X-UID: 1888 X-Keywords: NotJunk I would like the approximation methods of page 13 explained more clearly. From meyer@imap.theory.csail.mit.edu Mon Nov 28 23:02:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT42H5o025524 for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 23:02:17 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT3wFdI021982 for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 23:02:16 -0500 (EST) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAT3vw9K014618 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 28 Nov 2005 22:57:58 -0500 (EST) Message-ID: <000201c5f499$15955910$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 13 Comments Date: Mon, 28 Nov 2005 22:57:56 -0500 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 336 X-UID: 1889 X-Keywords: NonJunk NotJunk Will we be going over those horrible equations for confidence in class? They are still pretty confusing even in the notes. Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From meyer@imap.theory.csail.mit.edu Tue Nov 29 01:01:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT61A5o018236 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 01:01:10 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT618bP004349 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 01:01:09 -0500 (EST) Received: from christope8f0c6 (MACGREGOR-ONE-TWENTY-FOUR.MIT.EDU [18.239.5.124]) (authenticated bits=0) (User authenticated as yangc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAT6151v008182 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 01:01:06 -0500 (EST) Message-Id: <200511290601.jAT6151v008182@outgoing.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 13 Comments Date: Tue, 29 Nov 2005 01:03:06 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_002F_01C5F480.A8A39050" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcX0qpELJrydoAhpS0me67xMhEI4gg== X-Spam-Score: 0.179 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2330 X-UID: 1890 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_002F_01C5F480.A8A39050 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable For the derivation of the confidence interval stuff, why can we assume p = =3D =BD to continue the calculations for delta? Is it because delta is like the upper bound on the error, and not the error itself? =20 Thanks, Chris Yang=20 ------=_NextPart_000_002F_01C5F480.A8A39050 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

For the derivation of the confidence interval stuff, = why can we assume p =3D =BD to continue the calculations for delta?=A0 Is it = because delta is like the upper bound on the error, and not the error = itself?

 

Thanks,

Chris Yang

------=_NextPart_000_002F_01C5F480.A8A39050-- From meyer@imap.theory.csail.mit.edu Tue Nov 29 01:47:52 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT6lq5o026081 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 01:47:52 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAT6lp8B027797 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 01:47:51 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT6lpi4026773 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 01:47:51 -0500 (EST) Received: from [18.241.6.6] (NEW-TWO-SIXTY-ONE.MIT.EDU [18.241.6.6]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAT6lhok023583 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 01:47:44 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <59496C9D-DE34-4E43-9C8F-AFB9E97D5BD5@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: {Sayan} Weekly Email Comment Date: Tue, 29 Nov 2005 01:57:19 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 205 X-UID: 1891 X-Keywords: NonJunk NotJunk Hi, I didnt quite understand the section on polling and would like to have it gone over in class in more detail, especially the part about the confidence interval and its derivation. Thanks. Hamidou From meyer@imap.theory.csail.mit.edu Tue Nov 29 02:19:04 2005 Message-ID: <438C00E8.1040103@csail.mit.edu> Date: Tue, 29 Nov 2005 02:19:04 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Adriana Lopez CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Jelani] week 13 reading References: <438BA605.10505@mit.edu> In-Reply-To: <438BA605.10505@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 808 X-UID: 1892 X-Keywords: Variance crept in by accident; will be covered next week. I've revised the tutor problem to indicate this. Thanks for pointing this out. regards, A. Adriana Lopez wrote: > The section on polling was very confusing, I haven't quite understood > it yet. I would like this to be explained better in class. There are > a lot of variables and equations and it's hard to follow. I forgot > what the goal of all this was by the fourth paragraph, and had to > reread it a couple of times to understand why each equation brought us > closer to what we wanted. > Also, in the tutor problems, there was a question that included the > concept of variance, but this was not mentioned in the notes (or at > least I couldn't find it). This made it a lot more difficult to solve > the problem. > Adriana From meyer@imap.theory.csail.mit.edu Tue Nov 29 02:28:27 2005 Message-ID: <438C031A.4040302@csail.mit.edu> Date: Tue, 29 Nov 2005 02:28:26 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Sayan] Week 12 Comments References: <6.2.1.2.2.20051128140633.03e5fa00@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051128140633.03e5fa00@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 640 Status: RO X-UID: 1893 X-Keywords: Are you asking what independent events "look like"? Best answer I can give to that is the picture in section 4.3. By the way, happy to have you reviewing Notes 12, but this week's email is supposed to be about Notes 13. regards, A. Praveen Pamidimukkala wrote: > In section 4.3 of the Course Notes on Probability, disjointness is > mentioned in comparison to independence. I understand that > mathematically, when two events are disjoint, Pr{AintB} = 0 as opposed > to Pr{AintB} = Pr{A} x Pr{B}, which applies for independence. > However, in terms of events, how could this be described? > > Thanks, > Praveen Pamidimukkala > From ksindi@MIT.EDU Tue Nov 29 02:36:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT7ae5o008229 for <6042-probs@theory.csail.MIT.EDU>; Tue, 29 Nov 2005 02:36:40 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT7adnA015227 for <6042-probs@theory.csail.MIT.EDU>; Tue, 29 Nov 2005 02:36:39 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAT7aVID017167 for <6042-probs@theory.csail.MIT.EDU>; Tue, 29 Nov 2005 02:36:31 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAT7aV14015847; Tue, 29 Nov 2005 02:36:31 -0500 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 29 Nov 2005 02:36:31 -0500 Message-ID: <20051129023631.1lr183qjbe4ogwwo@webmail.mit.edu> Date: Tue, 29 Nov 2005 02:36:31 -0500 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [DAVID] Week 13 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1894 Content-Length: 208 X-Keywords: NonJunk NotJunk I didn't like the binomial sampling part. Maybe we could list the assumptions on it made in class. I also still don't understand what confidence levels are. I found the random varaibles bit to be very clear. From meyer@imap.theory.csail.mit.edu Tue Nov 29 01:17:29 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAT6HR5o019547 for <6042-staff@theory.csail.mit.edu>; Tue, 29 Nov 2005 01:17:27 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAT6HQ8B014631 for <6042-staff@theory.csail.mit.edu>; Tue, 29 Nov 2005 01:17:27 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAT6HQKk026034; Tue, 29 Nov 2005 01:17:26 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAT6HNok022508; Tue, 29 Nov 2005 01:17:23 -0500 (EST) Message-ID: <438BF263.4060305@mit.edu> Date: Tue, 29 Nov 2005 01:17:07 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kevin Wang CC: 6042-staff@theory.csail.mit.edu Subject: Re: TP 13.4 typo References: <6.2.3.4.2.20051128235648.029d64e0@po10.mit.edu> In-Reply-To: <6.2.3.4.2.20051128235648.029d64e0@po10.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.263 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,RCVD_IN_SORBS autolearn=no version=2.63 Status: RO Content-Length: 411 X-UID: 1895 X-Keywords: NotJunk Indeed, you are correct. Thanks for catching this. The TP has been adjusted accordingly. Let me know if you need to redo this problem. DS Kevin Wang wrote: > Hi, > > I think 8. has a typo: "The probability that the second voter chosen > will favor Gore, given that the *first *voter chosen is from the same > state as the first, may not equal /p/." > > It should be *second* right? > > Thanks, > Kevin From lye@MIT.EDU Tue Nov 29 11:25:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jATGPP5o011894 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 11:25:25 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jATGPNWJ018113 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 11:25:24 -0500 (EST) Received: from m66-080-5.mit.edu (M66-080-5.MIT.EDU [18.63.1.5]) (authenticated bits=56) (User authenticated as lye@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jATGPG5m027736 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 11:25:17 -0500 (EST) Received: (from lye@localhost) by m66-080-5.mit.edu (8.12.9) id jATGPGbh018478; Tue, 29 Nov 2005 11:25:16 -0500 (EST) Subject: [Sayan] Week 13 Comments From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 29 Nov 2005 11:25:16 -0500 Message-Id: <1133281516.18412.1.camel@m66-080-5.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1896 Content-Length: 188 X-Keywords: NonJunk NotJunk 3.3 The Numbers Game This was especially counterintuitive, and thus difficult. It seems strange that by guessing (assuming information you do not know), you would improve your chances. From pgroudas@MIT.EDU Tue Nov 29 16:11:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jATLBY5o000517 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 16:11:34 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jATLBWaX014976 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 16:11:32 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jATLBUHU013533 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 16:11:30 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jATLBUYH032127; Tue, 29 Nov 2005 16:11:30 -0500 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 29 Nov 2005 16:11:30 -0500 Message-ID: <20051129161130.qlhdifxhqzi84c00@webmail.mit.edu> Date: Tue, 29 Nov 2005 16:11:30 -0500 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: [HANSON] Week 13 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1897 Content-Length: 252 X-Keywords: NotJunk I found it difficult to really absorb everything past page 14, it just seemed that once it got to polling it started throwing so much math around that it was too much to get all at once. Hopefully this will all get cleared up in class. -Paul Groudas From nedzel@MIT.EDU Tue Nov 29 18:44:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jATNiT5o021409 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 18:44:29 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jATNiRRS017010 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 18:44:28 -0500 (EST) Received: from [18.96.7.69] (SIMMONS-FIVE-EIGHTY.MIT.EDU [18.96.7.69]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jATNiMlw007551 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 18:44:26 -0500 (EST) Message-ID: <438CE7F6.8000506@mit.edu> Date: Tue, 29 Nov 2005 18:44:54 -0500 From: "David A. Nedzel" User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 13 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1898 Content-Length: 195 X-Keywords: NonJunk NotJunk I'm fairly comfortable with this material, I've seen most of it in 6.041. It might be nice to have a refresher during lecture on the expectation & variance of different distributions. - David From brevzin@MIT.EDU Tue Nov 29 20:13:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU1Dv5o002952 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 20:13:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU1Dtr1020223 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 20:13:56 -0500 (EST) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU1DmIs028650 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 20:13:49 -0500 (EST) Message-Id: <6.1.2.0.1.20051129201038.019f4980@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Tue, 29 Nov 2005 20:13:52 -0500 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1899 Content-Length: 446 X-Status: A X-Keywords: NotJunk You guys go over a lot of definitions for expectation, but what about the more general one ? Let f: X --> R+ be a random variable, the expection is integral over X of f du, where u (mu) is the measure. Also, this entire section will be extremely weird since in 18.103 (Fourier Analysis, Measure Theory) we learn the measure theory side which has its own explanations and descriptions... and symbols. The overlap will be confusing. Barry From mwangi@MIT.EDU Tue Nov 29 20:26:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU1QC5o007957 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 20:26:12 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU1QAr1029479; Tue, 29 Nov 2005 20:26:10 -0500 (EST) Received: from IBM-1E80C73FC15.mit.edu (NEW-THREE-NINETY-NINE.MIT.EDU [18.241.6.144]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU1Q8pK002305 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 29 Nov 2005 20:26:09 -0500 (EST) Message-Id: <6.2.1.2.2.20051129201602.03ecde20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Tue, 29 Nov 2005 20:26:06 -0500 To: 6042-probs@theory.lcs.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 13 Comments Cc: mitras@MIT.EDU Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1900 Content-Length: 291 X-Keywords: NonJunk NotJunk Passage: 2 The Birthday Principle Page: 4 I found this passage most surprising because of the large probability of two people in a room with n people having the same birthday given that there are d days in a year. It seemed very counter-intuitive. Sincerely, Timothy M. Mwangi From bens@MIT.EDU Tue Nov 29 21:02:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU22i5o021065 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 21:02:44 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU22gr1023196 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 21:02:43 -0500 (EST) Received: from [18.221.0.117] ([18.221.0.117]) (authenticated bits=0) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU22aGc010316 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 21:02:36 -0500 (EST) Message-ID: <438D0827.6070508@mit.edu> Date: Tue, 29 Nov 2005 21:02:15 -0500 From: "Benjamin M. Schwartz" User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051014) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1901 Content-Length: 211 X-Keywords: NotJunk This number-picking game is very cool. I'm surprised that the best strategy isn't just guess that the numbers span 50. I guess it has to do with the envelope-giver guessing your strategy and working around it. From jehan@MIT.EDU Tue Nov 29 21:11:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU2B15o021547 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 21:11:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU2B0r1027936 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 21:11:00 -0500 (EST) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU2Ar6B011984 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 21:10:54 -0500 (EST) Message-Id: <5.2.1.1.2.20051129210818.00bc2650@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 29 Nov 2005 21:13:03 -0500 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 13 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1902 Content-Length: 79 X-Keywords: NonJunk NotJunk I thought the mean time to failure thing was pretty cool. no questions jehan From aston@MIT.EDU Tue Nov 29 22:02:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU32T5o005619 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 22:02:29 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU32Sr1029851 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 22:02:28 -0500 (EST) Received: from astonlaptop (NEW-THREE-TWENTY-SIX.MIT.EDU [18.241.6.71]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU32K8s022590 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 22:02:21 -0500 (EST) Message-Id: <200511300302.jAU32K8s022590@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 13 Comments Date: Tue, 29 Nov 2005 22:02:18 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0006_01C5F530.914B1760" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcX1WnmLayKZxAc1QOO1hsICJQY7mg== X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1903 Content-Length: 2343 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0006_01C5F530.914B1760 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Pages 12 and 13 mention a closed form formula for the binomial density function approximation, but I was extremely confused by the lack of rationalization for elements in the formula. I suppose we can take it on faith, but I feel kind of uncomfortable about that. - Aston ------=_NextPart_000_0006_01C5F530.914B1760 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Pages 12 and 13 mention a closed form formula for the binomial density function approximation, but I was extremely confused by = the lack of rationalization for elements in the formula. I suppose we can = take it on faith, but I feel kind of uncomfortable about = that.

 

         =    - Aston

------=_NextPart_000_0006_01C5F530.914B1760-- From hectorb@MIT.EDU Tue Nov 29 22:05:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU3515o005885 for <6042-probs@theory.csail.mit.edu>; Tue, 29 Nov 2005 22:05:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU34wr1001488; Tue, 29 Nov 2005 22:04:58 -0500 (EST) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU34orn023086 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 29 Nov 2005 22:04:50 -0500 (EST) Message-ID: <438D16D1.7020506@mit.edu> Date: Tue, 29 Nov 2005 22:04:49 -0500 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Week 13 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1904 Content-Length: 400 X-Keywords: NonJunk NotJunk 5.1 Equivalent Definitions of Expectation "There are some other ways of writing the definition of expectation. Sometimes using one of these other formulations can make computing an expectation a lot easier. One option is to group together all outcomes on which the random variable takes on the same value." p. 18-19 This section was not so clear and I would like to see it explained in lecture. From meyer@imap.theory.csail.mit.edu Tue Nov 29 22:17:32 2005 Message-ID: <438D19D2.2000502@csail.mit.edu> Date: Tue, 29 Nov 2005 22:17:38 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Barry Revzin Subject: Re: [Hanson] Reading References: <6.1.2.0.1.20051129201038.019f4980@po14.mit.edu> In-Reply-To: <6.1.2.0.1.20051129201038.019f4980@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1008 X-UID: 1905 X-Keywords: 18.103 is not a prereq for 6.042. I expect few 6.042 students go on to take it later, and the consideration of using consistent notation has never come up. Since all sample spaces in 6.042 are discrete (countable), we do fine using summations and avoiding the complexities of measure theory, most notably the existence of non-measurable sets. On the other hand, everything we do is consistent with the measure-theoretic perspective. I hope the conflicting notation does not burden you unduly. regards, A. Barry Revzin wrote: > You guys go over a lot of definitions for expectation, but what about > the more general one ? > > Let f: X --> R+ be a random variable, the expection is integral over X > of f du, where u (mu) is the measure. > > Also, this entire section will be extremely weird since in 18.103 > (Fourier Analysis, Measure Theory) we learn the measure theory side > which has its own explanations and descriptions... and symbols. The > overlap will be confusing. > > Barry > From avalys@MIT.EDU Tue Nov 29 22:34:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU3YC5o008028 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 22:34:12 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU3YBr1020400 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 22:34:11 -0500 (EST) Received: from [18.96.6.172] (SIMMONS-FOUR-TWENTY-SEVEN.MIT.EDU [18.96.6.172]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU3Y8pp000193 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 22:34:08 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <1B4ECB15-6AD2-4E75-BD75-8EBFF67433E6@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Required Reading Comments Date: Tue, 29 Nov 2005 22:34:07 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1906 Content-Length: 149 X-Keywords: NonJunk NotJunk I am a little bit unclear about how to actually apply the section on expected value to actual problems. Some examples would be nice. Alex Valys From ridell@MIT.EDU Tue Nov 29 22:46:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU3kh5o009201 for <6042-probs@theory.csail.MIT.EDU>; Tue, 29 Nov 2005 22:46:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU3kgr1029909 for <6042-probs@theory.csail.MIT.EDU>; Tue, 29 Nov 2005 22:46:42 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU3kevb002780 for <6042-probs@theory.csail.MIT.EDU>; Tue, 29 Nov 2005 22:46:40 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jAU3kemD015760; Tue, 29 Nov 2005 22:46:40 -0500 Received: from AP-SEVENTY.MIT.EDU (AP-SEVENTY.MIT.EDU [18.153.1.70]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 29 Nov 2005 22:46:40 -0500 Message-ID: <20051129224640.81zt0eycgv0kkwww@webmail.mit.edu> Date: Tue, 29 Nov 2005 22:46:40 -0500 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: reading 13, ta: hanson MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1907 Content-Length: 260 X-Keywords: NonJunk NotJunk Can you explain more about the probability density function on page 6? Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From fluff@MIT.EDU Tue Nov 29 22:51:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU3pK5o009706 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 22:51:20 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU3odr1002789 for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 22:50:39 -0500 (EST) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU3obYX003691 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 22:50:37 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <4FD036DE-F425-4DAC-9F5C-664D50EB4E6B@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 13 reading Date: Tue, 29 Nov 2005 22:48:03 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1908 Content-Length: 324 X-Keywords: NonJunk NotJunk Everything starting from the binomial density approximation onwards went completely over my head. I think I might just have a 10-page-per- day math comprehension limit. The expected value stuff seemed understandable with the six-sided die example, but I didn't get anything after that (starting from 5.1). ~Crystal From sil_03@MIT.EDU Tue Nov 29 23:14:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU4Ew5o016707 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 23:14:58 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU4EvVs017835 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 23:14:57 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU4EsDT008651 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 29 Nov 2005 23:14:55 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAU4EsAp012654; Tue, 29 Nov 2005 23:14:54 -0500 Received: from NEW-SIX-SIXTY-SIX.MIT.EDU (NEW-SIX-SIXTY-SIX.MIT.EDU [18.241.7.155]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 29 Nov 2005 23:14:54 -0500 Message-ID: <20051129231454.gjf23zjxvn4s80ck@webmail.mit.edu> Date: Tue, 29 Nov 2005 23:14:54 -0500 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 13 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1909 Content-Length: 210 X-Keywords: NonJunk NotJunk It is interesting that the population of the country has no effect on poll size! I don't see how. I couldn't really follow the Binomial Sampling theorem so it would be nice if we went over it again in class. From dowgun@MIT.EDU Wed Nov 30 00:00:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU50q5o004249 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 00:00:52 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU50oVs018961 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 00:00:50 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU50m3b017612 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 00:00:49 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jAU50mvp021786; Wed, 30 Nov 2005 00:00:48 -0500 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 00:00:48 -0500 Message-ID: <20051130000048.pr7oaai2tcg8wsc4@webmail.mit.edu> Date: Wed, 30 Nov 2005 00:00:48 -0500 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] ln13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1910 Content-Length: 404 X-Keywords: NotJunk Right, so this week's lecture notes introduced a lot of math that I can sort of follow, but I doubt if I'll be able to remember. Thus, my question for the week: do we need to remember how to derive the following formulas? e^(-n(n-1)/2d) The upper bound for PDF (#1 on page 13) The upper bound for CDF (bottom of 13) The formula for delta (#3 on page 15) and The binomial sampling equation. Thanks, Neil From ozcan@MIT.EDU Wed Nov 30 00:17:55 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU5Ht5o011114 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:17:55 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAU5Hsuq026577 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:17:54 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU5HrMN002167 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:17:53 -0500 (EST) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAU5Hkol006660 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:17:46 -0500 (EST) Message-Id: <5.2.1.1.2.20051130001547.02263a88@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 30 Nov 2005 00:17:34 -0500 To: 6042-probs@theory.csail.mit.edu From: Yasin Ozcan Subject: [hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jAU5Ht5o011114 Status: RO X-UID: 1911 Content-Length: 1827 X-Keywords: NonJunk NotJunk The section on confidence levels, and how we might interpret the confidence levels wrong, is quite interesting. Section 4: Confidence Levels Suppose a pollster uses a sample of 662 random voters to estimate the fraction of voters who prefer Clinton, and the pollster finds that 364 of them prefer Clinton. Its tempting, but sloppy, to say that this means: False Claim. With probability 0.95, the fraction, p, of voters who prefer Clinton is 364/6620.04. Since 364/662-0.04 > 0.50, there is a 95% chance that more than half the voters prefer Clinton. Whats objectionable about this statement is that it talks about the probability or chance that a real world fact is true, namely that the actual fraction, p, of voters favoring Clinton is more than 0.50. But p is what it is, and it simply makes no sense to talk about the probability that it is something else. For example, suppose p is actually 0.49; then its nonsense to ask about the probability that it is within 0.04 of 364/662 it simply isnt. A more careful summary of what we have accomplished goes this way: We have described a probabilistic procedure for estimating the value of the actual fraction, p. The probability that our estimation procedure will yield a value within 0.04 of p is 0.95. This is a bit of a mouthful, so special phrasing closer to the sloppy language is commonly used. The pollster would describe his conclusion by saying that At the 95% confidence level, the fraction of voters who prefer Clinton is 364/662 0.04. Its important to remember that confidence levels refer to the results of estimation procedures for real-world quantities. The real-world quantity being estimated is typically unknown, but fixed; it is not a random variable, so it makes no sense to talk about the probability that it has some property. From jjmonzon@MIT.EDU Wed Nov 30 00:34:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU5YF5o017894 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:34:15 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU5YEVs008611 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:34:14 -0500 (EST) Received: from JOSHDESKTOP.mit.edu (BURTON-ONE-FORTY-SIX.MIT.EDU [18.247.5.146]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU5Y6YU023326 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 00:34:07 -0500 (EST) Message-Id: <6.2.3.4.2.20051130002706.01d0d7e8@po9.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 30 Nov 2005 00:34:10 -0500 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: [David] - reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1912 Content-Length: 359 X-Keywords: NotJunk This subject matter is a bit complex but I like the examples given - some of them are a bit counterintuitive and confusing like the birthday principle. I liked the "making baby girls" example though - was funny. Josh Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From zacharyozer@gmail.com Wed Nov 30 00:46:24 2005 Return-Path: Received: from zproxy.gmail.com (zproxy.gmail.com [64.233.162.207]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU5kO5o024569 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 00:46:24 -0500 Received: by zproxy.gmail.com with SMTP id 13so342039nzn for <6042-probs@theory.lcs.mit.edu>; Tue, 29 Nov 2005 21:46:24 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=NN+FGaQWzQBs/hXj2c83BByFVfEu3JDtI2+BYqKHSyDbkqxrjslPtuY3iyYi9wariXJ3GDmxmONFJk2fPa7vrgriklZ1goY/MMlkGUd4U6sQlARgVMztubG5sLaNa2HbyJG91o0ydNRka21NTw/7zco/+ey9aX4yEqvpcXLA3XE= Received: by 10.64.220.11 with SMTP id s11mr3663412qbg; Tue, 29 Nov 2005 21:46:24 -0800 (PST) Received: by 10.65.152.13 with HTTP; Tue, 29 Nov 2005 21:46:24 -0800 (PST) Message-ID: <4f2613e50511292146q8dc62b7k4b5f45927a009b6f@mail.gmail.com> Date: Wed, 30 Nov 2005 00:46:24 -0500 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Tutor Comments: Mean time to failure LN 13 Pg 20 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jAU5kO5o024569 Status: RO X-UID: 1913 Content-Length: 619 X-Keywords: NotJunk I was just wondering how mean time to failure is affected by redundant systems, ie, what if I have 2 systems where one kicks in when the other fails. It seems like the probability of failure the same, since I'm still relying on an identical system, yet that doesn't make sense from a practical standpoint, since then there'd be little advantage to using redundant systems. Are they mulitiplicative? IE if the mean proabaility of failure of one system is 1/100, does the redundancy cause a mean probability of failure of (1/100)(1/100)? What about as the redundancy tends towards infinity? Just some thoughts, -zozer From bakster@MIT.EDU Wed Nov 30 00:54:10 2005 Return-Path: Received: from bexxxley.mit.edu (BEXXXLEY.MIT.EDU [18.246.1.238]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU5sA5o027136 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 00:54:10 -0500 Received: (from bakster@localhost) by bexxxley.mit.edu (8.12.9) id jAU5s70l019591; Wed, 30 Nov 2005 00:54:07 -0500 Date: Wed, 30 Nov 2005 00:54:07 -0500 (EST) From: Alexander G Bakst X-X-Sender: bakster@bexxxley To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 13 Comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1914 Content-Length: 118 X-Status: A X-Keywords: NonJunk NotJunk This wasn't implicit in the reading I thought: if x and y are two variables, is E[xy] = E[x] * E[y]? Alexander Bakst From vixen@MIT.EDU Wed Nov 30 00:59:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU5xQ5o027485 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 00:59:26 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU5xPVs022523 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 00:59:25 -0500 (EST) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU5xMdS026897 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 00:59:23 -0500 (EST) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 30 Nov 2005 00:57:16 -0500 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Subject: LN13 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1915 Content-Length: 71 X-Status: A X-Keywords: NotJunk I don't understand the difference between events and random variables. From meyer@imap.theory.csail.mit.edu Wed Nov 30 01:06:08 2005 Message-ID: <438D4155.9010904@csail.mit.edu> Date: Wed, 30 Nov 2005 01:06:13 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alexander G Bakst Subject: Re: [David] Week 13 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 315 X-UID: 1916 X-Keywords: only if X,y are independent counterexample when not independent: Let X=Y = +1 or -1 with equal probability (1/2). So E[X] = E[Y] = 0, but E[XY] = E[X^2] = 1. Alexander G Bakst wrote: > This wasn't implicit in the reading I thought: if x and y are two > variables, is E[xy] = E[x] * E[y]? > > Alexander Bakst From rshearer@MIT.EDU Wed Nov 30 01:07:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU67n5o003113 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:07:49 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU67mVs026797 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:07:48 -0500 (EST) Received: from [18.244.6.41] (SENIOR-TWO-NINETY-SIX.MIT.EDU [18.244.6.41]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU67bBO028033 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:07:42 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: [Jelani] TP13 Comments Date: Wed, 30 Nov 2005 01:07:35 -0500 X-Mailer: Apple Mail (2.623) X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1917 Content-Length: 140 X-Keywords: NonJunk NotJunk The reading didn't explain Expected Values very well at all, and I found myself confused when doing the tutor problems. - Rachel Shearer From meyer@imap.theory.csail.mit.edu Wed Nov 30 01:09:39 2005 Message-ID: <438D4229.6020807@csail.mit.edu> Date: Wed, 30 Nov 2005 01:09:45 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: Re: LN13 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 306 X-UID: 1918 X-Keywords: give me a hint where your understanding fails. for a start, an event is a set (of outcomes, that is, points in the sample space), while an RV is a function (from outcomes to whatever you want). regards, A. Amanda Seybold wrote: > I don't understand the difference between events and random variables. From rnjacobs@MIT.EDU Wed Nov 30 01:11:32 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6BW5o004097 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 01:11:32 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6BVuq018208 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 01:11:31 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6BVMN003817 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 01:11:31 -0500 (EST) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAU6BRok008936 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 01:11:28 -0500 (EST) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id jAU6BRIj002845; Wed, 30 Nov 2005 01:11:27 -0500 Message-Id: <200511300611.jAU6BRIj002845@white-meteo.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Reading response Date: Wed, 30 Nov 2005 01:11:27 -0500 From: r n jacobs X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1919 Content-Length: 168 X-Keywords: NonJunk NotJunk Nope, no comments. I still remember most of my 041. (A review of continuous probability would be nice but apparently will not be covered in 042...) - Robert Jacobs From sergiob@MIT.EDU Wed Nov 30 01:17:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6HS5o008719 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 01:17:28 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6HRVs001738 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 01:17:27 -0500 (EST) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-FORTY-SIX.MIT.EDU [18.244.5.46]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU6HNv8029099 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 01:17:24 -0500 (EST) Message-Id: <5.2.1.1.2.20051129231507.028c3a48@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 29 Nov 2005 23:17:21 -0700 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 13 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1920 Content-Length: 202 X-Keywords: NonJunk NotJunk I liked the part about confidence. It's a very useful concept in every-day experimental calculations. Probability doesn't cease to amaze me; intuition can be so misleading. Sergio Bacallado Group 1 From yaser@MIT.EDU Wed Nov 30 01:21:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6LV5o009181 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:21:31 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6LTVs003714 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:21:29 -0500 (EST) Received: from Charizard (NEW-SEVEN-THIRTY-ONE.MIT.EDU [18.241.7.220]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU6LNe5029473 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:21:23 -0500 (EST) Message-Id: <200511300621.jAU6LNe5029473@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: tp13 reading comments Date: Wed, 30 Nov 2005 01:21:25 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0066_01C5F54C.620520B0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcX1dkpdNbG+04S4To6Zc0pFYhTlqg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1921 Content-Length: 5323 X-Status: A X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0066_01C5F54C.620520B0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Hi David, As a whole, the reading was a bit dense with formulas, to the extent that I keep getting similar definitions mixed up (for instance the distinctions between the distributions). However, my question is: will we be concerned with the t, poisson, or chi distributions in terms of random variables? Thanks! _Yaser ------=_NextPart_000_0066_01C5F54C.620520B0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

As a whole, the reading was a bit = dense with formulas, to the extent that I keep getting similar definitions = mixed up (for instance the distinctions between the distributions). However, my = question is: will we be concerned with the t, poisson, or chi distributions in terms of random variables?


Thanks!

 

_Yaser

------=_NextPart_000_0066_01C5F54C.620520B0-- From ereid@MIT.EDU Wed Nov 30 01:28:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6SF5o013634 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:28:15 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6SEVs007208 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:28:14 -0500 (EST) Received: from [18.242.6.198] (NEXT-FOUR-FIFTY-THREE.MIT.EDU [18.242.6.198]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU6SCcG000155 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:28:12 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <49EAF327-2A3E-4CBD-8256-5E5E11D9E729@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Reading comments Date: Wed, 30 Nov 2005 01:28:09 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1922 Content-Length: 397 X-Status: A X-Keywords: NonJunk NotJunk I was initially surprised (and skeptical) about the numbers game strategy, but its proof makes sense. I guess my only question would be how to pick the "x" you hope is between L and H, but that's just a matter of personal opinion. :) Oh, and did you get my reading comments last week? You never responded, and the 6042-probs list was acting oddly when I tried to email it... Elizabeth From mpapi@MIT.EDU Wed Nov 30 01:28:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6Si5o013662 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:28:44 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6SgVs007504 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:28:43 -0500 (EST) Received: from [192.168.0.2] (MPAPI.MIT.EDU [18.239.4.219]) (authenticated bits=0) (User authenticated as mpapi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU6Sa1N000217 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:28:40 -0500 (EST) Subject: [Jelani] Week 13 comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 30 Nov 2005 01:28:35 -0500 Message-Id: <1133332115.8490.4.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.3 (2.2.3-2.fc4) Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1923 Content-Length: 320 X-Keywords: NonJunk NotJunk Section 3, particularly the calculations performed on pages 15 and 16 - I could definitely use a clearer explanation of polling and confidence levels in lecture. I also feel that the notes weren't very helpful with the last tutorial problem, but reading the answer to the problem after submitting helped a lot. - Matt From kevin08@MIT.EDU Wed Nov 30 01:46:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6kD5o019936 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:46:13 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6kCVs017058 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:46:12 -0500 (EST) Received: from kevlar.mit.edu (NEXT-THREE-FIFTY-SEVEN.MIT.EDU [18.242.6.102]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU6k9AE002004 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 01:46:10 -0500 (EST) Message-Id: <6.2.3.4.2.20051130014533.02a5f6f0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 30 Nov 2005 01:46:08 -0500 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1924 Content-Length: 120 X-Keywords: NonJunk NotJunk Hi, I was curious about where the entropy function H(a) came originated from (Section 3.4.2, Pg. 12). Thanks. Kevin From meyer@imap.theory.csail.mit.edu Wed Nov 30 01:54:55 2005 Message-ID: <438D4CC5.1060700@csail.mit.edu> Date: Wed, 30 Nov 2005 01:55:01 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: Re: LN13 References: <438D4229.6020807@csail.mit.edu> In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 832 X-UID: 1925 X-Keywords: Sure they are related: events correspond (via indicator variables defined in Notes 13) to zero-one valued RV's, so they are essentially a special case of RV's. Conversely, with an RV, R, and value v, there is an event [R=v] consisting of those outcomes w such that R(w)=v. So you could specify an RV by giving all the events [R=v]. Is it clearer now? regards, A. Amanda Seybold wrote: > It seems like they are expressing the same idea: creating a set of > outcomes with something in common. > >> give me a hint where your understanding fails. for a start, an event >> is a set (of outcomes, that is, points in the sample space), while an >> RV is a function (from outcomes to whatever you want). >> regards, A. >> >> Amanda Seybold wrote: >> >>> I don't understand the difference between events and random variables. From meyer@imap.theory.csail.mit.edu Wed Nov 30 01:11:50 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU6Bo5o004154 for ; Wed, 30 Nov 2005 01:11:50 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EhLC2-0004gr-2S for meyer@csail.mit.edu; Wed, 30 Nov 2005 01:11:50 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU6BmVs028827 for ; Wed, 30 Nov 2005 01:11:48 -0500 (EST) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU6Bf1s028494 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for ; Wed, 30 Nov 2005 01:11:41 -0500 (EST) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: In-Reply-To: <438D4229.6020807@csail.mit.edu> References: <438D4229.6020807@csail.mit.edu> Date: Wed, 30 Nov 2005 01:09:34 -0500 To: "Prof. Albert R. Meyer" From: Amanda Seybold Subject: Re: LN13 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO X-Status: A Content-Length: 415 X-UID: 1926 X-Keywords: NotJunk It seems like they are expressing the same idea: creating a set of outcomes with something in common. >give me a hint where your understanding fails. for a start, an >event is a set (of outcomes, that is, points in the sample space), >while an RV is a function (from outcomes to whatever you want). >regards, A. > >Amanda Seybold wrote: >>I don't understand the difference between events and random variables. From meyer@imap.theory.csail.mit.edu Wed Nov 30 02:00:22 2005 Message-ID: <438D4E0C.4080406@csail.mit.edu> Date: Wed, 30 Nov 2005 02:00:28 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Elizabeth Reid Subject: Re: [Hanson] Reading comments References: <49EAF327-2A3E-4CBD-8256-5E5E11D9E729@MIT.EDU> In-Reply-To: <49EAF327-2A3E-4CBD-8256-5E5E11D9E729@MIT.EDU> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 614 X-UID: 1927 X-Keywords: most recent email comment we have for you is 11/17. if you have a copy of last weeks, send it now. (It would help if you include the TP# in the subject line.) regards, A. Elizabeth Reid wrote: > I was initially surprised (and skeptical) about the numbers game > strategy, but its proof makes sense. I guess my only question would be > how to pick the "x" you hope is between L and H, but that's just a > matter of personal opinion. :) > > Oh, and did you get my reading comments last week? You never responded, > and the 6042-probs list was acting oddly when I tried to email it... > > Elizabeth From hkhall@MIT.EDU Wed Nov 30 02:10:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU7AK5o026614 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 02:10:20 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU7AIKa027484; Wed, 30 Nov 2005 02:10:18 -0500 (EST) Received: from [18.96.6.209] (SIMMONS-FOUR-SIXTY-FOUR.MIT.EDU [18.96.6.209]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU7AFtN003948 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Wed, 30 Nov 2005 02:10:16 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <6fe524b5cc90d96d530e7b06be446ee8@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: [David] Week 13 Comments Date: Wed, 30 Nov 2005 02:10:17 -0500 To: 6042-probs@theory.csail.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1928 Content-Length: 646 X-Keywords: NotJunk David- Well syntactically I am doing better this week. I am having some major problems though, primarily with approximating the binomial density function and the applications that follow. I understand the formula as it is just plug and chug, it is its roots that I do not understand, however. I would like to see a formal definition of where it comes from if it is not beyond the scope of the class. From there I really didn't understand any of the equivalencies used in determining the polling but that is probably because it is my first time through the material. I hope we go over some of this stuff in class tomorrow. -Harrison From rshroff@MIT.EDU Wed Nov 30 02:13:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU7Do5o027071 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 02:13:50 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU7DmKa029084 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 02:13:48 -0500 (EST) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU7DgsG004194 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 02:13:42 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id jAU7DgeW022350; Wed, 30 Nov 2005 02:13:42 -0500 Received: from NEXT-THREE-TWENTY.MIT.EDU (NEXT-THREE-TWENTY.MIT.EDU [18.242.6.65]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 02:13:42 -0500 Message-ID: <20051130021342.s448jmsvqhc8kgw4@webmail.mit.edu> Date: Wed, 30 Nov 2005 02:13:42 -0500 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Reading Assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.149 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1929 Content-Length: 122 X-Keywords: NonJunk NotJunk I was wondering if we could go over the approximation to the binomial distribution in some detail. Thanks, Rahul Shroff From lkini@MIT.EDU Wed Nov 30 02:17:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU7Ha5o030988 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 02:17:36 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU7HZKa000828 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 02:17:35 -0500 (EST) Received: from [18.239.6.94] (MACGREGOR-THREE-FORTY-NINE.MIT.EDU [18.239.6.94]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU7HWWW004486 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 02:17:33 -0500 (EST) Message-ID: <438D5200.1050608@mit.edu> Date: Wed, 30 Nov 2005 02:17:20 -0500 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 13 Lecture Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1930 Content-Length: 206 X-Keywords: NonJunk NotJunk Hi Hanson, I was a little confused about approximating cumulative binomial distribution functions, especially the caveat (Section 3.5, page 13). How does thinking it in complementary terms help? Lohith From jstritar@MIT.EDU Wed Nov 30 02:21:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU7L95o031553 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 02:21:09 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU7L8Ka002255 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 02:21:08 -0500 (EST) Received: from BAKER-THREE-NINETY-EIGHT.MIT.EDU (BAKER-THREE-NINETY-EIGHT.MIT.EDU [18.245.6.143]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU7L6aX004705 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 02:21:06 -0500 (EST) Subject: [Sayan] Reading comments From: Jon Stritar To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 30 Nov 2005 02:21:06 -0500 Message-Id: <1133335266.13472.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1931 Content-Length: 148 X-Keywords: NonJunk NotJunk 3.3 The Numbers game I thought the winning strategy for this was interesting. It is not very intuitive... but it still makes sense. Jon Stritar From dnreshef@MIT.EDU Wed Nov 30 02:46:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU7k95o002670 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 02:46:09 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU7k8Ka012896 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 02:46:08 -0500 (EST) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU7k2If006455 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 02:46:02 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id jAU7k2dA024166; Wed, 30 Nov 2005 02:46:02 -0500 Received: from MACGREGOR-EIGHTY-SIX.MIT.EDU (MACGREGOR-EIGHTY-SIX.MIT.EDU [18.239.5.86]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 02:46:02 -0500 Message-ID: <20051130024602.m7txa136ae80sowo@webmail.mit.edu> Date: Wed, 30 Nov 2005 02:46:02 -0500 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments 13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.11 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1932 Content-Length: 182 X-Keywords: I found the binomial distribution concepts and their applications (especially to polling) particularly interesting. Also, could we go over confidence levels in class? Thanks, -Dave From ctsims@MIT.EDU Wed Nov 30 03:34:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU8Yt5o021434 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 03:34:55 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU8YsKa003017 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 03:34:54 -0500 (EST) Received: from HowardRoark.mit.edu (EASTCAMPUS-NINE-TWENTY-SEVEN.MIT.EDU [18.238.6.160]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU8Yp31009221 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 03:34:52 -0500 (EST) Message-Id: <5.2.1.1.2.20051130031809.01ea6ee8@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 30 Nov 2005 03:32:11 -0500 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: [6042] Weekly reading problem - Clayton Sims Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1933 Content-Length: 190 X-Keywords: NonJunk NotJunk I found that the section on page 18 about expected values could use more examples about the use of expected values with variables. I would like to see that presented in class. -Clayton From arup@MIT.EDU Wed Nov 30 03:44:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU8ik5o022038 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 03:44:46 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU8ijKa006868 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 03:44:45 -0500 (EST) Received: from [18.247.5.179] (BURTON-ONE-SEVENTY-NINE.MIT.EDU [18.247.5.179]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU8iasE009664 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 03:44:38 -0500 (EST) Message-ID: <438D663B.1000605@mit.edu> Date: Wed, 30 Nov 2005 03:43:39 -0500 From: Arup Sarma User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Week 13 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1934 Content-Length: 197 X-Keywords: NotJunk Section 3.4.2: I'm kindof confused as to where the approximation for the binomial density function came from, so it would be nice if it were explained in a little more detail in lecture. |Arup| From bilodeau@MIT.EDU Wed Nov 30 04:03:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU9395o032050 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 04:03:09 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU938Ka014136 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 04:03:08 -0500 (EST) Received: from pbilodeau (MACGREGOR-TWO-TWENTY-ONE.MIT.EDU [18.239.5.221]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU936a4010510 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 04:03:06 -0500 (EST) Message-Id: <200511300903.jAU936a4010510@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [jelani] reading comments Date: Wed, 30 Nov 2005 04:02:48 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5F562.ED6FBB90" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcX1jNWZoMjh2+8rQ/K5DrefiWE59A== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1935 Content-Length: 2152 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5F562.ED6FBB90 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit For this week's reading the most confusing part was probably the part on confidence, or possibly the approximations for the binomial distributions. Beyond that a lot of this seems understandable. Peter Bilodeau ------=_NextPart_000_0000_01C5F562.ED6FBB90 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

For this week’s reading the most confusing part = was probably the part on confidence, or possibly the approximations for the binomial distributions.  Beyond that a lot of this seems = understandable.

 

Peter Bilodeau

------=_NextPart_000_0000_01C5F562.ED6FBB90-- From tonyng@MIT.EDU Wed Nov 30 04:46:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU9ks5o019424 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 04:46:54 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU9krKa001193 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 04:46:53 -0500 (EST) Received: from TNG.mit.edu (BURTON-NINETY-TWO.MIT.EDU [18.247.5.92]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU9ko0A012069 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 04:46:51 -0500 (EST) Message-Id: <5.2.1.1.2.20051130044258.02144200@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 30 Nov 2005 04:46:51 -0500 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments Week 13 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1936 Content-Length: 372 X-Keywords: NotJunk I found "The Numbers Game" (Section 3.3, starting on pg8) interesting. I doesn't seem intuitive at all that selecting a number at random can help make me more likely to win the game. It almost seems like it is the same as choosing one of the two numbers at random, but it's surprising that the math works out to show that it is in fact a winning strategy. - Tony Ng From kjhollen@MIT.EDU Wed Nov 30 04:48:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU9mM5o021188 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 04:48:22 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU9mLKa001754 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 04:48:21 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU9mEnc012106 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 04:48:15 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jAU9mE7s011240; Wed, 30 Nov 2005 04:48:14 -0500 Received: from dhcp-23-135.media.mit.edu (dhcp-23-135.media.mit.edu [18.85.23.135]) (User authenticated as kjhollen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 04:48:14 -0500 Message-ID: <20051130044814.v9fna3ft6ylcoo40@webmail.mit.edu> Date: Wed, 30 Nov 2005 04:48:14 -0500 From: Kate Hollenbach To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] numbers game MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1937 Content-Length: 51 X-Keywords: NonJunk NotJunk the numbers game is a neat example (p. 8-9). Kate From benlu@MIT.EDU Wed Nov 30 05:02:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUA2g5o005812 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 05:02:42 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUA2fYn006357 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 05:02:41 -0500 (EST) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUA2YKd012637 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 05:02:35 -0500 (EST) Message-ID: <438D78C1.8070701@mit.edu> Date: Wed, 30 Nov 2005 05:02:41 -0500 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051013) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 13 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1938 Content-Length: 207 X-Keywords: NotJunk My eyes definitely started to glaze over during the section on polling and confidence intervals. Also, the online tutor problem took me forever. It would be great to go over those topics in lecture. ~Ben From lana@MIT.EDU Wed Nov 30 06:50:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUBoJ5o030431 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 06:50:19 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUBoIYn016238 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 06:50:18 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUBoBXp017405 for <6042-probs@theory.csail.MIT.EDU>; Wed, 30 Nov 2005 06:50:12 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAUBoBv3019842; Wed, 30 Nov 2005 06:50:11 -0500 Received: from NEXT-FIVE-SIXTY-NINE.MIT.EDU (NEXT-FIVE-SIXTY-NINE.MIT.EDU [18.242.7.58]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 06:50:11 -0500 Message-ID: <20051130065011.sd7hi6fte84k8kcw@webmail.mit.edu> Date: Wed, 30 Nov 2005 06:50:11 -0500 From: Svetlana Goldenberg To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] TP 12 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1939 Content-Length: 171 X-Status: A X-Keywords: NonJunk NotJunk The part I found the most intriguing was the halting problem, and the intuitive feeling that it must have some kind of an answer. Can it be modeled by Markov chains? Lana From kromer@MIT.EDU Wed Nov 30 09:51:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUEpL5o003189 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 09:51:21 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUEpKrd027347 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 09:51:20 -0500 (EST) Received: from m56-129-21.mit.edu (M56-129-21.MIT.EDU [18.56.0.50]) (authenticated bits=56) (User authenticated as kromer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUEpDct029021 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 09:51:14 -0500 (EST) Received: (from kromer@localhost) by m56-129-21.mit.edu (8.12.9) id jAUEpDqa032749; Wed, 30 Nov 2005 09:51:13 -0500 Subject: [David] Email comments for reading From: Katherine Romer To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 30 Nov 2005 09:51:13 -0500 Message-Id: <1133362273.32582.2.camel@m56-129-21.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1940 Content-Length: 233 X-Keywords: NotJunk p.21 "If every couple follows the strategy of having children until they get a girl, what will eventually happen to the fraction of girls born in this world?" Will it remain the same (because each couple expects to have 2 children)? From veracarr@MIT.EDU Wed Nov 30 10:12:41 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFCf5o017167 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:12:41 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFCeP6007599 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:12:40 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFCdaT020862 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:12:39 -0500 (EST) Received: from [18.245.6.197] (BAKER-FOUR-FIFTY-TWO.MIT.EDU [18.245.6.197]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUFCaok005040 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:12:37 -0500 (EST) Message-ID: <438DC199.6040400@mit.edu> Date: Wed, 30 Nov 2005 10:13:29 -0500 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Random Variables, Distributions, and Expectations Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1941 Content-Length: 112 X-Keywords: NotJunk If we could cover when we would use each of the distributions and there importance, that would be very useful. From rian@MIT.EDU Wed Nov 30 10:20:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFKC5o022697 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:20:12 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFKBrd003291 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:20:11 -0500 (EST) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFK4nm013970 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:20:04 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <6532AE49-7541-4E9E-882C-F6F51C02FA3D@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rian Hunter Subject: [jelani] email comments Date: Wed, 30 Nov 2005 10:19:55 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1942 Content-Length: 258 X-Keywords: NotJunk In the part that is the "Analysis of the Winning Strategy" 3.3.2 what if you have different probabilities for each event, for instance not 50-50? I don't exactly understand the variable assignments for not 50-50 tree probabilities. Thanks rian hunter From cbossard@MIT.EDU Wed Nov 30 10:26:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFQ15o027229 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:26:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFPxrd010254 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:25:59 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFPr7I016712 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:25:54 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAUFPrKP026861; Wed, 30 Nov 2005 10:25:53 -0500 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 10:25:53 -0500 Message-ID: <20051130102553.cz8cvon92j9c0okw@webmail.mit.edu> Date: Wed, 30 Nov 2005 10:25:53 -0500 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 13 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1943 Content-Length: 95 X-Keywords: NonJunk NotJunk I'm really not sure what to ask. The distribution stuff seems a little tricky though. Cynthia From hmzhou@blackbird.csail.mit.edu Wed Nov 30 10:27:07 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFR75o027613; Wed, 30 Nov 2005 10:27:07 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAUFR7aq015328; Wed, 30 Nov 2005 10:27:07 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAUFR6Yv015325; Wed, 30 Nov 2005 10:27:07 -0500 Date: Wed, 30 Nov 2005 10:27:06 -0500 (EST) From: Hanson Zhou To: Lohith Kini cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 13 Lecture Notes In-Reply-To: <438D5200.1050608@mit.edu> Message-ID: References: <438D5200.1050608@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1944 Content-Length: 802 X-Status: A X-Keywords: NotJunk Hi...I think it's saying that in the case a > p, the formula does not hold (since the geometric sum does not hold). But if we look at tails instead of heads, then we can calculate 1-Pr[>=(1-a)n tails] to get the CDF we want(note that now the fraction is 1-a). The probability of a tail is 1-p and we have 1-a < 1-p by assumption. Then, we can do a similar analysis/formula to approximate Pr[>=(1-a)n tails]. You might want to check it or ask Prof. Meyer since lots of details are omitted here, but I think that's the right idea. -Hanson On Wed, 30 Nov 2005, Lohith Kini wrote: > Hi Hanson, > > I was a little confused about approximating cumulative binomial > distribution functions, especially the caveat (Section 3.5, page 13). > How does thinking it in complementary terms help? > > Lohith > From petek@MIT.EDU Wed Nov 30 10:29:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFTC5o031445 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:29:12 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFTArd013954 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:29:11 -0500 (EST) Received: from [128.30.5.207] (30-5-207.wireless.csail.mit.edu [128.30.5.207]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFT3t0018267 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:29:03 -0500 (EST) Message-ID: <438DC549.2080006@mit.edu> Date: Wed, 30 Nov 2005 10:29:13 -0500 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] This Week's Notes Content-Type: multipart/alternative; boundary="------------050103050002020207060103" X-Spam-Score: 1.928 X-Spam-Level: * (1.928) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1945 Content-Length: 1736 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. --------------050103050002020207060103 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit I'm up for hearing more on probability distributions and also about something that Ronnitt talked about last lecture to our breakout group. There is some complication in assigning a probability to whether a certain person prefers a candidate....something about it being either 1 or 0. Want to have that expanded a bit. -P -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------050103050002020207060103 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit I'm up for hearing more on probability distributions and also about something that Ronnitt talked about last lecture to our breakout group.  There is some complication in assigning a probability to whether a certain person prefers a candidate....something about it being either 1 or 0.  Want to have that expanded a bit. 

-P
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------050103050002020207060103-- From lmccart@MIT.EDU Wed Nov 30 10:38:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFbx5o002083 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:37:59 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFbwrd024913 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:37:58 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFbpI2022835 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:37:52 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jAUFbp4i005880; Wed, 30 Nov 2005 10:37:51 -0500 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 10:37:51 -0500 Message-ID: <20051130103751.ja0lxrqf2vww0kw0@webmail.mit.edu> Date: Wed, 30 Nov 2005 10:37:51 -0500 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.11 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1946 Content-Length: 125 X-Keywords: NotJunk I did not find anything particularly confusing about this reading. I found the Birthday Principle most interesting. -lauren From meyer@imap.theory.csail.mit.edu Wed Nov 30 10:40:25 2005 Message-ID: <438DC7E8.6060501@csail.mit.edu> Date: Wed, 30 Nov 2005 10:40:24 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Svetlana Goldenberg Subject: Re: [Sayan] TP 12 References: <20051130065011.sd7hi6fte84k8kcw@webmail.mit.edu> In-Reply-To: <20051130065011.sd7hi6fte84k8kcw@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 347 Status: RO X-UID: 1947 X-Keywords: This week it's Notes and TP 13. The Halting Problem is covered in 6.045 and 16.840 and, best of all :-) my course 6.044. regards, A. Svetlana Goldenberg wrote: >The part I found the most intriguing was the halting problem, and the intuitive >feeling that it must have some kind of an answer. Can it be modeled by Markov >chains? >Lana > > From cvnguyen@MIT.EDU Wed Nov 30 10:40:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFeX5o004093 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:40:33 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFeWrd028044 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:40:32 -0500 (EST) Received: from MULTIVAC (EASTCAMPUS-SIX-SIXTY.MIT.EDU [18.238.5.149]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFePtR024028 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:40:25 -0500 (EST) Message-ID: <002501c5f5c4$62ebc1a0$9505ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 13 comments Date: Wed, 30 Nov 2005 10:40:26 -0500 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.199 X-Spam-Level: * (1.199) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1948 Content-Length: 425 X-Keywords: NotJunk (page 21) If every couple has children until getting a girl, the fraction of girls in the world does not change (if we ignore factors that may affect children of different genders differently... disease, abortion, infanticide, etc.). This is interesting. Though the fraction remains the same, the fact that each nuclear family has one and only one girl may introduce some interesting social issues.... --Chieu Nguyen From meyer@imap.theory.csail.mit.edu Wed Nov 30 10:43:52 2005 Message-ID: <438DC8B8.5060607@csail.mit.edu> Date: Wed, 30 Nov 2005 10:43:52 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hanson Zhou CC: Lohith Kini Subject: Re: [Hanson] Week 13 Lecture Notes References: <438D5200.1050608@mit.edu> In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 1011 Status: RO X-UID: 1949 X-Keywords: yeah, that's right. the condition a>p is necessary technically, but because of the complement trick (look at tails), it is not a limitation. regards, A. Hanson Zhou wrote: >Hi...I think it's saying that in the case a > p, the formula does not hold >(since the geometric sum does not hold). But if we look at tails instead >of heads, then we can calculate 1-Pr[>=(1-a)n tails] to get the CDF we >want(note that now the fraction is 1-a). The probability of a tail is 1-p >and we have 1-a < 1-p by assumption. Then, we can do a similar >analysis/formula to approximate Pr[>=(1-a)n tails]. You might want to >check it or ask Prof. Meyer since lots of details are omitted here, but I >think that's the right idea. > >-Hanson > >On Wed, 30 Nov 2005, Lohith Kini wrote: > > > >>Hi Hanson, >> >>I was a little confused about approximating cumulative binomial >>distribution functions, especially the caveat (Section 3.5, page 13). >>How does thinking it in complementary terms help? >> >>Lohith >> >> >> From meyer@imap.theory.csail.mit.edu Wed Nov 30 05:52:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUAqk5o001112 for <6042-webmaster@theory.csail.mit.edu>; Wed, 30 Nov 2005 05:52:46 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUAqjYn024290 for <6042-webmaster@theory.csail.mit.edu>; Wed, 30 Nov 2005 05:52:45 -0500 (EST) Received: from SENIOR-THREE-TWENTY-ONE.MIT.EDU (SENIOR-THREE-TWENTY-ONE.MIT.EDU [18.244.6.66]) (authenticated bits=0) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUAqZst014501 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-webmaster@theory.csail.mit.edu>; Wed, 30 Nov 2005 05:52:38 -0500 (EST) Subject: [sayan] probability From: Jesus Medrano To: 6042-webmaster@theory.csail.mit.edu Content-Type: text/plain Date: Wed, 30 Nov 2005 05:52:32 +0000 Message-Id: <1133329953.10283.2.camel@localhost> Mime-Version: 1.0 X-Mailer: Evolution 2.2.3 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.695 X-Spam-Level: * (1.695) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 319 X-UID: 1950 X-Keywords: NotJunk I'm still uncertain about the theorem in the notes about the binomial sampling. What does this mean? Is it true that only 600 people are needed to be sampled out of any size n population to get an accurate view of the entire population. This does not seem very reasonable? Is this done in practice? Jesus Medrano From mukkala@MIT.EDU Wed Nov 30 10:48:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFmn5o006596 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 10:48:49 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFmlrd008682 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 10:48:47 -0500 (EST) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-FIFTY-TWO.MIT.EDU [18.235.1.152]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFmVg9028116 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 10:48:40 -0500 (EST) Message-Id: <6.2.1.2.2.20051130103802.01ec26e8@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 30 Nov 2005 10:48:31 -0500 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 13 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1951 Content-Length: 403 X-Keywords: NotJunk In section 3.4 of the Notes, the first sentence reads, "Of the more complex distributions, the binomial distribution is surely the most important in computer science." I understand that "an enormous number of analyses in computer science come down to proving that the tails of the binomial and similar distributions are very small," but what is an example of this? Thanks, Praveen Pamidimukkala From mracich@MIT.EDU Wed Nov 30 10:49:55 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFnt5o008144 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:49:55 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFnn1C010248 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:49:49 -0500 (EST) Received: from MACGREGOR-TWO-FORTY-THREE.MIT.EDU (MACGREGOR-TWO-FORTY-THREE.MIT.EDU [18.239.5.243]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUFnfMu017323 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:49:42 -0500 (EST) Subject: [Sayan] Comments for Course Notes, Week 13 (Random Variables, Distributions, and Expectation) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 30 Nov 2005 10:49:40 -0500 Message-Id: <1133365780.7903.6.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1952 Content-Length: 151 X-Keywords: NotJunk I found section 3, "Probability Distributions", (starting on page 6) confusing. I would appreciate it if this was explained further. Moira Racich From xiaoranz@MIT.EDU Wed Nov 30 10:54:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFs75o014625 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:54:07 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFs5rd015441 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:54:06 -0500 (EST) Received: from quickstation-lobby-16-1.mit.edu (QUICKSTATION-LOBBY-16-1.MIT.EDU [18.55.1.64]) (authenticated bits=56) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFrxwd000823 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 10:53:59 -0500 (EST) Received: (from xiaoranz@localhost) by quickstation-lobby-16-1.mit.edu (8.12.9) id jAUFrxLj022350; Wed, 30 Nov 2005 10:53:59 -0500 Subject: [Hanson] Week 13 Comments From: Xiaoran Zhang To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 30 Nov 2005 10:53:58 -0500 Message-Id: <1133366038.22264.2.camel@quickstation-lobby-16-1.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1953 Content-Length: 190 X-Keywords: NonJunk NotJunk The Birthday principle on page 4-5 is very surprising. The confidence levels is interesting in that it provides a good way to determine the accuracy of the estimations made. Xiaoran Zhang From vbrobbey@MIT.EDU Wed Nov 30 10:54:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFsc5o015418 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 10:54:38 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFsbrd016088 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 10:54:37 -0500 (EST) Received: from napoleon (NAPOLEON-BONAPARTE.MIT.EDU [18.18.3.59]) (authenticated bits=0) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFsTQm001057 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 10:54:30 -0500 (EST) Message-ID: <003901c5f5c6$587d65f0$3b031212@napoleon> From: "Valery Kwasi Brobbey" To: <6042-probs@theory.csail.mit.edu> Subject: [TA-name] Week 13 Comments Date: Wed, 30 Nov 2005 10:54:27 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0036_01C5F59C.6EA83D70" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.08 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1954 Content-Length: 1208 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0036_01C5F59C.6EA83D70 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Statement 8 on the last tutor problem tricked me. In addition to that I = didn't quite get the winning stategy for the numbers game. Aren't the = numbers that the other guy puts in the envelope completely independent = of the number x that I guess? ------=_NextPart_000_0036_01C5F59C.6EA83D70 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Statement 8 on the last tutor = problem tricked=20 me. In addition to that I didn't quite get the winning stategy for the = numbers=20 game. Aren't the numbers that the other guy puts in the envelope = completely=20 independent of the number x that I = guess?
------=_NextPart_000_0036_01C5F59C.6EA83D70-- From antonk@MIT.EDU Wed Nov 30 10:55:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUFtw5o017072 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:55:58 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUFtvrd017604 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:55:57 -0500 (EST) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUFtoDu001683 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 10:55:51 -0500 (EST) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [david] discussed more fully Date: Wed, 30 Nov 2005 10:55:46 -0500 Organization: Massachusetts Institute of Technology Message-ID: <005801c5f5c6$87855880$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0059_01C5F59C.9EAF5080" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcX1xobQxfEujCPYTnKW7uj3P5LTGQ== X-Spam-Score: 0.388 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1955 Content-Length: 3304 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0059_01C5F59C.9EAF5080 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi, Is it possible to discuss the differences between 2 approximations? : Binominal density function Cumulative Binominal distribution function. Thank you, Anton. ------=_NextPart_000_0059_01C5F59C.9EAF5080 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi,

Is it possible to discuss the differences between 2 = approximations?  :

Binominal density = function

Cumulative Binominal distribution = function.

 

Thank you,

 

Anton.

 

------=_NextPart_000_0059_01C5F59C.9EAF5080-- From jeffhoff@MIT.EDU Wed Nov 30 11:19:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGJA5o004459 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 11:19:10 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUGIPUs013110 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 11:19:09 -0500 (EST) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUG6d1w007090 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 11:06:40 -0500 (EST) Message-Id: <6.2.3.4.2.20051130110528.040ef808@hesiod> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 30 Nov 2005 11:06:38 -0500 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 13 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1956 Content-Length: 123 X-Keywords: NonJunk NotJunk Section 3.4 I found the binomial distribution a little confusing in general. could there be an extension done on this? From crowell@MIT.EDU Wed Nov 30 11:21:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGLZ5o005765 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 11:21:35 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUGLXU2018157 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 11:21:34 -0500 (EST) Received: from [18.238.6.42] (EASTCAMPUS-EIGHT-O-NINE.MIT.EDU [18.238.6.42]) (authenticated bits=0) (User authenticated as crowell@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUGLPBc014376 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 11:21:27 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <2A947BFA-FB4C-4B1A-AE4A-25A9751C18C4@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Robert Crowell Subject: [Hanson] Reading Comments Date: Wed, 30 Nov 2005 11:21:26 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1957 Content-Length: 178 X-Keywords: NotJunk I would like to see more about Expected Value in lecture (this is section 5.2 on page 20). I had some trouble on the tutor problem dealing with this subject. --Rob Crowell From hmzhou@blackbird.csail.mit.edu Wed Nov 30 11:37:42 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGbg5o008222; Wed, 30 Nov 2005 11:37:42 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAUGbgV0015991; Wed, 30 Nov 2005 11:37:42 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAUGbg9J015988; Wed, 30 Nov 2005 11:37:42 -0500 Date: Wed, 30 Nov 2005 11:37:42 -0500 (EST) From: Hanson Zhou To: Rebecca Idell cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: reading 13, ta: hanson In-Reply-To: <20051129224640.81zt0eycgv0kkwww@webmail.mit.edu> Message-ID: References: <20051129224640.81zt0eycgv0kkwww@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1958 Content-Length: 474 X-Keywords: NotJunk Well, most simply: PDF(x) = Pr[R=x]. R is the random variable and PDF(x) asks for the probability that the value of R turns out to be x. -Hanson On Tue, 29 Nov 2005, Rebecca Idell wrote: > > Can you explain more about the probability density function on page 6? > > Rebecca Idell > > -- > Rebecca Idell > Massachusetts Institute of Technology > Electrical Engineering and Computer Science, Class of 2007 > > 479 Commonwealth Avenue > Boston, MA 02215 > (617) 875-0889 > From kktyan@MIT.EDU Wed Nov 30 11:42:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGgu5o009218 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 11:42:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUGgtU2018202 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 11:42:55 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUGgmof025128 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 11:42:48 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAUGgmMV004734; Wed, 30 Nov 2005 11:42:48 -0500 Received: from BURTON-FOUR-TWENTY-THREE.MIT.EDU (BURTON-FOUR-TWENTY-THREE.MIT.EDU [18.247.6.168]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 11:42:48 -0500 Message-ID: <20051130114248.jbxdzc9hm2n4kok4@webmail.mit.edu> Date: Wed, 30 Nov 2005 11:42:48 -0500 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] (Belated) Reading Comments 13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1959 Content-Length: 610 X-Keywords: NotJunk I apologize for the late reading comments - I nearly forgot to do them, and only just realized that I hadn't done them last night as I'd meant to. In this reading, I was confused by the "Expected Value" sections (starting on page 18). I wasn't sure exactly how one could compute/calculate the expected value, or what the expected value really was, but then I reread the definition of the expected value and figured it out. For the most part, I understood the rest of the reading, though; only the section on expected values was confusing. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From hmzhou@blackbird.csail.mit.edu Wed Nov 30 11:45:29 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGjT5o009492; Wed, 30 Nov 2005 11:45:29 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAUGjTDL016066; Wed, 30 Nov 2005 11:45:29 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAUGjTI5016063; Wed, 30 Nov 2005 11:45:29 -0500 Date: Wed, 30 Nov 2005 11:45:29 -0500 (EST) From: Hanson Zhou To: Elizabeth Reid cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Reading comments In-Reply-To: <49EAF327-2A3E-4CBD-8256-5E5E11D9E729@MIT.EDU> Message-ID: References: <49EAF327-2A3E-4CBD-8256-5E5E11D9E729@MIT.EDU> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1960 Content-Length: 563 X-Keywords: NotJunk Hi...I'm pretty sure I forwarded your email last week to Prof. Meyer, though you might want to check. -Hanson On Wed, 30 Nov 2005, Elizabeth Reid wrote: > I was initially surprised (and skeptical) about the numbers game > strategy, but its proof makes sense. I guess my only question would > be how to pick the "x" you hope is between L and H, but that's just a > matter of personal opinion. :) > > Oh, and did you get my reading comments last week? You never > responded, and the 6042-probs list was acting oddly when I tried to > email it... > > Elizabeth > From hmzhou@blackbird.csail.mit.edu Wed Nov 30 11:50:56 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGou5o010283; Wed, 30 Nov 2005 11:50:56 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAUGouMf016122; Wed, 30 Nov 2005 11:50:56 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAUGouXj016119; Wed, 30 Nov 2005 11:50:56 -0500 Date: Wed, 30 Nov 2005 11:50:56 -0500 (EST) From: Hanson Zhou To: Zachary Adam Ozer cc: 6042-probs@theory.lcs.mit.edu Subject: Re: Tutor Comments: Mean time to failure LN 13 Pg 20 In-Reply-To: <4f2613e50511292146q8dc62b7k4b5f45927a009b6f@mail.gmail.com> Message-ID: References: <4f2613e50511292146q8dc62b7k4b5f45927a009b6f@mail.gmail.com> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1961 Content-Length: 932 X-Keywords: NotJunk I'm not sure I understand your question. If you have an identical backup system that turns on when the first one fails then the mean time to failure would seem to be 1/f + 1/f, where f is the failure probability of each system. -Hanson On Wed, 30 Nov 2005, Zachary Adam Ozer wrote: > I was just wondering how mean time to failure is affected by redundant > systems, ie, what if I have 2 systems where one kicks in when the > other fails. It seems like the probability of failure the same, since > I'm still relying on an identical system, yet that doesn't make sense > from a practical standpoint, since then there'd be little advantage to > using redundant systems. Are they mulitiplicative? IE if the mean > proabaility of failure of one system is 1/100, does the redundancy > cause a mean probability of failure of (1/100)(1/100)? What about as > the redundancy tends towards infinity? > > Just some thoughts, > > -zozer > > From hmzhou@blackbird.csail.mit.edu Wed Nov 30 11:57:55 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGvt5o013804; Wed, 30 Nov 2005 11:57:55 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAUGvtxb016186; Wed, 30 Nov 2005 11:57:55 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAUGvtRC016182; Wed, 30 Nov 2005 11:57:55 -0500 Date: Wed, 30 Nov 2005 11:57:55 -0500 (EST) From: Hanson Zhou To: Valery Kwasi Brobbey cc: 6042-probs@theory.csail.mit.edu Subject: Re: [TA-name] Week 13 Comments In-Reply-To: <003901c5f5c6$587d65f0$3b031212@napoleon> Message-ID: References: <003901c5f5c6$587d65f0$3b031212@napoleon> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1962 Content-Length: 625 X-Keywords: NotJunk Hi Valery...by TA-name you mean Hanson :). As for the winning strategy, it is the very fact that your selection of x is random that thwarts anything your opponent might do. So yes, your selection of x is independent, and this actually ends up helping you to do better than 50%. See me if this remains confusing. -Hanson On Wed, 30 Nov 2005, Valery Kwasi Brobbey wrote: > Statement 8 on the last tutor problem tricked me. In addition to that I > didn't quite get the winning stategy for the numbers game. Aren't the > numbers that the other guy puts in the envelope completely independent > of the number x that I guess? From ryan786@MIT.EDU Wed Nov 30 12:44:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUHiB5o031754 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 12:44:11 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUHi9U2000813 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 12:44:09 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUHi6MV021483 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 12:44:07 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAUHi6Je032199; Wed, 30 Nov 2005 12:44:06 -0500 Received: from PSK-SIXTEEN.MIT.EDU (PSK-SIXTEEN.MIT.EDU [18.217.1.16]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 30 Nov 2005 12:44:06 -0500 Message-ID: <20051130124406.svxufkihkww0s000@webmail.mit.edu> Date: Wed, 30 Nov 2005 12:44:06 -0500 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [sayan] 13 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1963 Content-Length: 252 X-Keywords: NonJunk NotJunk Apologies again for the late email, I have just now been able to read the pdf. The most difficult parts were the Probability Distributions (Section 3). I do think class today will clear it up though, as that is the topic of lecture (I think). -Ryan From mitras@theory.csail.mit.edu Wed Nov 30 13:06:42 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUI6g5o006143 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 13:06:42 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUI6evv017399; Wed, 30 Nov 2005 13:06:40 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUI6Wok016852; Wed, 30 Nov 2005 13:06:32 -0500 (EST) Message-ID: <438DEA25.1040205@theory.csail.mit.edu> Date: Wed, 30 Nov 2005 13:06:29 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "David A. Nedzel" CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 13 Comments References: <438CE7F6.8000506@mit.edu> In-Reply-To: <438CE7F6.8000506@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1964 Content-Length: 314 X-Keywords: NotJunk Expectation will be covered on Friday, and variance on next Wednesday. Best, sayan David A. Nedzel wrote: > I'm fairly comfortable with this material, I've seen most of it in > 6.041. It might be nice to have a refresher during lecture on the > expectation & variance of different distributions. > > - David From mitras@theory.csail.mit.edu Wed Nov 30 13:08:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUI8Q5o006452 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 13:08:26 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAUI8PEu003533 for <6042-probs@theory.csail.mit.edu>; Wed, 30 Nov 2005 13:08:25 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUI3VnT000063; Wed, 30 Nov 2005 13:03:31 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUI3Sok016662; Wed, 30 Nov 2005 13:03:29 -0500 (EST) Message-ID: <438DE970.7030309@theory.csail.mit.edu> Date: Wed, 30 Nov 2005 13:03:28 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Sayan] Week 12 Comments References: <6.2.1.2.2.20051128140633.03e5fa00@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051128140633.03e5fa00@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1965 Content-Length: 677 X-Keywords: NotJunk Events are sets of points in the sample space. Events are disjoint when these sets are disjoint. E.g, consider a 6-sided dice roll experiment. Define Event 1, A = even number comes up, B = odd number, C = prime number. Then, A and B are disjoint, A and C are not. -sayan Praveen Pamidimukkala wrote: > In section 4.3 of the Course Notes on Probability, disjointness is > mentioned in comparison to independence. I understand that > mathematically, when two events are disjoint, Pr{AintB} = 0 as opposed > to Pr{AintB} = Pr{A} x Pr{B}, which applies for independence. > However, in terms of events, how could this be described? > > Thanks, > Praveen Pamidimukkala > From mitras@theory.csail.mit.edu Wed Nov 30 13:20:15 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUIKF5o024388 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 13:20:15 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAUIKEEu024893 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 13:20:14 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUIKDnT000919; Wed, 30 Nov 2005 13:20:13 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUIK6ok017762; Wed, 30 Nov 2005 13:20:07 -0500 (EST) Message-ID: <438DED56.60101@theory.csail.mit.edu> Date: Wed, 30 Nov 2005 13:20:06 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Moira Racich CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Comments for Course Notes, Week 13 (Random Variables, Distributions, and Expectation) References: <1133365780.7903.6.camel@localhost.localdomain> In-Reply-To: <1133365780.7903.6.camel@localhost.localdomain> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1966 Content-Length: 236 X-Keywords: NotJunk Hope today's lecture is going to help. Best, Sayan Moira Racich wrote: >I found section 3, "Probability Distributions", (starting on page 6) >confusing. I would appreciate it if this was explained further. > >Moira Racich > > > From meyer@imap.theory.csail.mit.edu Wed Nov 30 11:45:30 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUGjS5o009479 for <6042-staff@theory.csail.mit.edu>; Wed, 30 Nov 2005 11:45:28 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUGjSvv013275; Wed, 30 Nov 2005 11:45:28 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUGjOMu021156; Wed, 30 Nov 2005 11:45:24 -0500 (EST) Message-ID: <438DD722.3020304@theory.csail.mit.edu> Date: Wed, 30 Nov 2005 11:45:22 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jesus Medrano , 6042 staff <6042-staff@theory.csail.mit.edu> Subject: Re: [sayan] probability References: <1133329953.10283.2.camel@localhost> In-Reply-To: <1133329953.10283.2.camel@localhost> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 765 X-UID: 1967 X-Keywords: NotJunk Hi, You have to be careful about phrasing these polling related probabilistic claims. The sample size of 600 is derived for a particular accuracy (0.04) and for a particular probability of error (5%). The surprising thing is that, the sample size required is independent of the actual size of the population. We'll do a similar problem in class today, hopefully it will clear up these ideas. Best, Sayan Jesus Medrano wrote: >I'm still uncertain about the theorem in the notes about the binomial >sampling. What does this mean? Is it true that only 600 people are >needed to be sampled out of any size n population to get an accurate >view of the entire population. This does not seem very reasonable? Is >this done in practice? > >Jesus Medrano > > From meyer@imap.theory.csail.mit.edu Wed Nov 30 02:01:47 2005 Message-ID: <438D4E61.5070509@csail.mit.edu> Date: Wed, 30 Nov 2005 02:01:53 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yaser Khan Subject: Re: David: tp13 reading comments References: <200511300621.jAU6LNe5029473@outgoing.mit.edu> In-Reply-To: <200511300621.jAU6LNe5029473@outgoing.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 474 X-UID: 1968 X-Keywords: poisson might get mentioned VERY briefly in the last week. no t or chi. regards, A. Yaser Khan wrote: > Hi David, > > > > As a whole, the reading was a bit dense with formulas, to the extent > that I keep getting similar definitions mixed up (for instance the > distinctions between the distributions). However, my question is: will > we be concerned with the t, poisson, or chi distributions in terms of > random variables? > > > Thanks! > > > > _Yaser > From dshin@MIT.EDU Wed Nov 30 14:09:03 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUJ935o022795 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 14:09:03 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAUJ92WT014770 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 14:09:02 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUJ926j003465; Wed, 30 Nov 2005 14:09:02 -0500 (EST) Received: from [192.168.1.5] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAUJ8sok020901; Wed, 30 Nov 2005 14:08:54 -0500 (EST) Message-ID: <438DF8C0.9020501@mit.edu> Date: Wed, 30 Nov 2005 14:08:48 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine Romer CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] Email comments for reading References: <1133362273.32582.2.camel@m56-129-21.mit.edu> In-Reply-To: <1133362273.32582.2.camel@m56-129-21.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1969 Content-Length: 542 X-Keywords: NotJunk Yes, that is correct. Another way to see it is to define an indicator variable G(x) for each baby x that equals 1 if x is a girl and 0 if x is a boy. Every baby x has E[G(x)] = 0.5, so by linearity of expectation the expected value of SUM(G(x)) is 0.5*population. DS Katherine Romer wrote: >p.21 "If every couple follows the strategy of having children until they >get a girl, what will eventually happen to the fraction of girls born in >this world?" >Will it remain the same (because each couple expects to have 2 >children)? > > From mdmurray@MIT.EDU Wed Nov 30 14:19:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAUJJ55o027440 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 14:19:05 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAUJJ4Wx022899 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 14:19:04 -0500 (EST) Received: from w20-575-15.mit.edu (W20-575-15.MIT.EDU [18.187.0.34]) (authenticated bits=56) (User authenticated as mdmurray@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAUJIu2w000931 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 30 Nov 2005 14:18:57 -0500 (EST) Received: (from mdmurray@localhost) by w20-575-15.mit.edu (8.12.9) id jAUJIuZK012066; Wed, 30 Nov 2005 14:18:56 -0500 Date: Wed, 30 Nov 2005 14:18:56 -0500 (EST) From: Michael D Murray To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading question Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1970 Content-Length: 279 X-Keywords: NotJunk Hey, Sorry for the delay, the reading was quite long to be done in two days. I am curious about the Bernoulli Distribution, do we need to know it in detail? There isn't alot of information in 3.1 and I don't really understand what the point of it is. Thanks, Michael Murray From shreyes19@gmail.com Wed Nov 30 19:34:38 2005 Return-Path: Received: from nproxy.gmail.com (nproxy.gmail.com [64.233.182.198]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB10Yb5o021450 for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 19:34:38 -0500 Received: by nproxy.gmail.com with SMTP id g2so49912nfe for <6042-probs@theory.lcs.mit.edu>; Wed, 30 Nov 2005 16:34:37 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=Mf8FlvaaN9781ED5bv0lmLml1BUBTOk52nPho1zfDy02zwk3dWbtWn1UxULEZicfvL9C4u1QxDLPXQBIgopKU4p247PaIEurdu8IO06E40840KZFHU4TeF8JRCsi6dEPu2/DpVxYGuQqtmcghgBHQUBKaQHo+ZzxTLHnGEQBnaY= Received: by 10.48.238.12 with SMTP id l12mr111979nfh; Wed, 30 Nov 2005 16:34:37 -0800 (PST) Received: by 10.48.161.8 with HTTP; Wed, 30 Nov 2005 16:34:37 -0800 (PST) Message-ID: <3c33fe380511301634q4d61bc5eo1cab3a2cc984f6e8@mail.gmail.com> Date: Wed, 30 Nov 2005 19:34:37 -0500 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 13 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_3830_21133891.1133397277067" Status: RO X-UID: 1971 Content-Length: 2210 X-Keywords: NotJunk ------=_Part_3830_21133891.1133397277067 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, Here are my comments on week 13's reading comments: My question deals with the the definition of Independence presented in the notes. The notes state on page 3: Random variables R1 and R2 are independent if for all x1 in the codomain of R1, and x2 in the codomain of R2, we have: Pr {R1 =3D x1 | R2 =3D x2} =3D Pr {R1 =3D x1} =B7 Pr {R2 =3D x2} . This definition makes sense to me, but my questions is whether knowing this is true implies Independence. From other classes (6.041), we are able to prove Independence by showing how the above equation holds, but there are some questions where the equation holds but intuitively R1 and R2 are not independent. I can't think of an example off the top of my head, but i kno= w for other formulas describing Independence (like E[XY] =3D E[X]E[Y]), provi= ng the equation does not necessarily prove Independence. Thanks, Shreyes group 7 ------=_Part_3830_21133891.1133397277067 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

Here are my comments on week 13's reading comments:

My question deals with the the definition of Independence presented in the = notes.  The notes state on page 3:
Random variables R1 and R2 are independent if for all x1 in the codomain of= R1, and x2 in the
codomain of R2, we have:
Pr {R1 =3D x1 | R2 =3D x2} =3D Pr {R1 =3D x1} =B7 Pr {R2 =3D x2} .
This definition makes sense to me, but my questions is whether knowing this is true implies Independence.  From other classes (6.041), we are able to prove Independence by showing how the above equation holds, but there are some questions where the equation holds but intuitively R1 and R2 are not independent.  I can't think of an example off the top of my head, but i know for other formulas describing Independence (like E[XY] =3D E[X]E[Y]), proving the equation does not necessarily prove Independence.

Thanks,
Shreyes
group 7
------=_Part_3830_21133891.1133397277067-- From mitras@theory.csail.mit.edu Thu Dec 1 17:28:19 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB1MSIXl016741 for <6042-probs@theory.lcs.mit.edu>; Thu, 1 Dec 2005 17:28:19 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB1MSHKQ022767; Thu, 1 Dec 2005 17:28:17 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB1MSHlA012477; Thu, 1 Dec 2005 17:28:17 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB1MSDMu003838; Thu, 1 Dec 2005 17:28:13 -0500 (EST) Message-ID: <438F78FD.6080208@theory.csail.mit.edu> Date: Thu, 01 Dec 2005 17:28:13 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Shreyes Seshasai CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 13 Reading Comments References: <3c33fe380511301634q4d61bc5eo1cab3a2cc984f6e8@mail.gmail.com> In-Reply-To: <3c33fe380511301634q4d61bc5eo1cab3a2cc984f6e8@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1972 Content-Length: 1972 X-Keywords: NotJunk Good question. A. X and Y are independent if and only if (<=>) P(X =x, Y = y) = for all x, y, P(X=x)P(Y=y). This is the definition of independence. B. X and Y are independent => E[XY] = E[X] E[Y], this is a one way implication. Here's a quick proof: E[XY] = \sum_{i,j} a_i b_j P[X = a_i, Y = b_j] = \sum_{i,j} a_i P[ X = a_i] b_j P[Y = b_j], since X and Y are independent. = \sum_i a_i P[X =a_i] \sum_j b_j P[Y = b_j] = E[X]E[Y] But the converse is not necessarily true, i.e., E[XY] = E[X]E[Y] does not imply independence of X Y. Here's an example: Let (X, Y) assume values (1,0), (0,1), (-1,0), and (0,-1) with equal probabilities 1/4. Then E[X] = E[Y] = 1 * 1/4 + (-1) * 1/4 = 0. E[XY] = 0. So, we do have E[XY] =E[X]E[Y]. Now lets see if they are independent. Choose x = 0, y = 0. P(X=0) = P(Y =0) = 1/2 whereas P(X=0, Y=0) = 0. So, P(X = 0, Y=0) is not = P(X=0)P(Y=0), i.e., they are not independent. But remember, E[X+Y] = E[X] + E[Y] regardless of independence. Best, Sayan Shreyes Seshasai wrote: > Hi, > > Here are my comments on week 13's reading comments: > > My question deals with the the definition of Independence presented in > the notes. The notes state on page 3: > Random variables R1 and R2 are independent if for all x1 in the > codomain of R1, and x2 in the > codomain of R2, we have: > Pr {R1 = x1 | R2 = x2} = Pr {R1 = x1} Pr {R2 = x2} . > This definition makes sense to me, but my questions is whether knowing > this is true implies Independence. From other classes (6.041), we are > able to prove Independence by showing how the above equation holds, > but there are some questions where the equation holds but intuitively > R1 and R2 are not independent. I can't think of an example off the > top of my head, but i know for other formulas describing Independence > (like E[XY] = E[X]E[Y]), proving the equation does not necessarily > prove Independence. > > Thanks, > Shreyes > group 7 From dangut@MIT.EDU Thu Dec 1 19:39:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB20d6Iu009822 for <6042-probs@theory.csail.MIT.EDU>; Thu, 1 Dec 2005 19:39:06 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB20d5Gi015781 for <6042-probs@theory.csail.MIT.EDU>; Thu, 1 Dec 2005 19:39:05 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB20d3S7004750 for <6042-probs@theory.csail.MIT.EDU>; Thu, 1 Dec 2005 19:39:04 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jB20d308019284; Thu, 1 Dec 2005 19:39:03 -0500 Received: from MACGREGOR-THREE-TWENTY-NINE.MIT.EDU (MACGREGOR-THREE-TWENTY-NINE.MIT.EDU [18.239.6.74]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 1 Dec 2005 19:39:03 -0500 Message-ID: <20051201193903.i5g5e2m91nogg04w@webmail.mit.edu> Date: Thu, 1 Dec 2005 19:39:03 -0500 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: [Hansen] TP13 Reading Response MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1973 Content-Length: 87 X-Keywords: NotJunk I'm still comfortable with the material, I think the birthday example was pretty cool. From meyer@imap.theory.csail.mit.edu Wed Nov 16 13:12:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAGIC75o032029 for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 13:12:07 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAGIC5YB016120; Wed, 16 Nov 2005 13:12:05 -0500 (EST) Received: from [18.96.6.248] (SIMMONS-FIVE-O-THREE.MIT.EDU [18.96.6.248]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAGIBmTx029706 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Wed, 16 Nov 2005 13:11:49 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=ISO-2022-JP; format=flowed Message-Id: <488a6fb141c55248b795e2c28e8cceb1@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: [David] Week 11 Comments Date: Wed, 16 Nov 2005 13:11:48 -0500 To: 6042-probs@theory.csail.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 553 X-UID: 1974 X-Keywords: NonJunk This is really a cool way to do counting problems, and really intuitive since you only have to worry about how to select each class of element for n elements independent of all the others. My only issue is that I don't always see the generating function . Like in the "Impossible" counting problem, I didn't see that the number of oranges was O(x) = 1 + x + x ^2 + x^3 + x^4 = (1 $B!](B x^5)/(1 $B!](B x) and I don't really know how you got there. That is probably something that comes with time however. This is really slick. -Harrison From meyer@imap.theory.csail.mit.edu Wed Nov 16 14:07:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAGJ775o006315 for <6042-probs@theory.csail.MIT.EDU>; Wed, 16 Nov 2005 14:07:07 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAGJ75YB017370 for <6042-probs@theory.csail.MIT.EDU>; Wed, 16 Nov 2005 14:07:06 -0500 (EST) Received: from m66-080-1.mit.edu (M66-080-1.MIT.EDU [18.63.1.1]) (authenticated bits=56) (User authenticated as lye@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAGJ72hh023705 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 16 Nov 2005 14:07:03 -0500 (EST) Received: (from lye@localhost) by m66-080-1.mit.edu (8.12.9) id jAGJ72JD022931; Wed, 16 Nov 2005 14:07:02 -0500 Subject: [Sayan] Week 11 comments From: Linda Ye To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 16 Nov 2005 14:07:02 -0500 Message-Id: <1132168022.22906.2.camel@m66-080-1.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 175 X-UID: 1975 X-Keywords: NonJunk Section 3.2 Finding a Closed Form for fib gen func I don't really understand why we split the denominators. Why does solving for the alphas give us the nth Fibonacci number? From meyer@imap.theory.csail.mit.edu Wed Nov 16 16:46:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAGLkv5o024930 for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 16:46:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAGLktYB023018 for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 16:46:56 -0500 (EST) Received: from [18.238.6.75] (EASTCAMPUS-EIGHT-FORTY-TWO.MIT.EDU [18.238.6.75]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAGLkq7J013219 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 16:46:52 -0500 (EST) Message-ID: <437BA8B1.2090306@mit.edu> Date: Wed, 16 Nov 2005 16:46:25 -0500 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 11 Comments Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 394 X-UID: 1976 X-Keywords: NonJunk On page 1, the lecture notes state "A generating function is a formal power series in the sense that we usually regard x as a placeholder rather than a number. Only in rare cases will we actually evaluate a generating function by letting x take a real number value, so we generally ignore the issue of convergence." In what cases might we actually evaluate a generating function? Jason. From meyer@imap.theory.csail.mit.edu Wed Nov 16 18:30:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAGNUK5o019715 for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 18:30:20 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAGNUJYB016924 for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 18:30:19 -0500 (EST) Received: from christope8f0c6 (MACGREGOR-ONE-TWENTY-FOUR.MIT.EDU [18.239.5.124]) (authenticated bits=0) (User authenticated as yangc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAGNUADm015510 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 16 Nov 2005 18:30:11 -0500 (EST) Message-Id: <200511162330.jAGNUADm015510@outgoing.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 11 Comments Date: Wed, 16 Nov 2005 18:32:07 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0066_01C5EADC.0CE44B80" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXrBfUkhsykDmfkSFKhOYrW+a1kZg== X-Spam-Score: 0.179 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2228 X-UID: 1977 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0066_01C5EADC.0CE44B80 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I don't really understand how the Convolution rule works. Will we be covering it in class on Friday? Also, the apples/oranges thing at the end is pretty cool. Thanks, Chris Yang ------=_NextPart_000_0066_01C5EADC.0CE44B80 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

I don’t really understand how the Convolution = rule works.  Will we be covering it in class on Friday?  Also, the = apples/oranges thing at the end is pretty cool.

 

Thanks,

Chris Yang

------=_NextPart_000_0066_01C5EADC.0CE44B80-- From hsoumare@MIT.EDU Wed Nov 16 21:52:36 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAH2qa5o025699 for <6042-probs@theory.lcs.mit.edu>; Wed, 16 Nov 2005 21:52:36 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAH2qZCE005738 for <6042-probs@theory.lcs.mit.edu>; Wed, 16 Nov 2005 21:52:35 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAH2qZM3014716 for <6042-probs@theory.lcs.mit.edu>; Wed, 16 Nov 2005 21:52:35 -0500 (EST) Received: from [18.241.6.25] (NEW-TWO-EIGHTY.MIT.EDU [18.241.6.25]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAH2qSMu013640 for <6042-probs@theory.lcs.mit.edu>; Wed, 16 Nov 2005 21:52:28 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <69C53F7D-7FEA-40EA-B7AD-27850D67CB59@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: {Sayan} Weekly Email Comment Date: Wed, 16 Nov 2005 22:01:49 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1978 Content-Length: 142 X-Keywords: NonJunk Hi I would like you to go over how the closed form of the nth coefficient of the fibonacci series was obtained on page 9. Thanks. Hamidou From xiaoranz@MIT.EDU Thu Nov 17 00:23:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAH5Nw5o009165 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 00:23:58 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAH5NvYB027223 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 00:23:57 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAH5Nu7V020678 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 00:23:56 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jAH5NucA018057; Thu, 17 Nov 2005 00:23:56 -0500 Received: from MCCORMICK-SEVEN-SIXTY.MIT.EDU (MCCORMICK-SEVEN-SIXTY.MIT.EDU [18.240.7.249]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 17 Nov 2005 00:23:56 -0500 Message-ID: <20051117002356.q2ggn1b0cdoo8sw4@webmail.mit.edu> Date: Thu, 17 Nov 2005 00:23:56 -0500 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Weekly Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1979 Content-Length: 503 X-Keywords: NonJunk I thought the generating functions are interesting in that they can help solve counting problems in a whole new way. The "Impossible" counting problem solution is truly awesome. ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From mdmurray@MIT.EDU Thu Nov 17 03:08:57 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAH88v5o004712 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 03:08:57 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAH88tCE021568; Thu, 17 Nov 2005 03:08:56 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAH88tM3024254; Thu, 17 Nov 2005 03:08:55 -0500 (EST) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAH88mMu026115; Thu, 17 Nov 2005 03:08:48 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <0D51FC0A-5E73-4CA7-846E-9E0334875C73@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu, 6042-probs@theory.lcs.mit.edu From: Michael Murray Subject: [Hanson] Generating Functions Notes Date: Thu, 17 Nov 2005 03:07:43 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1980 Content-Length: 494 X-Keywords: NonJunk Having never seen generating functions before I think the notes and lecture where a great introduction and everything so far is pretty clear. I like the use of whiteboard in lecture today, I thought it was much easier to follow than the slides and the lecture was at a much nicer pace than usual. I am however a little unclear in the explanation of the product rule. I don't really understand the table and I don't know why c_n = a_0*b_n + a_1*b_n-1 + a_2*b_n-2... Thanks, Michael From shreyes19@gmail.com Thu Nov 17 13:37:13 2005 Return-Path: Received: from nproxy.gmail.com (nproxy.gmail.com [64.233.182.194]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAHIbC5o029031 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 13:37:12 -0500 Received: by nproxy.gmail.com with SMTP id l36so489563nfa for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 10:37:12 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=lsgHD6RMcULSavVlis8wZ0uUjyKX8f6OdLpANjlqbpqZ0kzkq/w6E9u34t2dTwGpex3ic30GCgfF7RBHTBIu4z6mgn1PJUIqav30h88EjUB9O79O+lkkb9NTSt00Z2oIWbfDeFxiZu+q+98NyBXw/QpqOHD2dzOKZkp2SVxDEFQ= Received: by 10.48.229.10 with SMTP id b10mr440397nfh; Thu, 17 Nov 2005 10:37:12 -0800 (PST) Received: by 10.48.161.8 with HTTP; Thu, 17 Nov 2005 10:37:12 -0800 (PST) Message-ID: <3c33fe380511171037i4f862502kda5ec4d351eb5f3e@mail.gmail.com> Date: Thu, 17 Nov 2005 13:37:12 -0500 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 11 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_50059_899955.1132252632166" Status: RO X-UID: 1981 Content-Length: 1877 X-Keywords: NonJunk ------=_Part_50059_899955.1132252632166 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, That last "impossible" counting problem was cool. Funny though how the numbers seemed to cancel perfectly... It's still cool to know that the procedure works like that, even if the answer doesn't turn out as nice ever= y time. Since most of the first half of the notes has been done in lecture already, I'm guessing the rest will be done tomorrow. The question I have regarding the second section is in reference to the Convolution Rule (page 10). Is there an easy way to compute the generating function if the two sets A and = B are not disjoint? It's hard for me to even think of a common scenario when this is the case, but I'd be interested to see if there's an easy way someone has though of to do it. Thanks, Shreyes ------=_Part_50059_899955.1132252632166 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

That last "impossible" counting problem was cool.  Funny tho= ugh how the numbers seemed to cancel perfectly... It's still cool to know that the procedure works like that, even if the answer doesn't turn out as nice every time.
Since most of the first half of the notes has been done in lecture already, I'm guessing the rest will be done tomorrow.  The question I have regarding the second section is in reference to the Convolution Rule (page 10).  Is there an easy way to compute the generating function if the two sets A and B are not disjoint?  It's hard for me to even think of a common scenario when this is the case, but I'd be interested to see if there's an easy way someone has though of to do it.

Thanks,
Shreyes
------=_Part_50059_899955.1132252632166-- From hmzhou@blackbird.csail.mit.edu Thu Nov 17 16:18:19 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAHLIJ5o004898; Thu, 17 Nov 2005 16:18:19 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAHLII8n016097; Thu, 17 Nov 2005 16:18:18 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAHLIIt6016094; Thu, 17 Nov 2005 16:18:18 -0500 Date: Thu, 17 Nov 2005 16:18:18 -0500 (EST) From: Hanson Zhou To: Michael Murray cc: 6042-probs@theory.csail.mit.edu Subject: Re: [Hanson] Generating Functions Notes In-Reply-To: <0D51FC0A-5E73-4CA7-846E-9E0334875C73@mit.edu> Message-ID: References: <0D51FC0A-5E73-4CA7-846E-9E0334875C73@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1982 Content-Length: 1092 X-Keywords: The table is merely a multiplication table to show you the result of multiplying a term from each sum. From it you should be able to see which terms contribute to the coefficient of x^n in the final product. Specifically, each pair that multiplies to give x^n contributes to the coefficient of x^n in the final answer. The pairs that contribute are therefore: (0,n), (1, n-1), (2, n-2)... and hence the coefficient on x^n in the final product is: c_n = a_0*b_n + a_1*b_n-1 + a_2*b_n-2... See me if you are still confused. -Hanson On Thu, 17 Nov 2005, Michael Murray wrote: > Having never seen generating functions before I think the notes and > lecture where a great introduction and everything so far is pretty > clear. I like the use of whiteboard in lecture today, I thought it > was much easier to follow than the slides and the lecture was at a > much nicer pace than usual. > > I am however a little unclear in the explanation of the product rule. > I don't really understand the table and I don't know why > > c_n = a_0*b_n + a_1*b_n-1 + a_2*b_n-2... > > Thanks, > Michael > From ozcan@MIT.EDU Thu Nov 17 16:41:13 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAHLfD5o018719 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 16:41:13 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAHLfCv0013056 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 16:41:12 -0500 (EST) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAHLf4ol001023 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 16:41:05 -0500 (EST) Message-Id: <5.2.1.1.2.20051117163659.02394d48@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 17 Nov 2005 16:41:02 -0500 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [Hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1983 Content-Length: 208 X-Keywords: Section 3.2 as a whole, i.e. the construction of a closed form for the Fibonacci numbers: It is quite impressive to be able to code this much information into only one simple function, I am really amazed. From fluff@MIT.EDU Thu Nov 17 17:24:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAHMOJ5o010091 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 17:24:19 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAHMOHYW029150 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 17:24:18 -0500 (EST) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAHMOBk3018759 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 17:24:12 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 11 reading Date: Thu, 17 Nov 2005 17:22:08 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1984 Content-Length: 346 X-Keywords: NonJunk "Amazing!" It's still not entirely intuitive to me why the Convolution Rule works, but overall the reading was interesting, flowed well, and made sense. I liked how the reading was shorter and all on one topic, instead of like previous readings that were on a million different things all glommed onto some 25-page monstrosity. ~Crystal From mracich@MIT.EDU Thu Nov 17 18:25:56 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAHNPu5o029959 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 18:25:56 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAHNPuv0017621 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 18:25:56 -0500 (EST) Received: from MACGREGOR-FOUR-O-ONE.MIT.EDU (MACGREGOR-FOUR-O-ONE.MIT.EDU [18.239.6.146]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAHNPlMu018802 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 18:25:48 -0500 (EST) Subject: [Sayan] Comments for Course Notes, Week 11 (Generating Functions) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Thu, 17 Nov 2005 18:25:47 -0500 Message-Id: <1132269947.8084.22.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1985 Content-Length: 276 X-Keywords: I found section 3.2, "Finding a Closed Form", (starting on page 8) confusing, both in the notes and when it was covered in lecture. I would appreciate it if this was explained further. I think I would also find a brief partial fractions review very helpful. Moira Racich From ereid@MIT.EDU Thu Nov 17 18:36:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAHNaU5o031118 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 18:36:30 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAHNaTYW022105 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 18:36:29 -0500 (EST) Received: from [18.242.6.198] (NEXT-FOUR-FIFTY-THREE.MIT.EDU [18.242.6.198]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAHNaMYF006340 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 18:36:22 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <94C740B9-2FE5-4D4D-BC5D-6FD20BC9D421@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Reading Comments Date: Thu, 17 Nov 2005 18:36:10 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1986 Content-Length: 148 X-Keywords: NonJunk The reading makes sense, but I feel like I wouldn't be able to do it myself. Is there anywhere I can go that would help me practice this stuff? From dangut@MIT.EDU Thu Nov 17 19:13:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI0Dq5o018276 for <6042-probs@theory.csail.MIT.EDU>; Thu, 17 Nov 2005 19:13:52 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI0DpYW016474 for <6042-probs@theory.csail.MIT.EDU>; Thu, 17 Nov 2005 19:13:51 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI0Dn9K013316 for <6042-probs@theory.csail.MIT.EDU>; Thu, 17 Nov 2005 19:13:49 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAI0DnGM007198; Thu, 17 Nov 2005 19:13:49 -0500 Received: from MACGREGOR-FOUR-EIGHTY-THREE.MIT.EDU (MACGREGOR-FOUR-EIGHTY-THREE.MIT.EDU [18.239.6.228]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 17 Nov 2005 19:13:49 -0500 Message-ID: <20051117191349.qo4qd0bqx2os404s@webmail.mit.edu> Date: Thu, 17 Nov 2005 19:13:49 -0500 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: [Hansen] TP11 Reading Response MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1987 Content-Length: 160 X-Keywords: NonJunk These all make logical sense, but I was wondering if by chance you have a list or know where to find one so that I can review geometric summing? Thanks, Daniel From petek@MIT.EDU Thu Nov 17 20:53:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI1r85o010400 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 20:53:08 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI1r7YW012863 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 20:53:07 -0500 (EST) Received: from [18.194.1.37] (SN-THIRTY-SEVEN.MIT.EDU [18.194.1.37]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI1r5s5028547 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 20:53:05 -0500 (EST) Message-ID: <437D340D.1000309@mit.edu> Date: Thu, 17 Nov 2005 20:53:17 -0500 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] This Week's Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1988 Content-Length: 377 X-Keywords: NonJunk I'm still shaky on why a generating function is called what it is... Also need some more practice on counting with these things. It is kinda cool that you can solve some complex probs that way.... -P -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: From tylerw@MIT.EDU Thu Nov 17 21:10:04 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI2A35o022126 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 21:10:03 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI2A3v0023101 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 21:10:03 -0500 (EST) Received: from [18.246.1.160] (TYLERW.MIT.EDU [18.246.1.160]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAI2A0ok013801 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 21:10:01 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <3431AB96-1519-4D5A-9413-03E2898B828C@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.csail.mit.edu From: Tyler Williams Subject: [TA-name] Week 11 Comments Date: Thu, 17 Nov 2005 21:09:48 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1989 Content-Length: 60 X-Keywords: NonJunk I thought the impossible counting problem was interesting. From bilodeau@MIT.EDU Thu Nov 17 21:32:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI2Wv5o024360 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 21:32:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI2WuYW005191 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 21:32:56 -0500 (EST) Received: from pbilodeau (MACGREGOR-TWO-TWENTY-ONE.MIT.EDU [18.239.5.221]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI2Wmbc004247 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 21:32:48 -0500 (EST) Message-Id: <200511180232.jAI2Wmbc004247@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [jelani] reading comments Date: Thu, 17 Nov 2005 21:32:38 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5EBBE.6F45DBD0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXr6FeVrw3dzlYHRUCTZ2eGwvMXmQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1990 Content-Length: 2064 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5EBBE.6F45DBD0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit In this week's reading, part of what was hard to understand was the way the formula for the infinite sum was used to describe the series that generates the coefficients. Trying to think about what is being substituted for x and what operations happen on each term is a little difficult for me to grasp. ------=_NextPart_000_0000_01C5EBBE.6F45DBD0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

In this week’s reading, part of what was hard = to understand was the way the formula for the infinite sum was used to = describe the series that generates the coefficients.  Trying to think about = what is being substituted for x and what operations happen on each term is a = little difficult for me to grasp.

------=_NextPart_000_0000_01C5EBBE.6F45DBD0-- From jehan@MIT.EDU Thu Nov 17 21:33:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI2XV5o024721 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 21:33:32 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI2XUYW005459 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 21:33:30 -0500 (EST) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI2XSLZ004350 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 21:33:29 -0500 (EST) Message-Id: <5.2.1.1.2.20051117212717.00bc1968@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 17 Nov 2005 21:31:46 -0500 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 11 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1991 Content-Length: 63 X-Keywords: NonJunk I don't understand relating generating functions to counting. From dnreshef@MIT.EDU Thu Nov 17 21:36:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI2aY5o028373 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 21:36:34 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI2aWYW007236 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 21:36:32 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI2aQ0m004741 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 21:36:26 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAI2aQT1002449; Thu, 17 Nov 2005 21:36:26 -0500 Received: from MACGREGOR-EIGHTY-SIX.MIT.EDU (MACGREGOR-EIGHTY-SIX.MIT.EDU [18.239.5.86]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 17 Nov 2005 21:36:26 -0500 Message-ID: <20051117213626.gdig760cb4oc0ck8@webmail.mit.edu> Date: Thu, 17 Nov 2005 21:36:26 -0500 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading 11 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1992 Content-Length: 310 X-Keywords: NonJunk Of all of the course material we have covered thus far, this has been the most interesting, and I would like to get into it a lot more in class. I had a little bit of trouble with the section on counting with generating functions on page 9. Can we go over how to build generating functions that count? -Dave From hectorb@MIT.EDU Thu Nov 17 22:07:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI37C5o031052 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 22:07:12 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI379YW024840; Thu, 17 Nov 2005 22:07:09 -0500 (EST) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI376Be009319 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 17 Nov 2005 22:07:06 -0500 (EST) Message-ID: <437D4555.2070302@mit.edu> Date: Thu, 17 Nov 2005 22:07:01 -0500 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 11 Reading Comment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1993 Content-Length: 222 X-Keywords: NonJunk This week's reading was very straight forward; there wasn't anything I had a hard time understanding. I found the last section the most interesting, being able to generate functions for nasty counting problems. -Hector From icharny@MIT.EDU Thu Nov 17 22:08:33 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI38X5o031285 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 22:08:33 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAI38W5b021833 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 22:08:32 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI38WAe009343 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 22:08:32 -0500 (EST) Received: from [127.0.0.1] (EASTCAMPUS-NINE-EIGHTY-THREE.MIT.EDU [18.238.6.216]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAI38JMw028294 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 22:08:30 -0500 (EST) Message-ID: <437D45A4.3090007@mit.edu> Date: Thu, 17 Nov 2005 22:08:20 -0500 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 11 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1994 Content-Length: 98 X-Keywords: NonJunk I don't understand counting with functions (section 4, page 9). Please clarify this in lecture. From vixen@MIT.EDU Thu Nov 17 22:38:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI3c25o010006 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 22:38:02 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI3c1YW012690 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 22:38:01 -0500 (EST) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI3bvxW013920 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 22:37:59 -0500 (EST) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Thu, 17 Nov 2005 22:35:56 -0500 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Subject: LN11 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1995 Content-Length: 69 X-Keywords: I found the part about counting with generating functions confusing. From dowgun@MIT.EDU Thu Nov 17 22:58:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI3wa5o014963 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 22:58:36 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI3wZYW025233 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 22:58:35 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI3wSJq017238 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 17 Nov 2005 22:58:29 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAI3wSFn010054; Thu, 17 Nov 2005 22:58:28 -0500 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 17 Nov 2005 22:58:28 -0500 Message-ID: <20051117225828.bsxr9oqa0j0gw44g@webmail.mit.edu> Date: Thu, 17 Nov 2005 22:58:28 -0500 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] reading question MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1996 Content-Length: 553 X-Status: A X-Keywords: NonJunk So I realized that the example where we found the coefficiants for the Fibonacci sequence doesn't really provide a clean fomulaic way to find coefficiants given a closed form. I guess there's just always going to be some dirty math to do. Anyway, I was wondering if using the Taylor expansion to find the coefficiants was always a viable option. I mean, I realize that it would be a real pain to take the derivative of an expression say, thirty times, but it just seems like a more surefire way to get the answer if it really is foolproof. Thanks, Neil From brevzin@MIT.EDU Thu Nov 17 23:30:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI4Uh5o021455 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 23:30:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI4UgYW013921 for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 23:30:42 -0500 (EST) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI4UZwH022543 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 17 Nov 2005 23:30:36 -0500 (EST) Message-Id: <6.1.2.0.1.20051117232905.019f2b98@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Thu, 17 Nov 2005 23:30:35 -0500 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1997 Content-Length: 421 X-Keywords: For the convolution rule, I think it's really confusing that it's called the convolution rule. It really has nothing to do with the idea of a convolution from analysis, does it ? I'm really used to the convolution of f and g (f*g) being the integral of f(y)g(x-y)dy. Why IS it called that then ? Why not like... disjoint union rule ? Or Countable Additivity, which would be consistent with measure theory... Barry From tonyng@MIT.EDU Thu Nov 17 23:56:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI4uj5o023106 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 23:56:45 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI4uhYW028803 for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 23:56:44 -0500 (EST) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI4udHe026272 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 17 Nov 2005 23:56:41 -0500 (EST) Message-Id: <5.2.1.1.2.20051117235257.02142d78@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 17 Nov 2005 23:56:41 -0500 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments: Week 11 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1998 Content-Length: 350 X-Status: A X-Keywords: NonJunk I found the fruit counting problem ("An 'Impossible' Counting Problem, pg12) to be very interesting. I didn't understand the need for generating functions during the last lecture and it seemed excessively complicated to do, however, now I understand why it is necessary because it may be the only way we know how to solve some problems. - Tony From jacques@MIT.EDU Fri Nov 18 00:17:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI5HD5o032741 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 00:17:13 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI5HCYW010134 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 00:17:12 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI5H6Qr029343 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 00:17:06 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAI5H6xb017829; Fri, 18 Nov 2005 00:17:06 -0500 Received: from EASTCAMPUS-ONE-O-THREE-HUNDRED-FORTY-EIGHT.MIT.EDU (EASTCAMPUS-ONE-O-THREE-HUNDRED-FORTY-EIGHT.MIT.EDU [18.238.7.81]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 00:17:06 -0500 Message-ID: <20051118001706.zbxgn5qon6lw8ckg@webmail.mit.edu> Date: Fri, 18 Nov 2005 00:17:06 -0500 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 11 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1999 Content-Length: 311 X-Keywords: NonJunk Page 11-12. I never thought the Taylor Series, which I'd seen and used many times to approximate functions, could be related combinatorics (aka. most surprising passage). I actually thought generating functions were a difficult concept until I read this example, since it was a series I was so familiar with. From sergiob@MIT.EDU Fri Nov 18 00:25:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI5PQ5o001795 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:25:26 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI5PPYW014380 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:25:25 -0500 (EST) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI5PMuu000375 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:25:22 -0500 (EST) Message-Id: <5.2.1.1.2.20051117222122.00c5c020@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 17 Nov 2005 22:25:17 -0700 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 11 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2000 Content-Length: 362 X-Keywords: NonJunk I thought it was amazing how useful generating functions can be to solve counting problems. I would like to know if there is always an easy way to find a closed form for the nth term of a sequence, if you know the value of the generating function. That is, other than differentiating the function to get the Taylor expansion term. Sergio Bacallado Group A From jjmonzon@MIT.EDU Fri Nov 18 00:38:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI5cu5o006909 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:38:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI5ctYW021231 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:38:55 -0500 (EST) Received: from JOSHDESKTOP.mit.edu (BURTON-TWO-TWELVE.MIT.EDU [18.247.5.212]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI5cqof001946 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:38:54 -0500 (EST) Message-Id: <6.2.3.4.2.20051118003612.01e0a888@po9.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 18 Nov 2005 00:38:58 -0500 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: David - Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2001 Content-Length: 391 X-Keywords: NonJunk I'm a little confused on the convolution rule. Perhaps this would be explained a little more in class. Also, I didn't really understand how they obtained the answer in the last problem of the tutorial. Hope tomorrow's lecture clears this up. Josh Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From aston@MIT.EDU Fri Nov 18 00:51:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI5pt5o010259 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:51:55 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI5psYW027870 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:51:54 -0500 (EST) Received: from astonlaptop (NEW-SEVENTY-TWO.MIT.EDU [18.241.5.72]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI5pl9e003405 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 00:51:48 -0500 (EST) Message-Id: <200511180551.jAI5pl9e003405@outgoing.mit.edu> Reply-To: From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 11 Comments Date: Fri, 18 Nov 2005 00:51:47 -0500 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5EBDA.417239D0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXsBClqDP/jYmp1QGqsP8mEQisXug== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.778 X-Spam-Level: ** (2.778) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2002 Content-Length: 1335 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5EBDA.417239D0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I was really impressed by the fruit problem (pages 12 and 13), but I don't feel like the explanation for why the convolution is the right answer to problems like these (page 10) was a very convincing one. - Aston ------=_NextPart_000_0000_01C5EBDA.417239D0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable
I was = really=20 impressed by the fruit problem (pages 12 and 13), but I don't feel like = the=20 explanation for why the convolution is the right answer to problems like = these=20 (page 10) was a very convincing one.
 
    -=20 Aston
------=_NextPart_000_0000_01C5EBDA.417239D0-- From cwong08@MIT.EDU Fri Nov 18 01:34:01 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI6Y15o015454 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 01:34:01 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAI6Y05b001251 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 01:34:00 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI6Y0Ae015586 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 01:34:00 -0500 (EST) Received: from [127.0.0.1] (CWONG08.MIT.EDU [18.216.0.136]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAI6XmMv007563 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 01:33:53 -0500 (EST) Message-ID: <437D75C7.9000305@mit.edu> Date: Fri, 18 Nov 2005 01:33:43 -0500 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 11 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2003 Content-Length: 101 X-Keywords: NonJunk The part about convolution (pg 10) is confusing to me. I don't really get what it's trying to say. From scot@MIT.EDU Fri Nov 18 01:57:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI6vZ5o025168 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 01:57:35 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI6vYYW029724 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 01:57:34 -0500 (EST) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI6vPTM009350 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 01:57:27 -0500 (EST) Message-ID: <437D7B55.4000408@mit.edu> Date: Fri, 18 Nov 2005 01:57:25 -0500 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Comments Jelani Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2004 Content-Length: 267 X-Status: A X-Keywords: Hi, The biggest question I had from this week of notes was the reason that, while Prof. Meyer was on the board, he choose to round up on the sum, after erasing the second term. I was confused, and could not understand the conflicting explanations. Thanks, Scot From cbossard@MIT.EDU Fri Nov 18 03:03:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI83l5o020660 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 03:03:47 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI83kYW029169 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 03:03:46 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI83eUa013584 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 03:03:40 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAI83ed1018729; Fri, 18 Nov 2005 03:03:40 -0500 Received: from 1116host44.starwoodbroadband.com (1116host44.starwoodbroadband.com [12.146.116.44]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 03:03:40 -0500 Message-ID: <20051118030340.i5w8dncex73kocgg@webmail.mit.edu> Date: Fri, 18 Nov 2005 03:03:40 -0500 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 11 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2005 Content-Length: 737 X-Keywords: NonJunk I find this to be an interesting topic although I have trouble reproducing and creating these functions on my own they generally make sense to me when I see them explained. In section 2.5 on products I am not seeing/understanding how the terms of the diagonal were collected so I don't see why it's not aobo + a1b1 instead of aobn etc. Again with although the convolution rule makes some sense to me in section 4.2 I dont see how they are getting the numbers from the formula that was created. I can see that each is (1+x) and they should be multiplied. Intuively I understand that there's 1 way to choose 0 elements, 2 ways to choose 1, etc but I do not see how this information can be attained from the 1+2x+x^2 equation. Cynthia From kjhollen@MIT.EDU Fri Nov 18 03:05:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI85m5o020780 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 03:05:48 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI85lYW000154 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 03:05:47 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI85f4Q013703 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 03:05:41 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAI85fgN031472; Fri, 18 Nov 2005 03:05:41 -0500 Received: from c-65-96-174-14.hsd1.ma.comcast.net (c-65-96-174-14.hsd1.ma.comcast.net [65.96.174.14]) (User authenticated as kjhollen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 03:05:41 -0500 Message-ID: <20051118030541.5q6lwdoe1k0ockw8@webmail.mit.edu> Date: Fri, 18 Nov 2005 03:05:41 -0500 From: Kate Hollenbach To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] counting example MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.11 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2006 Content-Length: 113 X-Keywords: NonJunk the counting example for generating functions (p 12-13) is too cool!! hope we do some problems like it in class. From mpapi@MIT.EDU Fri Nov 18 03:31:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI8Vv5o022344 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 03:31:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI8VtYW011622 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 03:31:55 -0500 (EST) Received: from mpapi.MIT.EDU (MPAPI.MIT.EDU [18.239.4.219]) (authenticated bits=0) (User authenticated as mpapi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI8Vjge015001 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 03:31:45 -0500 (EST) Subject: [Jelani] week 11 comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 18 Nov 2005 03:33:27 -0500 Message-Id: <1132302808.31951.85.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2007 Content-Length: 300 X-Keywords: NonJunk Section 3.2, page 8. I could use a review or appendix or something regarding partial fractions. The examples are good, but once in a while there are functions where the numerators of the split terms are polynomials - anything more complex than the given examples I don't remember how to do. - Matt From veracarr@MIT.EDU Fri Nov 18 03:59:05 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI8x55o028025 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 03:59:05 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI8x4v0004487 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 03:59:04 -0500 (EST) Received: from [192.168.2.34] (c-67-176-37-51.hsd1.co.comcast.net [67.176.37.51]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAI8wuok000032 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 03:58:57 -0500 (EST) Message-ID: <437DB426.1010801@mit.edu> Date: Fri, 18 Nov 2005 03:59:50 -0700 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Generating Functions Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2008 Content-Length: 222 X-Keywords: NonJunk Genearting Functions pg 1 When trying to find the sequence for a generating function when you have written the function in closed form what's the best way to find the sequence? Do a taylor expansion on the closed form? From bakster@MIT.EDU Fri Nov 18 04:02:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI9295o028402 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 04:02:09 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI928YW024820 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 04:02:08 -0500 (EST) Received: from [18.246.6.125] (BEXLEY-THREE-EIGHTY.MIT.EDU [18.246.6.125]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI926rG016482 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 04:02:06 -0500 (EST) Message-ID: <437D988C.8020808@mit.edu> Date: Fri, 18 Nov 2005 04:02:04 -0500 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] Lecture Notes 11 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2009 Content-Length: 146 X-Keywords: NonJunk What kind of situations would require us to use values for x? In the readings it said that they are unusual or rare, I believe. Alexander Bakst From lmccart@MIT.EDU Fri Nov 18 04:19:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI9Jn5o030028 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 04:19:49 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI9JmYW002337 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 04:19:48 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI9Jfsp017236 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 04:19:41 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAI9JfYh002961; Fri, 18 Nov 2005 04:19:41 -0500 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 04:19:41 -0500 Message-ID: <20051118041941.dpepyf0a3wf4k048@webmail.mit.edu> Date: Fri, 18 Nov 2005 04:19:41 -0500 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] week 11 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2010 Content-Length: 118 X-Keywords: NonJunk I did not find anything particularly confusing in this reading, but it was a pretty good reading nonetheless -Lauren From mukkala@MIT.EDU Fri Nov 18 04:23:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI9N45o009796 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 04:23:04 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI9N2YW003746 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 04:23:02 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI9Mue9017375 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 04:22:56 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAI9MuI8003153; Fri, 18 Nov 2005 04:22:56 -0500 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as mukkala@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 04:22:55 -0500 Message-ID: <20051118042255.gnsdycackpmsk80c@webmail.mit.edu> Date: Fri, 18 Nov 2005 04:22:55 -0500 From: Praveen Pamidimukkala To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 11 Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2011 Content-Length: 229 X-Keywords: I'm not sure that I completely understand the last part of the reading. The application of generating functions to the fruit problem is kind of confusing. I hope to go over it in class on friday. Thanks, Praveen Pamidimukkala From kevin08@MIT.EDU Fri Nov 18 05:07:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIA7o5o007134 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 05:07:50 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIA7mYW022208 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 05:07:49 -0500 (EST) Received: from kevlar.mit.edu (NEXT-FIVE-EIGHTY-SIX.MIT.EDU [18.242.7.75]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIA7ekQ019043 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 05:07:41 -0500 (EST) Message-Id: <6.2.3.4.2.20051118050105.029324e0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 18 Nov 2005 05:07:35 -0500 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Notes 11 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2012 Content-Length: 237 X-Status: A X-Keywords: NonJunk Hi, I thought this reading was fairly straightforward and understood all of it. With that being said though, if Prof. Meyers could go over proving the bookkeeper rule (Section 4.3) in class, that would be instructive. Thanks, Kevin From antonk@MIT.EDU Fri Nov 18 05:56:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIAuK5o026391 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 05:56:20 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIAuJYW011817 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 05:56:19 -0500 (EST) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIAuCNl020825 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 05:56:13 -0500 (EST) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [david] needs more explanation Date: Fri, 18 Nov 2005 05:56:11 -0500 Organization: Massachusetts Institute of Technology Message-ID: <000501c5ec2e$b06336b0$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0006_01C5EC04.C78D2EB0" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXsLq5xzcUwNM7wQvaa4UevSlklXQ== X-Spam-Score: 0.758 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2013 Content-Length: 2554 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0006_01C5EC04.C78D2EB0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit HI, I had problems with coming up with the generating functions. Is there anything else that can be used apart from the sums? Anton. ------=_NextPart_000_0006_01C5EC04.C78D2EB0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

HI,

I had problems with coming up with the generating = functions.

Is there anything else that can be used apart from = the sums?

 

Anton.

 

------=_NextPart_000_0006_01C5EC04.C78D2EB0-- From arup@MIT.EDU Fri Nov 18 06:09:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIB9E5o027665 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 06:09:14 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIB9DYW016940 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 06:09:13 -0500 (EST) Received: from [18.247.7.36] (BURTON-FIVE-FORTY-SEVEN.MIT.EDU [18.247.7.36]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIB901h021311 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 06:09:08 -0500 (EST) Message-ID: <437DB623.2090907@mit.edu> Date: Fri, 18 Nov 2005 06:08:19 -0500 From: Arup Sarma User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Week 11 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2014 Content-Length: 126 X-Keywords: NonJunk Section 4.2: I think this should be covered in lecture since I wasn't sure about the choosing with repetition stuff. |Arup| From lkini@MIT.EDU Fri Nov 18 08:36:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIDad5o028758 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 08:36:39 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIDacYW000180 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 08:36:38 -0500 (EST) Received: from [18.239.6.100] (MACGREGOR-THREE-FIFTY-FIVE.MIT.EDU [18.239.6.100]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIDaUTh006684 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 08:36:31 -0500 (EST) Message-ID: <437DD8D5.3040909@mit.edu> Date: Fri, 18 Nov 2005 08:36:21 -0500 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 11 reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2015 Content-Length: 260 X-Keywords: NonJunk Hi Hanson, The most surprising thing I read in the lecture notes was how to use generating functions to count. This is immensely helpful for counting the larger, difficult counting questions such as that in the last few pages of the lecture notes. Lohith From kromer@MIT.EDU Fri Nov 18 09:17:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIEH65o026891 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 09:17:06 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIEH4YW029178 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 09:17:05 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIEGwb1017214 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 09:16:58 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAIEGwXs010211; Fri, 18 Nov 2005 09:16:58 -0500 Received: from BURTON-TWO-THIRTY-FIVE.MIT.EDU (BURTON-TWO-THIRTY-FIVE.MIT.EDU [18.247.5.235]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 09:16:58 -0500 Message-ID: <20051118091658.flzzeo34lb4ks0oo@webmail.mit.edu> Date: Fri, 18 Nov 2005 09:16:58 -0500 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2016 Content-Length: 337 X-Keywords: p. 10 "Informally, the only restrictions are that (1) the order in which items are selected is disregarded and (2) restrictions on the selection of items from sets A and B also apply in selecting items from A union B". Is there any way to use generating functions to count when the order in which items are selected matters? Katherine From ajshafer@MIT.EDU Fri Nov 18 09:21:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIELu5o031432 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 09:21:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIELtYW003685 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 09:21:55 -0500 (EST) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIELmk0018837 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 09:21:49 -0500 (EST) Message-ID: <000201c5ec4b$6bc71a40$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 11 Comments Date: Fri, 18 Nov 2005 09:21:51 -0500 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2017 Content-Length: 373 X-Keywords: NonJunk Generating functions still seem really hard. What does G(x) represent when we plug in a value for x? Is it only useful here for trying to derive the f(n) function? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From jstritar@MIT.EDU Fri Nov 18 09:34:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIEY15o001142 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 09:34:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIEXxYW015388 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 09:34:00 -0500 (EST) Received: from [18.245.6.234] (BAKER-FOUR-EIGHTY-NINE.MIT.EDU [18.245.6.234]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIEXpTD024084 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 09:33:53 -0500 (EST) Message-ID: <437DE64B.3070202@mit.edu> Date: Fri, 18 Nov 2005 09:33:47 -0500 From: Jon Stritar User-Agent: Thunderbird 1.5 (Windows/20051025) MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2018 Content-Length: 317 X-Keywords: NonJunk 3.1 Finding a Generating Function I thought this section was pretty interesting because I remember using a method involving matrices to obtain generating functions. I don't remember the exact procedure, but I am wondering if the two are related or simply different techniques for the same solution. Jon Stritar From meyer@imap.theory.csail.mit.edu Fri Nov 18 09:37:49 2005 Message-ID: <437DE73F.3060504@csail.mit.edu> Date: Fri, 18 Nov 2005 09:37:51 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kevin Wang Subject: Re: [Hanson] Notes 11 Comments References: <6.2.3.4.2.20051118050105.029324e0@po10.mit.edu> In-Reply-To: <6.2.3.4.2.20051118050105.029324e0@po10.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 464 Status: RO X-UID: 2019 X-Keywords: take another look at week 10 Monday, class problem 3, which we thought spelled out the Bookkeeper rule pretty thoroughly. Did you get a chance to do that problem in class? regards, A. Kevin Wang wrote: > Hi, > > I thought this reading was fairly straightforward and understood all > of it. With that being said though, if Prof. Meyers could go over > proving the bookkeeper rule (Section 4.3) in class, that would be > instructive. > > Thanks, > Kevin > From letrec@MIT.EDU Fri Nov 18 10:04:53 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIF4r5o031371 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:04:53 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAIF4p5b015630 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:04:52 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIF4pAe001116 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:04:51 -0500 (EST) Received: from [18.237.0.82] (TDCIP82.MIT.EDU [18.237.0.82]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIF4nok018042 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:04:49 -0500 (EST) Message-ID: <437DED8E.8020103@mit.edu> Date: Fri, 18 Nov 2005 10:04:46 -0500 From: Alton Torregano User-Agent: Debian Thunderbird 1.0.7 (X11/20051017) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 11 Comments Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2020 Content-Length: 524 X-Status: A X-Keywords: NonJunk This reading packet is one of the best written and most comprehensive. I have been introduced to the concept of generating functions before in Combinatorics, and the subject was never explained as succinctly and straightforward. In particular, the section on derivatives and Taylor's theorem are useful in gaining a slightly more in-depth and abstract understanding of the inner workings of the Generating Functions at this level of survey. Are we to cover exponential generating functions at all in this course? ~ Alton From kktyan@MIT.EDU Fri Nov 18 10:06:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIF6i5o031508 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 10:06:44 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIF6gYW016035 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 10:06:42 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIF6elX008391 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 10:06:40 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jAIF6eLr016249; Fri, 18 Nov 2005 10:06:40 -0500 Received: from BURTON-THREE-THIRTY-SIX.MIT.EDU (BURTON-THREE-THIRTY-SIX.MIT.EDU [18.247.6.81]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 10:06:40 -0500 Message-ID: <20051118100640.u6rpkk0lpe2scogs@webmail.mit.edu> Date: Fri, 18 Nov 2005 10:06:40 -0500 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments 11 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2021 Content-Length: 519 X-Keywords: I mostly understood everything in this reading, so I'd just like to note that the method to discover the Fibonacci sequence's generating function was quite fun and exciting. Overall, I'd like to submit the suggestion (for the reading, as Professor Meyer explained it better in class) how exactly to find the complex roots (alpha_1 and alpha_2, per se) and giving a review on partial fractions for those of us who don't quite remember how to do them. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From meyer@imap.theory.csail.mit.edu Fri Nov 18 10:19:12 2005 Message-ID: <437DF0F2.4060100@csail.mit.edu> Date: Fri, 18 Nov 2005 10:19:14 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Scot Frank Subject: Re: Comments Jelani References: <437D7B55.4000408@mit.edu> In-Reply-To: <437D7B55.4000408@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 800 Status: RO X-UID: 2022 X-Keywords: $Forwarded Actually Prof. Meyer was a little confused too :-) and when he thinks about it, he thinks /rounding/ (either way) is wrong. The right thing to do is choose the /nearest/ integer: The Fib formula looks like Term1 - Term2 where the absolute value of Term2 < 1/2. But since Term1 - Term2 is an integer (a Fib. number), it follows that the integer nearest to Term1 must be the final answer. So we can ignore Term2 and just pick the closest integer to Term1. Make sense now? Regards, A. Scot Frank wrote: > Hi, > > The biggest question I had from this week of notes was the reason > that, while Prof. Meyer was on the board, he choose to round up on the > sum, after erasing the second term. I was confused, and could not > understand the conflicting explanations. > > Thanks, > > Scot From meyer@imap.theory.csail.mit.edu Fri Nov 18 10:29:11 2005 Message-ID: <437DF349.2050701@csail.mit.edu> Date: Fri, 18 Nov 2005 10:29:13 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Tony Ng Subject: Re: [Jelani] Reading Comments: Week 11 References: <5.2.1.1.2.20051117235257.02142d78@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20051117235257.02142d78@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 734 Status: RO X-UID: 2023 X-Keywords: Even before the "amazing" fruit problem :-) , we used generating functions to find a formula, first for Fib nums, and then for another recursively specified sequence in the class prob. Didn't that illustrate the usefulness of gen funcs? .. or did you already know of some alternative way to find closed formulas for such sequences? regards, A. Tony Ng wrote: > I found the fruit counting problem ("An 'Impossible' Counting Problem, > pg12) to be very interesting. I didn't understand the need for > generating functions during the last lecture and it seemed excessively > complicated to do, however, now I understand why it is necessary > because it may be the only way we know how to solve some problems. > > - Tony > From meyer@imap.theory.csail.mit.edu Fri Nov 18 10:32:44 2005 Message-ID: <437DF41E.8050108@csail.mit.edu> Date: Fri, 18 Nov 2005 10:32:46 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Neil M Dowgun Subject: Re: [Hanson] reading question References: <20051117225828.bsxr9oqa0j0gw44g@webmail.mit.edu> In-Reply-To: <20051117225828.bsxr9oqa0j0gw44g@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 778 Status: RO X-UID: 2024 X-Keywords: Neil M Dowgun wrote: >So I realized that the example where we found the coefficiants for the Fibonacci >sequence doesn't really provide a clean fomulaic way to find coefficiants given >a closed form. > Actually, it does. What makes you think not? >I guess there's just always going to be some dirty math to do. >Anyway, I was wondering if using the Taylor expansion to find the coefficiants >was always a viable option. I mean, I realize that it would be a real pain to >take the derivative of an expression say, thirty times, but it just seems like >a more surefire way to get the answer if it really is foolproof. > > both Taylor expansion and the partial fraction method are useful, though there are problems where neither one works. >Thanks, Neil > > regards, A. From meyer@imap.theory.csail.mit.edu Fri Nov 18 10:35:44 2005 Message-ID: <437DF4D3.2000009@csail.mit.edu> Date: Fri, 18 Nov 2005 10:35:47 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alton Torregano Subject: Re: [Hanson] Week 11 Comments References: <437DED8E.8020103@mit.edu> In-Reply-To: <437DED8E.8020103@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 711 Status: RO X-UID: 2025 X-Keywords: glad you liked the Notes. No, we won't cover exponential gen funcs: ordinary ones are enough for a broad intro course like 6.042. regards, A. Alton Torregano wrote: >This reading packet is one of the best written and most comprehensive. I >have been introduced to the concept of generating functions before in >Combinatorics, and the subject was never explained as succinctly and >straightforward. > >In particular, the section on derivatives and Taylor's theorem are >useful in gaining a slightly more in-depth and abstract understanding of >the inner workings of the Generating Functions at this level of survey. >Are we to cover exponential generating functions at all in this course? > >~ Alton > > From meyer@imap.theory.csail.mit.edu Fri Nov 18 00:03:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAI53d5o024488 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 00:03:39 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAI53cYW002523 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 00:03:38 -0500 (EST) Received: from HowardRoark.mit.edu (EASTCAMPUS-EIGHT-THIRTY-EIGHT.MIT.EDU [18.238.6.71]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAI53aDr027191 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 00:03:36 -0500 (EST) Message-Id: <5.2.1.1.2.20051117223854.02036418@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 18 Nov 2005 00:01:19 -0500 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: [6.042] Reading Comments Lecture Notes 8 - Clayton Sims Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 420 X-UID: 2026 X-Keywords: NonJunk I felt like the material on page 7 was the most challenging. Generalizing the method of turning a sequence of coefficients into different Function Generators that are easily converted into a representation of the function is something that I feel I would benefit from practicing. Additionally, a quick review (5 mins.) of partial fractions in class would be nice, although not entirely necessary. -Clayton Sims From nedzel@MIT.EDU Fri Nov 18 10:42:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIFgn5o026043 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 10:42:49 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIFglYW021674 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 10:42:47 -0500 (EST) Received: from ThinkPadT43 (SIMMONS-ONE-NINETY-NINE.MIT.EDU [18.96.5.199]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIFgdSL024551 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 10:42:40 -0500 (EST) Message-Id: <200511181542.jAIFgdSL024551@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 11 Comments Date: Fri, 18 Nov 2005 10:42:33 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: 2.037 X-Spam-Level: ** (2.037) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2027 Content-Length: 80 X-Keywords: NonJunk I didn't understand the notes on using generating functions to count. - David From vbrobbey@MIT.EDU Fri Nov 18 10:49:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIFn05o030882 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:49:00 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIFmxYW028662 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:48:59 -0500 (EST) Received: from napoleon (NAPOLEON-BONAPARTE.MIT.EDU [18.18.3.59]) (authenticated bits=0) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIFmu52027362 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 10:48:57 -0500 (EST) Message-ID: <000c01c5ec57$95bd05b0$3b031212@napoleon> From: "Valery Kwasi Brobbey" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson]Week 11 comments Date: Fri, 18 Nov 2005 10:48:54 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0009_01C5EC2D.ABC42890" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -0.718 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2028 Content-Length: 1856 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0009_01C5EC2D.ABC42890 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Generating functions look very much like Taylor series expansion. I = actually used some of my knowledge of Taylor series to do some of the TP = problems. The lecture should probably have discussed the relationship = between the two. It's interesting how you can use generating functions = to count.=20 I like the lecture on Wednesday. I found that more interactive than = lectures with slides. However, I think the lecture was quite long. It = would have helped if we have the problem solving session right in the = middle. ------=_NextPart_000_0009_01C5EC2D.ABC42890 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Generating functions look very much = like Taylor=20 series expansion. I actually used some of my knowledge of Taylor series = to do=20 some of the TP problems. The lecture should probably have discussed the=20 relationship between the two. It's interesting how you can use = generating=20 functions to count.
I like the lecture on Wednesday. I = found that more=20 interactive than lectures with slides. However, I think the lecture was = quite=20 long. It would have helped if we have the problem solving session right = in the=20 middle.
 
------=_NextPart_000_0009_01C5EC2D.ABC42890-- From hmzhou@blackbird.csail.mit.edu Fri Nov 18 11:58:28 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIGwR5o030797; Fri, 18 Nov 2005 11:58:27 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAIGwRbf028621; Fri, 18 Nov 2005 11:58:27 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAIGwR7K028618; Fri, 18 Nov 2005 11:58:27 -0500 Date: Fri, 18 Nov 2005 11:58:27 -0500 (EST) From: Hanson Zhou To: Elizabeth Reid cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Reading Comments In-Reply-To: <94C740B9-2FE5-4D4D-BC5D-6FD20BC9D421@MIT.EDU> Message-ID: References: <94C740B9-2FE5-4D4D-BC5D-6FD20BC9D421@MIT.EDU> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2029 Content-Length: 352 X-Keywords: Most good texts on discrete mathematics should have sections on generating functions and exercises. Look around or ask Prof. Meyer for pointers. -Hanson On Thu, 17 Nov 2005, Elizabeth Reid wrote: > The reading makes sense, but I feel like I wouldn't be able to do it > myself. Is there anywhere I can go that would help me practice this > stuff? > From hmzhou@blackbird.csail.mit.edu Fri Nov 18 12:01:20 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIH1K5o031635; Fri, 18 Nov 2005 12:01:20 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAIH1KPq028665; Fri, 18 Nov 2005 12:01:20 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAIH1Ko0028662; Fri, 18 Nov 2005 12:01:20 -0500 Date: Fri, 18 Nov 2005 12:01:20 -0500 (EST) From: Hanson Zhou To: Daniel A Gutierrez cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hansen] TP11 Reading Response In-Reply-To: <20051117191349.qo4qd0bqx2os404s@webmail.mit.edu> Message-ID: References: <20051117191349.qo4qd0bqx2os404s@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2030 Content-Length: 398 X-Keywords: Well there is a formula for the geometric sum: 1+x+x^+...+x^n = (x^(n+1)-1)/(x-1). You can review from the course lecture notes or from just about any discrete math text. -Hanson On Thu, 17 Nov 2005, Daniel A Gutierrez wrote: > These all make logical sense, but I was wondering if by chance you have > a list or know where to find one so that I can review geometric summing? > Thanks, Daniel > From hmzhou@blackbird.csail.mit.edu Fri Nov 18 12:07:53 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIH7r5o032336; Fri, 18 Nov 2005 12:07:53 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jAIH7qbQ028727; Fri, 18 Nov 2005 12:07:52 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jAIH7qUt028724; Fri, 18 Nov 2005 12:07:52 -0500 Date: Fri, 18 Nov 2005 12:07:52 -0500 (EST) From: Hanson Zhou To: Barry Revzin cc: 6042-probs@theory.csail.mit.edu Subject: Re: [Hanson] Reading In-Reply-To: <6.1.2.0.1.20051117232905.019f2b98@po14.mit.edu> Message-ID: References: <6.1.2.0.1.20051117232905.019f2b98@po14.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2031 Content-Length: 721 X-Keywords: Well...and I'm purely speculating here: Integral f(y)g(n-y)dy, with f(y) = a_y x^y, and g(n-y) = b_(n-y) x^(n-y), would seem to be the continuous analogue of the discrete convolution we talk about here. Ask Prof. Meyer to make sure. -Hanson On Thu, 17 Nov 2005, Barry Revzin wrote: > For the convolution rule, I think it's really confusing that it's called > the convolution rule. It really has nothing to do with the idea of a > convolution from analysis, does it ? I'm really used to the convolution of > f and g (f*g) being the integral of f(y)g(x-y)dy. Why IS it called that > then ? Why not like... disjoint union rule ? Or Countable Additivity, which > would be consistent with measure theory... > > Barry > > From lana@MIT.EDU Fri Nov 18 12:23:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIHNv5o009032 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 12:23:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIHNt0C011117 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 12:23:56 -0500 (EST) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIHNsaC008964 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 12:23:54 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id jAIHNs3o002445; Fri, 18 Nov 2005 12:23:54 -0500 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 12:23:54 -0500 Message-ID: <20051118122354.ujookdbnpiskw0s4@webmail.mit.edu> Date: Fri, 18 Nov 2005 12:23:54 -0500 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] TP11 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2032 Content-Length: 177 X-Keywords: NonJunk I was wondering if it is possible to have a function that can not be expressed through a generating function or if 2 generating functions can result in the same expression Lana From ksindi@MIT.EDU Fri Nov 18 13:47:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIIl05o028638 for <6042-probs@theory.csail.MIT.EDU>; Fri, 18 Nov 2005 13:47:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIIkx0C024536 for <6042-probs@theory.csail.MIT.EDU>; Fri, 18 Nov 2005 13:46:59 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAIIkr0X008754 for <6042-probs@theory.csail.MIT.EDU>; Fri, 18 Nov 2005 13:46:53 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAIIkrIv030446; Fri, 18 Nov 2005 13:46:53 -0500 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 18 Nov 2005 13:46:53 -0500 Message-ID: <20051118134653.j9kik70v0asks8cc@webmail.mit.edu> Date: Fri, 18 Nov 2005 13:46:53 -0500 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [David] Week 11 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2033 Content-Length: 153 X-Keywords: I found it conceptiually confusing of how one treats generating functions only formally without worry of convergence. However Im getting the hang of it. From dshin@MIT.EDU Fri Nov 18 17:16:28 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMGS5o010910 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 17:16:28 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMGRVi009673 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 18 Nov 2005 17:16:27 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMGQBa022368; Fri, 18 Nov 2005 17:16:26 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMGOok021449; Fri, 18 Nov 2005 17:16:24 -0500 (EST) Message-ID: <437E5286.3050200@mit.edu> Date: Fri, 18 Nov 2005 17:15:34 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] Email comments for reading References: <20051118091658.flzzeo34lb4ks0oo@webmail.mit.edu> In-Reply-To: <20051118091658.flzzeo34lb4ks0oo@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2034 Content-Length: 860 X-Keywords: Interesting question! I spent a bit of time thinking about whether generating functions could be used in this type of situation, and I'm leaning towards an answer of "no". The reason is that multiplication is commutative: x^a*x^b = x^{ab}. It doesn't seem possible to encode order-dependent information with this order-independent operation (multiplication). Maybe there exists some crazy method using non-commutative algebraic structures, but I am highly doubtful. DS Katherine A Romer wrote: >p. 10 "Informally, the only restrictions are that (1) the order in which items >are selected is disregarded and (2) restrictions on the selection of items from >sets A and B also apply in selecting items from A union B". > >Is there any way to use generating functions to count when the order in which >items are selected matters? > >Katherine > > From dshin@MIT.EDU Fri Nov 18 17:40:52 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMeq5o015999 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 17:40:52 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMeoOp023578; Fri, 18 Nov 2005 17:40:50 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMelMu004586; Fri, 18 Nov 2005 17:40:47 -0500 (EST) Message-ID: <437E583D.9010003@mit.edu> Date: Fri, 18 Nov 2005 17:39:57 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alexander Bakst CC: 6042-probs@theory.lcs.mit.edu, 6042 staff <6042-staff@theory.csail.mit.edu> Subject: Re: [David] Lecture Notes 11 Comments References: <437D988C.8020808@mit.edu> In-Reply-To: <437D988C.8020808@mit.edu> Content-Type: multipart/alternative; boundary="------------020405060808050509060906" X-Spam-Score: 1.218 X-Spam-Level: * (1.218) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2035 Content-Length: 4414 X-Keywords: This is a multi-part message in MIME format. --------------020405060808050509060906 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Alexander Bakst wrote: > What kind of situations would require us to use values for x? In the > readings it said that they are unusual or rare, I believe. There are a few really ingenious applications of generating functions that involve using values for x. Typically, the plugging-in-for-x step is to reach some sort of contradiction: you have two generating functions that are supposed to be equal, but they don't agree at a particular value of x. Unfortunately, most examples are fairly complex (pun somewhat intended). I tried to think of a simple example, but this is the best I could come up with: ----- *Problem*: Suppose we have a collection of infinite arithmetic sequences (like {1,4,7,10,...} or {10,12,14,16,...}) where each sequence has difference term larger than 1. Suppose that every natural number occurs EXACTLY once among the sequences. Prove that two of the sequences have the same difference term. *Solution*: Suppose not. Let the sequences be S_i = {a_i + n*b_i} for i=1,2,... Then, since all the difference terms are distinct, the values of b_i are all distinct. Assume without loss of generality that b_1 is the smallest of the b_i. Sequence S_i can be represented by a sequence of 0's and 1's, which can be represented by the generating function x^{a_i}/(1-x^{b_i}). The natural numbers can be represented by a sequence of all 1's, which can be represented by the generating function 1/(1-x). Since each natural number occurs exactly once among the sequences, we have that 1/(1-x) = SUM_i (x^{a_i}/(1-x^{b_i})) Let x be close to a (b_1)-th root of unity. I won't give a rigorous proof, but it's true that the right hand side blows up while the left hand side doesn't. A contradiction. ----- Hopefully you find that somewhat cool. DS --------------020405060808050509060906 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Alexander Bakst wrote:
What kind of situations would require us to use values for x? In the readings it said that they are unusual or rare, I believe.
There are a few really ingenious applications of generating functions that involve using values for x.  Typically, the plugging-in-for-x step is to reach some sort of contradiction:  you have two generating functions that are supposed to be equal, but they don't agree at a particular value of x. 

Unfortunately, most examples are fairly complex (pun somewhat intended).  I tried to think of a simple example, but this is the best I could come up with:

-----

Problem:  Suppose we have a collection of infinite arithmetic sequences (like {1,4,7,10,...} or {10,12,14,16,...}) where each sequence has difference term larger than 1.  Suppose that every natural number occurs EXACTLY once among the sequences.  Prove that two of the sequences have the same difference term. 

Solution:  Suppose not.  Let the sequences be S_i = {a_i + n*b_i} for i=1,2,...  Then, since all the difference terms are distinct, the values of b_i are all distinct.  Assume without loss of generality that b_1 is the smallest of the b_i. 

Sequence S_i can be represented by a sequence of 0's and 1's, which can be represented by the generating function x^{a_i}/(1-x^{b_i}).  The natural numbers can be represented by a sequence of all 1's, which can be represented by the generating function 1/(1-x).  Since each natural number occurs exactly once among the sequences, we have that

1/(1-x) = SUM_i (x^{a_i}/(1-x^{b_i}))

Let x be close to a (b_1)-th root of unity.  I won't give a rigorous proof, but it's true that the right hand side blows up while the left hand side doesn't.  A contradiction. 

-----

Hopefully you find that somewhat cool. 

DS --------------020405060808050509060906-- From dshin@MIT.EDU Fri Nov 18 17:43:52 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMhq5o020866 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:43:52 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMhpVi022551 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:43:51 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMhoBa023540; Fri, 18 Nov 2005 17:43:51 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMhmok022831; Fri, 18 Nov 2005 17:43:48 -0500 (EST) Message-ID: <437E58F2.5060402@mit.edu> Date: Fri, 18 Nov 2005 17:42:58 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 11 Comments References: <488a6fb141c55248b795e2c28e8cceb1@mit.edu> In-Reply-To: <488a6fb141c55248b795e2c28e8cceb1@mit.edu> Content-Type: text/plain; charset=ISO-2022-JP Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2036 Content-Length: 668 X-Keywords: Glad you're liking this stuff. Do you see why the oranges thing works now? DS Harrison King Hall wrote: > This is really a cool way to do counting problems, and really > intuitive since you only have to worry about how to select each class > of element for n elements independent of all the others. My only issue > is that I don't always see the generating function . Like in the > "Impossible" counting problem, I didn't see that the number of oranges > was O(x) = 1 + x + x ^2 + x^3 + x^4 = (1 $B!](B x^5)/(1 $B!](B x) and I don't > really know how you got there. That is probably something that comes > with time however. This is really slick. > -Harrison > From dshin@MIT.EDU Fri Nov 18 17:47:59 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMlx5o024327 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:47:59 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMlwVi029077 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:47:58 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMlwBa023675; Fri, 18 Nov 2005 17:47:58 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMlook023000; Fri, 18 Nov 2005 17:47:50 -0500 (EST) Message-ID: <437E59E4.6050200@mit.edu> Date: Fri, 18 Nov 2005 17:47:00 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason Juang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 11 Comments References: <437BA8B1.2090306@mit.edu> In-Reply-To: <437BA8B1.2090306@mit.edu> Content-Type: multipart/alternative; boundary="------------080504060501080501020402" X-Spam-Score: 2.569 X-Spam-Level: ** (2.569) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2037 Content-Length: 5064 X-Keywords: This is a multi-part message in MIME format. --------------080504060501080501020402 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 7bit Jason Juang wrote: > In what cases might we actually evaluate a generating function? There are a few really ingenious applications of generating functions that involve using values for x. Typically, the plugging-in-for-x step is to reach some sort of contradiction: you have two generating functions that are supposed to be equal, but they don't agree at a particular value of x. Unfortunately, most examples are fairly complex (pun somewhat intended). I tried to think of a simple example, but this is the best I could come up with: ----- *Problem*: Suppose we have a collection of infinite arithmetic sequences (like {1,4,7,10,...} or {10,12,14,16,...}) where each sequence has difference term larger than 1. Suppose that every natural number occurs EXACTLY once among the sequences. Prove that two of the sequences have the same difference term. *Solution*: Suppose not. Let the sequences be S_i = {a_i + n*b_i} for i=1,2,... Then, since all the difference terms are distinct, the values of b_i are all distinct. Assume without loss of generality that b_1 is the smallest of the b_i. Sequence S_i can be represented by a sequence of 0's and 1's, which can be represented by the generating function x^{a_i}/(1-x^{b_i}). The natural numbers can be represented by a sequence of all 1's, which can be represented by the generating function 1/(1-x). Since each natural number occurs exactly once among the sequences, we have that 1/(1-x) = SUM_i (x^{a_i}/(1-x^{b_i})) Let x be close to a (b_1)-th complex root of unity. I won't give a rigorous proof, but it's true that the right hand side blows up while the left hand side doesn't. (It relies on the fact that b_1 is distinct from all the other b_i...) A contradiction. ----- Hopefully you find that somewhat cool. If you want a really cool challenge, try your hand at this: Problem: Suppose you have a tiling of an 8x8 chessboard with 21 trinominos (3x1 rectangles) and one monomino (1x1 rectangle). Find all possible positions for the monomino. Believe it or not, it can be done with (finite) generating functions! It involves cubic roots of unity... DS --------------080504060501080501020402 Content-Type: text/html; charset=windows-1252 Content-Transfer-Encoding: 8bit Jason Juang wrote:
In what cases might we actually evaluate a generating function?
There are a few really ingenious applications of generating functions that involve using values for x. Typically, the plugging-in-for-x step is to reach some sort of contradiction: you have two generating functions that are supposed to be equal, but they don't agree at a particular value of x.

Unfortunately, most examples are fairly complex (pun somewhat intended). I tried to think of a simple example, but this is the best I could come up with:

-----

Problem: Suppose we have a collection of infinite arithmetic sequences (like {1,4,7,10,...} or {10,12,14,16,...}) where each sequence has difference term larger than 1. Suppose that every natural number occurs EXACTLY once among the sequences. Prove that two of the sequences have the same difference term.

Solution: Suppose not. Let the sequences be S_i = {a_i + n*b_i} for i=1,2,... Then, since all the difference terms are distinct, the values of b_i are all distinct. Assume without loss of generality that b_1 is the smallest of the b_i.

Sequence S_i can be represented by a sequence of 0's and 1's, which can be represented by the generating function x^{a_i}/(1-x^{b_i}). The natural numbers can be represented by a sequence of all 1's, which can be represented by the generating function 1/(1-x). Since each natural number occurs exactly once among the sequences, we have that

1/(1-x) = SUM_i (x^{a_i}/(1-x^{b_i}))

Let x be close to a (b_1)-th complex root of unity. I won't give a rigorous proof, but it's true that the right hand side blows up while the left hand side doesn't. (It relies on the fact that b_1 is distinct from all the other b_i...) A contradiction.

-----

Hopefully you find that somewhat cool.

If you want a really cool challenge, try your hand at this:

Problem: Suppose you have a tiling of an 8x8 chessboard with 21 trinominos (3x1 rectangles) and one monomino (1x1 rectangle). Find all possible positions for the monomino.

Believe it or not, it can be done with (finite) generating functions! It involves cubic roots of unity...

DS --------------080504060501080501020402-- From dshin@MIT.EDU Fri Nov 18 17:48:17 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMmH5o024338 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:48:17 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMmGVi029580 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:48:16 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMmFBa023684; Fri, 18 Nov 2005 17:48:15 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMm8ok023012; Fri, 18 Nov 2005 17:48:08 -0500 (EST) Message-ID: <437E59F6.8050503@mit.edu> Date: Fri, 18 Nov 2005 17:47:18 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 11 Comments References: <200511162330.jAGNUADm015510@outgoing.mit.edu> In-Reply-To: <200511162330.jAGNUADm015510@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2038 Content-Length: 253 X-Keywords: Did class help with convolution? DS Chris Yang wrote: > I dont really understand how the Convolution rule works. Will we be > covering it in class on Friday? Also, the apples/oranges thing at the > end is pretty cool. > > Thanks, > > Chris Yang > From dshin@MIT.EDU Fri Nov 18 17:49:55 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMnt5o024445 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:49:55 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMnrOp023843; Fri, 18 Nov 2005 17:49:53 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMnook023098; Fri, 18 Nov 2005 17:49:50 -0500 (EST) Message-ID: <437E5A5C.7010005@mit.edu> Date: Fri, 18 Nov 2005 17:49:00 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sergio Bacallado CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 11 Comments References: <5.2.1.1.2.20051117222122.00c5c020@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20051117222122.00c5c020@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2039 Content-Length: 539 X-Keywords: Taylor expansion is sure to work. If your function is a quotient of polynomials, then partial fractions often can do the trick. DS Sergio Bacallado wrote: > I thought it was amazing how useful generating functions can be to > solve counting problems. I would like to know if there is always an > easy way to find a closed form for the nth term of a sequence, if you > know the value of the generating function. That is, other than > differentiating the function to get the Taylor expansion term. > > Sergio Bacallado > Group A > From dshin@MIT.EDU Fri Nov 18 17:53:34 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMrY5o024623 for <6042-probs@theory.lcs.mit.edu>; Fri, 18 Nov 2005 17:53:34 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMrWOp024036; Fri, 18 Nov 2005 17:53:32 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMrTMu005130; Fri, 18 Nov 2005 17:53:29 -0500 (EST) Message-ID: <437E5B37.6070903@mit.edu> Date: Fri, 18 Nov 2005 17:52:39 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: cbossard@MIT.EDU CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 11 reading comments References: <20051118030340.i5w8dncex73kocgg@webmail.mit.edu> In-Reply-To: <20051118030340.i5w8dncex73kocgg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2040 Content-Length: 969 X-Keywords: It would have been great if you had come to lecture today - we covered much of these topics in class. I would recommend coming to office hours if you are still having difficulty. DS cbossard@MIT.EDU wrote: >I find this to be an interesting topic although I have trouble >reproducing and creating these functions on my own they generally >make sense to me when I see them explained. In section 2.5 on >products I am not seeing/understanding how the terms of the diagonal >were collected so I don't see why it's not aobo + a1b1 instead of >aobn etc. Again with although the convolution rule makes some sense >to me in section 4.2 I dont see how they are getting the numbers from >the formula that was created. I can see that each is (1+x) and they >should be multiplied. Intuively I understand that there's 1 way to >choose 0 elements, 2 ways to choose 1, etc but I do not see how this >information can be attained from the 1+2x+x^2 equation. > >Cynthia > > From dshin@MIT.EDU Fri Nov 18 17:54:01 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMs15o024680 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:54:01 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMrxVi008807 for <6042-probs@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:54:00 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMrxBa023952; Fri, 18 Nov 2005 17:53:59 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMrqok023380; Fri, 18 Nov 2005 17:53:52 -0500 (EST) Message-ID: <437E5B4E.6040406@mit.edu> Date: Fri, 18 Nov 2005 17:53:02 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-probs@theory.csail.mit.edu Subject: Re: David - Reading Comments References: <6.2.3.4.2.20051118003612.01e0a888@po9.mit.edu> In-Reply-To: <6.2.3.4.2.20051118003612.01e0a888@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2041 Content-Length: 464 X-Keywords: did class help? DS Joshua Jen C. Monzon wrote: > I'm a little confused on the convolution rule. Perhaps this would be > explained a little more in class. Also, I didn't really understand how > they obtained the answer in the last problem of the tutorial. Hope > tomorrow's lecture clears this up. > > Josh > > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From meyer@imap.theory.csail.mit.edu Fri Nov 18 17:40:53 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAIMeq5o015997 for <6042-staff@theory.csail.mit.edu>; Fri, 18 Nov 2005 17:40:52 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAIMeoOp023578; Fri, 18 Nov 2005 17:40:50 -0500 (EST) Received: from [192.168.1.4] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAIMelMu004586; Fri, 18 Nov 2005 17:40:47 -0500 (EST) Message-ID: <437E583D.9010003@mit.edu> Date: Fri, 18 Nov 2005 17:39:57 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alexander Bakst CC: 6042-probs@theory.lcs.mit.edu, 6042 staff <6042-staff@theory.csail.mit.edu> Subject: Re: [David] Lecture Notes 11 Comments References: <437D988C.8020808@mit.edu> In-Reply-To: <437D988C.8020808@mit.edu> Content-Type: multipart/alternative; boundary="------------020405060808050509060906" X-Spam-Score: 1.218 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,HTML_MESSAGE autolearn=ham version=2.63 Status: RO Content-Length: 4414 X-UID: 2042 X-Keywords: This is a multi-part message in MIME format. --------------020405060808050509060906 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Alexander Bakst wrote: > What kind of situations would require us to use values for x? In the > readings it said that they are unusual or rare, I believe. There are a few really ingenious applications of generating functions that involve using values for x. Typically, the plugging-in-for-x step is to reach some sort of contradiction: you have two generating functions that are supposed to be equal, but they don't agree at a particular value of x. Unfortunately, most examples are fairly complex (pun somewhat intended). I tried to think of a simple example, but this is the best I could come up with: ----- *Problem*: Suppose we have a collection of infinite arithmetic sequences (like {1,4,7,10,...} or {10,12,14,16,...}) where each sequence has difference term larger than 1. Suppose that every natural number occurs EXACTLY once among the sequences. Prove that two of the sequences have the same difference term. *Solution*: Suppose not. Let the sequences be S_i = {a_i + n*b_i} for i=1,2,... Then, since all the difference terms are distinct, the values of b_i are all distinct. Assume without loss of generality that b_1 is the smallest of the b_i. Sequence S_i can be represented by a sequence of 0's and 1's, which can be represented by the generating function x^{a_i}/(1-x^{b_i}). The natural numbers can be represented by a sequence of all 1's, which can be represented by the generating function 1/(1-x). Since each natural number occurs exactly once among the sequences, we have that 1/(1-x) = SUM_i (x^{a_i}/(1-x^{b_i})) Let x be close to a (b_1)-th root of unity. I won't give a rigorous proof, but it's true that the right hand side blows up while the left hand side doesn't. A contradiction. ----- Hopefully you find that somewhat cool. DS --------------020405060808050509060906 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Alexander Bakst wrote:
What kind of situations would require us to use values for x? In the readings it said that they are unusual or rare, I believe.
There are a few really ingenious applications of generating functions that involve using values for x.  Typically, the plugging-in-for-x step is to reach some sort of contradiction:  you have two generating functions that are supposed to be equal, but they don't agree at a particular value of x. 

Unfortunately, most examples are fairly complex (pun somewhat intended).  I tried to think of a simple example, but this is the best I could come up with:

-----

Problem:  Suppose we have a collection of infinite arithmetic sequences (like {1,4,7,10,...} or {10,12,14,16,...}) where each sequence has difference term larger than 1.  Suppose that every natural number occurs EXACTLY once among the sequences.  Prove that two of the sequences have the same difference term. 

Solution:  Suppose not.  Let the sequences be S_i = {a_i + n*b_i} for i=1,2,...  Then, since all the difference terms are distinct, the values of b_i are all distinct.  Assume without loss of generality that b_1 is the smallest of the b_i. 

Sequence S_i can be represented by a sequence of 0's and 1's, which can be represented by the generating function x^{a_i}/(1-x^{b_i}).  The natural numbers can be represented by a sequence of all 1's, which can be represented by the generating function 1/(1-x).  Since each natural number occurs exactly once among the sequences, we have that

1/(1-x) = SUM_i (x^{a_i}/(1-x^{b_i}))

Let x be close to a (b_1)-th root of unity.  I won't give a rigorous proof, but it's true that the right hand side blows up while the left hand side doesn't.  A contradiction. 

-----

Hopefully you find that somewhat cool. 

DS --------------020405060808050509060906-- From minilek@gmail.com Sat Nov 19 17:20:33 2005 Return-Path: Received: from zproxy.gmail.com (zproxy.gmail.com [64.233.162.206]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAJMKX5o003467 for <6042-probs@theory.lcs.mit.edu>; Sat, 19 Nov 2005 17:20:33 -0500 Received: by zproxy.gmail.com with SMTP id l8so476019nzf for <6042-probs@theory.lcs.mit.edu>; Sat, 19 Nov 2005 14:20:33 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:cc:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=qPN9JR04KWTogW9W8UMysXNVovZD+C9cg68Z0DLTg4QZJr6iaK1XZAD0NfapH0b6iC50ic5wO1QcaY2gT/ntIc6ORumXg17ZJmSsTai/d116IYLmsfqKnD85uF8s1Fr+AyAb3rvf0tYM4S1ndOl3g5KuBuSTsdUajH+pqwqjBYQ= Received: by 10.64.199.9 with SMTP id w9mr1173433qbf; Sat, 19 Nov 2005 14:20:33 -0800 (PST) Received: by 10.64.150.8 with HTTP; Sat, 19 Nov 2005 14:20:33 -0800 (PST) Message-ID: Date: Sat, 19 Nov 2005 17:20:33 -0500 From: Jelani Nelson Sender: minilek@gmail.com To: David Shin Subject: Re: [David] Lecture Notes 11 Comments Cc: Alexander Bakst , 6042-probs@theory.lcs.mit.edu, 6042 staff <6042-staff@theory.csail.mit.edu> In-Reply-To: <437E583D.9010003@mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline References: <437D988C.8020808@mit.edu> <437E583D.9010003@mit.edu> Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jAJMKX5o003467 Status: RO X-UID: 2043 Content-Length: 219 X-Keywords: >> The natural numbers can be represented by a sequence of all 1's, which can be >> represented by the generating function 1/(1-x) Aren't the natural numbers represented by x times the derivative of 1/(1-x)? -Jelani From dshin@MIT.EDU Sat Nov 19 17:23:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAJMNh5o007207 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 19 Nov 2005 17:23:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAJMNf8R029318; Sat, 19 Nov 2005 17:23:41 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAJMNegd007256; Sat, 19 Nov 2005 17:23:40 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAJMNeDX025206; Sat, 19 Nov 2005 17:23:40 -0500 Received: from W20-575-87.MIT.EDU (W20-575-87.MIT.EDU [18.187.1.248]) (User authenticated as dshin830@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sat, 19 Nov 2005 17:23:40 -0500 Message-ID: <20051119172340.nyecjozz2coo8800@webmail.mit.edu> Date: Sat, 19 Nov 2005 17:23:40 -0500 From: David Shin To: Jelani Nelson Cc: Alexander Bakst , 6042-probs@theory.lcs.MIT.EDU, 6042 staff <6042-staff@theory.csail.MIT.EDU> Subject: Re: [David] Lecture Notes 11 Comments References: <437D988C.8020808@mit.edu> <437E583D.9010003@mit.edu> In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2044 Content-Length: 531 X-Keywords: With this application, I'm using using coefficients of 1 and 0 to represent a subset of the natural numbers. For example, the subset {0,2,3} corresponds to the sequence <1,0,1,1,0,0,0...> and thus to the function 1+x^2+x^3. Sorry if it wasn't clear. DS Quoting Jelani Nelson : >>> The natural numbers can be represented by a sequence of all 1's, > which can be >>> represented by the generating function 1/(1-x) > > Aren't the natural numbers represented by x times the derivative of 1/(1-x)? > > -Jelani > From meyer@imap.theory.csail.mit.edu Sat Nov 19 17:23:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAJMNh5o007205 for <6042-staff@theory.csail.MIT.EDU>; Sat, 19 Nov 2005 17:23:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAJMNf8R029318; Sat, 19 Nov 2005 17:23:41 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAJMNegd007256; Sat, 19 Nov 2005 17:23:40 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAJMNeDX025206; Sat, 19 Nov 2005 17:23:40 -0500 Received: from W20-575-87.MIT.EDU (W20-575-87.MIT.EDU [18.187.1.248]) (User authenticated as dshin830@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sat, 19 Nov 2005 17:23:40 -0500 Message-ID: <20051119172340.nyecjozz2coo8800@webmail.mit.edu> Date: Sat, 19 Nov 2005 17:23:40 -0500 From: David Shin To: Jelani Nelson Cc: Alexander Bakst , 6042-probs@theory.lcs.MIT.EDU, 6042 staff <6042-staff@theory.csail.MIT.EDU> Subject: Re: [David] Lecture Notes 11 Comments References: <437D988C.8020808@mit.edu> <437E583D.9010003@mit.edu> In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 531 X-UID: 2045 X-Keywords: With this application, I'm using using coefficients of 1 and 0 to represent a subset of the natural numbers. For example, the subset {0,2,3} corresponds to the sequence <1,0,1,1,0,0,0...> and thus to the function 1+x^2+x^3. Sorry if it wasn't clear. DS Quoting Jelani Nelson : >>> The natural numbers can be represented by a sequence of all 1's, > which can be >>> represented by the generating function 1/(1-x) > > Aren't the natural numbers represented by x times the derivative of 1/(1-x)? > > -Jelani > From meyer@imap.theory.csail.mit.edu Sat Nov 19 17:20:34 2005 Return-Path: Received: from wproxy.gmail.com (wproxy.gmail.com [64.233.184.206]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAJMKX5o003472 for <6042-staff@theory.csail.mit.edu>; Sat, 19 Nov 2005 17:20:33 -0500 Received: by wproxy.gmail.com with SMTP id i12so589364wra for <6042-staff@theory.csail.mit.edu>; Sat, 19 Nov 2005 14:20:33 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:cc:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=qPN9JR04KWTogW9W8UMysXNVovZD+C9cg68Z0DLTg4QZJr6iaK1XZAD0NfapH0b6iC50ic5wO1QcaY2gT/ntIc6ORumXg17ZJmSsTai/d116IYLmsfqKnD85uF8s1Fr+AyAb3rvf0tYM4S1ndOl3g5KuBuSTsdUajH+pqwqjBYQ= Received: by 10.64.199.9 with SMTP id w9mr1173433qbf; Sat, 19 Nov 2005 14:20:33 -0800 (PST) Received: by 10.64.150.8 with HTTP; Sat, 19 Nov 2005 14:20:33 -0800 (PST) Message-ID: Date: Sat, 19 Nov 2005 17:20:33 -0500 From: Jelani Nelson Sender: minilek@gmail.com To: David Shin Subject: Re: [David] Lecture Notes 11 Comments Cc: Alexander Bakst , 6042-probs@theory.lcs.mit.edu, 6042 staff <6042-staff@theory.csail.mit.edu> In-Reply-To: <437E583D.9010003@mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline References: <437D988C.8020808@mit.edu> <437E583D.9010003@mit.edu> Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jAJMKX5o003472 Status: RO Content-Length: 219 X-UID: 2046 X-Keywords: >> The natural numbers can be represented by a sequence of all 1's, which can be >> represented by the generating function 1/(1-x) Aren't the natural numbers represented by x times the derivative of 1/(1-x)? -Jelani From juang@MIT.EDU Mon Oct 3 18:46:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j93MkNw0004518 for <6042-probs@theory.csail.mit.edu>; Mon, 3 Oct 2005 18:46:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j93MkM4u019890 for <6042-probs@theory.csail.mit.edu>; Mon, 3 Oct 2005 18:46:22 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j93MkFuB012547 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 3 Oct 2005 18:46:15 -0400 (EDT) Message-ID: <4341B493.5010504@mit.edu> Date: Mon, 03 Oct 2005 18:45:39 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2047 Content-Length: 167 X-Keywords: NonJunk I was most surprised by section 5.3 (Classifying Polyhedra) on page 14 of the reading. The proof that only 5 regular polyhedra exist was enlightening. Jason Juang. From rnjacobs@MIT.EDU Mon Oct 3 20:36:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j940aZw0026615 for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 20:36:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j940aZmx026365 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 20:36:35 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j940aXok003749 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 20:36:33 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j940aXUv024591; Mon, 3 Oct 2005 20:36:33 -0400 Message-Id: <200510040036.j940aXUv024591@white-meteo.mit.edu> To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading response Date: Mon, 03 Oct 2005 20:36:32 -0400 From: r n jacobs X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2048 Content-Length: 346 X-Keywords: NonJunk It took a while for me to find the definition of "matching" since it was only boldface in a paragraph. Are the tutor questions specifically supposed to reflect the reading? It seems strange that the material was half already covered and half required doing the reading. Otherwise, everything seems straightforward this week. - Robert Jacobs From benlu@MIT.EDU Mon Oct 3 22:16:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j942GOw0012374 for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 22:16:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j942GM0M029415 for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 22:16:22 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j942GG1B022907 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 22:16:16 -0400 (EDT) Message-ID: <4341E5F1.5070306@mit.edu> Date: Mon, 03 Oct 2005 22:16:17 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 5 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2049 Content-Length: 347 X-Keywords: NonJunk Hey Jelani, I was delighted to see the formal definitions and comparisons between graphs, trees, cycles, and paths. I've only worked with them while implementing abstract data types, so it's kind of cool to see them as formal mathematical concepts. I guess I wish I had taken this class before 6.034 and 6.170. Maybe even before 6.001. ~Ben From pgroudas@MIT.EDU Mon Oct 3 23:14:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j943Epw0017536 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 23:14:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j943En1i004809 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 23:14:49 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j943ElFf004308 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 23:14:47 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j943Ek2V009584; Mon, 3 Oct 2005 23:14:46 -0400 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 3 Oct 2005 23:14:46 -0400 Message-ID: <20051003231446.pvxuc6vde1us480c@webmail.mit.edu> Date: Mon, 3 Oct 2005 23:14:46 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments for wed, oct 5 TA Hanson MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.351 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2050 Content-Length: 291 X-Keywords: NonJunk I am interested in the trapping mentioned on page 6. Where it says that we shouldn't start with an n-edge graphe and then add an edge, but do the shrink-down, grow-back method. It mentions that we'll see an example in class, but I'd like something more formal in the notes. -Paul Groudas From ksindi@MIT.EDU Mon Oct 3 23:40:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j943exw0021023 for <6042-probs@theory.csail.MIT.EDU>; Mon, 3 Oct 2005 23:40:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j943ev1i020432 for <6042-probs@theory.csail.MIT.EDU>; Mon, 3 Oct 2005 23:40:58 -0400 (EDT) Received: from w20-575-2.mit.edu (W20-575-2.MIT.EDU [18.187.0.21]) (authenticated bits=56) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j943epWN009396 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Mon, 3 Oct 2005 23:40:51 -0400 (EDT) Received: (from ksindi@localhost) by w20-575-2.mit.edu (8.12.9) id j943epH5009341; Mon, 3 Oct 2005 23:40:51 -0400 Subject: [David] Week 5 Comments From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 03 Oct 2005 23:40:51 -0400 Message-Id: <1128397251.8343.18.camel@w20-575-2.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2051 Content-Length: 472 X-Keywords: NonJunk "In graph terms, our goal is to find a matching for the girls; that is, a subset of edges such that exactly one edge is incident to each girl and at most one edge is incident to each boy.For example, here is one possible matching for the girls:" I found this definition of matching to be confusing and the example not so helpful. I likes Eurler's formula and noticed the analogy of it with de Rham or chech cohomology. It woul'd be fun to see more than one proof of it. From sheldons@MIT.EDU Tue Oct 4 00:13:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j944Dnw0026042 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 00:13:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j944Dl1i008729 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 00:13:47 -0400 (EDT) Received: from [10.0.0.2] (209-6-159-26.c3-0.smr-ubr3.sbo-smr.ma.cable.rcn.com [209.6.159.26]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j944DeNE015205 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 00:13:41 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) References: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Sheldon Chan Subject: [David] Week 5 Comments Date: Tue, 4 Oct 2005 00:13:38 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.734) X-Spam-Score: 0.501 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2052 Content-Length: 264 X-Keywords: NonJunk I found Page 14, Section 5.3 Classifying Polyhedra to be most interesting. The concept of discovering polyhedra, based on specific restrictions and Euler's formula, is interesting, given it is shown to exist mathematically, prior to visualizing it. Sheldon From ereid@MIT.EDU Tue Oct 4 01:21:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j945Liw0002593 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 01:21:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j945Lg1i015424 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 01:21:43 -0400 (EDT) Received: from [18.242.6.96] (NEXT-THREE-FIFTY-ONE.MIT.EDU [18.242.6.96]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j945LU1t024443 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 01:21:40 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <5F196C36-0B0A-4F42-A808-0CCB751DBBA3@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] reading comment Date: Tue, 4 Oct 2005 01:21:26 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.081 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2053 Content-Length: 199 X-Keywords: NonJunk I think it might be helpful if we could go over classifying polyhedra in class. I didn't quite follow the math, although if I have time I'll look at it tomorrow when I'm less tired... Elizabeth From cwong08@MIT.EDU Tue Oct 4 01:30:50 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j945Uow0004056 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 01:30:50 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j945UnIk003460 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 01:30:49 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j945Ueol013568 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 01:30:46 -0400 (EDT) Message-ID: <43421374.9040704@mit.edu> Date: Tue, 04 Oct 2005 01:30:28 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [TA-name] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.944 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2054 Content-Length: 147 X-Keywords: NonJunk On page 10, when defining bipartite graphs, it is unclear what an "odd cycle" is. I also don't really understand the polyhedra part on page 14. From shauni@MIT.EDU Tue Oct 4 02:04:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9464vw0008088 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 02:04:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9464qAK005490 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 02:04:55 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j945v8Va027745 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 01:57:08 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j945v8m1020221; Tue, 4 Oct 2005 01:57:08 -0400 Received: from SIMMONS-FOUR-THIRTY-TWO.MIT.EDU (SIMMONS-FOUR-THIRTY-TWO.MIT.EDU [18.96.6.177]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 01:57:08 -0400 Message-ID: <20051004015708.9yq7mgg4d1k4c84k@webmail.mit.edu> Date: Tue, 4 Oct 2005 01:57:08 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Reading comments week 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2055 Content-Length: 188 X-Keywords: NonJunk Planar Graphs (pp. 11-12) I'd like to have the section on planar graphs explained more fully in lecture; when can a graph be planar and what are the special properties of a planar graph? From mike_a@MIT.EDU Tue Oct 4 02:38:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j946cow0014670 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 02:38:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j946cmAF021124 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 02:38:48 -0400 (EDT) Received: from [18.238.6.106] (EASTCAMPUS-EIGHT-SEVENTY-THREE.MIT.EDU [18.238.6.106]) (authenticated bits=0) (User authenticated as mike_a@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j946ckf8000647 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 02:38:47 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: mike anderson Subject: [David] Weekly reading comments Date: Tue, 4 Oct 2005 02:38:45 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.23 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2056 Content-Length: 244 X-Status: A X-Keywords: NonJunk I am a little confused by theorem 4.2 (page 10), regarding bipartite graphs and odd cycles. Does odd cycles mean there are an odd number of paths? this doesn't seem to agree w/ the diagram, and there is no proof for this theorem. -mike From hectorb@MIT.EDU Tue Oct 4 02:50:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j946ocw0016106 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 02:50:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j946oaAF026289; Tue, 4 Oct 2005 02:50:36 -0400 (EDT) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j946oT6W001371 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 4 Oct 2005 02:50:29 -0400 (EDT) Message-ID: <4340D452.4060103@mit.edu> Date: Mon, 03 Oct 2005 02:48:50 -0400 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.01 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2057 Content-Length: 534 X-Keywords: NonJunk "A 1994 University of Chicago study entitled The Social Organization of Sexuality found that on average men have 74% more opposite-gender partners than women, confirming a view that men are more promiscuous. But whoever reported this finding was lying or confused!" -pg 1 I don't think the reporters were lying. The data was probably gathered by surveys, and the ones lying were the women. The men might have exaggerated a little, but women never admit to having multiple sexual partners. This is why the study was incorrect. From rehughes@MIT.EDU Tue Oct 4 03:55:16 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j947tGw0023254 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:16 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j947tEEY001747 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:15 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j947tE9j024191 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:14 -0400 (EDT) Received: from [18.244.6.55] (SENIOR-THREE-TEN.MIT.EDU [18.244.6.55]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j947t2ok018101 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:02 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <156e4dedbd1233a85dd168b76bee12c6@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.csail.mit.edu From: Richard Hughes Subject: [Julani] Week 5 Comments Date: Tue, 4 Oct 2005 03:55:02 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2058 Content-Length: 300 X-Keywords: NonJunk I find the leap of logic in section 5.3 (the classifying polyhedra) and the history of the Pythagoras cult quite interesting. I'd love to hear more about how the ancients went through this basic series of important logical deductions and what we can (and can't) learn from them. Richard Hughes From lye@MIT.EDU Tue Oct 4 07:40:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94BeZw0023112 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 07:40:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94BeYYf027392 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 07:40:34 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94BeSuk015851 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 07:40:28 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j94BeSgk028766; Tue, 4 Oct 2005 07:40:28 -0400 Received: from RANDOM-FOUR-FORTY.MIT.EDU (RANDOM-FOUR-FORTY.MIT.EDU [18.243.6.185]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 07:40:28 -0400 Message-ID: <20051004074028.ryue99jas8w04gkg@webmail.mit.edu> Date: Tue, 4 Oct 2005 07:40:28 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2059 Content-Length: 287 X-Status: A X-Keywords: NonJunk Section 2.1, pg. 5: "Simple cycles will be cycles that don't cross themselves." How are non-simple cycles defined? If a non-simple cycle were taken as a subgraph, would there be a graph it is always isomorphic to? (e.g. for the simple graph, it is always isomorphic to C_n) Lunduo Ye From hkhall@MIT.EDU Tue Oct 4 08:22:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94CMew0030918 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 08:22:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94CMcHS020600; Tue, 4 Oct 2005 08:22:38 -0400 (EDT) Received: from [18.96.6.59] (SIMMONS-THREE-FOURTEEN.MIT.EDU [18.96.6.59]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94CMVkI022224 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Tue, 4 Oct 2005 08:22:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: LN5 Reading Comments [David] Date: Tue, 4 Oct 2005 08:24:31 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2060 Content-Length: 502 X-Keywords: NonJunk This is actually the first section that I feel really comfortable with, probably because there can exist a concrete visualization in the form of a graph. The only part of the reading that started to give me trouble was the section on translating planar graphs into polyhedra. While Euler's Formula is straightforward I was having difficulty proving that there were only 5 that satisfied the criterion (right?). Anyway, I found this section to be cake otherwise, at least I hope so. -Harrison From mukkala@MIT.EDU Tue Oct 4 09:28:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94DSWw0009047 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:28:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94DSVHS012493 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:28:31 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94DSOGw011902 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:28:25 -0400 (EDT) Message-Id: <6.2.1.2.2.20051004091356.01d77828@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Tue, 04 Oct 2005 09:28:35 -0400 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2061 Content-Length: 677 X-Status: A X-Keywords: NonJunk In section 5.2, it says: "Every edge on a cycle in the graph borders two faces and is traversed by each of the two cycles bounding these two faces, and not by any other face boundary. If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one 'going' and the other 'coming'), and not by any other face boundary." I thought every edge touched two faces. I understand that in this case you were probably referring to edges that touched the outside face. However, I still don't understand how the edge is traversed twice by the cycled in both directions. Why can't it just be traversed once? Thanks, Praveen Pamidimukkala From ajshafer@MIT.EDU Tue Oct 4 09:48:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Dmiw0015499 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:48:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94DmgHS001860 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:48:42 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94Dmdbr020763 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:48:40 -0400 (EDT) Message-ID: <000c01c5c8ea$61468080$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Tue, 4 Oct 2005 09:49:01 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.135 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2062 Content-Length: 361 X-Keywords: NonJunk I was confused by the classifying polyhedra section. Does this depend on some properties of projection? Can this be applied to higher dimensions? Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From mitras@drake.csail.mit.edu Tue Oct 4 14:33:11 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94IXAw0005405; Tue, 4 Oct 2005 14:33:11 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94IXAjQ017885; Tue, 4 Oct 2005 14:33:10 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94IXAJG017884; Tue, 4 Oct 2005 14:33:10 -0400 Date: Tue, 4 Oct 2005 14:33:10 -0400 From: Sayan Mitra Message-Id: <200510041833.j94IXAJG017884@drake.csail.mit.edu> To: 6042-probs@theory.csail.MIT.EDU, shauni@MIT.EDU Subject: Re: [Sayan] Reading comments week 5 Status: RO X-UID: 2063 Content-Length: 1054 X-Keywords: NonJunk A graph is planar if there exists a drawing of the graph (on a flat sheet of paper) such that no two of its edges interset. There are many nice properties of planar graph including Euler's formula (5.1). We will probably see more properties in class exercises. -s. >From shauni@MIT.EDU Tue Oct 4 02:04:57 2005 Date: Tue, 4 Oct 2005 01:57:08 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Reading comments week 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 474 Content-Length: 188 X-Keywords: NonJunk Planar Graphs (pp. 11-12) I'd like to have the section on planar graphs explained more fully in lecture; when can a graph be planar and what are the special properties of a planar graph? From mitras@drake.csail.mit.edu Tue Oct 4 14:40:48 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Iemw0008285; Tue, 4 Oct 2005 14:40:48 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94Iem2L017935; Tue, 4 Oct 2005 14:40:48 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94IemDZ017934; Tue, 4 Oct 2005 14:40:48 -0400 Date: Tue, 4 Oct 2005 14:40:48 -0400 From: Sayan Mitra Message-Id: <200510041840.j94IemDZ017934@drake.csail.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU, lye@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Status: RO X-UID: 2064 Content-Length: 1277 X-Keywords: NonJunk Definition of simple cycles: Consider a cycle C represented by a sequence of vertices v_1,...,v_n. If no two vertices in the sequence are equal other than v_1 = v_n, then C is simple. If there exists v_i, v_j, j \notin {1,n}, such that v_i = v_j, then C is a nonsimple cycle. I did not understand the last part of the question. Are you asking if a nonsimple cycles always contain simple cycles ? That is true. -s. >From lye@MIT.EDU Tue Oct 4 07:40:36 2005 Date: Tue, 4 Oct 2005 07:40:28 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 478 Content-Length: 287 X-Keywords: NonJunk Section 2.1, pg. 5: "Simple cycles will be cycles that don't cross themselves." How are non-simple cycles defined? If a non-simple cycle were taken as a subgraph, would there be a graph it is always isomorphic to? (e.g. for the simple graph, it is always isomorphic to C_n) Lunduo Ye From mitras@drake.csail.mit.edu Tue Oct 4 14:48:22 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94ImMw0009174; Tue, 4 Oct 2005 14:48:22 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94ImM6q017976; Tue, 4 Oct 2005 14:48:22 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94ImMbu017975; Tue, 4 Oct 2005 14:48:22 -0400 Date: Tue, 4 Oct 2005 14:48:22 -0400 From: Sayan Mitra Message-Id: <200510041848.j94ImMbu017975@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, mukkala@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Status: RO X-UID: 2065 Content-Length: 1714 X-Keywords: NonJunk This is probably best explained with a quick figure; however, let me try. You are right, the special case is when the edges are on a outer face. Think of a tree hanging from a bounded face. Indeed, every edge on this tree is also traversed twice; once in each direction. If you draw the circle that traverses this outer face (in which the tree lies). You will notice that every branch of this tree is traversed twice by the circle, once going up and once coming down. -s. >From mukkala@MIT.EDU Tue Oct 4 09:28:32 2005 X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Tue, 04 Oct 2005 09:28:35 -0400 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 480 Content-Length: 677 X-Keywords: In section 5.2, it says: "Every edge on a cycle in the graph borders two faces and is traversed by each of the two cycles bounding these two faces, and not by any other face boundary. If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one 'going' and the other 'coming'), and not by any other face boundary." I thought every edge touched two faces. I understand that in this case you were probably referring to edges that touched the outside face. However, I still don't understand how the edge is traversed twice by the cycled in both directions. Why can't it just be traversed once? Thanks, Praveen Pamidimukkala From dangut@MIT.EDU Tue Oct 4 15:52:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94JqLw0025324 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 15:52:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94JqJ85004831 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 15:52:19 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94JqCCC001537 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 15:52:12 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j94JqC0M009445; Tue, 4 Oct 2005 15:52:12 -0400 Received: from MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU (MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU [18.239.7.231]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 15:52:12 -0400 Message-ID: <20051004155212.soy36pqy00gss00k@webmail.mit.edu> Date: Tue, 4 Oct 2005 15:52:12 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: Wed Oct 5 Reading assignment reply, R13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.556 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2066 Content-Length: 111 X-Status: A X-Keywords: I'm okay with this material, a sidenote question though is what is the difference between a Lemma and a proof? From petek@MIT.EDU Tue Oct 4 16:10:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KAbw0027404 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:10:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KAa85023702 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:10:36 -0400 (EDT) Received: from [18.51.6.17] (HAYDEN-TWO-SEVENTY-TWO.MIT.EDU [18.51.6.17]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KAXqY009544 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:10:34 -0400 (EDT) Message-ID: <4342E1BF.4010003@mit.edu> Date: Tue, 04 Oct 2005 16:10:39 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Comments for This Week Content-Type: multipart/alternative; boundary="------------040509040406040309010603" X-Spam-Score: -2.059 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2067 Content-Length: 2075 X-Keywords: NonJunk This is a multi-part message in MIME format. --------------040509040406040309010603 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit I'm having difficulty with Euler's formula and getting it into my head. Should I memorize it? Or should I understand it's concepts? Also.... I was going to comment on Theorem 4.2 being unsupported, until I found a great way to think of it. A graph can only truly be bipartite if it's chromatic number is 2 (I hope I have the vocab right). If there's an odd cycle, some adjacent nodes would need to be colored the same... (A B A --- cycle of length 3 has AA (no good )) -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------040509040406040309010603 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit I'm having difficulty with Euler's formula and getting it into my head.  Should I memorize it?  Or should I understand it's concepts?


Also....

I was going to comment on Theorem 4.2 being unsupported, until I found a great way to think of it.  A graph can only truly be bipartite if it's chromatic number is 2 (I hope I have the vocab right).  If there's an odd cycle, some adjacent nodes would need to be colored the same...   (A B A --- cycle of length 3 has AA (no good ))  
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------040509040406040309010603-- From alisonc@MIT.EDU Tue Oct 4 16:11:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KBew0027511 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:11:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KBd85024789 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:11:39 -0400 (EDT) Received: from [18.238.5.231] (EASTCAMPUS-SEVEN-FORTY-TWO.MIT.EDU [18.238.5.231]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KBatC010003 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:11:37 -0400 (EDT) Message-ID: <4342E1FD.2070109@mit.edu> Date: Tue, 04 Oct 2005 16:11:41 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Lecture notes 5 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2068 Content-Length: 401 X-Status: A X-Keywords: NonJunk The initial example is interesting to me because if the researcher's methods were sound, then what it *really* proves is that men are much more likely to exaggerate their number of partners (or women to under-estimate). The section on coloring graphs is also really interesting: such a simple way of representing things and yet still complex in the method of doing it and the information gained. From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:37:37 2005 Message-ID: <4342E817.1080700@csail.mit.edu> Date: Tue, 04 Oct 2005 16:37:43 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: sayan mitra Subject: Re: [Sayan] Week 5 Comments References: <6.2.1.2.2.20051004091356.01d77828@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051004091356.01d77828@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 965 Status: RO X-UID: 2069 X-Keywords: Think of a face that has a single edge "hanging" off its boundary. A cycle around the boundary has to go back & forth on the edge to get back to where it starts. The animation on the web illustrates this. Regards, A. Praveen Pamidimukkala wrote: > In section 5.2, it says: > > "Every edge on a cycle in the graph borders two faces and is > traversed by each of the two cycles bounding these two faces, and not by > any other face boundary. If an edge touches only one face, then it is > traversed twice by the cycle that borders the face (one 'going' and the > other 'coming'), and not by any other face boundary." > > I thought every edge touched two faces. I understand that in this case > you were probably referring to edges that touched the outside face. > However, I still don't understand how the edge is traversed twice by the > cycled in both directions. Why can't it just be traversed once? > > Thanks, > Praveen Pamidimukkala > From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:46:27 2005 Message-ID: <4342EA27.7070602@csail.mit.edu> Date: Tue, 04 Oct 2005 16:46:31 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: mike anderson CC: David Shin Subject: Re: [David] Weekly reading comments References: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> In-Reply-To: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 443 Status: RO X-UID: 2070 X-Keywords: "odd" means "odd length" (number of edge traversals) in this context; for simple cycles, this is the same as the number of distinct vertices on it. regards, A. mike anderson wrote: > I am a little confused by theorem 4.2 (page 10), regarding bipartite > graphs and odd cycles. Does odd cycles mean there are an odd number of > paths? this doesn't seem to agree w/ the diagram, and there is no proof > for this theorem. > > > -mike From meyer@imap.theory.csail.mit.edu Tue Oct 04 14:25:31 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94IPUw0003896; Tue, 4 Oct 2005 14:25:30 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94IPU5K017838; Tue, 4 Oct 2005 14:25:30 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94IPUgL017837; Tue, 4 Oct 2005 14:25:30 -0400 Date: Tue, 4 Oct 2005 14:25:30 -0400 From: Sayan Mitra Message-Id: <200510041825.j94IPUgL017837@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, cwong08@MIT.EDU Subject: Re: [TA-name] Week 5 Comments Cc: 6042-staff@theory.csail.mit.edu Status: RO Content-Length: 993 X-UID: 2071 X-Keywords: NonJunk An odd cycle is a cycle with odd number of edges. In the polyhedra part, we are deriving some constraints on the number of vertices, edges, and faces of regular polyhedra, using Euler's formula. Which part exactly did you not understand ? -sayan >From cwong08@MIT.EDU Tue Oct 4 01:30:50 2005 Date: Tue, 04 Oct 2005 01:30:28 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [TA-name] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.944 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 473 Content-Length: 147 X-Keywords: On page 10, when defining bipartite graphs, it is unclear what an "odd cycle" is. I also don't really understand the polyhedra part on page 14. From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:43:11 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j94Kh9w0001547 for <6042-staff@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:43:10 -0400 Received: (qmail 99189 invoked from network); 4 Oct 2005 20:43:09 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 4 Oct 2005 20:43:08 -0000 Message-ID: <4342E962.6030903@csail.mit.edu> Date: Tue, 04 Oct 2005 16:43:14 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Linda Ye CC: 6042-staff@theory.csail.mit.edu Subject: Re: [Sayan] Week 5 Comments References: <20051004074028.ryue99jas8w04gkg@webmail.mit.edu> In-Reply-To: <20051004074028.ryue99jas8w04gkg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 634 X-UID: 2072 X-Keywords: I'm not sure that there is a most useful general def of "cycle", so I'm implicitly letting it be any path that begins and ends where it starts. You can't define it as isomorphic to a subgraph: how would you distinguish the simple cycle C_n from the nonsimple cycle that went around C_n twice? regards, A. Linda Ye wrote: > Section 2.1, pg. 5: > > "Simple cycles will be cycles that don't cross themselves." > > How are non-simple cycles defined? If a non-simple cycle were taken as a > subgraph, would there be a graph it is always isomorphic to? (e.g. for the > simple graph, it is always isomorphic to C_n) > > Lunduo Ye From kevin08@MIT.EDU Tue Oct 4 16:51:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KpHw0002803 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:51:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Kow85003807 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:50:58 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-ONE-SIXTY-THREE.MIT.EDU [18.242.5.163]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KorJ7026731 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:50:55 -0400 (EDT) Message-Id: <6.2.3.4.2.20051004164841.01e52848@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Tue, 04 Oct 2005 16:50:51 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.001 X-Spam-Level: * (1.001) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2073 Content-Length: 236 X-Keywords: NonJunk Hi, It seems to me that bipartite graphs are really important, but they're not discussed too heavily in the notes. Could you add an example or two of how they're applied to solve "common" computer science questions? Thanks, Kevin From avalys@MIT.EDU Tue Oct 4 16:53:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KrBw0002932 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:53:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KqJ85005031 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:52:19 -0400 (EDT) Received: from [18.96.5.183] (SIMMONS-ONE-EIGHTY-THREE.MIT.EDU [18.96.5.183]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KqDEd027320 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:52:13 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <248E756A-4D81-4C26-92D3-C276999202D0@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Required reading comments... Date: Tue, 4 Oct 2005 16:52:15 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2074 Content-Length: 195 X-Keywords: NonJunk I don't really have anything significant to say, the reading was fairly straightforward. The sections on Euler's Formula and polyhedrons were a bit difficult to follow, though. Alex Valys From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:53:42 2005 Message-ID: <4342EBDC.1060808@csail.mit.edu> Date: Tue, 04 Oct 2005 16:53:48 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alison Cichowlas CC: Hanson M Zhou Subject: Re: [Hanson] Lecture notes 5 comments References: <4342E1FD.2070109@mit.edu> In-Reply-To: <4342E1FD.2070109@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 604 Status: RO X-UID: 2075 X-Keywords: Alternatively, it could be that the researchers used a sample that had 74% more women, though how they would select such a skewed sample is puzzling. regards, A. Alison Cichowlas wrote: > The initial example is interesting to me because if the researcher's > methods were sound, then what it *really* proves is that men are much > more likely to exaggerate their number of partners (or women to > under-estimate). The section on coloring graphs is also really > interesting: such a simple way of representing things and yet still > complex in the method of doing it and the information gained. From hmzhou@blackbird.csail.mit.edu Tue Oct 4 17:01:20 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94L1Kw0004550; Tue, 4 Oct 2005 17:01:20 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j94L1JUA019203; Tue, 4 Oct 2005 17:01:20 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j94L1JwK019200; Tue, 4 Oct 2005 17:01:19 -0400 Date: Tue, 4 Oct 2005 17:01:19 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu, 6042-staff@theory.csail.mit.edu Subject: Re: [Hanson] Week 5 Comments In-Reply-To: <6.2.3.4.2.20051004164841.01e52848@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051004164841.01e52848@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2076 Content-Length: 443 X-Keywords: NonJunk Bipartite matchings are widely applicable. They actually use matching algorithms to assign med students to hospitals or something like that. -Hanson On Tue, 4 Oct 2005, Kevin Wang wrote: > Hi, > > It seems to me that bipartite graphs are really important, but > they're not discussed too heavily in the notes. Could you add an > example or two of how they're applied to solve "common" computer > science questions? > > Thanks, > Kevin > > From meyer@imap.theory.csail.mit.edu Tue Oct 04 17:01:51 2005 Return-Path: Received: from smtp014.mail.yahoo.com (smtp014.mail.yahoo.com [216.136.173.58]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j94L1mw0004654 for <6042-staff@theory.csail.mit.edu>; Tue, 4 Oct 2005 17:01:49 -0400 Received: (qmail 43242 invoked from network); 4 Oct 2005 21:01:48 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp014.mail.yahoo.com with SMTP; 4 Oct 2005 21:01:47 -0000 Message-ID: <4342EDC2.8010304@csail.mit.edu> Date: Tue, 04 Oct 2005 17:01:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: David Shin CC: 6042-staff@theory.csail.mit.edu Subject: Re: [David] Weekly reading comments References: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> <4342EA27.7070602@csail.mit.edu> <4342EAF5.7030806@mit.edu> In-Reply-To: <4342EAF5.7030806@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1064 X-UID: 2077 X-Keywords: good question. i thought to cc 6042-staff selectively on answers that I thought might be of general interest; i think I've only done this one or twice so far this week. i have also been cc'ing the specific TA named by the student when I reply (as well as filing my reply in probs), tho i wasn't sure if this was a good thing to do. Shall I stop cc'ing you and just leave my replies in the probs file? regards, A. David Shin wrote: > Professor Meyer, > > Is there a reason you are CC'ing 6042-staff or the TA's instead of > 6042-probs? > > Prof. Albert R. Meyer wrote: > >> "odd" means "odd length" (number of edge traversals) in this context; >> for simple cycles, this is the same as the number of distinct vertices >> on it. >> regards, A. >> >> mike anderson wrote: >> >>> I am a little confused by theorem 4.2 (page 10), regarding bipartite >>> graphs and odd cycles. Does odd cycles mean there are an odd number >>> of paths? this doesn't seem to agree w/ the diagram, and there is no >>> proof for this theorem. >>> >>> >>> -mike >> >> From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:50:40 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Kodw0002736 for ; Tue, 4 Oct 2005 16:50:39 -0400 Received: from south-station-annex.mit.edu ([18.72.1.2]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EMtkF-0008Nm-K7 for meyer@csail.mit.edu; Tue, 04 Oct 2005 16:50:39 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j94KocsS029501 for ; Tue, 4 Oct 2005 16:50:38 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KocSF024732 for ; Tue, 4 Oct 2005 16:50:38 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j94KoGok001100 for ; Tue, 4 Oct 2005 16:50:17 -0400 (EDT) Message-ID: <4342EAF5.7030806@mit.edu> Date: Tue, 04 Oct 2005 16:49:57 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Prof. Albert R. Meyer" Subject: Re: [David] Weekly reading comments References: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> <4342EA27.7070602@csail.mit.edu> In-Reply-To: <4342EA27.7070602@csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO X-Status: A Content-Length: 591 X-UID: 2078 X-Keywords: Professor Meyer, Is there a reason you are CC'ing 6042-staff or the TA's instead of 6042-probs? Prof. Albert R. Meyer wrote: > "odd" means "odd length" (number of edge traversals) in this context; > for simple cycles, this is the same as the number of distinct vertices > on it. > regards, A. > > mike anderson wrote: > >> I am a little confused by theorem 4.2 (page 10), regarding bipartite >> graphs and odd cycles. Does odd cycles mean there are an odd number >> of paths? this doesn't seem to agree w/ the diagram, and there is no >> proof for this theorem. >> >> >> -mike > From arup@MIT.EDU Tue Oct 4 17:24:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94LOnw0009829 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:24:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94LOm2e003074 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:24:48 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-TWO-TWENTY-SEVEN.MIT.EDU [18.247.5.227]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94LOiNa009331 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:24:45 -0400 (EDT) Message-Id: <6.2.3.4.2.20051004172123.047bf878@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Tue, 04 Oct 2005 17:24:41 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Week 5 Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.045 X-Spam-Level: * (1.045) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2079 Content-Length: 299 X-Keywords: Section 2.5, page 6 (and used later as well): I think I don't completely understand the "shrink-down, grow-back" style of induction. It seems like it assumes P(n+1) before proving it. Also, what are examples of the logical errors that can occur if this style of argument isn't used? |Arup| From yaser@MIT.EDU Tue Oct 4 17:54:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94LsQw0018482 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:54:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94LsP2e027001 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:54:25 -0400 (EDT) Received: from Charizard (NEW-SIX-EIGHTY-THREE.MIT.EDU [18.241.7.172]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94LsMXg019580 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:54:22 -0400 (EDT) Message-Id: <200510042154.j94LsMXg019580@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: week 5 reading question Date: Tue, 4 Oct 2005 17:54:20 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0078_01C5C90C.A5C85F90" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXJLixKBLADFgtcQGuzIpTlXRtBpg== X-Spam-Score: 0.7 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2080 Content-Length: 5964 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0078_01C5C90C.A5C85F90 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi David, When it comes to k-coloring, is there a direct relation between k-coloring of vertices (as outlined in the reading) and the more general map-coloring problem? I.e., on the tutor problem there was a graph that had a chromatic value of 3, but would have a chromatic value of 2 if we were attempting to fill in edges or areas rather than vertices. Are areas between edges EVER considered legitimate or useful in such graphs? Thanks! Take care, _Yaser ------=_NextPart_000_0078_01C5C90C.A5C85F90 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

When it comes to k-coloring, is = there a direct relation between k-coloring of vertices (as outlined in the = reading) and the more general map-coloring problem? I.e., on the tutor problem there = was a graph that had a chromatic value of 3, but would have a chromatic value = of 2 if we were attempting to fill in edges or areas rather than vertices. Are = areas between edges EVER considered legitimate or useful in such = graphs?

 

Thanks! =


Take care,

 

_Yaser

------=_NextPart_000_0078_01C5C90C.A5C85F90-- From adnaan@MIT.EDU Tue Oct 4 17:55:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Ltkw0018539 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 17:55:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Ltj2e027978 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 17:55:45 -0400 (EDT) Received: from m12-182-12.mit.edu (M12-182-12.MIT.EDU [18.19.0.43]) (authenticated bits=56) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94LtgAr019988 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 17:55:43 -0400 (EDT) Received: (from adnaan@localhost) by m12-182-12.mit.edu (8.12.9) id j94LtgGY003447; Tue, 4 Oct 2005 17:55:42 -0400 Subject: Week 5 comments-Hanson From: Adnaan N Jiwaji To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 04 Oct 2005 17:55:42 -0400 Message-Id: <1128462942.3180.4.camel@m12-182-12.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2081 Content-Length: 47 X-Keywords: The concept of a bipartite graph is confusing From natalia3@MIT.EDU Tue Oct 4 18:11:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94MBxw0020951 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 18:11:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94MBw2e009596 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 18:11:58 -0400 (EDT) Received: from hayden-1.mit.edu (HAYDEN-1.MIT.EDU [18.51.1.31]) (authenticated bits=56) (User authenticated as natalia3@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94MBmrG024178 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 18:11:49 -0400 (EDT) Received: (from natalia3@localhost) by hayden-1.mit.edu (8.12.9) id j94MBmMA020998; Tue, 4 Oct 2005 18:11:48 -0400 Subject: [Jelani] Week 5 Comments From: Natalia N Chernenko To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 04 Oct 2005 18:11:48 -0400 Message-Id: <1128463908.20774.11.camel@hayden-1.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2082 Content-Length: 219 X-Keywords: NonJunk It would be nice to discuss planar graphs (Section 5, pp. 11-13) more in lecture, especially Euler's formula (Section 5.1). I found the derivation of the formula a little confusing (induction proof). Natalia Chernenko From dnreshef@MIT.EDU Tue Oct 4 18:59:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Mxmw0027491 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 18:59:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Mxl2e013327 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 18:59:47 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94MxF76005997 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 18:59:15 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j94MxFgC022215; Tue, 4 Oct 2005 18:59:15 -0400 Received: from M66-064-09.MIT.EDU (M66-064-09.MIT.EDU [18.63.4.61]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 18:59:15 -0400 Message-ID: <20051004185915.x4fwuy4o38ws0ckg@webmail.mit.edu> Date: Tue, 4 Oct 2005 18:59:15 -0400 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments for 10/5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2083 Content-Length: 260 X-Keywords: NonJunk Hey, This weeks reading was actually more interesting than usual, and I am really interested in learning more about graph theory. Hall's marriage thm on page 15 was a little confusing. Also, I wouldn't mind discussing connectedness further in class. -Dave From aston@MIT.EDU Tue Oct 4 19:20:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94NK8w0030978 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:20:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94NK72e026668 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:20:07 -0400 (EDT) Received: from aston (BOOKX.MIT.EDU [18.241.0.191]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NK4i1009988 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:20:05 -0400 (EDT) Message-Id: <200510042320.j94NK4i1009988@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: Email comments Date: Tue, 4 Oct 2005 19:20:11 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_007B_01C5C918.A4515980" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 thread-index: AcXJOirdtoR8vEvLRjiOxlM0LBMM8g== X-Spam-Score: 1.722 X-Spam-Level: * (1.722) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2084 Content-Length: 2251 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_007B_01C5C918.A4515980 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I found pages 14 and 15, the explanation of the limitations on regular 3d shapes, to be theoretically very cool and interesting, but I got lost when the equations started appearing, even after a couple read throughs. - Aston ------=_NextPart_000_007B_01C5C918.A4515980 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I found pages 14 and 15, the explanation of the = limitations on regular 3d shapes, to be theoretically very cool and interesting, but = I got lost when the equations started appearing, even after a couple read = throughs.

 

         =    - Aston

------=_NextPart_000_007B_01C5C918.A4515980-- From ridell@MIT.EDU Tue Oct 4 19:24:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94NO2w0001692 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 19:24:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94NO12e029068 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 19:24:01 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NNsCl010622 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 19:23:54 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j94NNsUs028364; Tue, 4 Oct 2005 19:23:54 -0400 Received: from STRATTON-THREE-SIXTY-SEVEN.MIT.EDU (STRATTON-THREE-SIXTY-SEVEN.MIT.EDU [18.187.6.112]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 19:23:54 -0400 Message-ID: <20051004192354.o5g222c08lo0o8c0@webmail.mit.edu> Date: Tue, 4 Oct 2005 19:23:54 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: week 5 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2085 Content-Length: 352 X-Keywords: NonJunk I am confused about the colorale graphs, and the theorem described on page 10. Could you explain more about how a graph with degree at most k is (k+1)-colorable. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From kktyan@MIT.EDU Tue Oct 4 19:31:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94NVXw0003556 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 19:31:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94NVV2e003901 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 19:31:32 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NVO8P012178 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 19:31:24 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j94NVOAd027956; Tue, 4 Oct 2005 19:31:24 -0400 Received: from BURTON-FOUR-THIRTY-FOUR.MIT.EDU (BURTON-FOUR-THIRTY-FOUR.MIT.EDU [18.247.6.179]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 19:31:24 -0400 Message-ID: <20051004193124.wo2hpl958jcwc84o@webmail.mit.edu> Date: Tue, 4 Oct 2005 19:31:24 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading 5 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2086 Content-Length: 1026 X-Keywords: I found the part about k-connected vertices most confusing. Namely, the passage on page 5: Definition 2.6. Two vertices in a graph are k-connected if they remain connected in any subgraph obtained by deleting k-1 edges. A graph is k-connected if every pair of its vertices are k-connected. How do you know what "k" is? What I mean to say is - to test for k-connectivity, you have to delete k-1 edges and see if the two vertices in question are still conected. I assume that this refers just to the edges that exist along some path between the two vertices, but I still don't quite understand how "k" is derived. Like, in Figure 1 on page 3, they say that B and E are 2-connected, which means you could delete 1 edge between the two vertices and still be connected, correct? But couldn't you delete the three edges (B-C) (C-D) and (D-E) and still be connected? I'm sorry, this was very long. I'm just confused over the definition of k-connectivity. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From bilodeau@MIT.EDU Tue Oct 4 19:50:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Notw0005696 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:50:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Noa2e015220 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:50:36 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THIRTY.MIT.EDU [18.239.5.30]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NoXBo015901 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:50:33 -0400 (EDT) Message-Id: <200510042350.j94NoXBo015901@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Comments [jelani] Date: Tue, 4 Oct 2005 19:50:26 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C91C.DE4183F0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJPmS638Nbv5hsQQaD++88v/IpRQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.419 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2087 Content-Length: 3141 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C91C.DE4183F0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Reading comments week 5: In this weeks reading, the section about using Euler's rule for planar graphs to examine regular polyhedra makes sense, but it seems to me that a figure with enough faces cannot be viewed through one face so that all edges are within it. I find this a little confusing, but I guess its just another logical leap based on geometrical intuition. Peter Bilodeau ------=_NextPart_000_0000_01C5C91C.DE4183F0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Reading comments week 5:

In this weeks reading, the section about using = Euler’s rule for planar graphs to examine regular polyhedra makes sense, but it = seems to me that a figure with enough faces cannot be viewed through one face = so that all edges are within it.  I find this a little confusing, but I = guess its just another logical leap based on geometrical = intuition.

 

Peter Bilodeau

------=_NextPart_000_0000_01C5C91C.DE4183F0-- From hzhou@MIT.EDU Tue Oct 4 20:00:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9500Ew0007831 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:00:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9500Dkj020157 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:00:13 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-TWO-EIGHTY-THREE.MIT.EDU [18.247.6.28]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9500AWg017553 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:00:11 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004195656.02a9d758@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 20:00:14 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: email Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.055 X-Spam-Level: * (1.055) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2088 Content-Length: 455 X-Keywords: NonJunk Hi, I didn't really have any questions concerning the lecture notes, but the one example brought up in class was somewhat confusing. Prof Meyer said that the survey that stated 75% more guys had sexual interactions than girls is false. However, if you view guys/girls as vertices, the survey is simply saying that 75% more of the guys vertices has edges compared to girls vertices with edges? Thus the survey could have some validity?? - Steve From fluff@MIT.EDU Tue Oct 4 20:03:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95037w0008030 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:03:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95035kj021772 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:03:06 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95031Jo018009 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:03:01 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8BC5300F-BC1D-48A6-88A8-24CBAFC63466@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 5 reading Date: Tue, 4 Oct 2005 20:02:57 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2089 Content-Length: 791 X-Keywords: NonJunk Overall, I thought the reading was pretty understandable and for once had enough examples. I didn't get this thing in the proof of Euler's Formula (pg. 12). In the base case, it says that a graph with nothing but a single vertex has one face, the "outside" face. What the? How can something with no edges have faces? Also, I have a comment about something on page 6, in the second paragraph after Corollary 2.10. It talks about taking an n+1 graph, then generalizing about an n subgraph to generalize about the n+1 one in induction. This is something that we needed in the pset that was due on Monday (on the last problem). Suggestion: maybe in the future, this shrink-down/grow-back paradigm in induction with graphs can be introduced before the pset is due. ~Crystal From dowgun@MIT.EDU Tue Oct 4 20:10:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j950APw0008461 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:10:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j950AOkj026506 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:10:24 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j950AMlI019192 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:10:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j950AMYQ019786; Tue, 4 Oct 2005 20:10:22 -0400 Received: from PLP-FORTY-SIX.MIT.EDU (PLP-FORTY-SIX.MIT.EDU [18.218.1.46]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 20:10:22 -0400 Message-ID: <20051004201022.2x0to8rcggg8gsow@webmail.mit.edu> Date: Tue, 4 Oct 2005 20:10:22 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2090 Content-Length: 907 X-Keywords: NonJunk I was most interested in the section about planar embeddings (page 12) because I was sure that the notes were going to prove the idea behind many a child's logic puzzle - that all embedded graphs are 4-colorable. Surprisingly they did not! This left me to wonder if this theorem has ever actually been proven, although I know from experience that it is almost certainly true. In fact, the coloring of the faces of a planar embeddign never came up at all, so I thought maybe it would be better to replace the embedding with a graph of vertices where each vertex stands in for a face, and each edge crosses an edge that it is replacing. It is a feature of this graph that no edges will ever cross each other. Therefore, this is ANOTHER planar embedding, so it follows that if the faces of planar embeddings are 4-colorable, THEN THE VERTICES ARE TOO. Has anyone ever tried to prove THIS theorem? Neil Dowgun From crowell@MIT.EDU Tue Oct 4 20:44:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j950iZw0011422 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:44:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j950iYkj016309 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:44:34 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j950iW9G025125 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:44:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j950iW2U004130; Tue, 4 Oct 2005 20:44:32 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 20:44:32 -0400 Message-ID: <20051004204432.jnvky8j8wbxes44w@webmail.mit.edu> Date: Tue, 4 Oct 2005 20:44:32 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 5 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2091 Content-Length: 665 X-Keywords: I did not understand the proof of lemma 5.2 on page 13 (specifically the line in the proof that states 'If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one ?going? and the other ?coming?), and not by any other face boundary.' I am also not clear on part 1 of the proof of the Marriage theorem on page 16 (I don't understand how "In this case, we have some latitude: we pair an arbitrary girl with a boy she likes and send them both away" ensures that we don't violate the marriage condition (if, say, G1 likes B1 and G2 likes B1, B2, B3 and we first pair G2 with B1 and send them "on their way"). -Rob Crowell From vixen@MIT.EDU Tue Oct 4 20:53:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j950rIw0011967 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 20:53:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j950rHkj021328 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 20:53:17 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j950r904026587 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 20:53:10 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Tue, 4 Oct 2005 20:52:14 -0400 To: 6042-probs@theory.csail.mit.edu From: Amanda Seybold Subject: [Jelani] LN5 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2092 Content-Length: 141 X-Keywords: I was confused by the formal statement of Hall's Theorem on page 17. I would also like to see algorithms for matching in a bipartite graph. From yangc@MIT.EDU Tue Oct 4 21:18:58 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j951Ivw0013668 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:58 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j951IuNJ008422 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:56 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j951IuIY001811 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:56 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j951InMv011661 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:53 -0400 (EDT) Message-Id: <200510050118.j951InMv011661@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 5 Comments Date: Tue, 4 Oct 2005 21:18:45 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000A_01C5C929.346CB9F0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJSrsoqClu6QFGQyKK3y8HwBpGOQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 0.872 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2093 Content-Length: 2322 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_000A_01C5C929.346CB9F0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit On page 17, it says that there are very efficient algorithms for doing the Hall's Theorem matching. What are some examples of those? Also, what sorts of problems can be reduced to finding a matching? Thanks, Chris Yang ------=_NextPart_000_000A_01C5C929.346CB9F0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

On page 17, it says that there are very efficient = algorithms for doing the Hall’s Theorem matching.  What are some = examples of those?  Also, what sorts of problems can be reduced to finding a matching?  =

 

Thanks,

Chris Yang

------=_NextPart_000_000A_01C5C929.346CB9F0-- From hsoumare@MIT.EDU Tue Oct 4 22:04:07 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95247w0020051 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:07 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j95246NJ023812 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:06 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95245IY002799 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:06 -0400 (EDT) Received: from [18.241.6.193] (NEW-FOUR-FORTY-EIGHT.MIT.EDU [18.241.6.193]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95241Mu013525 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:01 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: [Sayan] Weekly Email Comment Date: Tue, 4 Oct 2005 22:12:35 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2094 Content-Length: 259 X-Status: A X-Keywords: NonJunk I found the passage on sex in America to be the most surprising because I never thought that you could prove mathematically, that the only reason men are more promiscuous than women is because of the differences in the population size. Hamidou Soumare From ozcan@MIT.EDU Tue Oct 4 22:05:22 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9525Mw0020194 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:05:22 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9525MOv014950 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:05:22 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9525Eol013757 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:05:14 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004220040.028accd0@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 22:05:02 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2095 Content-Length: 491 X-Keywords: Page 9, in Coloring Graphs: We can recast this scheduling problem as a question about coloring the vertices of a graph. Create a vertex for each course with a final exam. Put an edge between two vertices if some student is taking both courses. The above paragraph, and the section it comes from, explains how to model the scheduling problem with graphs. I was quite amazed (and surprised) how such a complex problem is modeled in a very simple and comprehensive way. best, Yasin From icharny@mit.edu Tue Oct 4 22:10:52 2005 Return-Path: Received: from auathena (EASTCAMPUS-SIX-NINETY-EIGHT.MIT.EDU [18.238.5.187]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j952Apw0020615 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 22:10:51 -0400 Received: (from icharny@localhost) by auathena (8.12.9) id j952AnIj010271; Tue, 4 Oct 2005 22:10:49 -0400 Subject: [Sayan] Week 5 Comments From: Isaac E Charny To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 04 Oct 2005 22:10:48 -0400 Message-Id: <1128478249.24973.16.camel@auathena> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 Status: RO X-UID: 2096 Content-Length: 123 X-Keywords: NonJunk I was very surprised that the faces, edges, and vertices were related by the simple equation v - e + f = 2. ~Isaac Charny From jjmonzon@MIT.EDU Tue Oct 4 22:15:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j952Ffw0020907 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:15:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j952Fdkj009018 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:15:39 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-TWO-TWELVE.MIT.EDU [18.247.5.212]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j952FKHI012253 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:15:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004221138.020ed008@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 22:15:24 -0400 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2097 Content-Length: 370 X-Keywords: I found the Hall's Marriage Theorem a bit confusing and I hope this would be discussed better in lecture. I just found it hard to read the notation the notes uses and a layman's explanation would be very much appreciated. Josh Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From brevzin@MIT.EDU Tue Oct 4 22:36:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j952aYw0025987 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:36:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j952aXkj021282 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:36:33 -0400 (EDT) Received: from BARRY.mit.edu ([18.218.1.202]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j952aPpO016637 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:36:26 -0400 (EDT) Message-Id: <6.1.2.0.1.20051004223238.019d6ce8@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Tue, 04 Oct 2005 22:36:33 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2098 Content-Length: 269 X-Keywords: I found the classifying polyhedra section very interesting, since I never knew that you could actually prove that there are only 5. I just took it for granted that they were there. Also, the proof was of Euler's formula was pretty hot. I enjoyed reading it. Barry From dshin@MIT.EDU Tue Oct 4 23:01:28 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9531Rw0028488 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:01:27 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9531Qjv017175 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:01:26 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j952w38U003989; Tue, 4 Oct 2005 22:58:03 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j952vrMu015392; Tue, 4 Oct 2005 22:57:53 -0400 (EDT) Message-ID: <43434111.8040108@mit.edu> Date: Tue, 04 Oct 2005 22:57:21 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yaser Khan CC: 6042-probs@theory.lcs.mit.edu Subject: Re: David: week 5 reading question References: <200510042154.j94LsMXg019580@outgoing.mit.edu> In-Reply-To: <200510042154.j94LsMXg019580@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------030405050302080903060109" X-Spam-Score: -2.448 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2099 Content-Length: 7118 X-Keywords: This is a multi-part message in MIME format. --------------030405050302080903060109 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Yes, you are touching upon some very deep concepts with this line of thought. Map coloring and "vertex coloring" are equivalent in some sense: given some map (say, a map of the USA), construct a graph G that has a vertex for each region in the map (US states), and an edge between u and v if the corresponding regions border each other. map <---> graph region <---> vertex border <---> edge Then, coloring the regions in the map is equivalent to coloring the vertices of the graph. DS Yaser Khan wrote: > Hi David, > > > > When it comes to k-coloring, is there a direct relation between > k-coloring of vertices (as outlined in the reading) and the more > general map-coloring problem? I.e., on the tutor problem there was a > graph that had a chromatic value of 3, but would have a chromatic > value of 2 if we were attempting to fill in edges or areas rather than > vertices. Are areas between edges EVER considered legitimate or useful > in such graphs? > > > > Thanks! > > > Take care, > > > > _Yaser > --------------030405050302080903060109 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Yes, you are touching upon some very deep concepts with this line of thought.

Map coloring and "vertex coloring" are equivalent in some sense:  given some map (say, a map of the USA), construct a graph G that has a vertex for each region in the map (US states), and an edge between u and v if the corresponding regions border each other.

map <---> graph
region <---> vertex
border <---> edge

Then, coloring the regions in the map is equivalent to coloring the vertices of the graph.

DS

Yaser Khan wrote:

Hi David,

 

When it comes to k-coloring, is there a direct relation between k-coloring of vertices (as outlined in the reading) and the more general map-coloring problem? I.e., on the tutor problem there was a graph that had a chromatic value of 3, but would have a chromatic value of 2 if we were attempting to fill in edges or areas rather than vertices. Are areas between edges EVER considered legitimate or useful in such graphs?

 

Thanks!


Take care,

 

_Yaser

--------------030405050302080903060109-- From moscicki@MIT.EDU Tue Oct 4 23:05:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9535Ww0028989 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:05:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9535R6s007650 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:05:27 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9535OhC022313 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:05:24 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9535OTN022801; Tue, 4 Oct 2005 23:05:24 -0400 Received: from NEW-THREE-SIXTY-SEVEN.MIT.EDU (NEW-THREE-SIXTY-SEVEN.MIT.EDU [18.241.6.112]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:05:24 -0400 Message-ID: <20051004230524.kwm3677bu70gwso4@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:05:24 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: week 5 comment MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_3ch3plf2khds" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 3.14 X-Spam-Level: *** (3.14) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2100 Content-Length: 1159 X-Keywords: NonJunk This message is in MIME format. --=_3ch3plf2khds Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit attached --=_3ch3plf2khds Content-Type: text/plain; charset=ISO-8859-1; name="week5comment.txt" Content-Disposition: attachment; filename="week5comment.txt" Content-Transfer-Encoding: 7bit Back in the realm of intution. All the stuff before Coloring Graphs was covered in class. It was clear and I think I understood it. Coloring- an interesting way of looking at graphs. Also, the general method of shrink-down grow-back I think would have been useful for the previous problem set. Its nice to have a general method of using induction on graphs. Before I read about that, it seemed like something that would be really hard to do. Bipartite Graphs are understandable. Planar Graphs are a bit fuzzy in my mind. I definitely appreciate the applications to PCB design though. Euler's formula is fine, I have a feeling that will be important anyway. It might have been nice to state Hall's Theorem formally first so the reader can keep it in mind as we go through the example. --=_3ch3plf2khds-- From lana@MIT.EDU Tue Oct 4 23:18:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953Icw0030888 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:18:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953Ib6s015573 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:18:37 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953IYh3025001 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:18:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j953IYHW022942; Tue, 4 Oct 2005 23:18:34 -0400 Received: from NEXT-SEVEN-THIRTEEN.MIT.EDU (NEXT-SEVEN-THIRTEEN.MIT.EDU [18.242.7.202]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:18:33 -0400 Message-ID: <20051004231833.fe7j5cx0guosgk0o@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:18:33 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.165 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2101 Content-Length: 331 X-Keywords: NonJunk I was wondering if there was a way of proving that a graph cannot be planar without Euler's Formula (for example the two graphs used in the reading: the complete graph on 5 vertices and the complete bipartite graph on 6 vertices). Is there an algorithm (other than Euler's formula) of showing that a graph cannot be planar? Lana From tonyng@MIT.EDU Tue Oct 4 23:19:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953JOw0030917 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:19:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953JM6s015897 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:19:23 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953JDqZ025072 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:19:15 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004231521.020f4678@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 23:19:28 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2102 Content-Length: 463 X-Keywords: I found Coloring Graphs (section 4, page 9) most surprising. I always thought that graphs were used to visualize relations between vertices. However, I never realized it was possible to assign colors and get other useful applications such as the finals scheduling situation (I have seen assigning numbers to edges to represent distances, but I've never seen assigning properties to the vertices themselves and getting something useful from it). - Tony Ng From aka_kame@MIT.EDU Tue Oct 4 23:21:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953Ltw0031104 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:21:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953Ls6s017471 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:21:54 -0400 (EDT) Received: from [18.234.1.87] (DP-EIGHTY-SEVEN.MIT.EDU [18.234.1.87]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953LldY025539 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:21:47 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <382973AE-959B-446E-9671-C62830B1DF67@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-1--148549695 From: Akari Kameyama Subject: Week 5 comments Date: Tue, 4 Oct 2005 23:21:45 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 0.501 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2103 Content-Length: 3422 X-Keywords: NonJunk --Apple-Mail-1--148549695 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed "We can regard the corners and edges of these polyhedra as the vertices and edges of a planar graph. (This is another logical leap based on geometric intuition.)" So far we have only discussed "graphs" from a 2-dimensional perspective. Although 2D graphs can be "flexed" (for lack of a better word) to become 3D, it feels as though there exists/should exist a whole different realm of possibilities in the 3D world which cannot be translated into 2D. For example, the dog and house situation, could be done in a 3D representation yes? --Apple-Mail-1--148549695 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
"We can regard the corners and edges of these = polyhedra as the vertices and edges of a planar=A0
graph. (This is = another logical leap based on geometric = intuition.)"

So far we have = only discussed "graphs" from a 2-dimensional perspective.=A0 Although 2D = graphs can be "flexed" (for lack of a better word) to become 3D, it = feels as though there exists/should exist a whole different realm of = possibilities in the 3D world which cannot be translated into 2D.=A0 For = example, the dog and house situation, could be done in a 3D = representation yes?


= --Apple-Mail-1--148549695-- From jehan@MIT.EDU Tue Oct 4 23:28:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953Suw0031624 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:28:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953Ss6s021594 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:28:54 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953SlwX026902 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:28:48 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004232442.00bbaed0@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 23:30:08 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2104 Content-Length: 66 X-Keywords: NonJunk I didn't really understand the proof for euler's formula. jehan From kjhollen@MIT.EDU Tue Oct 4 23:33:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953XMw0032282 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:33:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953XL6s024226 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:33:21 -0400 (EDT) Received: from MACGREGOR-FOUR-O-FOUR.MIT.EDU (MACGREGOR-FOUR-O-FOUR.MIT.EDU [18.239.6.149]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953XB7c027714 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:33:14 -0400 (EDT) Subject: [Hanson] clarification of Hall's Theorem From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Tue, 04 Oct 2005 23:33:11 -0400 Message-Id: <1128483191.30474.4.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 2.544 X-Spam-Level: ** (2.544) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2105 Content-Length: 418 X-Keywords: NonJunk Hello, I understand what Hall's Theorem (p. 15-17) is and it's pretty neat -- it seems pretty straight forward how to use it to find a mapping between different subsets in graphs, and I can think of how it would be useful in a computer program. However, I struggled a bit with the tutor problem on how to know when you can't apply Hall's Theorem. I think I get it, but it would be nice to see another example. Kate From meyer@imap.theory.csail.mit.edu Tue Oct 04 23:43:21 2005 Message-ID: <43434BDF.8080203@csail.mit.edu> Date: Tue, 04 Oct 2005 23:43:27 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hamidou Soumare CC: 6042-staff@theory.csail.mit.edu Subject: Re: [Sayan] Weekly Email Comment References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 813 Status: RO X-UID: 2106 X-Keywords: It's a misunderstanding to say "men are more promiscuous ...because..." The relative fraction of sexual partners has nothing to do with promiscuity; it just shows how misleading averages can be. For example, suppose there was a population of 100 mean and 1000 woman, and each man was married to one woman, (leaving 900 unmarried women). Now the men have on average 10 times as many partners (man average = 0.1, woman average 0.01), but I don't think you'd want to claim anybody was promiscuous. regards, A. Hamidou Soumare wrote: > I found the passage on sex in America to be the most surprising because > I never thought that you could prove mathematically, that the only > reason men are more promiscuous than women is because of the > differences in the population size. > > Hamidou Soumare From rshroff@MIT.EDU Tue Oct 4 23:49:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953niw0001460 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:49:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953nh6s003483 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:49:43 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953nfvP000680 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:49:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j953nfiq031584; Tue, 4 Oct 2005 23:49:41 -0400 Received: from NEXT-SIXTY-SEVEN.MIT.EDU (NEXT-SIXTY-SEVEN.MIT.EDU [18.242.5.67]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:49:41 -0400 Message-ID: <20051004234941.22n59d76b1xc8ckw@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:49:41 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David]E-mail Comments for assigned reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2107 Content-Length: 163 X-Keywords: NonJunk Reference: Sections 3.2 and 4.2 I was a bit confused about the idea of spanning trees and Bipartite graphs, and would like to go over it in class. -Rahul Shroff From minilek@theory.csail.mit.edu Tue Oct 4 23:51:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953pMw0001590 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:51:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953pJ6s004542; Tue, 4 Oct 2005 23:51:19 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953pCpC000975 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 4 Oct 2005 23:51:16 -0400 (EDT) Message-ID: <43434DC7.8090809@theory.csail.mit.edu> Date: Tue, 04 Oct 2005 23:51:35 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Pete Kruskall Subject: Re: [Jelani] Comments for This Week References: <4342E1BF.4010003@mit.edu> In-Reply-To: <4342E1BF.4010003@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.529 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2108 Content-Length: 1182 X-Keywords: NonJunk >> A graph can only truly be bipartite if it's chromatic number is 2 (I hope I have the vocab right). This is correct, but this is true by definition (a graph is defined to be bipartite if it's chromatic number is two). As for Euler's formula, it would be best to try to understand its proof. Try to read it over and digest it -- it may take a few times reading over. Let me know if you still have questions about it after we cover it in lecture. -Jelani Pete Kruskall wrote: > I'm having difficulty with Euler's formula and getting it into my > head. Should I memorize it? Or should I understand it's concepts? > > > Also.... > > I was going to comment on Theorem 4.2 being unsupported, until I found > a great way to think of it. A graph can only truly be bipartite if > it's chromatic number is 2 (I hope I have the vocab right). If > there's an odd cycle, some adjacent nodes would need to be colored the > same... (A B A --- cycle of length 3 has AA (no good )) > >-- >Pete Kruskall >28 The Fenway >Boston, MA 02215 > >::::::::::::::::::::::::::::: >617.536.9925 :::Sigma Nu::::: >508.843.5861 ::::Cell Phone:: >::::http://tege.mit.edu:::::: > > > From minilek@theory.csail.mit.edu Tue Oct 4 23:56:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953uSw0001859 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:56:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953uP6s007501; Tue, 4 Oct 2005 23:56:25 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953uBPD001826 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 4 Oct 2005 23:56:18 -0400 (EDT) Message-ID: <43434EF1.1060008@theory.csail.mit.edu> Date: Tue, 04 Oct 2005 23:56:33 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Arup Sarma Subject: Re: [Jelani] Week 5 Reading Comments References: <6.2.3.4.2.20051004172123.047bf878@po14.mit.edu> In-Reply-To: <6.2.3.4.2.20051004172123.047bf878@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2109 Content-Length: 749 X-Keywords: NonJunk Arup, P(n+1) isn't being assumed before proving it. In the inductive step, we're looking at a graph with k vertices and n+1 edges. We take he subgraph obtained by removing one of the edges arbitrarily. This subgraph has n edges, so P(n) holds on that subgraph. Now we reason about what happens when we put back the edge that was removed and show that P(n) --> P(n+1). Does this clear things up? -Jelani Arup Sarma wrote: > Section 2.5, page 6 (and used later as well): I think I don't > completely understand the "shrink-down, grow-back" style of > induction. It seems like it assumes P(n+1) before proving it. Also, > what are examples of the logical errors that can occur if this style > of argument isn't used? > > |Arup| > From minilek@theory.csail.mit.edu Wed Oct 5 00:00:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9540Qw0003013 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:00:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9540O6s009426; Wed, 5 Oct 2005 00:00:24 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9540Lne002458 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:00:22 -0400 (EDT) Message-ID: <43434FEC.1030001@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:00:44 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Crystal Chao Subject: Re: [Jelani] week 5 reading References: <8BC5300F-BC1D-48A6-88A8-24CBAFC63466@mit.edu> In-Reply-To: <8BC5300F-BC1D-48A6-88A8-24CBAFC63466@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.564 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2110 Content-Length: 1115 X-Keywords: NonJunk The faces are the connected regions the graph's edges divide the plane into. Since there are no edges in a single-vertex graph, the entire plane is in the same connected region, and this is the "outside face" the base case is referring to. Does that clear things up? Crystal Chao wrote: > Overall, I thought the reading was pretty understandable and for once > had enough examples. I didn't get this thing in the proof of Euler's > Formula (pg. 12). In the base case, it says that a graph with nothing > but a single vertex has one face, the "outside" face. What the? How > can something with no edges have faces? > > Also, I have a comment about something on page 6, in the second > paragraph after Corollary 2.10. It talks about taking an n+1 graph, > then generalizing about an n subgraph to generalize about the n+1 one > in induction. This is something that we needed in the pset that was > due on Monday (on the last problem). Suggestion: maybe in the future, > this shrink-down/grow-back paradigm in induction with graphs can be > introduced before the pset is due. > > ~Crystal From minilek@theory.csail.mit.edu Wed Oct 5 00:04:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9544Nw0003268 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:04:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9544K6s011414; Wed, 5 Oct 2005 00:04:20 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9544Itb003041 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:04:18 -0400 (EDT) Message-ID: <434350D9.3080905@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:04:41 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Natalia N Chernenko Subject: Re: [Jelani] Week 5 Comments References: <1128463908.20774.11.camel@hayden-1.mit.edu> In-Reply-To: <1128463908.20774.11.camel@hayden-1.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.576 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2111 Content-Length: 511 X-Keywords: NonJunk The proof is somewhat tricky, and may take reading over a couple times (especially the second case). Try to read it over some more to really digest the proof. If you still find some parts of the proof confusing though, just let me know. -Jelani Natalia N Chernenko wrote: >It would be nice to discuss planar graphs (Section 5, pp. 11-13) more in >lecture, especially Euler's formula (Section 5.1). I found the >derivation of the formula a little confusing (induction proof). > >Natalia Chernenko > > From lkini@MIT.EDU Wed Oct 5 00:10:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954A2w0003645 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 00:10:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954A06s014378 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 00:10:00 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9549vq7004352 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 00:09:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9549vmP001390; Wed, 5 Oct 2005 00:09:57 -0400 Received: from MACGREGOR-TWO-FORTY.MIT.EDU (MACGREGOR-TWO-FORTY.MIT.EDU [18.239.5.240]) (User authenticated as lkini@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 00:09:57 -0400 Message-ID: <20051005000957.osf3cx3xuhog400g@webmail.mit.edu> Date: Wed, 5 Oct 2005 00:09:57 -0400 From: Lohith G Kini To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Difficulty with reading lecture notes for week 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.879 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2112 Content-Length: 254 X-Keywords: NonJunk Hi Hanson, I am having trouble understanding the proof for lemma 5.2 on page 13 of the lecture notes. I understand the mathematics that leads off from statement (1) but I don't understand the explanation behind the original inequality 2e >= 3f. Lohith From minilek@theory.csail.mit.edu Wed Oct 5 00:12:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954Cww0003776 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:12:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Cu6s015926; Wed, 5 Oct 2005 00:12:56 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954CjKT004810 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:12:49 -0400 (EDT) Message-ID: <434352D4.8060102@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:13:08 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: Re: [Jelani] LN5 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2113 Content-Length: 937 X-Keywords: NonJunk >> I was confused by the formal statement of Hall's Theorem on page 17 We have some bipartite graph, so we can partition the vertices into "the left side" (L) and "the right side" (R), such that all edges have one endpoint in L and one endpoint in R. A matching for L is a function f from vertices of L to vertices of R such that no two vertices of L get mapped to the same thing, and if f(x) = y, then the edge x--y exists. Hall's Theorem is saying a matching for L exists if and only if for every subset S of L, |N(S)| >= |S|. N(S) is the set of vertices that are not in S but are adjacent to something in S (so, since S is a subset of L, N(S) will be a subset of R -- do you see why?). Let me know if you are still confused after lecture. -Jelani Amanda Seybold wrote: > I was confused by the formal statement of Hall's Theorem on page 17. I > would also like to see algorithms for matching in a bipartite graph. From iyzhang@MIT.EDU Wed Oct 5 00:13:56 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954Dtw0003847 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:55 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j954Dsjv024507 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Ds8U005560 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:54 -0400 (EDT) Received: from [18.239.7.4] (MACGREGOR-FIVE-FIFTEEN.MIT.EDU [18.239.7.4]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j954Dmok018413 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:51 -0400 (EDT) Message-ID: <434352EF.7050302@mit.edu> Date: Wed, 05 Oct 2005 00:13:35 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2114 Content-Length: 56 X-Keywords: I would like planar graphs explain more. thanks, irene From minilek@theory.csail.mit.edu Wed Oct 5 00:17:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954Htw0004149 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:17:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Hr6s018748; Wed, 5 Oct 2005 00:17:54 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954HmWC005553 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:17:51 -0400 (EDT) Message-ID: <43435403.1050606@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:18:11 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: r n jacobs Subject: Re: [Jelani] Reading response References: <200510040036.j940aXUv024591@white-meteo.mit.edu> In-Reply-To: <200510040036.j940aXUv024591@white-meteo.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.564 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2115 Content-Length: 547 X-Keywords: NonJunk The tutor questions are supposed to reflect the reading and provide practice applying what you just learned. Is your complaint that it isn't doing this? -Jelani r n jacobs wrote: >It took a while for me to find the definition of "matching" since it >was only boldface in a paragraph. > >Are the tutor questions specifically supposed to reflect the reading? >It seems strange that the material was half already covered and half >required doing the reading. > >Otherwise, everything seems straightforward this week. > > - Robert Jacobs > > From mpapi@MIT.EDU Wed Oct 5 00:36:28 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954aSw0008614 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:36:28 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j954aRjv016087 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:36:27 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Yj8U005998 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:34:45 -0400 (EDT) Received: from MACGREGOR-THREE-FORTY-SIX.MIT.EDU (MACGREGOR-THREE-FORTY-SIX.MIT.EDU [18.239.6.91]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j954YgMu018462 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:34:42 -0400 (EDT) Subject: [Jelani] Week 5 Comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 05 Oct 2005 00:34:41 -0400 Message-Id: <1128486881.7375.18.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.425 X-Spam-Level: * (1.425) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2116 Content-Length: 383 X-Keywords: NonJunk Page 17, section 6.1 & theorem 6.2, the formal statement of Hall's theorem. The marriage analogy is pretty useful in getting to understand the theorem, and the proof is good. I think it would also be helpful to further elaborate on applications of the theorem (and perhaps some not-so-obvious applications, since they seem to arise from time to time for different things). -- Matt From dshin@MIT.EDU Wed Oct 5 00:43:47 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954hlw0010006 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:43:47 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j954hkjv023440 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:43:46 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954hj8U006169; Wed, 5 Oct 2005 00:43:46 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j954hhok019215; Wed, 5 Oct 2005 00:43:43 -0400 (EDT) Message-ID: <434359E3.8090704@mit.edu> Date: Wed, 05 Oct 2005 00:43:15 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 5 comments References: <382973AE-959B-446E-9671-C62830B1DF67@mit.edu> In-Reply-To: <382973AE-959B-446E-9671-C62830B1DF67@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2117 Content-Length: 1460 X-Keywords: Think about it this way: suppose you took a "2D" graph on a piece of paper, and placed a coin on each vertex, and glued string between coins wherever there was an edge. Just by lifting some of the coins up in the air, you'd be making this graph "3D"! Now, take some "3D" graph, "frozen in time and space", and imagine placing a floating coin at each vertex, and gluing a string between coins wherever there is an edge. Then, bam, "unfreeze" time, and gravity pulls the coins (and strings) down to the ground. And presto, you have a "2D" graph! All of the concepts we've studied so far (matchings, cycles, simple paths, connected components) stay the same in these two graphs. If you could understand this little thought experiment, you'd see that 2D graphs and 3D graphs really are the same thing - there is no such thing as a "2D" or "3D" graph! Akari Kameyama wrote: > "We can regard the corners and edges of these polyhedra as the > vertices and edges of a planar > graph. (This is another logical leap based on geometric intuition.)" > > So far we have only discussed "graphs" from a 2-dimensional > perspective. Although 2D graphs can be "flexed" (for lack of a better > word) to become 3D, it feels as though there exists/should exist a > whole different realm of possibilities in the 3D world which cannot be > translated into 2D. For example, the dog and house situation, could > be done in a 3D representation yes? > > From antonk@MIT.EDU Wed Oct 5 00:46:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954kKw0010151 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:46:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954kJ6s003055 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:46:19 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954kGUH009627 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:46:16 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 5 Comments Date: Wed, 5 Oct 2005 00:46:14 -0400 Organization: Massachusetts Institute of Technology Message-ID: <009301c5c967$b801d730$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0094_01C5C946.30F03730" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXJZ7cubGrNNXxNTNufAnHQiIk85w== X-Spam-Score: -1.046 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2118 Content-Length: 2180 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0094_01C5C946.30F03730 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Is it possible to go over planar graphs? I understand the faces and the math behind it, but I do not understand why we need it and the proof is a bit hard. Anton. ------=_NextPart_000_0094_01C5C946.30F03730 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Is it possible to go over planar = graphs?

I understand the faces and the math behind it, but I = do not understand why we need it and the proof is a bit = hard.

 

Anton.

------=_NextPart_000_0094_01C5C946.30F03730-- From clintonb@MIT.EDU Wed Oct 5 00:54:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954s5w0011951 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:54:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954s46s006973 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:54:04 -0400 (EDT) Received: from ccbibmx30 (NEW-EIGHTY-TWO.MIT.EDU [18.241.5.82]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954rtuj010555 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:53:55 -0400 (EDT) Message-Id: <200510050453.j954rtuj010555@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Jelani: Week 5 Comments Date: Wed, 5 Oct 2005 00:53:26 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_005B_01C5C947.323BD490" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXJaLiPDplbT+1qQdiWpdm+G6fcJA== X-Spam-Score: 1.15 X-Spam-Level: * (1.15) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2119 Content-Length: 3510 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_005B_01C5C947.323BD490 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I actually understood most of the reading this time. I need a little more time to study the bipartite section. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_005B_01C5C947.323BD490 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

I actually understood most of the reading this time. = I need a little more time to study the bipartite = section.

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_005B_01C5C947.323BD490-- From jstritar@MIT.EDU Wed Oct 5 01:10:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955A6w0012956 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 01:10:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955A46s014500 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 01:10:05 -0400 (EDT) Received: from [18.245.6.98] (BAKER-THREE-FIFTY-THREE.MIT.EDU [18.245.6.98]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955A18N012612 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 01:10:02 -0400 (EDT) Message-ID: <43436028.1080700@mit.edu> Date: Wed, 05 Oct 2005 01:10:00 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.054 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2120 Content-Length: 224 X-Keywords: NonJunk 5.3 Classifying Polyhedra I thought this section was really interesting. I have never thought of polyhedrans as planar graphs and it was pretty surprising. It took a few minutes of visualizing to see the connection. Jon From dshin@MIT.EDU Wed Oct 5 01:14:04 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955E4w0013177 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:14:04 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j955E3jv021497 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:14:03 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955E38U006736; Wed, 5 Oct 2005 01:14:03 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j955Dxok020061; Wed, 5 Oct 2005 01:13:59 -0400 (EDT) Message-ID: <434360FC.6030702@mit.edu> Date: Wed, 05 Oct 2005 01:13:32 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 5 Comments References: <200510050118.j951InMv011661@outgoing-legacy.mit.edu> In-Reply-To: <200510050118.j951InMv011661@outgoing-legacy.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2121 Content-Length: 1195 X-Keywords: It is a little difficult to describe those algorithms with the limited machinery we have developed in the course thus far. If you would like, I could show you how these algorithms work during OH or before/after class. As for problems that reduce to finding a matching, there are lots! One application: medical schools use (bipartite) matching algorithms to accept applicants to try to make sure everybody gets in somewhere. A common variant on the matching problem is to find a max/min-weight matching (when the edges have real-valued weights). This is often a useful subroutine for many problems, including some variants of the traveling salesman problem (TSP). In min-TSP, a salesman has to visit a bunch of cities (i.e., vertices on a graph) and wants to minimize the total amount of time he has to spend. (Matching is not used in min-TSP, but it is used in max-TSP). Hope that gives you a flavor of some of the uses. DS Chris Yang wrote: > On page 17, it says that there are very efficient algorithms for doing > the Halls Theorem matching. What are some examples of those? Also, > what sorts of problems can be reduced to finding a matching? > > Thanks, > > Chris Yang > From lmccart@MIT.EDU Wed Oct 5 01:25:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955Plw0014079 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:25:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955Pk6s021856 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:25:46 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955PdOG014147 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:25:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j955Pdax010348; Wed, 5 Oct 2005 01:25:39 -0400 Received: from KS-ONE-TWENTY-EIGHT.MIT.EDU (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 01:25:39 -0400 Message-ID: <20051005012539.cck5vdbdhdfvoso8@webmail.mit.edu> Date: Wed, 5 Oct 2005 01:25:39 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 5 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2122 Content-Length: 159 X-Keywords: NonJunk I am still a little confused about the actual definition of a connected component. Both his defintion in class and the reading were somewhat unclear. -lauren From ryan786@MIT.EDU Wed Oct 5 01:27:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955Row0014148 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:27:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955Rn6s022819 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:27:49 -0400 (EDT) Received: from ryoung (PSK-SIX.MIT.EDU [18.217.1.6]) (authenticated bits=0) (User authenticated as ryan786@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955Rk7x014339 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:27:47 -0400 (EDT) Message-Id: <200510050527.j955Rk7x014339@outgoing.mit.edu> Reply-To: From: "ryan" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 01:27:48 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C94B.FF2CBF60" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJbYWLaqr8h/Z0Ts6GE4a5JVJBJA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.375 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2123 Content-Length: 2797 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C94B.FF2CBF60 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit This was by far and away my favorite reading assignment thus far. I liked how there was more of an emphasis on visual representation and less of an emphasis on technical logic (although admittedly it is a necessary evil in this class). I especially liked the section on Hall's Marriage Theorem, as I had never seen it before, but once I read it, it made perfect sense. I guess I still have only a shaky hold on connected components, but that's probably more from our class Monday than the reading tonight. -Ryan Young ------=_NextPart_000_0000_01C5C94B.FF2CBF60 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

This was by far and away my favorite reading = assignment thus far.  I liked how there was more of an emphasis on visual = representation and less of an emphasis on technical logic (although admittedly it is a = necessary evil in this class).  I especially liked the section on = Hall’s Marriage Theorem, as I had never seen it before, but once I read it, it made = perfect sense.  I guess I still have only a shaky hold on connected = components, but that’s probably more from our class Monday than the reading = tonight.

 

-Ryan Young

------=_NextPart_000_0000_01C5C94B.FF2CBF60-- From aeon@MIT.EDU Wed Oct 5 01:29:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955TFw0014269 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:29:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955TD6s023508 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:29:13 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955TB9C014499 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:29:11 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j955TBuK026914; Wed, 5 Oct 2005 01:29:11 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:29:11 -0600 Message-ID: <20051004232911.lw6n5nvxu4u8088o@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:29:11 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: LN5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.686 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2124 Content-Length: 683 X-Keywords: NonJunk Page 7, "Tree Properties" Just about everything here can be applied to data network topology, so I found it to be particularly useful. I.e., k-connectedness describes the robustness and fault tolerance of a network, spanning trees give the cheapest setup possible, etc. In general, I just found this to be cool. Page 9, "Coloring Graphs" This idea was surprising to me. I didn't even think about "coloring" a graph. What I wonder is how a program could be written to assign colors and then make sure all nodes are different without individually checking them all... it must sort the graph somehow, but it's not immediately apparent how that sort would be........ ;) - John Marrero From sriaz@MIT.EDU Wed Oct 5 02:23:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956NGw0025950 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:23:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j956NFIs014676 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:23:15 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j956NCZb018556 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:23:13 -0400 (EDT) Message-ID: <43437152.8010903@mit.edu> Date: Wed, 05 Oct 2005 02:23:14 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.881 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2125 Content-Length: 264 X-Keywords: NonJunk One of the issues I had with this week's reading, and the tutor problems, was the questions regarding Graph Coloring. In Section 4, coloring graphs are gone over, but I would have liked to see more detail as that would have helped with the TP. Thanks Sameer From shreyes19@gmail.com Wed Oct 5 02:25:48 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.195]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956Pmw0026554 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:25:48 -0400 Received: by xproxy.gmail.com with SMTP id s9so68595wxc for <6042-probs@theory.lcs.mit.edu>; Tue, 04 Oct 2005 23:25:43 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=B7fMoyo9D6KFNBxZ6hu8asWOwiqR+DM2BJazZbCR6JAE8mFRNTpBOYHlK3Gh/HeLXcNIKsdcKNwoVCjyBB8YWruUCzeWYulaNPvGzHrC9rwXq886Kdx3NL/sXyAt61GIMudKxeFFZ1Kw7Lilqg0j0rObnuPJhXPJPTscNjGjNKg= Received: by 10.70.59.17 with SMTP id h17mr304424wxa; Tue, 04 Oct 2005 23:25:43 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Tue, 4 Oct 2005 23:25:43 -0700 (PDT) Message-ID: <3c33fe380510042325p4122fdfeqfcb19cf04f6c805d@mail.gmail.com> Date: Wed, 5 Oct 2005 02:25:43 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 5 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_23205_25177894.1128493543150" Status: RO X-UID: 2126 Content-Length: 2036 X-Keywords: NonJunk ------=_Part_23205_25177894.1128493543150 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, The part in this week's reading I found most surprising/interesting was the section on polyhedra. "Thus, we can regard the corners and edges of these polyhedra as the vertices and edges of a planar graph. (This is another logical leap based on geometric intuition.) This means Euler's formula for planar graphs can help guide our search for regular polyhedra." (Page 14). = I just thought it was an interesting way to approach the problem of finding all regular polyhedra, and was surprised to see that translating the proble= m into 2D planar graphs made it much simpler to visualize and inspect. I also thought it was interesting to be able to define a regular polyhedra in term= s of its qualities v, e, and f, without regard at first to its possible shape= . Thanks, Shreyes Seshasai Group 7 ------=_Part_23205_25177894.1128493543150 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

The part in this week's reading I found most surprising/interesting was the= section on polyhedra. 
"Thus, we can regard the corners and edges of these polyhedra as the v= ertices and edges of a planar
graph. (This is another logical leap based on geometric intuition.) This me= ans Euler's formula for
planar graphs can help guide our search for regular polyhedra." (Page 14).  I just thought it was an interesting way to approach the problem of finding all regular polyhedra, and was surprised to see that translating the problem into 2D planar graphs made it much simpler to visualize and inspect.  I also thought it was interesting to be able to define a regular polyhedra in terms of its qualities v, e, and f, without regard at first to its possible shape.

Thanks,
Shreyes Seshasai
Group 7
------=_Part_23205_25177894.1128493543150-- From bens@MIT.EDU Wed Oct 5 02:31:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956Vpw0026850 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 02:31:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j956VnIs018156 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 02:31:49 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j956Vh4V019107 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 02:31:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j956Vgib009113; Wed, 5 Oct 2005 02:31:42 -0400 Received: from LSLATER.MIT.EDU (LSLATER.MIT.EDU [18.221.0.171]) (User authenticated as bens@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 02:31:42 -0400 Message-ID: <20051005023142.dci3ppmec1wkcgg0@webmail.mit.edu> Date: Wed, 5 Oct 2005 02:31:42 -0400 From: Benjamin M Schwartz To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Week 5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2127 Content-Length: 64 X-Keywords: NonJunk I cannot understand Euler's formula (p. 12) this late at night. From mracich@MIT.EDU Wed Oct 5 02:44:56 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956iuw0029142 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:56 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j956isvr018666 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j956is25008681 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:54 -0400 (EDT) Received: from MACGREGOR-THREE-SIXTY-TWO.MIT.EDU (MACGREGOR-THREE-SIXTY-TWO.MIT.EDU [18.239.6.107]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j956ijMu022160 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:45 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 5 (Graphs) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 05 Oct 2005 02:44:42 -0400 Message-Id: <1128494682.23545.11.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2128 Content-Length: 202 X-Keywords: NonJunk I found section 5.3, Classifying Polyhedra, (starting on page 14) interesting, but also kind of confusing. I would appreciate it if this was reviewed in greater depth during lecture. Moira Racich From bakster@MIT.EDU Wed Oct 5 03:20:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j957KRw0001754 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 03:20:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j957KQIs006783 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 03:20:26 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j957KIhW021666 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 03:20:19 -0400 (EDT) Message-ID: <43437EB2.5040400@mit.edu> Date: Wed, 05 Oct 2005 03:20:18 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2129 Content-Length: 134 X-Keywords: NonJunk I just wanted to make sure that we go over planar graphs in class. The tutor problem set didn't seem to cover them. Alexander Bakst From scot@MIT.EDU Wed Oct 5 04:25:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j958PJw0011207 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 04:25:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j958PIIs028941 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 04:25:18 -0400 (EDT) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j958PE7V024038 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 04:25:15 -0400 (EDT) Message-ID: <43438DD7.3030903@mit.edu> Date: Wed, 05 Oct 2005 04:24:55 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: week 5 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.903 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2130 Content-Length: 343 X-Status: A X-Keywords: NonJunk It was unclear at my table during last lecture whether the fault of the sex statistics had to do with people mis-reporting, or the statistical analysis. From page 5, and lecture, I'm still confused on what we are using for the term of connectedness, and from the in-class problem are we saying it is connected if there is any simple path? From zev@MIT.EDU Wed Oct 5 04:48:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j958mMw0013774 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 04:48:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j958mKIs007107 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 04:48:21 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j958mDh4024800 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 04:48:13 -0400 (EDT) Message-ID: <4343934C.70107@mit.edu> Date: Wed, 05 Oct 2005 04:48:12 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 5 comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.073 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2131 Content-Length: 405 X-Keywords: -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 I like the use of diagrams and examples in these notes: they made the sections much more clear. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDQ5NMlO3j8HLL0+4RAjLYAKDx7fTI1VLmaqoMms/G4TPpUFA9ywCg087+ UJLmVVexJMVbuhHB3ASi46I= =iPX7 -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Wed Oct 05 06:14:58 2005 Message-ID: <4343A7A9.2050703@csail.mit.edu> Date: Wed, 05 Oct 2005 06:15:05 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Scot Frank Subject: Re: week 5 References: <43438DD7.3030903@mit.edu> In-Reply-To: <43438DD7.3030903@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 1098 Status: RO X-UID: 2132 X-Keywords: two vertices are connected iff there is a path that begins at one and ends at the other. if there's any path, there will also be a simple one. I don't know what the fault was in the sex statistics -- I just know that the claim that in the averages for the whole population, men have 74% more partners than women, is impossible. One possible explanation is that the men exaggerated their number of partners (and the researchers mistakenly believed them). Another explanation is that the researchers tested a sample of men and women, presumably including all the partners of everybody in the sample, and wound up with 74% more women in the sample. regards, A. P.S. Remember to put your TA's name in the subject. Scot Frank wrote: > It was unclear at my table during last lecture whether the fault of the > sex statistics had to do with people mis-reporting, or the statistical > analysis. From page 5, and lecture, I'm still confused on what we are > using for the term of connectedness, and from the in-class problem are > we saying it is connected if there is any simple path? From medrano@MIT.EDU Wed Oct 5 06:40:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Ae1w0026621 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 06:40:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95AdxPG015695 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 06:39:59 -0400 (EDT) Received: from grumpy-fuzzball.mit.edu (GRUMPY-FUZZBALL.MIT.EDU [18.7.16.79]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95AdvbP028997 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 06:39:57 -0400 (EDT) Received: (from medrano@localhost) by grumpy-fuzzball.mit.edu (8.12.9) id j95AdvWp004207; Wed, 5 Oct 2005 06:39:57 -0400 (EDT) Date: Wed, 5 Oct 2005 06:39:56 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]comments on reading Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2133 Content-Length: 257 X-Keywords: NonJunk I'm am a bit confused about coloring. I thought that any map of the USA (a graph) is colorable with at least 4 colors, but theorem 4.1 says that a graph with maximum degree at most k is (k+1)-colorable. Can it be less than (k+1) colorable? Jesus Medrano From nedzel@MIT.EDU Wed Oct 5 07:06:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95B6Cw0029327 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 07:06:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95B68PG026114 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 07:06:08 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SEVENTY-SEVEN.MIT.EDU [18.96.5.77]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95B64Qj000616 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 07:06:04 -0400 (EDT) Message-Id: <200510051106.j95B64Qj000616@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 07:06:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.272 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2134 Content-Length: 72 X-Keywords: NonJunk In section 5.1, what is the significance of planer embedding? - David From mdmurray@MIT.EDU Wed Oct 5 08:54:09 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Cs9w0011662 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 08:54:09 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Cs9Zq027521 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 08:54:09 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95Cs1ok005202 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 08:54:02 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-7--114245286 From: Michael Murray Subject: week 5 Date: Wed, 5 Oct 2005 08:53:29 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.098 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2135 Content-Length: 8744 X-Status: A X-Keywords: --Apple-Mail-7--114245286 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed On pg. 13: " Otherwise, G has at least one cycle. Select a spanning tree and an =20 edge u=E2=80=94v in the cycle, but not in the tree. (The spanning tree can not contain all edges in =20 the cycle, since trees are acyclic.) Removing u=E2=80=94v merges the two faces on either side = of =20 the edge and leaves a graph G0 with only e edges and some number of vertices v and faces =20 f. Graph G0 is connected, because there is a path between every pair of vertices =20 within the spanning tree. So v=E2=88=92e+f =3D 2 by the induction assumption P(e). Thus, the = original =20 graph G had v vertices, e+1 edges, and f +1 faces. Since v=E2=88=92(e+1)+(f +1) =3D v=E2=88=92e+f = =3D 2, P(e=20 +1) is again true. Otherwise, G has at least one cycle. Select a spanning tree and an =20 edge u=E2=80=94v in the cycle, but not in the tree. (The spanning tree can not contain all edges in =20 the cycle, since trees are acyclic.) Removing u=E2=80=94v merges the two faces on either side = of =20 the edge and leaves a graph G0 with only e edges and some number of vertices v and faces =20 f. Graph G0 is connected, because there is a path between every pair of vertices =20 within the spanning tree. So v=E2=88=92e+f =3D 2 by the induction assumption P(e). Thus, the = original =20 graph G had v vertices, e+1 edges, and f +1 faces. Since v=E2=88=92(e+1)+(f +1) =3D v=E2=88=92e+f = =3D 2, P(e=20 +1) is again true. " I did not understand this part of the proof or Euler's formula. Why =20 do we take a spanning tree and a cycle not in the tree? How do we =20 argue from there? This proof confused me. Michael Murray= --Apple-Mail-7--114245286 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
On pg. 13:

" Otherwise, G has = at least one cycle. Select a spanning tree and an edge u=E2=80=94v in = the cycle,
but not in the tree. (The spanning tree can = not contain all edges in the cycle, since trees
are acyclic.) = Removing u=E2=80=94v merges the two faces on either side of the edge and = leaves
a graph G0 with only e edges and some number of vertices v and faces f. = Graph G0 = is
connected, because there is a path between = every pair of vertices within the spanning tree.
So v=E2=88=92e+f =3D= 2 by the induction assumption P(e). Thus, the original graph G had v = vertices,
e+1 edges, and f +1 faces. Since v=E2=88=92(e+1= )+(f +1) =3D v=E2=88=92e+f =3D 2, P(e+1) is again = true.
Otherwise, G has at least one cycle. Select a = spanning tree and an edge u=E2=80=94v in the = cycle,
but not in the tree. (The spanning tree can = not contain all edges in the cycle, since trees
are acyclic.) = Removing u=E2=80=94v merges the two faces on either side of the edge and = leaves
a graph G0 with only e edges and some number of vertices v and faces f. = Graph G0 = is
connected, because there is a path between = every pair of vertices within the spanning tree.
So v=E2=88=92e+f =3D= 2 by the induction assumption P(e). Thus, the original graph G had v = vertices,
e+1 edges, and f +1 faces. Since v=E2=88=92(e+1= )+(f +1) =3D v=E2=88=92e+f =3D 2, P(e+1) is again true. = "=C2=A0


I did not = understand this part of the proof or Euler's formula.=C2=A0 Why do we = take a spanning tree and a cycle not in the tree? How do we argue from = there? This proof confused me.

Michael = Murray
= --Apple-Mail-7--114245286-- From a_lopez@MIT.EDU Wed Oct 5 08:57:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Cvnw0012202 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 08:57:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Cvmcq024959 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 08:57:48 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95CvkOw019949 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 08:57:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j95CvkpG002479; Wed, 5 Oct 2005 08:57:46 -0400 Received: from MDMURRAY.MIT.EDU (MDMURRAY.MIT.EDU [18.234.0.116]) (User authenticated as a_lopez@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 08:57:46 -0400 Message-ID: <20051005085746.ikssu33okdmskc8w@webmail.mit.edu> Date: Wed, 5 Oct 2005 08:57:46 -0400 From: a_lopez@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: week 5 readings MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 1.031 X-Spam-Level: * (1.031) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2136 Content-Length: 642 X-Status: A X-Keywords: NonJunk "Suppose the graph is drawn in the plane with f faces. Every edge on a cycle in the graph borders two faces and is traversed once by each of the two cycles bounding these two faces, and not by any other face boundary. If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one ?going? and the other ?coming?), and not by any other face boundary. So the total number of traversals of any edge by the face boundaries is exactly two." pg. 13 I did not understand this proof at all. I got lost after the second sentence. I've read it a couple of times but I still don't understand it. Adriana Lopez From kromer@MIT.EDU Wed Oct 5 09:14:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DEUw0014873 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:14:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DETcq006943 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:14:29 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DEQ7D025332 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:14:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j95DEQKN009922; Wed, 5 Oct 2005 09:14:26 -0400 Received: from BURTON-TWO-O-TWO.MIT.EDU (BURTON-TWO-O-TWO.MIT.EDU [18.247.5.202]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:14:26 -0400 Message-ID: <20051005091426.v8b03pef1lccoowo@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:14:26 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2137 Content-Length: 423 X-Keywords: NonJunk p. 15 "Checking the finitely-many cases that remain turns up five solutions. For each valid combination of n and m, we can compute the associated number of vertices v, edges e, and faces f. And polyhedra with these properties do actually exist." I found it surprising that there are only 5 regular polyhedra, and that all the regular polyhedra computed to be possible with Euler's formula actually exist. Katherine Romer From hejing85@MIT.EDU Wed Oct 5 09:14:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DElw0014890 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:14:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DEkcq007212 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:14:46 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DEeR0025402 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:14:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j95DEdu9003251; Wed, 5 Oct 2005 09:14:39 -0400 Received: from NEW-ONE-O-SEVEN.MIT.EDU (NEW-ONE-O-SEVEN.MIT.EDU [18.241.5.107]) (User authenticated as hejing85@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:14:39 -0400 Message-ID: <20051005091439.yfo57e4xc7k84ogs@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:14:39 -0400 From: Jing He To: 6042-probs@theory.csail.MIT.EDU Subject: [David] week 5 comments MIME-Version: 1.0 Content-Type: text/plain; charset=GB2312 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2138 Content-Length: 102 X-Keywords: NonJunk The part of the readings that talked about Euler's Formula and the polyhedra was interesting. --Jing From jeffhoff@MIT.EDU Wed Oct 5 09:16:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DGPw0015036 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 09:16:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DGOcq008647 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 09:16:24 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DG4V7025932 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 09:16:05 -0400 (EDT) Message-Id: <5.2.1.1.2.20051005090947.04f56990@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 05 Oct 2005 09:18:06 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 2.544 X-Spam-Level: ** (2.544) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2139 Content-Length: 255 X-Keywords: NonJunk Page 9 - Section 4 Coloring graphs I understand coloring graphs and sort of how to do them. Is there a trick or something to doing them easier instead of "brute force"? Can we go over it more in class? Group 3 / C Jeffrey D. Hoff jeffhoff@mit.edu From manosdefierro@gmail.com Wed Oct 5 09:18:14 2005 Return-Path: Received: from wproxy.gmail.com (wproxy.gmail.com [64.233.184.196]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DIDw0015123 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:18:13 -0400 Received: by wproxy.gmail.com with SMTP id i16so64297wra for <6042-probs@theory.lcs.mit.edu>; Wed, 05 Oct 2005 06:18:08 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=P6st3JL5FiJbWXwcDfYxLL6WtWQZFlYRi2ufPb5spfj6mO0/u31TjuucuoGFehmpzY0dtCRR7c1jnw229ho9crtx5ze9815Myl+GcOJEiAxe4y/Ad7Yt6g91EwrRYLrep/ombAnZuHjeE8/1WSXik1DnYbJ5SJ3hzwAu2J6zugk= Received: by 10.54.125.17 with SMTP id x17mr361464wrc; Wed, 05 Oct 2005 06:18:08 -0700 (PDT) Received: by 10.54.151.13 with HTTP; Wed, 5 Oct 2005 06:18:08 -0700 (PDT) Message-ID: <2bac7c5d0510050618j193063fctbed2df1b8f902008@mail.gmail.com> Date: Wed, 5 Oct 2005 09:18:08 -0400 From: Mario Marrufo Reply-To: joeyrufo@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: required e-mail comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_12635_14485148.1128518288571" Status: RO X-UID: 2140 Content-Length: 1079 X-Keywords: NonJunk ------=_Part_12635_14485148.1128518288571 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found this week's reading assignment much easier to digest than last week's assignment. That said, there may be some stuff I've missed. For example, I'm not sure I remember the precise definition of what a "degree" is (mentioned in section 1, pages 1 and 2). Another section I wouldn't mind if was covered in lecture would be section 5.2 (page 13). ------=_Part_12635_14485148.1128518288571 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found this week's reading assignment much easier to digest than last week's assignment. That said, there may be some stuff I've missed. For example, I'm not sure I remember the precise definition of what a "degree" is (mentioned in section 1, pages 1 and 2). Another sect= ion I wouldn't mind if was covered in lecture would be section 5.2 (page 13). ------=_Part_12635_14485148.1128518288571-- From sil_03@MIT.EDU Wed Oct 5 09:32:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DWow0017108 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:32:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DWncq022786 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:32:49 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DWgIw002313 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:32:42 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j95DWgKK028715; Wed, 5 Oct 2005 09:32:42 -0400 Received: from NEW-TWO-THIRTY-EIGHT.MIT.EDU (NEW-TWO-THIRTY-EIGHT.MIT.EDU [18.241.5.238]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:32:42 -0400 Message-ID: <20051005093242.j5pudxwhaa0444cs@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:32:42 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2141 Content-Length: 166 X-Keywords: I thought the section on classifying plyhedra on page 14-15 was interesting, but I would like to know what other kinds of problems you can solve using faces. Silvia From zacharyozer@gmail.com Wed Oct 5 09:35:05 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.205]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DZ4w0017316 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:35:04 -0400 Received: by xproxy.gmail.com with SMTP id s8so100673wxc for <6042-probs@theory.lcs.mit.edu>; Wed, 05 Oct 2005 06:34:59 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=EIH8xV8HTuUj+zVkot9G7Or7KwEkp1vTSJ/gTJSZzekbaOXGvbPQw/LVLpEjFmDOesOfzEKiiaPNYhx7BrAg7KXJatg3Bkpg25AksHqA2wrLdzmrJPKwgefZ+hITzQ7jYjizUdTKKZ++8OeayKp5B1Vf3B2Bs2P1vI4LC+qIEzM= Received: by 10.70.43.2 with SMTP id q2mr404859wxq; Wed, 05 Oct 2005 06:34:59 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Wed, 5 Oct 2005 06:34:59 -0700 (PDT) Message-ID: <4f2613e50510050634pfe780d5o774a3a60dbf3dc6d@mail.gmail.com> Date: Wed, 5 Oct 2005 09:34:59 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j95DZ4w0017316 Status: RO X-UID: 2142 Content-Length: 403 X-Keywords: My question is about k-colorable graphs: In 6.034 we learned that any map (which can be represented as a graph) could be colored with at most 4 colors. There was no rigid proof, although, various demonstrations led us to beleive that this was true. Why then is the upper limit on chromatic number k+1 for a graph with maximum degree k, and not k+1 for graphs less than 3, and 4 for all others? -zozer From lmccart@MIT.EDU Wed Oct 5 09:43:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dhlw0017786 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:43:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Dhkcq002507 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:43:46 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DhifI007547 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:43:44 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j95Dhiea007575; Wed, 5 Oct 2005 09:43:44 -0400 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:43:44 -0400 Message-ID: <20051005094344.hlc0xsqogey8o8c8@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:43:44 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] other question MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2143 Content-Length: 881 X-Keywords: NonJunk I keep reading these emails about normalizing scores for psets over tas, but I just don't really understand what is meant by that. I'm confused why when I do my psets with my friend and we have essentially the same answers, he gets 35 while I get a 23. There was one problem where I referenced a proof we did in class and got 1/10 while he got 9/10 for doing the same thing. I don't understand why you guys don't talk about acceptable answers before grading psets. I find everything in this class to be pretty ambiguous. It's always unclear to me what has been proven and what has not and what we're allowed to assume. Also, it's even more difficult for me when the professor puts up incorrect information on slides and frequently makes mistakes while explaining things. It seems like with some preplanning many of thse problems could be avoided. Thank you, Lauren McCarthy From miki_tnd@MIT.EDU Wed Oct 5 09:47:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dlhw0017970 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:47:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Dlfcq006084 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:47:42 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DlZhG009320 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:47:35 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j95DlZ4u005794; Wed, 5 Oct 2005 09:47:35 -0400 Received: from PIKA-SEVENTY-NINE.MIT.EDU (PIKA-SEVENTY-NINE.MIT.EDU [18.214.1.79]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:47:35 -0400 Message-ID: <20051005094735.t6qyyb5cq4e84s44@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:47:35 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: (sayan) reading comment 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 2.727 X-Spam-Level: ** (2.727) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2144 Content-Length: 167 X-Keywords: NonJunk I've always found the proof of Euler's classification of polyhedrons unclear. On pg.14 of the reading, it would be nice if that was explained by someone in person. From nancyk@MIT.EDU Wed Oct 5 09:47:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dlmw0017975 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:47:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Dllcq006140; Wed, 5 Oct 2005 09:47:47 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95Dler9009343; Wed, 5 Oct 2005 09:47:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j95DlehD009762; Wed, 5 Oct 2005 09:47:40 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:47:40 -0400 Message-ID: <20051005094740.2nrmctpbgy0owsss@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:47:40 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 5 reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2145 Content-Length: 476 X-Keywords: NonJunk Hi, On page 12, under section 5.1 "Euler's Formula," the last sentence of the first paragraph says that only one point (a single, common endpoint) can appear on two of the curves. Since each vertex must be assigned to a distinct point in the plane, it seems like that last sentence should specify that the "single, common endpoint" must be the vertex from which the edge started, since the ending point cannot have 2 or more edges incident to it. Is this right? Nancy Keuss From veracarr@MIT.EDU Wed Oct 5 09:50:34 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DoYw0018183 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:50:34 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DoXZq029659 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:50:33 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95DoQok008931 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:50:26 -0400 (EDT) Message-ID: <4343F680.8090600@mit.edu> Date: Wed, 05 Oct 2005 09:51:28 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Graphs Reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2146 Content-Length: 271 X-Keywords: NonJunk Planar Graphs (pg 11) Is there an easy way to determine whether the graph can be planer. They gave two examples where if one edge was to be removed that it would make the graph planar. I understand the definitions but is there an easy way to determine this property? From mitras@drake.csail.mit.edu Wed Oct 5 09:55:46 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dtjw0019348; Wed, 5 Oct 2005 09:55:46 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j95Dtj1P009188; Wed, 5 Oct 2005 09:55:45 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j95Dtj1g009187; Wed, 5 Oct 2005 09:55:45 -0400 Date: Wed, 5 Oct 2005 09:55:45 -0400 From: Sayan Mitra Message-Id: <200510051355.j95Dtj1g009187@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, nedzel@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Cc: 6042-staff@theory.csail.mit.edu Status: RO X-UID: 2147 Content-Length: 1215 X-Keywords: NonJunk I presume you are asking about the significance of the term "planar embedding" here and not the class of planar graphs. An embedding is a representation of a topological object, manifold, graph, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. A graph embedding is a particular drawing of a graph (with sometimes added constraint that the embedding be planar, i.e., has no crossing edges). While the underlying object is independent of the embedding, a clever choice of embedding can lead to particularly illuminating diagrams. I took these definitions from Mathworld. -S. >From nedzel@MIT.EDU Wed Oct 5 07:06:12 2005 From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 07:06:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.272 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 In section 5.1, what is the significance of planer embedding? - David From mitras@drake.csail.mit.edu Wed Oct 5 10:00:12 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95E0Cw0019620; Wed, 5 Oct 2005 10:00:12 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j95E0CA3009221; Wed, 5 Oct 2005 10:00:12 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j95E0CGm009220; Wed, 5 Oct 2005 10:00:12 -0400 Date: Wed, 5 Oct 2005 10:00:12 -0400 From: Sayan Mitra Message-Id: <200510051400.j95E0CGm009220@drake.csail.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU, medrano@MIT.EDU Subject: Re: [Sayan]comments on reading Cc: 6042-probs@theory.csail.mit.edu Status: RO X-UID: 2148 Content-Length: 1048 X-Keywords: NonJunk Yes. Consider a star graph with a central vertex v with k adjacent vertices, and all these other (not v) vertices are adjacent to v only. This graph has max degree k, but it is 2 colorable. In general, you have seen from the lecture notes that a tree is always 2 colorable irrespective of the maximum degree. -S. >From medrano@MIT.EDU Wed Oct 5 06:40:01 2005 Date: Wed, 5 Oct 2005 06:39:56 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]comments on reading MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 555 Content-Length: 257 X-Keywords: I'm am a bit confused about coloring. I thought that any map of the USA (a graph) is colorable with at least 4 colors, but theorem 4.1 says that a graph with maximum degree at most k is (k+1)-colorable. Can it be less than (k+1) colorable? Jesus Medrano From cvnguyen@MIT.EDU Wed Oct 5 10:16:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EGWw0022983 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:16:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EGVcq002399 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:16:31 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-ONE-O-FOUR-HUNDRED-TWENTY.MIT.EDU [18.238.7.153]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EGNAx022038 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:16:23 -0400 (EDT) Message-ID: <001901c5c9b7$5ece8c80$9907ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 5 comments Date: Wed, 5 Oct 2005 10:16:24 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MIMEOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -3.142 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2149 Content-Length: 296 X-Keywords: NonJunk (page 12, section 5.1) I found the discussion of Euler's formula particularly interesting. I remember reading somewhere a few years ago that there were at least fifteen different proofs of it because it was such a profound result that people sought alternate elegant proofs. --Chieu Nguyen From mrivas03@MIT.EDU Wed Oct 5 10:38:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EcHw0027601 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:38:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EcGcq022707 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:38:16 -0400 (EDT) Received: from el-ternero.mit.edu (TDCIP95.MIT.EDU [18.237.0.95]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EcCXS001947 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:38:14 -0400 (EDT) Message-Id: <6.2.3.4.2.20051005103051.02e6aaf8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 05 Oct 2005 10:33:54 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: [Sayan] Week 5 Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2150 Content-Length: 206 X-Keywords: NonJunk pg 12. Section Euler's Formula. A rerun of Euler's Formula and creating Polyhedra would be helpful if included during lecture: It's proof by induction and how to catalog polyhedra. Thanks, Manuel Rivas From vbrobbey@MIT.EDU Wed Oct 5 10:42:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EgZw0028060 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 10:42:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EgYcq026886 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 10:42:34 -0400 (EDT) Received: from newhouse-6.mit.edu (NEWHOUSE-6.MIT.EDU [18.241.2.253]) (authenticated bits=56) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EgUow004217 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 10:42:31 -0400 (EDT) Received: (from vbrobbey@localhost) by newhouse-6.mit.edu (8.12.9) id j95EgUr0003197; Wed, 5 Oct 2005 10:42:30 -0400 Subject: [David] Week 5 Comments From: Valery K Brobbey To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 05 Oct 2005 10:42:30 -0400 Message-Id: <1128523350.3022.8.camel@newhouse-6.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: -0.529 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2151 Content-Length: 621 X-Keywords: NonJunk What the University of Chicago study probably meant was that 74% of the time it's men to look for opposite gender partners. Even by common sense their assertion that men have on average 74% more opposite gender partners is ridiculous. I was impressed about how the idea of colouring a graph can be used to solve problems such as scheduling. I guess the housing department uses Hall's Marriage Theorem, or something similar to assign houses to freshmen. Of course this is more complicated because you have to think of preferences. I understood the concept of planar graphs but I got lost in Classifying Polyhedra section. From rian@MIT.EDU Wed Oct 5 10:58:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EwOw0031299 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 10:58:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EwMcq012681 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 10:58:22 -0400 (EDT) Received: from m12-182-9.mit.edu (M12-182-9.MIT.EDU [18.19.0.40]) (authenticated bits=56) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EwJjT011603 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 10:58:20 -0400 (EDT) Received: (from rian@localhost) by m12-182-9.mit.edu (8.12.9) id j95EwJQ6029727; Wed, 5 Oct 2005 10:58:19 -0400 Subject: (jelani) required reading comments From: rian To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 05 Oct 2005 10:58:19 -0400 Message-Id: <1128524299.28520.2.camel@m12-182-9.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 3.64 X-Spam-Level: *** (3.64) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2152 Content-Length: 212 X-Keywords: NonJunk i felt that i had a good grasp of the material in this week's reading. nothing was really surprising or difficult, and i'm impartial to anything being discussed more fully in class honestly. thanks. Rian Hunter From meyer@imap.theory.csail.mit.edu Wed Oct 05 11:04:49 2005 Message-ID: <4343EB96.6050802@csail.mit.edu> Date: Wed, 05 Oct 2005 11:04:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: a_lopez@MIT.EDU Subject: Re: week 5 readings References: <20051005085746.ikssu33okdmskc8w@webmail.mit.edu> In-Reply-To: <20051005085746.ikssu33okdmskc8w@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 854 X-UID: 2153 X-Keywords: Take a look at the animation for this (link on course main page, also in the Appendix for Week 5 Notes). Otherwise, wait till Fri lecture. regards, A a_lopez@MIT.EDU wrote: > "Suppose the graph is drawn in the plane with f faces. Every edge on a cycle in > the graph > borders two faces and is traversed once by each of the two cycles bounding these > two faces, and > not by any other face boundary. If an edge touches only one face, then it is > traversed twice by > the cycle that borders the face (one ?going? and the other ?coming?), > and not by any other face > boundary. So the total number of traversals of any edge by the face boundaries > is exactly two." > > pg. 13 > > I did not understand this proof at all. I got lost after the second sentence. > I've read it a couple of times but I still don't understand it. > > Adriana Lopez From meyer@imap.theory.csail.mit.edu Wed Oct 05 11:08:07 2005 Message-ID: <4343EC5F.9050003@csail.mit.edu> Date: Wed, 05 Oct 2005 11:08:15 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Michael Murray Subject: Re: week 5 References: In-Reply-To: Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 1793 X-UID: 2154 X-Keywords: Yeah, I don't care for it either, but doing it more carefully, as in the Appendix, may not be easier to understand. looking at the planar drawing animation (link on the course main page) may help. regards, A. Michael Murray wrote: > On pg. 13: > > " Otherwise, G has at least one cycle. Select a spanning tree and an > edge u—v in the cycle, > but not in the tree. (The spanning tree can not contain all edges in the > cycle, since trees > are acyclic.) Removing u—v merges the two faces on either side of the > edge and leaves > a graph G0 with only e edges and some number of vertices v and faces f. > Graph G0 is > connected, because there is a path between every pair of vertices within > the spanning tree. > So v−e+f = 2 by the induction assumption P(e). Thus, the original graph > G had v vertices, > e+1 edges, and f +1 faces. Since v−(e+1)+(f +1) = v−e+f = 2, P(e+1) is > again true. > Otherwise, G has at least one cycle. Select a spanning tree and an edge > u—v in the cycle, > but not in the tree. (The spanning tree can not contain all edges in the > cycle, since trees > are acyclic.) Removing u—v merges the two faces on either side of the > edge and leaves > a graph G0 with only e edges and some number of vertices v and faces f. > Graph G0 is > connected, because there is a path between every pair of vertices within > the spanning tree. > So v−e+f = 2 by the induction assumption P(e). Thus, the original graph > G had v vertices, > e+1 edges, and f +1 faces. Since v−(e+1)+(f +1) = v−e+f = 2, P(e+1) is > again true. " > > > I did not understand this part of the proof or Euler's formula. Why do > we take a spanning tree and a cycle not in the tree? How do we argue > from there? This proof confused me. > > Michael Murray From meyer@imap.theory.csail.mit.edu Wed Oct 05 09:55:46 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dtjw0019348; Wed, 5 Oct 2005 09:55:46 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j95Dtj1P009188; Wed, 5 Oct 2005 09:55:45 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j95Dtj1g009187; Wed, 5 Oct 2005 09:55:45 -0400 Date: Wed, 5 Oct 2005 09:55:45 -0400 From: Sayan Mitra Message-Id: <200510051355.j95Dtj1g009187@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, nedzel@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Cc: 6042-staff@theory.csail.mit.edu Status: RO Content-Length: 1215 X-UID: 2155 X-Keywords: NonJunk I presume you are asking about the significance of the term "planar embedding" here and not the class of planar graphs. An embedding is a representation of a topological object, manifold, graph, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. A graph embedding is a particular drawing of a graph (with sometimes added constraint that the embedding be planar, i.e., has no crossing edges). While the underlying object is independent of the embedding, a clever choice of embedding can lead to particularly illuminating diagrams. I took these definitions from Mathworld. -S. >From nedzel@MIT.EDU Wed Oct 5 07:06:12 2005 From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 07:06:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.272 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 In section 5.1, what is the significance of planer embedding? - David From meyer@imap.theory.csail.mit.edu Wed Oct 05 00:48:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954lww0010245 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:47:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954lv6s003849 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:47:57 -0400 (EDT) Received: from [18.240.5.57] (MCCORMICK-FIFTY-SEVEN.MIT.EDU [18.240.5.57]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954lnN2009799 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:47:50 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <0e70597c2530cc20e602a3471ad75a49@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6.042-online-tutor@theory.csail.mit.edu From: Rachel Shearer Subject: [Jelani] Week 5 Comment Date: Wed, 5 Oct 2005 00:47:48 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -2.099 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 288 X-UID: 2156 X-Keywords: NonJunk The section on Euler's Formula gave me a little bit of difficulty. It's not that I don't understand it when I'm reading it, it's just that I don't think I'd be able to explain it to anyone else...if that makes any sense at all. I think listening to lecture will help though. Rachel From kkdb@MIT.EDU Wed Oct 5 11:23:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95FNUw0001780 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 11:23:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95FEw7p026862 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 11:23:29 -0400 (EDT) Received: from w20-575-52.mit.edu (W20-575-52.MIT.EDU [18.187.0.71]) (authenticated bits=56) (User authenticated as kkdb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95F6UY1015314 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 11:06:31 -0400 (EDT) Received: (from kkdb@localhost) by w20-575-52.mit.edu (8.12.9) id j95F6Ufv028763; Wed, 5 Oct 2005 11:06:30 -0400 (EDT) Subject: David: Email Comments for Graphs From: Kaustuv De Biswas To: 6042-probs@theory.csail.MIT.EDU Content-Type: multipart/alternative; boundary="=-wIewF5uZghRLutvtfPYl" Date: Wed, 05 Oct 2005 11:06:29 -0400 Message-Id: <1128524789.28702.5.camel@w20-575-52.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.077 X-Spam-Level: * (1.077) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2157 Content-Length: 918 X-Keywords: NonJunk --=-wIewF5uZghRLutvtfPYl Content-Type: text/plain Content-Transfer-Encoding: 7bit The Induction proof for K-Coloring of Graph was a little unclear. p(1) says 1 vertex graph is 1-colorable and then you are assuming p(n) as n-vertex graph is at most k-colorable. the relationship between n and k are not very clear in the proof. Kaustuv. --=-wIewF5uZghRLutvtfPYl Content-Type: text/html; charset=utf-8 The Induction proof for K-Coloring of Graph was a little unclear. p(1) says 1 vertex graph is 1-colorable and then you are assuming p(n) as n-vertex graph is at most k-colorable. the relationship between n and k are not very clear in the proof.
Kaustuv. --=-wIewF5uZghRLutvtfPYl-- From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:09:54 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95H9sw0028108; Wed, 5 Oct 2005 13:09:54 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95H9s0h026774; Wed, 5 Oct 2005 13:09:54 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95H9sxM026771; Wed, 5 Oct 2005 13:09:54 -0400 Date: Wed, 5 Oct 2005 13:09:53 -0400 (EDT) From: Hanson Zhou To: Karena Tyan cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Reading 5 comments In-Reply-To: <20051004193124.wo2hpl958jcwc84o@webmail.mit.edu> Message-ID: References: <20051004193124.wo2hpl958jcwc84o@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2158 Content-Length: 1337 X-Keywords: k-connected here refers to the connectivity of the graph, not between a specific pair. While it is true that deleting those three edges will not disconnect B and E, it DOES disconnect D from the rest of the graph. -Hanson On Tue, 4 Oct 2005, Karena Tyan wrote: > I found the part about k-connected vertices most confusing. Namely, the passage > on page 5: > > Definition 2.6. Two vertices in a graph are k-connected if they remain connected > in any subgraph obtained by deleting k-1 edges. A graph is k-connected if > every pair of its vertices are k-connected. > > How do you know what "k" is? What I mean to say is - to test for > k-connectivity, you have to delete k-1 edges and see if the two vertices in > question are still conected. I assume that this refers just to the edges that > exist along some path between the two vertices, but I still don't quite > understand how "k" is derived. Like, in Figure 1 on page 3, they say that B > and E are 2-connected, which means you could delete 1 edge between the two > vertices and still be connected, correct? But couldn't you delete the three > edges (B-C) (C-D) and (D-E) and still be connected? > > I'm sorry, this was very long. I'm just confused over the definition of > k-connectivity. > > - Karena > > -- > 410 Memorial Drive > Cambridge, MA 02139 > (585)957-5923 > From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:17:33 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HHWw0028579; Wed, 5 Oct 2005 13:17:32 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HHWwb026814; Wed, 5 Oct 2005 13:17:32 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HHWUs026811; Wed, 5 Oct 2005 13:17:32 -0400 Date: Wed, 5 Oct 2005 13:17:32 -0400 (EDT) From: Hanson Zhou To: Lohith G Kini cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Difficulty with reading lecture notes for week 5 In-Reply-To: <20051005000957.osf3cx3xuhog400g@webmail.mit.edu> Message-ID: References: <20051005000957.osf3cx3xuhog400g@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2159 Content-Length: 621 X-Keywords: Ok...it's a bit tricky. Go to office hours to get this cleared up. Essentially, we are counting edges as we traverse the boundary of a face. Since each edge is traversed twice(once for each face that it bounds), the total is 2e, and this is at least 3f since each face has at least 3 bounding edges. -Hanson On Wed, 5 Oct 2005, Lohith G Kini wrote: > Hi Hanson, > > I am having trouble understanding the proof for lemma 5.2 on page 13 of the > lecture notes. I understand the mathematics that leads off from statement (1) > but I don't understand the explanation behind the original inequality 2e >= 3f. > > Lohith > From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:22:52 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HMqw0029991; Wed, 5 Oct 2005 13:22:52 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HMq2O026846; Wed, 5 Oct 2005 13:22:52 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HMqwR026843; Wed, 5 Oct 2005 13:22:52 -0400 Date: Wed, 5 Oct 2005 13:22:52 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 5 reading assignment In-Reply-To: <20051005094740.2nrmctpbgy0owsss@webmail.mit.edu> Message-ID: References: <20051005094740.2nrmctpbgy0owsss@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2160 Content-Length: 730 X-Keywords: Not sure what technicality you are referring to...it is simply that two edges drawn in the plane may not cross except at possibly the common vertex to which they are both incident. -Hanson On Wed, 5 Oct 2005, Nancy L Keuss wrote: > Hi, > > On page 12, under section 5.1 "Euler's Formula," the last sentence of the first > paragraph says that only one point (a single, common endpoint) can appear on > two of the curves. Since each vertex must be assigned to a distinct point in > the plane, it seems like that last sentence should specify that the "single, > common endpoint" must be the vertex from which the edge started, since the > ending point cannot have 2 or more edges incident to it. Is this right? > > Nancy Keuss > From scholtz@MIT.EDU Wed Oct 5 13:39:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HdLw0032089 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 13:39:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95HdK6W013510 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 13:39:20 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95HdGQ1019059 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 13:39:16 -0400 (EDT) Message-Id: <200510051739.j95HdGQ1019059@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 5 Comments Date: Wed, 5 Oct 2005 13:39:17 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C9B2.2F02C700" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJ07Vh2SxhJZJWQTWnRbkn6vWKOA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.856 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2161 Content-Length: 1714 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C9B2.2F02C700 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I would like to have Hall's Marriage Theorem (p 15-16) discussed more fully in lecture. ------=_NextPart_000_0000_01C5C9B2.2F02C700 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I would like to have Hall’s Marriage = Theorem (p 15-16) discussed more fully in lecture.

------=_NextPart_000_0000_01C5C9B2.2F02C700-- From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:39:59 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Hdww0032237; Wed, 5 Oct 2005 13:39:59 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HdwYv026938; Wed, 5 Oct 2005 13:39:58 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HdwiU026935; Wed, 5 Oct 2005 13:39:58 -0400 Date: Wed, 5 Oct 2005 13:39:58 -0400 (EDT) From: Hanson Zhou To: Vera Carr cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Graphs Reading In-Reply-To: <4343F680.8090600@mit.edu> Message-ID: References: <4343F680.8090600@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2162 Content-Length: 632 X-Keywords: Good question. Somewhat beyond the scope of this course though. Kuratowski's theorem characterizes planar graphs as those without a K3,3 or a K5 minor, and there might be algorithms to test planarity along those lines. I am not aware of them though it seems like there would be research done on this. -Hanson On Wed, 5 Oct 2005, Vera Carr wrote: > Planar Graphs (pg 11) > > Is there an easy way to determine whether the graph can be planer. They > gave two examples where if one edge was to be removed that it would make > the graph planar. I understand the definitions but is there an easy way > to determine this property? > From cbossard@MIT.EDU Wed Oct 5 13:46:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HkBw0001462 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 13:46:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Hk96W020094; Wed, 5 Oct 2005 13:46:09 -0400 (EDT) Received: from w20-575-48.mit.edu (W20-575-48.MIT.EDU [18.187.0.67]) (authenticated bits=56) (User authenticated as cbossard@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95Hk1GM021748 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 13:46:02 -0400 (EDT) Received: (from cbossard@localhost) by w20-575-48.mit.edu (8.12.9) id j95Hk1Nu024949; Wed, 5 Oct 2005 13:46:01 -0400 (EDT) Subject: [David] Week 5 reading comments From: Cynthia C Bossard To: 6042-probs@theory.lcs.MIT.EDU Content-Type: multipart/alternative; boundary="=-voWHfZDfmwBAnk2MDAn3" Date: Wed, 05 Oct 2005 13:46:01 -0400 Message-Id: <1128534361.24876.2.camel@w20-575-48.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.129 X-Spam-Level: * (1.129) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2163 Content-Length: 838 X-Keywords: NonJunk --=-voWHfZDfmwBAnk2MDAn3 Content-Type: text/plain Content-Transfer-Encoding: 7bit Sections 5.1 and 5.2 I am having some trouble understanding the proofs, but I think I get the concept Cynthia Sorry about the time for some reason I looked at the wrong date and thought this was due Friday. --=-voWHfZDfmwBAnk2MDAn3 Content-Type: text/html; charset=utf-8 Sections 5.1 and 5.2
I am having some trouble understanding the proofs, but I think I get the concept
Cynthia

Sorry about the time for some reason I looked at the wrong date and thought this was due Friday. --=-voWHfZDfmwBAnk2MDAn3-- From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:51:15 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HpFw0001841; Wed, 5 Oct 2005 13:51:15 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HpFHZ027005; Wed, 5 Oct 2005 13:51:15 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HpFsv027002; Wed, 5 Oct 2005 13:51:15 -0400 Date: Wed, 5 Oct 2005 13:51:15 -0400 (EDT) From: Hanson Zhou To: Hanson Zhou cc: Vera Carr , 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Graphs Reading In-Reply-To: Message-ID: References: <4343F680.8090600@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2164 Content-Length: 804 X-Keywords: See Hopcroft and Tarjan, "Efficient Planarity Testing", 1974...and possibly many others. -Hanson On Wed, 5 Oct 2005, Hanson Zhou wrote: > Good question. Somewhat beyond the scope of this course though. > Kuratowski's theorem characterizes planar graphs as those without a K3,3 > or a K5 minor, and there might be algorithms to test planarity along those > lines. I am not aware of them though it seems like there would be > research done on this. > > -Hanson > > On Wed, 5 Oct 2005, Vera Carr wrote: > > > Planar Graphs (pg 11) > > > > Is there an easy way to determine whether the graph can be planer. They > > gave two examples where if one edge was to be removed that it would make > > the graph planar. I understand the definitions but is there an easy way > > to determine this property? > > > From dshin@MIT.EDU Wed Oct 5 17:05:51 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95L5pw0006813 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 17:05:51 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95L5nDX019163; Wed, 5 Oct 2005 17:05:49 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95L5jok008960; Wed, 5 Oct 2005 17:05:45 -0400 (EDT) Message-ID: <43444027.9090608@mit.edu> Date: Wed, 05 Oct 2005 17:05:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kaustuv De Biswas CC: 6042-probs@theory.csail.mit.edu Subject: Re: David: Email Comments for Graphs References: <1128524789.28702.5.camel@w20-575-52.mit.edu> In-Reply-To: <1128524789.28702.5.camel@w20-575-52.mit.edu> Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2165 Content-Length: 429 X-Keywords: k is considered fixed. You induct on n. For example, say k = 10. P(1): Every 1-vertex graph is 10-colorable. Is that clear now? Kaustuv De Biswas wrote: > The Induction proof for K-Coloring of Graph was a little unclear. p(1) > says 1 vertex graph is 1-colorable and then you are assuming p(n) as > n-vertex graph is at most k-colorable. the relationship between n and > k are not very clear in the proof. > Kaustuv. From dshin@MIT.EDU Wed Oct 5 17:16:20 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95LGKw0008518 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 17:16:20 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j95LGJZf025407; Wed, 5 Oct 2005 17:16:19 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95LGIc5008023; Wed, 5 Oct 2005 17:16:18 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95LGAok009461; Wed, 5 Oct 2005 17:16:10 -0400 (EDT) Message-ID: <43444298.4020109@mit.edu> Date: Wed, 05 Oct 2005 17:16:08 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Lauren McCarthy CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [david] other question References: <20051005094344.hlc0xsqogey8o8c8@webmail.mit.edu> In-Reply-To: <20051005094344.hlc0xsqogey8o8c8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2166 Content-Length: 1212 X-Keywords: Lauren McCarthy wrote: >I keep reading these emails about normalizing scores for psets over tas, but I >just don't really understand what is meant by that. > Sorry about the confusion - what is meant is that the scores will be normalized at the END of the term, when final grades are computed. The raw scores you receive on your psets are not normalized. The staff did have a meeting and decided we would try harder to coordinate grading rubrics in the future. >I'm confused why when I do >my psets with my friend and we have essentially the same answers, he gets 35 >while I get a 23. There was one problem where I referenced a proof we did in >class and got 1/10 while he got 9/10 for doing the same thing. > If the proof you are referring to is the subset-take-away game from the in-class problems, let me assure you that there is absolutely no way to use what was proved in class to prove problem 4 from pset 2. If it is something else, please bring this problem in to office hours and I will be glad to look it over. If your friend received points he did not deserve, there is not much we can do about that, but if you did not receive points that you deserve, this will be fixed. DS From dshin@MIT.EDU Wed Oct 5 17:27:53 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95LRrw0011101 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 17:27:53 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95LRoDX020268; Wed, 5 Oct 2005 17:27:50 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95LRjMu009655; Wed, 5 Oct 2005 17:27:46 -0400 (EDT) Message-ID: <4344454F.9000106@mit.edu> Date: Wed, 05 Oct 2005 17:27:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Valery K Brobbey CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 5 Comments References: <1128523350.3022.8.camel@newhouse-6.mit.edu> In-Reply-To: <1128523350.3022.8.camel@newhouse-6.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2167 Content-Length: 1119 X-Keywords: There are some really interesting problems that arise with bipartite matching with preference lists (something you allude to in your comments). In this problem, you have a fully connected bipartite graph, and each vertex has a "preference list" - i.e., a ranking of the vertices on the other side. As you can imagine, this is used in all sorts of applications (perhaps housing freshmen is one of them). By the way, your TA is Hanson, not me. DS Valery K Brobbey wrote: >What the University of Chicago study probably meant was that 74% of the >time it's men to look for opposite gender partners. Even by common sense >their assertion that men have on average 74% more opposite gender >partners is ridiculous. >I was impressed about how the idea of colouring a graph can be used to >solve problems such as scheduling. I guess the housing department uses >Hall's Marriage Theorem, or something similar to assign houses to >freshmen. Of course this is more complicated because you have to think >of preferences. >I understood the concept of planar graphs but I got lost in Classifying >Polyhedra section. > > From mike_a@MIT.EDU Mon Oct 10 15:52:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9AJqdw0003484 for <6042-probs@theory.lcs.mit.edu>; Mon, 10 Oct 2005 15:52:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9AJqcIJ022651 for <6042-probs@theory.lcs.mit.edu>; Mon, 10 Oct 2005 15:52:38 -0400 (EDT) Received: from [18.238.6.106] (EASTCAMPUS-EIGHT-SEVENTY-THREE.MIT.EDU [18.238.6.106]) (authenticated bits=0) (User authenticated as mike_a@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9AJqVCY009139 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 10 Oct 2005 15:52:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <6B542EA4-FBA3-4B14-8D15-ACD34C0AF4C4@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: mike anderson Subject: weekly reading Date: Mon, 10 Oct 2005 15:52:30 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2168 Content-Length: 101 X-Keywords: NonJunk is the reading coming out soon? (i'm trying to schedule my course work for the week) thanks, mike From juang@MIT.EDU Wed Oct 12 19:19:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9CNJd6s032084 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 19:19:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9CNJbk7008911 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 19:19:38 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9CNJaBw019763 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 19:19:36 -0400 (EDT) Message-ID: <434D99FE.8060500@mit.edu> Date: Wed, 12 Oct 2005 19:19:26 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2169 Content-Length: 347 X-Keywords: NonJunk I was surprised at the seemingly obvious vulnerability in Turing's code (on page 16). (Of course, it's obvious _now_...) I would like to see some discussion of why we should be confident that RSA actually works (other than "nobody's found a vulnerability yet"), since a lot of stuff would break were a vulnerability to be found. Jason Juang. From rehughes@MIT.EDU Wed Oct 12 20:39:12 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D0dC6s008643 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:12 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9D0dBxW019097 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:12 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D0dBe0017777 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:11 -0400 (EDT) Received: from [18.244.6.55] (SENIOR-THREE-TEN.MIT.EDU [18.244.6.55]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9D0d7ok028201 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:07 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <6a8d0069d7ea59fe3f71e174d7f2db4a@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Richard Hughes Subject: [Jelani] Date: Wed, 12 Oct 2005 20:39:06 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2170 Content-Length: 298 X-Keywords: NonJunk I liked the famous questions in number theory. The fact that every even number less than 10^16 can be written as the sum of two primes is just crazy-amazing. Richard Hughes P.S.: The answer to the last problem in the online tutor is wrong, I'm sure of it. 3780 is definitely not 3^3*5*7^2. From brevzin@MIT.EDU Wed Oct 12 20:43:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D0hs6s009309 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 20:43:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D0hrCc010701 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 20:43:53 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D0hgUl005107 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 20:43:48 -0400 (EDT) Message-Id: <6.1.2.0.1.20051012202630.01af0880@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Wed, 12 Oct 2005 20:42:59 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: Comment to Hanson, regarding my pset Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.058 X-Spam-Level: * (1.058) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2171 Content-Length: 1081 X-Keywords: NonJunk So, on my last pset, I showed that the chain of prime numbers fits the requirement because # primes = pi(x) > x / lnx > x / log2x for all x >= 11 and then showed that for x < 11 the case still holds. Hanson said this was an open problem and was only asymptotically true and gave me 0/3. I want my 3 points back. Here is a graph of pi(x) compared to x / lnx: http://mathworld.wolfram.com/images/eps-gif/PrimePi_1001.gif You can clearly see how the red graph of x /lnx is significantly less than pi(x), and for the lower numbers where it is unclear... x pi(x) x/lnx 2 1 2.88 3 2 2.73 4 2 2.88 5 3 3.11 6 3 3.35 7 4 3.60 8 4 3.85 9 4 4.10 10 4 4.34 11 5 4.59 12 5 4.83 13 6 5.07 14 6 5.30 15 6 5.54 16 6 5.77 17 7 6.00 18 7 6.23 19 8 6.45 20 8 6.68 ... 30 10 8.82 50 15 12.78 ratio: pi(x) / (x / logx) 100 25 21.7 1.15 1000 168 145 1.15 10^4 1229 1086 1.13 10^5 9592 8684 1.10 10^10 455052511,434294482 1.05 etc. While it IS true that pi(x) / (x / logx) --> 1 as x --> infinity, pi(x) remains larger always. Can i have my 3 points back now? Thank you, Barry From rnjacobs@MIT.EDU Wed Oct 12 21:28:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D1SZ6s013952 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 21:28:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D1SVCc015033 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 21:28:31 -0400 (EDT) Received: from m38-370-17.mit.edu (M38-370-17.MIT.EDU [18.107.0.36]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D1SOmq013399 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 21:28:25 -0400 (EDT) Received: (from rnjacobs@localhost) by m38-370-17.mit.edu (8.12.9) id j9D1SOQo029380; Wed, 12 Oct 2005 21:28:24 -0400 (EDT) Message-Id: <200510130128.j9D1SOQo029380@m38-370-17.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Reading response Date: Wed, 12 Oct 2005 21:28:24 -0400 From: r n jacobs X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2172 Content-Length: 58 X-Keywords: NonJunk Everything this week made sense to me. - Robert Jacobs From icharny@MIT.EDU Wed Oct 12 22:22:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D2Mr6s018792 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 22:22:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D2MqCc026764 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 22:22:52 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D2Mj5L024664 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 22:22:45 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9D2Mj1r012334; Wed, 12 Oct 2005 22:22:45 -0400 Received: from 65-78-6-83.c3-0.ned-ubr2.sbo-ned.ma.cable.rcn.com (65-78-6-83.c3-0.ned-ubr2.sbo-ned.ma.cable.rcn.com [65.78.6.83]) (User authenticated as icharny@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 12 Oct 2005 22:22:45 -0400 Message-ID: <20051012222245.4qwrqk3coo6os0og@webmail.mit.edu> Date: Wed, 12 Oct 2005 22:22:45 -0400 From: icharny@MIT.EDU To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Week 6 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2173 Content-Length: 117 X-Keywords: NonJunk I didn't really understand Turing's 2nd Code (page 18, section 8), and the use of modular encryption. ~Isaac Charny From vixen@MIT.EDU Wed Oct 12 22:45:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D2jY6s021944 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:45:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D2jXCc013660 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:45:33 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D2jTgP029133 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:45:31 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 12 Oct 2005 22:44:37 -0400 To: 6042-probs@theory.csail.mit.edu From: Amanda Seybold Subject: [Jelani] LN6 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.385 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2174 Content-Length: 158 X-Keywords: The explanation of how and why multiplicative inverses work to decrypt messages (pg. 20) went by a little fast. Perhaps in class it could be demonstrated. From ajshafer@MIT.EDU Wed Oct 12 22:56:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D2u46s023222 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:56:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D2tiCc021072 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:55:44 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D2tbLj001051 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:55:37 -0400 (EDT) Message-ID: <000201c5cfa1$98fc4100$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 6 Comments Date: Wed, 12 Oct 2005 22:55:40 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2175 Content-Length: 397 X-Keywords: NonJunk I really liked the section on cryptography. Can we do an example of RSA cryptgraphy in class using two very small primes? Also, how does this relate to elliptic curve cryptography? Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From dshin@MIT.EDU Wed Oct 12 23:38:59 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D3cx6s025656 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 23:38:59 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D3cvCe004644; Wed, 12 Oct 2005 23:38:57 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9D3crok004712; Wed, 12 Oct 2005 23:38:54 -0400 (EDT) Message-ID: <434DD6C3.2020401@mit.edu> Date: Wed, 12 Oct 2005 23:38:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason Juang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 6 Comments References: <434D99FE.8060500@mit.edu> In-Reply-To: <434D99FE.8060500@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2176 Content-Length: 1754 X-Keywords: A very legitimate question, and the lack of justification provided by the mathematical community can be troubling indeed. Right now, our confidence in RSA is pretty much simply, "nobody's cracked it yet". Sadly, the best justification we can give is: 1. If you can factor, you can crack RSA. 2. So, we think cracking RSA is as hard as factoring. (even though the reverse direction of 1 is what really matters) 3. We think factoring is hard. In fact, people HAVE found a slight vulnerability - the RSA function actually "leaks" information. To give you a rough idea of what this vulnerability is, note that if M is a square, so is M^d. Since RSA takes M to M^d (mod n), it "preserves squares" in some sense. So, if you can detect if a number is a square efficiently, you can gain information from M from M^d (mod n). That was a bit hand-wavish, because the real vulnerability involves some pretty advanced number theory. There's something called a Jacobi symbol (a value either +1 or -1), and the Jacobi symbol of M is always equal to the Jacobi symbol of M^d when d is odd. Also, the Jacobi symbol can be computed efficiently. So, by computing the JS of the encrypted message M^d (mod n), you can gain information about M. If you find this stuff interesting, I'd suggest thinking about course like 6.875 (Cryptography) in the future. DS Jason Juang wrote: > I was surprised at the seemingly obvious vulnerability in Turing's > code (on page 16). (Of course, it's obvious _now_...) I would like to > see some discussion of why we should be confident that RSA actually > works (other than "nobody's found a vulnerability yet"), since a lot > of stuff would break were a vulnerability to be found. > > Jason Juang. From iyzhang@MIT.EDU Thu Oct 13 02:21:58 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D6Lw6s008486 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:58 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9D6Lvo1024183 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:57 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D6LuOW026020 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:56 -0400 (EDT) Received: from [18.239.6.35] (MACGREGOR-TWO-NINETY.MIT.EDU [18.239.6.35]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9D6LmMu007163 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:48 -0400 (EDT) Message-ID: <434DFCF0.4000304@mit.edu> Date: Thu, 13 Oct 2005 02:21:36 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2177 Content-Length: 86 X-Keywords: I would like to hear more about interesting problems in number theory. thanks, Irene From medrano@MIT.EDU Thu Oct 13 12:20:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DGKH6s005457 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 12:20:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DGKFRV016939 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 12:20:16 -0400 (EDT) Received: from quick-statalib3.mit.edu (QUICK-STATALIB3.MIT.EDU [18.50.1.77]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DGK7OP020364 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 12:20:08 -0400 (EDT) Received: (from medrano@localhost) by quick-statalib3.mit.edu (8.12.9) id j9DGK7Zo002403; Thu, 13 Oct 2005 12:20:07 -0400 Date: Thu, 13 Oct 2005 12:20:07 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]Number Theory Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2178 Content-Length: 375 X-Keywords: NonJunk I don't really know what to write for this section after reading. I have taken a course in number theory and I understand the material about it. This was a good refresher for me. For a moment I had forgotten how to comptuer the Euler-phi function of a composite number with powers of primes as factors. If difficulties arise during class I will let you know. Jesus From ridell@MIT.EDU Thu Oct 13 12:50:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DGoM6s011475 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 12:50:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DGoLRV020798 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 12:50:21 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DGoJvj002285 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 12:50:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9DGoJNZ013686; Thu, 13 Oct 2005 12:50:19 -0400 Received: from STRATTON-FOUR-SEVENTY-EIGHT.MIT.EDU (STRATTON-FOUR-SEVENTY-EIGHT.MIT.EDU [18.187.6.223]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 12:50:19 -0400 Message-ID: <20051013125019.zqdu835wokgk4ksg@webmail.mit.edu> Date: Thu, 13 Oct 2005 12:50:19 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: tp 6 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2179 Content-Length: 293 X-Keywords: NonJunk Can you explain more about corollary 8.3 on page 20. I am having trouble understanding the sequence. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From lye@MIT.EDU Thu Oct 13 13:07:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DH7j6s015026 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 13:07:45 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DH7hRV010607 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 13:07:43 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DH7buB009288 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 13:07:37 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9DH7b7R003408; Thu, 13 Oct 2005 13:07:37 -0400 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 13:07:37 -0400 Message-ID: <20051013130737.kug7w8m5rofks8k0@webmail.mit.edu> Date: Thu, 13 Oct 2005 13:07:37 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 6 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2180 Content-Length: 211 X-Keywords: NonJunk pg. 12, the Prime Number Theorem: "The prime number theorem gives an approximate answer [equation]" I've heard this before but never saw it proved. Could you do the proof in lecture (or outline it briefly)? From cwong08@mit.edu Thu Oct 13 13:43:49 2005 Return-Path: Received: from w20-575-35.MIT.EDU (W20-575-35.MIT.EDU [18.187.0.54]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DHhm6s022099 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 13:43:49 -0400 Received: (from cwong08@localhost) by w20-575-35.MIT.EDU (8.12.9) id j9DHhmn2006544; Thu, 13 Oct 2005 13:43:48 -0400 Subject: [TA-name] Week 6 Comments From: Christopher Wong To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 13 Oct 2005 13:43:48 -0400 Message-Id: <1129225428.6489.1.camel@w20-575-35> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 Status: RO X-UID: 2181 Content-Length: 109 X-Keywords: Euler's Theorem on page 26 was what I found the most difficult passage. I don't really understand it at all. From bakster@MIT.EDU Thu Oct 13 13:52:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DHqo6s022543 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 13:52:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DHpxRV026491 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 13:51:59 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DHpv9b026594 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 13:51:57 -0400 (EDT) Message-ID: <434E9EB9.6050402@mit.edu> Date: Thu, 13 Oct 2005 13:51:53 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] TP6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2182 Content-Length: 85 X-Keywords: NonJunk What's the difference between the modulo and rem functions? Thanks. Alexander Bakst From hzhou@MIT.EDU Thu Oct 13 15:28:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DJSg6s008126 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 15:28:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DJSeSo015586 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 15:28:40 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-SEVENTY-ONE.MIT.EDU [18.247.5.71]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DJSawU007325 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 15:28:37 -0400 (EDT) Message-Id: <5.2.1.1.2.20051013152724.02a98f40@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 13 Oct 2005 15:28:36 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: reading question Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.05 X-Spam-Level: * (1.05) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2183 Content-Length: 330 X-Keywords: NonJunk Hi, I was wondering if there is an easier way to do the last problem of the tutorials. I did it be finding the factors of 3780, which are 2 3 5 and 7, and then subtracting out all numbers that are factors of 2 3 5 and 7 from 3780, which became tedious at the end when i had to account for repeat counting. thanks - Steve From petek@MIT.EDU Thu Oct 13 16:10:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DKAB6s016068 for <"6042-probs"@theory.lcs.mit.edu>; Thu, 13 Oct 2005 16:10:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DKA9So003364 for <"6042-probs"@theory.lcs.mit.edu>; Thu, 13 Oct 2005 16:10:10 -0400 (EDT) Received: from [18.51.5.140] (HAYDEN-ONE-FORTY.MIT.EDU [18.51.5.140]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DKA2rZ024255 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <"6042-probs"@theory.lcs.mit.edu>; Thu, 13 Oct 2005 16:10:07 -0400 (EDT) Message-ID: <434EBF1A.1010901@mit.edu> Date: Thu, 13 Oct 2005 16:10:02 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "6.042 Reading Comments" <"6042-probs"@theory.lcs.mit.edu> Subject: [Jelani] Comments on Number Theory LN Content-Type: multipart/alternative; boundary="------------090007000108090102010602" X-Spam-Score: -2.059 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2184 Content-Length: 1937 X-Keywords: NonJunk This is a multi-part message in MIME format. --------------090007000108090102010602 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Everything was fine up until modulus proofs, which became kinda tricky since I've always grown up with definition of modulo = rem. I have circled 8.4's 2nd step in its proof, I think for the reason that I don't understand Lemma 7.2 as well as I should. I'm going to reread the notes tonight to see if I get a better grasp, and hope that the lectures supplement the material enough to get me a good grasp on all of this. -Pete -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------090007000108090102010602 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Everything was fine up until modulus proofs, which became kinda tricky since I've always grown up with definition of modulo = rem.  I have circled 8.4's 2nd step in its proof, I think for the reason that I don't understand Lemma 7.2 as well as I should.  I'm going to reread the notes tonight to see if I get a better grasp, and hope that the lectures supplement the material enough to get me a good grasp on all of this.

-Pete
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------090007000108090102010602-- From benlu@MIT.EDU Thu Oct 13 17:19:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DLJ86s026974 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 17:19:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DLJ6j5012357 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 17:19:06 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DLIxKs019914 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 17:19:00 -0400 (EDT) Message-ID: <434ECF53.1080601@mit.edu> Date: Thu, 13 Oct 2005 17:19:15 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 6 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.073 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2185 Content-Length: 529 X-Keywords: "On the other hand, inverses generally do not exist over the integers. For example, 7 can not be multiplied by another integer to give 1. Surprisingly, multiplicative inverses do exist when we're working modulo a prime number. For example, if we're working modulo 5, then 3 is a multiplicative inverse of 7." ~ pg19 Whoa. Whoa. Still hurting my brain. In fact, I think most of the things about modular arithmetic are still hurting my brain. Like division only working for relatively prime factors and such. Ouch. ~Ben Lu From dowgun@MIT.EDU Thu Oct 13 17:26:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DLQU6s027652 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 17:26:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DLQTj5019449 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 17:26:29 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DLQQeW022219 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 17:26:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9DLQQw8014641; Thu, 13 Oct 2005 17:26:26 -0400 Received: from W20-575-34.MIT.EDU (W20-575-34.MIT.EDU [18.187.0.53]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 17:26:26 -0400 Message-ID: <20051013172626.mhi7ko9t63ccgk0w@webmail.mit.edu> Date: Thu, 13 Oct 2005 17:26:26 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: ln6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2186 Content-Length: 710 X-Keywords: NonJunk Wow. So... I hope we're going over Modulo operations a lot in class. My main quesiton is this (pretend the equals signs have three lines) - if 9 = 2 (mod 7) is our expression, is this expressed as "9 mod 7 equals 2" or "9 equals 2 mod 7". Both are certainly true statements, but it becomes confusing when you refer to multiplicative inverses, since 9 and 2 do not have the same multiplicative inverse ... wait they do! yikes. Is this always true? Probably. Anyway, I dont think it would be fair to put a subject that we are just learning the class before a test on the test, especially because the only applications of Number Theory seem to involve causing confusion (among people you are at war with). Neil From alisonc@MIT.EDU Thu Oct 13 19:55:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DNtY6s016063 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 19:55:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DNtWZl019807 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 19:55:33 -0400 (EDT) Received: from [18.58.6.141] (PIERCE-THREE-NINETY-SIX.MIT.EDU [18.58.6.141]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DNs6Zx023250 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 19:54:07 -0400 (EDT) Message-ID: <434EF3A4.5030200@mit.edu> Date: Thu, 13 Oct 2005 19:54:12 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Comments on lecture notes #6 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2187 Content-Length: 456 X-Keywords: NonJunk It's always interesting to see direct connections between abstract math stuff and very real things like online security. It's really interesting to see the huge variety of ways that it's possible for a code to really fail: there's so many places where potential weaknesses could occur even in seemingly-secure schemes (that it's almost difficult not to be a bit paranoid about all the information floating around out there in so-far-unbroken codes). From ozcan@MIT.EDU Thu Oct 13 20:34:05 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E0Y56s018678 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:34:05 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E0Y4Rr012940 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:34:05 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E0Y2ol026993 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:34:03 -0400 (EDT) Message-Id: <5.2.1.1.2.20051013203037.029c2c10@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 13 Oct 2005 20:33:49 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2188 Content-Length: 1421 X-Keywords: The first 2 paragraphs of section 8 were amazing. I realized how theoretical math could be used in real life and this increases my motivation to study. Quotation (Section 8): In 1940 France had fallen before Hitler's army, and Britain alone stood against the Nazis in western Europe. British resistance depended on a steady flow of supplies brought across the north Atlantic from the United States by convoys of ships. These convoys were engaged in a cat-and-mouse game with German "U-boat" submarines, which prowled the Atlantic, trying to sink supply ships and starve Britain into submission. The outcome of this struggle pivoted on a balance of information: could the Germans locate convoys better than the Allies could locate U-boats or vice versa? Germany lost. But a critical reason behind Germany's loss was made public only in 1974: the British had broken Germany's naval code, Enigma. Through much of the war, the Allies were able to route convoys around German submarines by listening into German communications. The British government didn't explain how Enigma was broken until 1996. When the analysis was finally released (by the US), the author was none other than Alan Turing. In 1939 he had joined the secret British codebreaking effort at Bletchley Park. There, he played a central role in cracking the German's Enigma code and thus in preventing Britain from falling into Hitler's hands. From ereid@MIT.EDU Thu Oct 13 20:59:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E0xe6s022663 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:59:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E0xdZj002436 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:59:39 -0400 (EDT) Received: from [18.242.5.227] (NEXT-TWO-TWENTY-SEVEN.MIT.EDU [18.242.5.227]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E0xW95003663 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:59:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <47C17CD4-3BC1-4A64-AC64-E81B9B54EE6A@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Week 6 reading comment Date: Thu, 13 Oct 2005 20:59:24 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.088 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2189 Content-Length: 142 X-Keywords: NonJunk I'm still a little confused as to where the prime number theorem came from, and how exactly it was used for the prime-number-in-e question. From sil_03@MIT.EDU Thu Oct 13 21:06:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E1626s023103 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 21:06:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E160Zj006684 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 21:06:01 -0400 (EDT) Received: from w20-575-32.mit.edu (W20-575-32.MIT.EDU [18.187.0.51]) (authenticated bits=56) (User authenticated as sil_03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E15wi5004702 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 21:05:59 -0400 (EDT) Received: (from sil_03@localhost) by w20-575-32.mit.edu (8.12.9) id j9E15wNL019466; Thu, 13 Oct 2005 21:05:58 -0400 (EDT) Subject: [David] questions From: Silvia F Baptista To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 13 Oct 2005 21:05:58 -0400 Message-Id: <1129251958.19267.5.camel@w20-575-32.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2190 Content-Length: 241 X-Keywords: NonJunk I'm really confused about mod, congruence, and rem. I would like to see more examples. Also, on the bottom of page 19 it says p|(1-tk) and therefore tk=1 (mod P). Shouldn't it be 1=tk (mod p) since a=b (mod n) implies n|(a-b)??? ~Silvia From mdmurray@MIT.EDU Thu Oct 13 21:53:14 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E1rE6s030902 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:14 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9E1rDBs018463 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:13 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E1rCjw003124 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:12 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E1r5ok029747 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:05 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <5F83572A-10B3-43DE-AF8F-D4E174313546@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Michael Murray Subject: [Hanson] Reading assignment - week 6 Date: Thu, 13 Oct 2005 21:52:53 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2191 Content-Length: 316 X-Keywords: NonJunk I found the reading assignment very enjoyable and interesting this week, although after reading though it I am still unclear as to the difference between rem and mod, they seem to be able to be used interchangeably and yet there is a distinction made at the beginning of the reading. Thanks, Michael Murray From xiaoranz@MIT.EDU Thu Oct 13 21:58:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E1wL6s031413 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 21:58:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E1wKZj015258 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 21:58:20 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E1wIwT014280 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 21:58:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9E1wIwp028728; Thu, 13 Oct 2005 21:58:18 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 21:58:18 -0400 Message-ID: <20051013215818.mpamimfkb6hc8g0g@webmail.mit.edu> Date: Thu, 13 Oct 2005 21:58:18 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 6 Comments References: In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2192 Content-Length: 350 X-Keywords: NonJunk I found section 8 (page 18-22) about the Turing code most interesting/surprising because in highschool I did a project on cryptography and came across RSA/Turing code but didn't fully understand the concept behind it before. Reading about it in association with what we learnt in number theory is refreshing and interesting. Xiaoran (Sharon) Zhang From a_lopez@MIT.EDU Thu Oct 13 22:13:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2Dj6s001686 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:13:45 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2DgZj027294 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:13:43 -0400 (EDT) Received: from [18.239.6.154] (MACGREGOR-FOUR-O-NINE.MIT.EDU [18.239.6.154]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2DZYt017120 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:13:36 -0400 (EDT) Message-ID: <434F1455.1070401@mit.edu> Date: Thu, 13 Oct 2005 22:13:41 -0400 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: week 6 reading [revised] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2193 Content-Length: 673 X-Keywords: NonJunk On pg. 27: "We can find a multiplicative inverses using Euler's theorem as we did with Fermat's theorem: if k is relatively prime to n, then k^(phi(n) -1) is a multiplicative inverse of k modulo n." Can we also find an inverse for k modulo n by finding a linear combination of k and n that equals 1, that is, find integers s and t such that ks + nt = 1? We can find such integers using "the pulverizer" (we know they exist because gcd(k, n) = 1). (let a =c b (mod n) denot a is congruent to b modulo n) Since ks + nt = 1, then ks + nt =c 1 (mod n) so ks =c 1 (mod n) and s is a multiplicative inverse of k modulo n. Which method is most efficient? Adriana From jjmonzon@MIT.EDU Thu Oct 13 22:20:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2KC6s002724 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:20:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2KAZj002475 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:20:10 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-TWO-TWELVE.MIT.EDU [18.247.5.212]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2K3ee018367 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:20:04 -0400 (EDT) Message-Id: <6.2.3.4.2.20051013221456.01e81438@po9.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 13 Oct 2005 22:20:10 -0400 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2194 Content-Length: 317 X-Keywords: NonJunk I found the RSA to be very cool because I finally saw a connection between 6.042 and computer science. It would be really great if RSA is explained well in class. Josh Monzon Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From jstritar@MIT.EDU Thu Oct 13 22:26:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2Q06s003738 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:26:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2PxZj006801 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:25:59 -0400 (EDT) Received: from [18.245.6.110] (BAKER-THREE-SIXTY-FIVE.MIT.EDU [18.245.6.110]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2Pop7019447 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:25:52 -0400 (EDT) Message-ID: <434F172C.7060001@mit.edu> Date: Thu, 13 Oct 2005 22:25:48 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2195 Content-Length: 230 X-Keywords: NonJunk 7. Modular Arithmetic I thought the section on Modular Arithmetic was pretty hard to understand, probably because it was not very familiar. I also thought the sections on encoding/decoding were really interesting. Jon Stritar From hsoumare@MIT.EDU Thu Oct 13 22:27:37 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2Rb6s003902 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:37 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2RaBs001466 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:36 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2RZjw003996 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:35 -0400 (EDT) Received: from [18.241.6.193] (NEW-FOUR-FORTY-EIGHT.MIT.EDU [18.241.6.193]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E2RSok001174 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:28 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <41C83F1A-5B1D-4F20-9690-9141156DA9F3@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: {Sayan} Weekly Email Comment Date: Thu, 13 Oct 2005 22:36:10 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2196 Content-Length: 221 X-Keywords: NonJunk Hi, I would like the passage at the top of pg 20 to be explained in lecture because I don't totally see how the inverse of the key and the remainder function are used to decode the message. Thanks. Hamidou Soumare From bens@MIT.EDU Thu Oct 13 22:30:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2UB6s004414 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:30:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2U9Zj009825 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:30:09 -0400 (EDT) Received: from [18.221.0.117] ([18.221.0.117]) (authenticated bits=0) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2U2Xb020183 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:30:07 -0400 (EDT) Message-ID: <434F1812.7010508@mit.edu> Date: Thu, 13 Oct 2005 22:29:38 -0400 From: "Benjamin M. Schwartz" User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050723) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2197 Content-Length: 88 X-Keywords: page 27: So what do phi() and inverses have to do with RSA? Will I find out in lecture? From hectorb@MIT.EDU Thu Oct 13 23:06:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E36E6s010492 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:06:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E36B4S004678; Thu, 13 Oct 2005 23:06:11 -0400 (EDT) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E369is026104 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 13 Oct 2005 23:06:10 -0400 (EDT) Message-ID: <434F209F.4050400@mit.edu> Date: Thu, 13 Oct 2005 23:06:07 -0400 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2198 Content-Length: 272 X-Keywords: NonJunk "In 1977 at MIT, Ronald Rivest, Adi Shamir, and Leonard Adleman proposed a highly secure cryptosystem (called RSA) based on number theory. Despite decades of attack, no significant weakness has been found." - pg 23 Amazing, would like to hear more about this in class. From fluff@MIT.EDU Thu Oct 13 23:27:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3RA6s014737 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:27:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3R94S019937 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:27:09 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E3R2Mo029605 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:27:02 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <1514E291-B632-485C-AEE8-E0AE5F84BAFE@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [jelani] week 6 reading Date: Thu, 13 Oct 2005 23:27:38 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2199 Content-Length: 378 X-Keywords: NonJunk Um, holy crap. This reading was SO FRICKIN' LONG. I thought the narrative parts and the cryptographic applications were extremely interesting, and were definitely a nice fast-read break in between the proofs (since following proofs is slow reading). However, I am almost certain I did not retain anything past page 20, i.e. all the modular arithmetic. Geez. ~Crystal From clintonb@MIT.EDU Thu Oct 13 23:34:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3YI6s015448 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:34:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3YG4S024996 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:34:17 -0400 (EDT) Received: from ccbibmx30 (NEW-ONE-FORTY-EIGHT.MIT.EDU [18.241.5.148]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E3YDnQ000806 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:34:14 -0400 (EDT) Message-Id: <200510140334.j9E3YDnQ000806@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Jelani: Week 6 Comments Date: Thu, 13 Oct 2005 23:34:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0057_01C5D04E.9798D7D0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXQcB3U7NONQDfxRs20YLXwKo89KQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.344 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2200 Content-Length: 3370 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0057_01C5D04E.9798D7D0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The Die Hard jug problem is pretty neat. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0057_01C5D04E.9798D7D0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

The Die Hard jug problem is pretty neat. =

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_0057_01C5D04E.9798D7D0-- From brevzin@MIT.EDU Thu Oct 13 23:38:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3cJ6s015797 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:38:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3cI4S027790 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:38:18 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E3cB9O001389 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:38:15 -0400 (EDT) Message-Id: <6.1.2.0.1.20051013233355.01aac540@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Thu, 13 Oct 2005 23:38:14 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.057 X-Spam-Level: * (1.057) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2201 Content-Length: 908 X-Keywords: NonJunk I don't understand the necessity of the difference between rem and mod. They mean preciesly the same thing, don't they ? I think it's much more confusing to have to write proofs out with the rem's, such as with the proof of Euler's thm. I mean, clearly A rem N = A mod N... always, so why the difference ? And I actually used the AKS algorithm over the summer in dealing with encryption software, it is a surprisingly simple algorithm that takes a surprising amount of time to actually get to work right. I was especially a fan of the step toward the beginning that is something to the effect of If N is some a^b, return composite. Took me a while to figure out how to write that one efficiently. Also used RSA, which might become obsolete with quantum computing since it's security is solely based on the fact that factoring takes AGES. Might also happen if somebody solves Riemann... ? Barry From mracich@MIT.EDU Thu Oct 13 23:48:28 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3mS6s017197 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:48:28 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3mSqd018519 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:48:28 -0400 (EDT) Received: from MACGREGOR-THREE-THIRTY-NINE.MIT.EDU (MACGREGOR-THREE-THIRTY-NINE.MIT.EDU [18.239.6.84]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E3mKMu000282 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:48:20 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 6 (Introduction to Number Theory) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Thu, 13 Oct 2005 23:48:19 -0400 Message-Id: <1129261699.5576.38.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2202 Content-Length: 225 X-Keywords: I found section 7, Modular Arithmetic, (starting on page 16) kind of confusing. I would appreciate it if this was reviewed in greater depth during lecture (and I would appreciate seeing some more examples). Moira Racich From mpapi@MIT.EDU Fri Oct 14 00:03:39 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E43c6s019965 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 00:03:38 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E43cqd018840 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 00:03:38 -0400 (EDT) Received: from MACGREGOR-TWO-O-FOUR.MIT.EDU (MACGREGOR-TWO-O-FOUR.MIT.EDU [18.239.5.204]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E43UMu000842 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 00:03:30 -0400 (EDT) Subject: [Jelani] Week 6 Comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 14 Oct 2005 00:03:29 -0400 Message-Id: <1129262610.7478.41.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2203 Content-Length: 278 X-Keywords: NonJunk Section 8.3 on page 21, on Fermat's Theorem; I found that I had to read it over a few times for it to click. Since the theorem has some relevant and interesting applications, I'd like to see it explained further in class, particularly the section on finding inverses. -- Matt From aston@MIT.EDU Fri Oct 14 00:32:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E4W26s022605 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 00:32:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E4W04S007434 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 00:32:00 -0400 (EDT) Received: from aston (ASTON.MIT.EDU [18.241.3.49]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E4VvJk009896 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 00:31:58 -0400 (EDT) Message-Id: <200510140431.j9E4VvJk009896@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [TA-name] Week 6 Comments Date: Fri, 14 Oct 2005 00:31:51 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_003F_01C5D056.AD2A69D0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXQeDKi4kK9s/8iTNO18FdWkiaDYA== X-Spam-Score: 0.275 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2204 Content-Length: 2536 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_003F_01C5D056.AD2A69D0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The multiplicative inverse modulo primes (page 19) is a concept I've not heard of before, and seems pretty cool. I missed, however, how we dropped the rem on page 20 in proving that m* x k^-1 is congruent to m (mod p) - Aston ------=_NextPart_000_003F_01C5D056.AD2A69D0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

The multiplicative inverse modulo primes (page 19) is = a concept I’ve not heard of before, and seems pretty = cool.

 

I missed, however, how we dropped the rem on page 20 = in proving that m* x k^-1 is congruent to m (mod = p)

 

         =    - Aston

------=_NextPart_000_003F_01C5D056.AD2A69D0-- From dangut@MIT.EDU Fri Oct 14 00:36:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E4a56s022870 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 00:36:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E4a44S010054 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 00:36:04 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E4ZwWG010394 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 00:35:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9E4Zwfn005476; Fri, 14 Oct 2005 00:35:58 -0400 Received: from MACGREGOR-ONE-O-NINE.MIT.EDU (MACGREGOR-ONE-O-NINE.MIT.EDU [18.239.5.109]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 00:35:58 -0400 Message-ID: <20051014003558.c1ovs3i5ff7ooc4w@webmail.mit.edu> Date: Fri, 14 Oct 2005 00:35:58 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: TP6 response R13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2205 Content-Length: 135 X-Keywords: NonJunk I think the pulverizing method for getting the coefficients is neat, other than that, most of this isn't too unfamiliar for this week. From sergiob@MIT.EDU Fri Oct 14 01:08:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E58B6s027053 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:08:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E58A4S002243 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:08:10 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E57tsT014543 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:08:04 -0400 (EDT) Message-Id: <5.2.1.1.2.20051013230108.03181ef0@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 13 Oct 2005 23:07:50 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 1.045 X-Spam-Level: * (1.045) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9E58B6s027053 Status: RO X-UID: 2206 Content-Length: 330 X-Keywords: I found Turings life the most interesting part of the lecture notes. His contributions were incredibly seminal, and the way that his life ended is very sad :( . He devoted his life to making theories about computing, when the most efficient computers were still mechanical. The impact of his work is mind-boggling. Sergio From dshin@MIT.EDU Fri Oct 14 01:20:48 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E5Km6s029914 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:20:48 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5Kkqd020391; Fri, 14 Oct 2005 01:20:47 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E5KhMu003327; Fri, 14 Oct 2005 01:20:43 -0400 (EDT) Message-ID: <434F401E.7070301@mit.edu> Date: Fri, 14 Oct 2005 01:20:30 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Silvia F Baptista CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] questions References: <1129251958.19267.5.camel@w20-575-32.mit.edu> In-Reply-To: <1129251958.19267.5.camel@w20-575-32.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2207 Content-Length: 424 X-Keywords: Are you confused about their definitions, or their applications? tk=1 (mod p) is the same as 1=tk (mod p). This is because p|n iff p|-n. DS Silvia F Baptista wrote: >I'm really confused about mod, congruence, and rem. I would like to see >more examples. > >Also, on the bottom of page 19 it says p|(1-tk) and therefore tk=1 (mod >P). Shouldn't it be 1=tk (mod p) since a=b (mod n) implies n|(a-b)??? > >~Silvia > > From dshin@MIT.EDU Fri Oct 14 01:33:25 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E5XP6s030633 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 01:33:25 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5XOH2022787 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 01:33:24 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5XMWh009048; Fri, 14 Oct 2005 01:33:22 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E5XJok007436; Fri, 14 Oct 2005 01:33:19 -0400 (EDT) Message-ID: <434F4312.3080009@mit.edu> Date: Fri, 14 Oct 2005 01:33:06 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alexander Bakst CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] TP6 Comments References: <434E9EB9.6050402@mit.edu> In-Reply-To: <434E9EB9.6050402@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2208 Content-Length: 439 X-Keywords: a rem b is the remainder when dividing a by b. Note that rem is a function (maps integers to integers) In this course, mod is not a function. We write a = b (mod n) if n divides a-b. Note that 2 = 7 (mod 5), but 2 rem 5 does not equal 7. On the other hand, 7 = 2 (mod 5), and 7 rem 5 = 2. Hope that helps. DS Alexander Bakst wrote: > What's the difference between the modulo and rem functions? > > Thanks. > Alexander Bakst From dshin@MIT.EDU Fri Oct 14 01:35:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E5ZZ6s031621 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 01:35:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5ZXqd020657; Fri, 14 Oct 2005 01:35:33 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E5ZPMu003994; Fri, 14 Oct 2005 01:35:25 -0400 (EDT) Message-ID: <434F4391.7090202@mit.edu> Date: Fri, 14 Oct 2005 01:35:13 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Steven Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: reading question References: <5.2.1.1.2.20051013152724.02a98f40@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20051013152724.02a98f40@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2209 Content-Length: 670 X-Keywords: Yes, there is a function called Euler's phi function defined in the notes. phi(n) = # of integers from 1 to n relatively prime to n. The notes give an efficient way of computing phi. Go back to the tutor and click "Check Answers" to see what the computation looks like for the tutor problem. DS Steven Zhou wrote: > Hi, > > I was wondering if there is an easier way to do the last problem of > the tutorials. I did it be finding the factors of 3780, which are 2 3 > 5 and 7, and then subtracting out all numbers that are factors of 2 3 > 5 and 7 from 3780, which became tedious at the end when i had to > account for repeat counting. thanks > > - Steve > From shauni@MIT.EDU Fri Oct 14 02:09:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E69g6s004472 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 02:09:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E69fs7008843 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 02:09:41 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E69dCp020113 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 02:09:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9E69da6007774; Fri, 14 Oct 2005 02:09:39 -0400 Received: from SIMMONS-ONE-SEVENTY-NINE.MIT.EDU (SIMMONS-ONE-SEVENTY-NINE.MIT.EDU [18.96.5.179]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 02:09:39 -0400 Message-ID: <20051014020939.rjsfhf0dacf4kcwg@webmail.mit.edu> Date: Fri, 14 Oct 2005 02:09:39 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] week 6 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.287 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2210 Content-Length: 133 X-Keywords: NonJunk I thought the most interesting part of this week's reading was the history of Alan Turing and codebreaking during WWII (p.18, 23) :) From sriaz@MIT.EDU Fri Oct 14 02:11:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6Bq6s004811 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:11:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6Bps7010124 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:11:51 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6BnEk020285 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:11:50 -0400 (EDT) Message-ID: <434F4C28.6030108@mit.edu> Date: Fri, 14 Oct 2005 02:11:52 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Readings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2211 Content-Length: 253 X-Keywords: NonJunk One of the things I found interesting on this week's reading was the discussion of the RSA algorithm (Section 10, page 23). This is something I had covered while taking 18.310, but the lecture notes here provided a greater breakground to the issue. From lkini@MIT.EDU Fri Oct 14 02:13:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6DX6s005001 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:13:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6DVs7011267 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:13:32 -0400 (EDT) Received: from [18.239.6.97] (MACGREGOR-THREE-FIFTY-TWO.MIT.EDU [18.239.6.97]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6DSMr020392 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:13:30 -0400 (EDT) Message-ID: <434F4C87.6050906@mit.edu> Date: Fri, 14 Oct 2005 02:13:27 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 6 Lecture Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2212 Content-Length: 274 X-Keywords: NonJunk Hi Hanson, The only question I had after reading the lecture notes was the proof of Corollary 8.3. I don't understand why the first sequence must contain all of the numbers from 0 to p-1. I would appreciate it if you could go explain that a little more. Thanks, Lohith From cvnguyen@MIT.EDU Fri Oct 14 02:16:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6Gs6s005201 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:16:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6Grs7013263 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:16:53 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-FIVE-TWENTY-EIGHT.MIT.EDU [18.238.5.17]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6Gox8020635 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:16:51 -0400 (EDT) Message-ID: <010b01c5d086$e026c740$9907ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 6 comments Date: Fri, 14 Oct 2005 02:16:54 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MIMEOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.627 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2213 Content-Length: 236 X-Keywords: I enjoyed the discussion of the role of Turing's code during World War II and of Turing's life. It's always beneficial to see how the subject material relates to real life and how people have applied it in the past. --Chieu Nguyen From lana@MIT.EDU Fri Oct 14 02:19:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6Jm6s005272 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:19:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6Jls7014883 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:19:47 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6JeRt020815 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:19:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9E6JeS6028416; Fri, 14 Oct 2005 02:19:40 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 02:19:40 -0400 Message-ID: <20051014021940.zfum59m9ktw80wo0@webmail.mit.edu> Date: Fri, 14 Oct 2005 02:19:40 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2214 Content-Length: 253 X-Keywords: NonJunk The part that I found most interesting was Euler's theorem and everything that was related to Phi. The function phi in general seems to be a somewhat unusual function because there does not seem to be an empirical way of directly deriving it. Lana 6 From ryan786@MIT.EDU Fri Oct 14 02:40:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6em6s006559 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:40:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6eks7027729 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:40:47 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6ehLP022252 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:40:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9E6ehM5020489; Fri, 14 Oct 2005 02:40:43 -0400 Received: from PSK-NINETY-ONE.MIT.EDU (PSK-NINETY-ONE.MIT.EDU [18.217.1.91]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 02:40:43 -0400 Message-ID: <20051014024043.hf6627c5xps8cg8c@webmail.mit.edu> Date: Fri, 14 Oct 2005 02:40:43 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [sayan] email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2215 Content-Length: 392 X-Keywords: NonJunk I really enjoyed the brief background on Alan Turing. Stuff like that helps liven up the more boring parts of the reading. One question: what is the large, bold pi shape on the right hand side of Riemann Hypothesis equation? The same as a summation, but a multiplication? Sorry I didn't pick up on it. No real complaints, aside from the fact that it was really really long. -Ryan Young From tonyng@MIT.EDU Fri Oct 14 02:46:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6k16s006800 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:46:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6jxs7001047 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:45:59 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6ju4A022605 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:45:57 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014024111.02103400@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 02:45:59 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2216 Content-Length: 302 X-Keywords: NonJunk I found the RSA encryption method (starting in pg 23 section 10 on Arithmetic with an Arbitrary Modulus) very interesting. It surprised me that it is possible to use a public key and still have the system be completely secure. I am hoping lecture will give more details on how and why this works. From yangc@MIT.EDU Fri Oct 14 03:06:57 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E76v6s008430 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:06:57 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E76vUP022777 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:06:57 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E76jMv007659 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:06:50 -0400 (EDT) Message-Id: <200510140706.j9E76jMv007659@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] TP6 Comments Date: Fri, 14 Oct 2005 03:06:39 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00CA_01C5D06C.4BE52140" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXQjdKZZVEjLPehSJyNAcPKA/OeKg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2217 Content-Length: 2215 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_00CA_01C5D06C.4BE52140 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit You mention this amazingly simple algorithm for primality testing (twice mentioned, in fact) - but neglect to state how this amazing algorithm works. I guess I'll just ask now - how does this amazingly simple algorithm work? -Chris Yang ------=_NextPart_000_00CA_01C5D06C.4BE52140 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

You mention this amazingly simple algorithm for = primality testing (twice mentioned, in fact) – but neglect to state how this amazing algorithm works.  I guess I’ll just ask now – = how does this amazingly simple algorithm work?

 

-Chris Yang

------=_NextPart_000_00CA_01C5D06C.4BE52140-- From scot@MIT.EDU Fri Oct 14 03:12:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7CY6s009380 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:12:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7CXs7016301 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:12:33 -0400 (EDT) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7CLba024081 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:12:22 -0400 (EDT) Message-ID: <434F5A55.9030505@mit.edu> Date: Fri, 14 Oct 2005 03:12:21 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: this week's comments (Jelani) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.186 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2218 Content-Length: 357 X-Keywords: Hi, I had a question about page 12. This may be simple but when I read it I don't directly see why primes gradually taper off. Are you saying that there become fewer and fewer as the number of digits increase? I thought that the number of primes was infinite. I feel so far the class up to this point in the reading has been sufficient. thanks, scot From zev@MIT.EDU Fri Oct 14 03:26:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7Qu6s010106 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:26:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7Qts7024147 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:26:55 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7QlMN024822 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:26:52 -0400 (EDT) Message-ID: <434F5DB6.10207@mit.edu> Date: Fri, 14 Oct 2005 03:26:46 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 6 comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.239 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2219 Content-Length: 676 X-Keywords: NonJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 The solution to 6.5 has a typo. It claims (4-2) * (27-9) * (5-1) * (49-7) = 864 it means (4-2) * (27-9) * (5-1) * (7-1) = 864 In general, the notes were much more entertaining than those in the past. However, there was a lot of information thrown at us. I didn't really understand the more advanced modular athithmetic sections or the statements about sequences. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDT122lO3j8HLL0+4RAnqRAJ4ilWz8pdIJAun63SgqrKU1RsOUOwCfXAsD 3ofMK3UG1deLlXzVO3ZmdG8= =1NpS -----END PGP SIGNATURE----- From tylerw@MIT.EDU Fri Oct 14 03:30:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7Uv6s010601 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:30:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7Uts7026434 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:30:56 -0400 (EDT) Received: from [18.246.6.207] (BEXLEY-FOUR-SIXTY-TWO.MIT.EDU [18.246.6.207]) (authenticated bits=0) (User authenticated as tylerw@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7UnE3025025 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:30:50 -0400 (EDT) Message-ID: <434F5EAE.4000303@mit.edu> Date: Fri, 14 Oct 2005 03:30:54 -0400 From: Tyler Williams User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2220 Content-Length: 331 X-Keywords: NonJunk Hey, so my comment for this week about the reading was that I really enjoyed how the reading was interspersed with history about Turing, and RSA, etc. This made the reading much more interesting to read, and the whole thing about RSA was cool because it was practical. I hope there are future readings also like this... Tyler From scholtz@MIT.EDU Fri Oct 14 03:32:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7WX6s010715 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:32:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7WVs7027200 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:32:32 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7WPsY025080 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:32:26 -0400 (EDT) Message-Id: <200510140732.j9E7WPsY025080@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 6 Comments Date: Fri, 14 Oct 2005 03:32:34 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5D06F.EAEC9E50" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXQkXFj9rSBNhdSQE2K7O58PL35sw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.593 X-Spam-Level: ** (2.593) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2221 Content-Length: 1810 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5D06F.EAEC9E50 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I would like to have the Riemann Hypothesis discussed more fully in lecture. I don't understand what it means or why it's so famous. ------=_NextPart_000_0000_01C5D06F.EAEC9E50 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I would like to have the Riemann Hypothesis discussed more fully in lecture. I don’t understand what it means = or why it’s so famous.

------=_NextPart_000_0000_01C5D06F.EAEC9E50-- From kjhollen@MIT.EDU Fri Oct 14 03:35:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7Z26s011063 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:35:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7Z1s7028558 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:35:01 -0400 (EDT) Received: from MACGREGOR-TWO-NINETY-NINE.MIT.EDU (MACGREGOR-TWO-NINETY-NINE.MIT.EDU [18.239.6.44]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7Ysa0025168 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:34:55 -0400 (EDT) Subject: [Hanson] about rsa and number theory From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 14 Oct 2005 03:34:54 -0400 Message-Id: <1129275294.29833.3.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2222 Content-Length: 184 X-Keywords: NonJunk I am interested to see how the RSA decoding works in class tomorrow. It might also be nice to quickly go over how it works, because I think I understand but I am not completely sure. From aeon@MIT.EDU Fri Oct 14 03:40:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7eE6s011605 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 03:40:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7eDs7001548 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 03:40:13 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7eB3W025439 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 03:40:11 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9E7eBds031296; Fri, 14 Oct 2005 03:40:11 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 01:40:11 -0600 Message-ID: <20051014014011.y61phbs76w0g8oww@webmail.mit.edu> Date: Fri, 14 Oct 2005 01:40:11 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: LN6 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.671 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2223 Content-Length: 403 X-Keywords: NonJunk Page 11, "The Fundamental Theorem of Arithmetic" This makes sense and all, but I fail to see where it was applied after it was introduced, exactly.. how does it involve the material afterwards? Page 23, "Arithmetic with an Arbitrary Modulus Great, but very abstract. A simple example with actual values and strings would help reinforce the ideas behind the RSA protocol, if possible. - John Marrero From dnreshef@MIT.EDU Fri Oct 14 03:42:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7gC6s011741 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 03:42:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7gBs7002633 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 03:42:11 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7g9ir025534 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 03:42:09 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9E7g9eE019651; Fri, 14 Oct 2005 03:42:09 -0400 Received: from MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU (MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU [18.239.7.33]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 03:42:09 -0400 Message-ID: <20051014034209.hkert02mh9kokgsk@webmail.mit.edu> Date: Fri, 14 Oct 2005 03:42:09 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: HANSON: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 3.14 X-Spam-Level: *** (3.14) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2224 Content-Length: 182 X-Keywords: NonJunk Hey, This week's reading was awesome! I found some of the things regarding GCDs a little confusing. Things like question 3 on the tutor. Also, RSA was a little confusing. -Dave From rshearer@MIT.EDU Fri Oct 14 03:46:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7kd6s012249 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:46:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7kcs7005245 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:46:38 -0400 (EDT) Received: from [18.240.5.57] (MCCORMICK-FIFTY-SEVEN.MIT.EDU [18.240.5.57]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7kZSa025765 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:46:36 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) In-Reply-To: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> References: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Rachel Shearer Subject: [Jelani] Week 6 Comment Date: Fri, 14 Oct 2005 03:46:28 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: -1.346 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2225 Content-Length: 299 X-Keywords: NonJunk I didn't really understand how to use multiplicative inverses in decryption using Turing's code (with modular arithmetic). I mean, supposedly you can "recover the original message" by multiplying by the inverse of the key but I don't understand how that math works (page 20). Rachel Shearer From yaser@MIT.EDU Fri Oct 14 04:11:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E8BD6s014876 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 04:11:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E8BCs7018851 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 04:11:12 -0400 (EDT) Received: from Charizard (NEW-ONE-THIRTY-SEVEN.MIT.EDU [18.241.5.137]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E8B970026828 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 04:11:09 -0400 (EDT) Message-Id: <200510140811.j9E8B970026828@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: reading email for 0ct 14 Date: Fri, 14 Oct 2005 04:11:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0015_01C5D075.49CF98A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXQltBO6EZ2T2xxSsCKqEl3jRJAKg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.178 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2226 Content-Length: 5494 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0015_01C5D075.49CF98A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi David, I think I get most of the reading (that is, understand it enough to follow lecture), but I would definitely like it if we could go over in detail how the RSA encryption technique actually works. It looks nice & dandy on paper, but seems almost a bit too simple to be true. Perhaps a real-life example would be useful. Thanks! _Yaser ------=_NextPart_000_0015_01C5D075.49CF98A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

I think I get most of the reading = (that is, understand it enough to follow lecture), but I would definitely like = it if we could go over in detail how the RSA encryption technique actually = works. It looks nice & dandy on paper, but seems almost a bit too simple to be = true. Perhaps a real-life example would be = useful…

 

Thanks!

_Yaser

------=_NextPart_000_0015_01C5D075.49CF98A0-- From adnaan@MIT.EDU Fri Oct 14 04:25:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E8PA6s015731 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 04:25:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E8P9s7026409 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 04:25:09 -0400 (EDT) Received: from AMMAR (NEW-FOUR-SIXTY-SEVEN.MIT.EDU [18.241.6.212]) (authenticated bits=0) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E8P2Mp027350 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 04:25:02 -0400 (EDT) Message-ID: <000d01c5d098$c7284a40$d406f112@AMMAR> From: "Adnaan" To: <6042-probs@theory.csail.mit.edu> Subject: week reading - hanson Date: Fri, 14 Oct 2005 04:25:03 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000A_01C5D077.3F985280" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.952 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2227 Content-Length: 790 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_000A_01C5D077.3F985280 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable the reading section on turning code is confusing ------=_NextPart_000_000A_01C5D077.3F985280 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
the reading section on turning code is=20 confusing
------=_NextPart_000_000A_01C5D077.3F985280-- From kevin08@MIT.EDU Fri Oct 14 05:05:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E95U6s019975 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:05:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E95TJD017528 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:05:29 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-THREE-SEVENTY-EIGHT.MIT.EDU [18.242.6.123]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E95Mlk028796 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:05:23 -0400 (EDT) Message-Id: <6.2.3.4.2.20051014050241.01ea38b8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 14 Oct 2005 05:05:20 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Number Theory Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2228 Content-Length: 305 X-Keywords: NotJunk I had a question about Section 8.4, page 22 where we are computing the key k. I don't see how the second line of the congruence (which uses "part 2 of Lemma 7.2") follows from the first line and part 2 of Lemma 7.2: specifically how (mk rem p) becomes just mk. Does it use part 1 of Lemma 2? -Kevin From mrivas03@MIT.EDU Fri Oct 14 05:14:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E9EV6s020967 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:14:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E9ETJD022286 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:14:29 -0400 (EDT) Received: from el-ternero.mit.edu (TDCIP95.MIT.EDU [18.237.0.95]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E9EMVw029093 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:14:22 -0400 (EDT) Message-Id: <6.2.3.4.2.20051014051147.02e2cb08@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 14 Oct 2005 05:14:00 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: [Sayan] Week 6 Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.726 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2229 Content-Length: 397 X-Keywords: NonJunk NotJunk Hello Sayan, Just wanted to point out that from the reading Section 6.1 page 15 was extremely interesting to me (Turing's Code). If possible, it would be great to go over this in class. The question of ensuring that two numbers are prime and the security of turing's code and how it was possible to break it makes the concepts in the rest of the reading very interesting. Thanks, Manuel From jeffhoff@MIT.EDU Fri Oct 14 05:16:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E9Gn6s021161 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 05:16:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E9GlJD023533 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 05:16:47 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E9GdYp029192 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 05:16:41 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014051503.0281c650@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 05:16:38 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2230 Content-Length: 235 X-Keywords: NonJunk NotJunk Section 3.4 The Pulverizer This section cleared up what we went over in class. In class several of us were confused as to how the pulverizer actually helped and gave us the final answer, but now it's clear with the substitution. From cbossard@MIT.EDU Fri Oct 14 07:12:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EBCS6s003923 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 07:12:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EBCRJD027180 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 07:12:27 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EBCPSO004418 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 07:12:25 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9EBCPRe010817; Fri, 14 Oct 2005 07:12:25 -0400 Received: from MCCORMICK-TWO-FOURTEEN.MIT.EDU (MCCORMICK-TWO-FOURTEEN.MIT.EDU [18.240.5.214]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 07:12:25 -0400 Message-ID: <20051014071225.mzg9o7npi1sgwgws@webmail.mit.edu> Date: Fri, 14 Oct 2005 07:12:25 -0400 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 6 Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.839 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2231 Content-Length: 165 X-Keywords: NonJunk NotJunk I found this whole thing a little confusing. Probably the biggest issue I have is with mod. I keep forgetting how to find mod and how it differs from rem. Cynthia From kromer@MIT.EDU Fri Oct 14 09:10:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDAx6s018844 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:10:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDAvbg020232 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:10:57 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDAuVH024065 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:10:56 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9EDAtqM018446; Fri, 14 Oct 2005 09:10:55 -0400 Received: from BURTON-TWO-O-TWO.MIT.EDU (BURTON-TWO-O-TWO.MIT.EDU [18.247.5.202]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 09:10:55 -0400 Message-ID: <20051014091055.bqpu4dsbamo8s4kc@webmail.mit.edu> Date: Fri, 14 Oct 2005 09:10:55 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.486 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2232 Content-Length: 259 X-Keywords: NotJunk p. 12 "As a rule of thumb, about 1 integer out of every ln x in the vicinity of x is prime." I find it strange (and vert interesting) that it's possible to find and prove an average distribution of primes, even though primes are very erratically distributed. From letrec@MIT.EDU Fri Oct 14 09:39:50 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDdo6s021485 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:39:50 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDdoDi001797 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:39:50 -0400 (EDT) Received: from [18.237.1.57] (TDC-FIFTY-SEVEN.MIT.EDU [18.237.1.57]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9EDdhok025686 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:39:43 -0400 (EDT) Message-ID: <434FB51E.3010202@mit.edu> Date: Fri, 14 Oct 2005 09:39:42 -0400 From: Alton Torregano User-Agent: Debian Thunderbird 1.0.7 (X11/20051010) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: 6042-student: Hanson, Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2233 Content-Length: 303 X-Keywords: NonJunk NotJunk Notes are very thorough, I found the section on the Nazi's/Turing to be a useful historical review/mathematical refresher as I was presented with the material. However, I felt the notes could use slightly more information on RSA encryption (though I assume it will mostly be covered in slides). ~alton From veracarr@MIT.EDU Fri Oct 14 09:47:42 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDlg6s022167 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:47:42 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDlgDi002104 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:47:42 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9EDldMu023512 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:47:40 -0400 (EDT) Message-ID: <434FD35A.2010003@mit.edu> Date: Fri, 14 Oct 2005 09:48:42 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Intro to Number Theory References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2234 Content-Length: 513 X-Keywords: NonJunk NotJunk Pg. 25 - Theorem 10.1 What i don't understand is if you have, for example, 27 as the number you want to find the relative primes of by part 1 of Theorem 10.1 you can way f(27)=f(3*3*3)=f(3)*f(3)*f(3) and we know for 3 that is has 2 relative primes (2 and 1) so we can say f(3)*f(3)*f(3)=2*2*2? I realize that by using part 2 (where p is a prime...) of Theorm 10.1 you get the right answer but I don't understand why you can't multiply the number of relative primes that each factor has as noted in part 1? From fgreen@MIT.EDU Fri Oct 14 09:55:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDto6s023053 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:55:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDtnbg004367 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:55:49 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDtksM010327 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:55:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9EDtkNN005705; Fri, 14 Oct 2005 09:55:46 -0400 Received: from EASTCAMPUS-SIX-FIFTY-FOUR.MIT.EDU (EASTCAMPUS-SIX-FIFTY-FOUR.MIT.EDU [18.238.5.143]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 09:55:46 -0400 Message-ID: <20051014095546.ijfr3ppvzqgo8wc8@webmail.mit.edu> Date: Fri, 14 Oct 2005 09:55:46 -0400 From: Forrest O Green To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Reading Comment Forrest Green MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2235 Content-Length: 323 X-Keywords: NonJunk NotJunk "An amazingly simple, yet efficient method was finally discovered in 2002 by Agrawal, Kayal, and Saxena." >From page 3. I would very much like to hear an explination of the AKS primality test algorithm that would qualify as simple. I will not be in lecture on Friday, but I may ask in person next Wen. -Forrest Green From nedzel@MIT.EDU Fri Oct 14 09:59:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDx66s023215 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 09:59:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDx5bg007617 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 09:59:05 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-FOUR-NINETY-SEVEN.MIT.EDU [18.96.6.242]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDx3dW011658 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 09:59:03 -0400 (EDT) Message-Id: <200510141359.j9EDx3dW011658@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 6 Comments Date: Fri, 14 Oct 2005 09:58:58 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.909 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2236 Content-Length: 220 X-Keywords: NonJunk NotJunk I don't have any reading questions at the moment; I enjoyed the story behind the lecture notes. Are there any sample exams that I can study for Monday, or should I just review class problems and problem sets? - David From miki_tnd@MIT.EDU Fri Oct 14 09:59:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDx96s023220 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:59:09 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDx8bg007633 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:59:08 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDx6Tb011673 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:59:06 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9EDx6oq023829; Fri, 14 Oct 2005 09:59:06 -0400 Received: from PIKA-ONE-O-NINE.MIT.EDU (PIKA-ONE-O-NINE.MIT.EDU [18.214.1.109]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 09:59:06 -0400 Message-ID: <20051014095906.7bb0av9wmrv48scw@webmail.mit.edu> Date: Fri, 14 Oct 2005 09:59:06 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: sayan reading assignment 6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2237 Content-Length: 289 X-Keywords: NotJunk Wow, so that's what the billboard down mass ave is about.... Anyway, I found most of the number theory topics interesting, and would like to know the proof for the phi function described on pg 25. I'm also interested in the significance of the rienmann hypothesis - what's the big deal? From zacharyozer@gmail.com Fri Oct 14 10:20:16 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.207]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEKF6s027343 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:20:15 -0400 Received: by xproxy.gmail.com with SMTP id t10so452165wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 07:20:15 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=Q/qeh8cXNSxUHnMDMGTHihyqOdBVBUolrmvpk04oQbjN9K1iWwiZbWvWiHaANVg4G1rAwAE8wVevb1BIlrUkYQku3BDtpPmhrvdR9R3uTVnfFE05pIstpQXPjl2xKPgZCpXYgzRWquzqwId3zxsyFEiryer7HOAE+KqR1WhtUuk= Received: by 10.70.16.16 with SMTP id 16mr1179043wxp; Fri, 14 Oct 2005 07:20:15 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Fri, 14 Oct 2005 07:20:15 -0700 (PDT) Message-ID: <4f2613e50510140720w68babb78q28dda12d6b5c7545@mail.gmail.com> Date: Fri, 14 Oct 2005 10:20:15 -0400 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Proof for Theorem 3.1 Page 6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9EEKF6s027343 Status: RO X-UID: 2238 Content-Length: 208 X-Keywords: NonJunk NotJunk I found the first convoluted statement to be "Thus, in particular, gdc(a,b) divides m and so gcd(a,b) <= m." It is not entirely clear to me why this is true based on the previous steps of the proof. -zozer From ctsims@MIT.EDU Fri Oct 14 10:31:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEV56s029484 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:31:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEV4bg011337 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:31:04 -0400 (EDT) Received: from HowardRoark.mit.edu ([18.90.7.228]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEV2Uv024685 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:31:03 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014101900.0280bd00@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 10:29:33 -0400 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: Reading Comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2239 Content-Length: 274 X-Keywords: NonJunk NotJunk I would like The implications of the Riemann Hypothesis (page 24) explained in further detail in lecture. The Hypothesis seems to have some implications to the phi function, but I could not determine which connection was supposed to be drawn from the reading. -Clayton From aka_kame@MIT.EDU Fri Oct 14 10:41:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEfh6s031032 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 10:41:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEfgbg022261 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 10:41:42 -0400 (EDT) Received: from [18.58.6.191] (PIERCE-FOUR-FORTY-SIX.MIT.EDU [18.58.6.191]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEfWJE029199 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 10:41:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-10-669836657 From: Akari Kameyama Subject: Week 6 Comment Date: Fri, 14 Oct 2005 10:41:31 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.331 X-Spam-Level: * (1.331) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2240 Content-Length: 7085 X-Keywords: NonJunk NotJunk --Apple-Mail-10-669836657 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed In the RSA scheme 3. Select an integer e such that gcd(e, (p =E2=88=92 1)(q =E2=88=92 1)) = =3D 1. The public key is the pair (e, n). This should be distributed widely. 4. Compute d such that de =E2=89=A1 1 (mod (p =E2=88=92 1)(q =E2=88=92 = 1)). The secret key is the pair (d,n). This should be kept hidden! If we know that d is derived from de =E2=89=A1 1, then since e is =20 distributed widely, doesnt this give away d? I guess I am missing something here... --Apple-Mail-10-669836657 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
In the RSA scheme

3. Select an = integer e = such = that gcd(e, (p=C2=A0=E2=88=92 1)(q =E2=88=92 1)) =3D 1.=C2=A0
The public key is the pair (e, n). This should be distributed = widely.=C2=A0
4. Compute d such that de=C2=A0=E2=89=A1 1 (mod (p =E2=88=92 1)(q =E2=88=92 1)).=C2=A0
The = secret = key is the = pair (d,n). This should = be kept hidden!

If we know that d = is derived from=C2=A0de=C2=A0=E2=89=A1=C2=A01, then since e is distributed widely, doesnt = this give away d?=C2=A0=C2=A0
I guess I am missing something = here...
= --Apple-Mail-10-669836657-- From mukkala@MIT.EDU Fri Oct 14 10:47:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EElY6s000361 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:47:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EElXbg028740 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:47:33 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EElRi1001713 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:47:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EElRs6022235; Fri, 14 Oct 2005 10:47:27 -0400 Received: from AP-SIXTY-FOUR.MIT.EDU (AP-SIXTY-FOUR.MIT.EDU [18.153.1.64]) (User authenticated as mukkala@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:47:27 -0400 Message-ID: <20051014104727.ommfax2zre3pk4ww@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:47:27 -0400 From: Praveen Pamidimukkala To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 6 Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2241 Content-Length: 490 X-Keywords: NotJunk In section 10.3 entitled "Euler's Theorem," Lemma 10.4 states, "Suppose n is a positive integer and k is relatively prime to n. Let k1,...,kr denote all the integers relatively prime to n in the range 0<=ki Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEnK6s000618 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:49:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEnIbg000424 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:49:18 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEnG12002439 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:49:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EEnGCD022396; Fri, 14 Oct 2005 10:49:16 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:49:16 -0400 Message-ID: <20051014104916.1zga4x3jni80o0ks@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:49:16 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2242 Content-Length: 288 X-Keywords: NotJunk As I'm sure will be convered in friday's lecture, I am having a bit of trouble getting my mind around congruences (page 16 section 7). I understand the definition, but I probably just need more practice working with them in order to develop an intuitive grasp on them. -Rob Crowell From kktyan@MIT.EDU Fri Oct 14 10:52:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEqo6s001359 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:52:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEqmbg004188 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:52:48 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEqkV0003905 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:52:47 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9EEqk6G030042; Fri, 14 Oct 2005 10:52:46 -0400 Received: from BURTON-THREE-SIXTY-SEVEN.MIT.EDU (BURTON-THREE-SIXTY-SEVEN.MIT.EDU [18.247.6.112]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:52:46 -0400 Message-ID: <20051014105246.gjpieggicl1c8wk4@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:52:46 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading 6 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.542 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2243 Content-Length: 714 X-Status: A X-Keywords: NotJunk I actually found most of the concepts in the reading understandable after rereading a few times. I only have a question which seems trivial in the grand scheme of things: What exactly are the subtle but profound shades of difference between modulo and remainder? i.e. y(mod x) and y rem x? (if I put the first term in correct form). On page 2, it says that "n rem d is the remainder when n is divided by d". But isn't modulo also the remainder when n is divided by d? I'm sorry; it's really a small, trite question, but it's bugging me, as we often see rem and mod used in similar contexts and I'm not really sure what the difference is. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From moscicki@MIT.EDU Fri Oct 14 10:56:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEuK6s002454 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:56:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEuJbg008069 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:56:19 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEuGqK005418 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:56:17 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EEuGW4023077; Fri, 14 Oct 2005 10:56:16 -0400 Received: from NONE-FOUR-SEVENTEEN.MIT.EDU (NONE-FOUR-SEVENTEEN.MIT.EDU [18.54.6.162]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:56:16 -0400 Message-ID: <20051014105616.mxoxcp0b3pwss0wo@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:56:16 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: week 6 comments MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_736bs5v5j668" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2244 Content-Length: 1716 X-Keywords: NotJunk This message is in MIME format. --=_736bs5v5j668 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit wrote them last night and then realized i never e-mailed them. --=_736bs5v5j668 Content-Type: text/plain; charset=ISO-8859-1; name="week6comment.txt" Content-Disposition: attachment; filename="week6comment.txt" Content-Transfer-Encoding: 7bit Number theory... I have always been weak in it. Since I know someone actually reads this stuff, I can comment on an unrelated thing: A lot of the time on my homework I get the comment "this is the right reasoning for the proof its just poorly written". I'm not the only one that gets that. Might be something to address. Divisors, divison algorithm: clear enough. Euclids algorithm: clear after a little thought. I think this stuff will require a lot of practice, but it will be fine after that, and I think I don't speak only for myself. Its a little different way of thinking. The Pulverizer: extension of Euclid's Algorithm. Fundamental Theorem of Arithmetic: knew it before but took it for granted. Reading about it is pretty cool :) Cryptography- extremely interesting. Modulo arithmetic- had the good fortune to have dealt with this stuff before in the context of building divisibility tests for some arbitrary number. Turing 2.0- really cool! i think this is how such encryption programs as PGP works? Multiplicative Inverse: It is followable so it seems clear but it remains to be seen whether it is repeatable or not. To answer my own pgp question- i guess not. The part about permuting the sequence is even more cool. This has been the most interesting reading so far! --=_736bs5v5j668-- From lmccart@MIT.EDU Fri Oct 14 11:06:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EF676s004115 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 11:06:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EF5hbg018549 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 11:05:43 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EF5a8m010083 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 11:05:37 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9EF5aVY021021; Fri, 14 Oct 2005 11:05:36 -0400 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 11:05:36 -0400 Message-ID: <20051014110536.5kvnz9esi8g8gco8@webmail.mit.edu> Date: Fri, 14 Oct 2005 11:05:36 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.401 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2245 Content-Length: 106 X-Keywords: NotJunk I didn't find anything in the reading very confusing but I found the code stuff most interesting. -lauren From antonk@MIT.EDU Fri Oct 14 11:12:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFCs6s004898 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:12:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EFCrHx023273 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:12:53 -0400 (EDT) Received: from silencer ([18.234.0.138]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EFCk1A013184 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:12:46 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] wouldlike to be discussed Date: Fri, 14 Oct 2005 11:12:45 -0400 Organization: Massachusetts Institute of Technology Message-ID: <000701c5d0d1$bb9e6770$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0008_01C5D0B0.348CC770" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXQ0bsBOz7io9UORdSB2Ng4/ugNDQ== X-Spam-Score: -1.214 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2246 Content-Length: 2343 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0008_01C5D0B0.348CC770 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The prime number theorem and the solution to the google problem. I understand the approximation but does this mean that the first X consecutive numbers will be prime? Anton. ------=_NextPart_000_0008_01C5D0B0.348CC770 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

The prime number theorem and the solution to the = google problem.

I understand the approximation but does this mean = that the first X consecutive numbers will be prime?

 

Anton.

 

------=_NextPart_000_0008_01C5D0B0.348CC770-- From nancyk@MIT.EDU Fri Oct 14 11:16:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFGe6s005752 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 11:16:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EFAYQ7021317; Fri, 14 Oct 2005 11:16:39 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EF6pxc010672; Fri, 14 Oct 2005 11:06:52 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EF6pmt024262; Fri, 14 Oct 2005 11:06:51 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 11:06:51 -0400 Message-ID: <20051014110651.ld5tvwsmnf4848og@webmail.mit.edu> Date: Fri, 14 Oct 2005 11:06:51 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 6 reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2247 Content-Length: 265 X-Keywords: NonJunk NotJunk Hi, I was confused about Theorem 3.1 under the section "Linear Combinations and the GCD." I don't know if I understand why it refers to a "positive linear combination." Does this not mean that each of the coefficients of a and b are positive? Thanks. Nancy Keuss From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:29:30 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFTU6s009549; Fri, 14 Oct 2005 11:29:30 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFTUk0003210; Fri, 14 Oct 2005 11:29:30 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFTTND003207; Fri, 14 Oct 2005 11:29:30 -0400 Date: Fri, 14 Oct 2005 11:29:29 -0400 (EDT) From: Hanson Zhou To: Aston Motes cc: 6042-probs@theory.csail.mit.edu Subject: Re: [TA-name] Week 6 Comments In-Reply-To: <200510140431.j9E4VvJk009896@outgoing.mit.edu> Message-ID: References: <200510140431.j9E4VvJk009896@outgoing.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2248 Content-Length: 365 X-Keywords: NotJunk See Lemma 7.2. By TA-name, you mean "Hanson" :). -Hanson On Fri, 14 Oct 2005, Aston Motes wrote: > The multiplicative inverse modulo primes (page 19) is a concept I've not > heard of before, and seems pretty cool. > > > > I missed, however, how we dropped the rem on page 20 in proving that m* x > k^-1 is congruent to m (mod p) > > > > - Aston > > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:32:45 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFWj6s010136; Fri, 14 Oct 2005 11:32:45 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFWj4e003247; Fri, 14 Oct 2005 11:32:45 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFWj2L003244; Fri, 14 Oct 2005 11:32:45 -0400 Date: Fri, 14 Oct 2005 11:32:45 -0400 (EDT) From: Hanson Zhou To: Lohith Kini cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 6 Lecture Notes In-Reply-To: <434F4C87.6050906@mit.edu> Message-ID: References: <434F4C87.6050906@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2249 Content-Length: 636 X-Keywords: NotJunk The first sequence has all those numbers because 1. there are p numbers in the sequence 2. they are all different mod p The proof shows how to prove the latter. Essentially, you use cancellation and proof by contradiction to show distinctness. If this is still confusing, go to office hours. -Hanson On Fri, 14 Oct 2005, Lohith Kini wrote: > Hi Hanson, > > The only question I had after reading the lecture notes was the proof of > Corollary 8.3. I don't understand why the first sequence must contain > all of the numbers from 0 to p-1. I would appreciate it if you could go > explain that a little more. > > Thanks, > Lohith > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:36:13 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFaC6s010736; Fri, 14 Oct 2005 11:36:12 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFaCsB003283; Fri, 14 Oct 2005 11:36:12 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFaCF5003280; Fri, 14 Oct 2005 11:36:12 -0400 Date: Fri, 14 Oct 2005 11:36:12 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Number Theory Reading Comments In-Reply-To: <6.2.3.4.2.20051014050241.01ea38b8@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051014050241.01ea38b8@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2250 Content-Length: 372 X-Keywords: NotJunk yes, part 1. On Fri, 14 Oct 2005, Kevin Wang wrote: > I had a question about Section 8.4, page 22 where we are computing > the key k. I don't see how the second line of the congruence (which > uses "part 2 of Lemma 7.2") follows from the first line and part 2 of > Lemma 7.2: specifically how (mk rem p) becomes just mk. Does it use > part 1 of Lemma 2? > > -Kevin > > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:40:53 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFer6s011430; Fri, 14 Oct 2005 11:40:53 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFerx5003331; Fri, 14 Oct 2005 11:40:53 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFerC1003328; Fri, 14 Oct 2005 11:40:53 -0400 Date: Fri, 14 Oct 2005 11:40:53 -0400 (EDT) From: Hanson Zhou To: Vera Carr cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Intro to Number Theory In-Reply-To: <434FD35A.2010003@mit.edu> Message-ID: References: <434FD35A.2010003@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2251 Content-Length: 825 X-Keywords: NotJunk I'm not sure I understand this question...f(27) cannot be decomposed in the way you have written. f(27)=f(3^3), which we know how to compute. f(ab)=f(a)f(b) for a and b that are relatively prime. You are not meant to understand why this works yet. -Hanson On Fri, 14 Oct 2005, Vera Carr wrote: > Pg. 25 - Theorem 10.1 > > What i don't understand is if you have, for example, 27 as the number > you want to find the relative primes of by part 1 of Theorem 10.1 you > can way f(27)=f(3*3*3)=f(3)*f(3)*f(3) and we know for 3 that is has 2 > relative primes (2 and 1) so we can say f(3)*f(3)*f(3)=2*2*2? I realize > that by using part 2 (where p is a prime...) of Theorm 10.1 you get the > right answer but I don't understand why you can't multiply the number of > relative primes that each factor has as noted in part 1? > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:42:47 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFgl6s011868; Fri, 14 Oct 2005 11:42:47 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFglPA003352; Fri, 14 Oct 2005 11:42:47 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFgl4Z003349; Fri, 14 Oct 2005 11:42:47 -0400 Date: Fri, 14 Oct 2005 11:42:47 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 6 reading assignment In-Reply-To: <20051014110651.ld5tvwsmnf4848og@webmail.mit.edu> Message-ID: References: <20051014110651.ld5tvwsmnf4848og@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2252 Content-Length: 430 X-Keywords: NotJunk Positive linear combination refers to a linear combination that evaluates to a positive number. -Hanson On Fri, 14 Oct 2005, Nancy L Keuss wrote: > Hi, > > I was confused about Theorem 3.1 under the section "Linear Combinations and the > GCD." I don't know if I understand why it refers to a "positive linear > combination." Does this not mean that each of the coefficients of a and b are > positive? Thanks. > > Nancy Keuss > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 12:10:22 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EGAM6s021410; Fri, 14 Oct 2005 12:10:22 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EGAMqZ003746; Fri, 14 Oct 2005 12:10:22 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EGAMAd003743; Fri, 14 Oct 2005 12:10:22 -0400 Date: Fri, 14 Oct 2005 12:10:22 -0400 (EDT) From: Hanson Zhou To: Zachary Adam Ozer cc: 6042-probs@theory.lcs.mit.edu Subject: Re: Proof for Theorem 3.1 Page 6 In-Reply-To: <4f2613e50510140720w68babb78q28dda12d6b5c7545@mail.gmail.com> Message-ID: References: <4f2613e50510140720w68babb78q28dda12d6b5c7545@mail.gmail.com> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2253 Content-Length: 367 X-Keywords: NotJunk What is confusing here? Note that m is a positive number, so if a|m then clearly a<=m. -Hanson On Fri, 14 Oct 2005, Zachary Adam Ozer wrote: > I found the first convoluted statement to be "Thus, in particular, > gdc(a,b) divides m and so gcd(a,b) <= m." It is not entirely clear to > me why this is true based on the previous steps of the proof. > > -zozer > > From dshin@MIT.EDU Fri Oct 14 13:37:54 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EHbs6s009543 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 13:37:54 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EHbq3w012175; Fri, 14 Oct 2005 13:37:52 -0400 (EDT) Received: from [192.168.1.3] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9EHbjMu008437; Fri, 14 Oct 2005 13:37:45 -0400 (EDT) Message-ID: <434FECDA.5060305@mit.edu> Date: Fri, 14 Oct 2005 13:37:30 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 6 Comment References: In-Reply-To: Content-Type: multipart/alternative; boundary="------------010406080202030305080709" X-Spam-Score: 2.82 X-Spam-Level: ** (2.82) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2254 Content-Length: 6401 X-Status: A X-Keywords: NotJunk This is a multi-part message in MIME format. --------------010406080202030305080709 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Yes, we know de = 1 (mod (p-1)(q-1)), but we don't know what (p-1)(q-1) is. DS Akari Kameyama wrote: > In the RSA scheme > > 3. Select an integer e such that gcd(e, (p − 1)(q − 1)) = 1. > The public key is the pair (e, n). This should be distributed widely. > 4. Compute d such that de ≡ 1 (mod (p − 1)(q − 1)). > The secret key is the pair (d,n). This should be kept hidden! > > If we know that d is derived from de ≡ 1, then since e is distributed > widely, doesnt this give away d? > I guess I am missing something here... --------------010406080202030305080709 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: 8bit Yes, we know de = 1 (mod (p-1)(q-1)), but we don't know what (p-1)(q-1) is.

DS

Akari Kameyama wrote:
In the RSA scheme

3. Select an integer e such that gcd(e, (p − 1)(q − 1)) = 1. 
The public key is the pair (e, n). This should be distributed widely. 
4. Compute d such that de ≡ 1 (mod (p − 1)(q − 1)). 
The secret key is the pair (d,n). This should be kept hidden!

If we know that d is derived from de ≡ 1, then since e is distributed widely, doesnt this give away d?  
I guess I am missing something here...
--------------010406080202030305080709-- From meyer@imap.theory.csail.mit.edu Fri Oct 14 14:06:54 2005 Message-ID: <434FF3BF.1040206@csail.mit.edu> Date: Fri, 14 Oct 2005 14:06:55 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Karena Tyan Subject: Re: [Hanson] Reading 6 Comments References: <20051014105246.gjpieggicl1c8wk4@webmail.mit.edu> In-Reply-To: <20051014105246.gjpieggicl1c8wk4@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 787 Status: RO X-UID: 2255 X-Keywords: you're not alone on remainder vs mod; will be clarified in lecture today. regards, A. Karena Tyan wrote: >I actually found most of the concepts in the reading understandable after >rereading a few times. I only have a question which seems trivial in the grand >scheme of things: > >What exactly are the subtle but profound shades of difference between modulo and >remainder? i.e. y(mod x) and y rem x? (if I put the first term in correct >form). On page 2, it says that "n rem d is the remainder when n is divided by >d". But isn't modulo also the remainder when n is divided by d? I'm sorry; >it's really a small, trite question, but it's bugging me, as we often see rem >and mod used in similar contexts and I'm not really sure what the difference >is. > >- Karena > > > > From shreyes19@gmail.com Fri Oct 14 14:11:39 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.195]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIBd6s016764 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 14:11:39 -0400 Received: by xproxy.gmail.com with SMTP id s7so498091wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:11:38 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=SylmuW63EZoyMu4FP8Cs2/riDBmplVyHEdsjzHK5Cqa0AVUYUJt8+d91U1gdaPf5aiBJLB6vo7p/1Xk0VnP1QT5ximzAZQCrbc4gZydI43jEybNwRzWgyiupnaE7wolsECqyaQp0BNIGgfsRdoZiIE64gdUwqd/+LxQrios9Lnc= Received: by 10.70.129.10 with SMTP id b10mr1340399wxd; Fri, 14 Oct 2005 11:11:38 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Fri, 14 Oct 2005 11:11:38 -0700 (PDT) Message-ID: <3c33fe380510141111w556d188ag6c919e8a014e5ab6@mail.gmail.com> Date: Fri, 14 Oct 2005 14:11:38 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 6 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_25948_28406409.1129313498754" Status: RO X-UID: 2256 Content-Length: 1523 X-Keywords: NonJunk NotJunk ------=_Part_25948_28406409.1129313498754 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, Here are my comments for the week 6 reading: The part that still remains a little unclear to me is the difference in use between rem and mod. On page 18, it makes the following assertions: 1. a = =3D (a rem n) (mod n) 2. a =3D b (mod n) if and only if (a rem n) =3D (b rem n) These make sense to me, but I'm still confused on when to use mod and when to use rem. For example, Lemma 8.2 uses mod, while the corrollary 8.3 you got from it uses rem. Are there specific cases where one or the other is preferred? Thanks, Shreyes Seshasai Group 7 ------=_Part_25948_28406409.1129313498754 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

Here are my comments for the week 6 reading:

The part that still remains a little unclear to me is the difference in use between rem and mod.  On page 18, it makes the following assertions: 1. a =3D (a rem n) (mod n)
2. a =3D b (mod n) if and only if (a rem n) =3D (b rem n)
These make sense to me, but I'm still confused on when to use mod and when to use rem.  For example, Lemma 8.2 uses mod, while the corrollary 8.3 you got from it uses rem.  Are there specific cases where one or the other is preferred?

Thanks,
Shreyes Seshasai
Group 7
------=_Part_25948_28406409.1129313498754-- From hkhall@MIT.EDU Fri Oct 14 14:26:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIQQ6s020089 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 14:26:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EIQOYa029226 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 14:26:24 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EIQH5r027396 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 14:26:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EIQHoN010631; Fri, 14 Oct 2005 14:26:17 -0400 Received: from PLP-TWELVE.MIT.EDU (PLP-TWELVE.MIT.EDU [18.218.1.12]) (User authenticated as hkhall@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 14:26:17 -0400 Message-ID: <20051014142617.7qezvz5mrcgs400c@webmail.mit.edu> Date: Fri, 14 Oct 2005 14:26:17 -0400 From: Harrison K Hall To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 6 comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2257 Content-Length: 428 X-Keywords: NonJunk NotJunk David- Sorry this has taken me so long to send. My parents came into town and I just lost track of time. I am doing the TP now as well. Hopefully I can still get some credit out of it. The part of the reading that I found most difficult is Fermat's Theorem. Even though I have seen it before I still Don't really understand its proof. I understand how it can be useful, but not its workings. Thanks for understanding, Harrison From minilek@theory.csail.mit.edu Fri Oct 14 14:47:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIl06s023597 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 14:47:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EIkvYa020966; Fri, 14 Oct 2005 14:46:57 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EIkk07006053 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Fri, 14 Oct 2005 14:46:50 -0400 (EDT) Message-ID: <434FFD27.5010401@theory.csail.mit.edu> Date: Fri, 14 Oct 2005 14:47:03 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051011) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Jelani] LN6 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2258 Content-Length: 634 X-Keywords: NonJunk NotJunk The idea was that if I wanted to send a message to you, I'd represent the message as a number, m, and send you mk (mod p). p would be public knowledge, and k would be our secret. You could then decrypt mk by multiplying by k^{-1} (the inverse of k). Later a vulnerability was discussed where if an adversary knew m mod p and mk mod p, they could figure out the secret, k using Fermat's Little Theorem. Are things clearer now? -Jelani Amanda Seybold wrote: > The explanation of how and why multiplicative inverses work to decrypt > messages (pg. 20) went by a little fast. Perhaps in class it could be > demonstrated. From minilek@theory.csail.mit.edu Fri Oct 14 14:50:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIoe6s023992 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 14:50:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EIobYa024525; Fri, 14 Oct 2005 14:50:37 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EIoVMU007431 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Fri, 14 Oct 2005 14:50:31 -0400 (EDT) Message-ID: <434FFE09.3050604@theory.csail.mit.edu> Date: Fri, 14 Oct 2005 14:50:49 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051011) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Benjamin M. Schwartz" CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Jelani] reading References: <434F1812.7010508@mit.edu> In-Reply-To: <434F1812.7010508@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.11 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2259 Content-Length: 317 X-Keywords: NonJunk NotJunk Look at step 4 where the receiver computes d such that de = 1 (mod (p-1)(q-1)) (note (p-1)(q-1) is phi(n)). Think about how the receiver would compute such a d given that he knows phi(n). Benjamin M. Schwartz wrote: > page 27: > So what do phi() and inverses have to do with RSA? > Will I find out in lecture? From meyer@imap.theory.csail.mit.edu Fri Oct 14 11:10:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFAb6s004728 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 11:10:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EFAYI3021317 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 11:10:36 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EF9TAF011792 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 11:09:30 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014110901.00bfedd8@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 11:09:29 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.923 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 67 X-UID: 2260 X-Keywords: NonJunk NotJunk I found the description of RSA encoding very interesting. Jehan From meyer@imap.theory.csail.mit.edu Mon Oct 17 08:30:24 2005 Message-ID: <43539962.3010305@csail.mit.edu> Date: Mon, 17 Oct 2005 08:30:26 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez Subject: Re: Wed Oct 5 Reading assignment reply, R13 References: <20051004155212.soy36pqy00gss00k@webmail.mit.edu> In-Reply-To: <20051004155212.soy36pqy00gss00k@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 660 Status: RO X-UID: 2261 X-Keywords: In case you haven't gotten this straight already: A "Lemma," like a Theorem or Corollary, asserts the truth of some mathematical statement. A "proof" is an explanation of why the asserted statement is true. Important statements are generally labeled "Theorems." An important statement that is a special case of, or follows directly from, a Theorem is called a "Corollary" of that Theorem. A statement whose importance comes mainly from its later use in proving a Theorem is called a "Lemma." Regards, A. Daniel A Gutierrez wrote: > I'm okay with this material, a sidenote question though is what is the > difference between a Lemma and a proof? From juang@MIT.EDU Fri Oct 21 22:12:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9M2Ctd0002364 for <6042-probs@theory.csail.mit.edu>; Fri, 21 Oct 2005 22:12:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9M2Csow000709 for <6042-probs@theory.csail.mit.edu>; Fri, 21 Oct 2005 22:12:54 -0400 (EDT) Received: from [18.238.5.31] (EASTCAMPUS-FIVE-FORTY-TWO.MIT.EDU [18.238.5.31]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9M2Cp2M025157 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 21 Oct 2005 22:12:52 -0400 (EDT) Message-ID: <4359A010.6030908@mit.edu> Date: Fri, 21 Oct 2005 22:12:32 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 7 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2262 Content-Length: 342 X-Keywords: NonJunk NotJunk On Page 14 of the reading: "Theorem 5.8. The Mating Algorithm marries every boy to his optimal mate and every girl to her pessimal mate." This is surprising. (Well, it's not _that_ surprising, and it's only surprising before the proof is presented. But still, mildly surprising.) Is there a practical "gender-equal" algorithm? Jason. From lmccart@MIT.EDU Fri Oct 21 23:47:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9M3lrd0013679 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 21 Oct 2005 23:47:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9M3lqow015804 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 21 Oct 2005 23:47:52 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9M3loF8003938 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 21 Oct 2005 23:47:50 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9M3loHF002287; Fri, 21 Oct 2005 23:47:50 -0400 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 21 Oct 2005 23:47:50 -0400 Message-ID: <20051021234750.c0zpnoc756lw84oo@webmail.mit.edu> Date: Fri, 21 Oct 2005 23:47:50 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 7 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2263 Content-Length: 459 X-Status: A X-Keywords: NotJunk The part of the reading that I found most confusing was the 'extended euclidean algorithm'. I think I understand the concept but a brief review of the idea in class would be helpful. Also, would it be possible to switch groups? It's really difficult to work with that asian kid with braces, I'm not sure of his name, because he talks to me like I'm stupid. When I have the correct answer and he does not, he refuses to listen to my ideas. thanks, lauren From dshin@MIT.EDU Sat Oct 22 01:11:20 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9M5BKd0025151; Sat, 22 Oct 2005 01:11:20 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9M5BJaP018400; Sat, 22 Oct 2005 01:11:19 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9M5BJKr000781; Sat, 22 Oct 2005 01:11:19 -0400 (EDT) Received: from [192.168.1.5] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9M5BCok012576; Sat, 22 Oct 2005 01:11:12 -0400 (EDT) Message-ID: <4359C9EE.7070505@mit.edu> Date: Sat, 22 Oct 2005 01:11:10 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason Juang CC: 6042-probs@theory.csail.mit.edu, 6042 staff <6042-staff@theory.csail.mit.edu> Subject: Re: [David] Week 7 Comments References: <4359A010.6030908@mit.edu> In-Reply-To: <4359A010.6030908@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2264 Content-Length: 1705 X-Keywords: NotJunk A very good question. I do not know of algorithms that "naturally" induce a "gender-equal" marriage. However, it is usually possible to construct marriages that are neither male-optimal nor female-optimal. There is a nice result that the set of ALL stable marriages form what is called a "distribute lattice": roughly, this means that given any two stable marriages M and M', you can generate two other stable marriages via some operations. The word "lattice" should hint at this idea - the geometric analogue would be like taking two vectors v and v' and generating two other vectors (v+v' and v-v'). In particular, if we have M: male-optimal marriage M': female-optimal marriage we can generate: M ^ M': matching obtainined by giving each woman the worse of her partners in the two matchings M v M': matching obtainined by giving each woman the better of her partners in the two matchings Usually, M ^ M' and M v M' will neither be male-optimal nor female-optimal. (You might want to prove to yourself that these are still stable marriages) For even more of a challege, you might want to show that these stable marriage operations are distributive (hence, "distributive lattice"): 1. A ^ (B v C) = (A ^ B) v (A ^ C) 2. A v (B ^ C) = (A v B) ^ (A v C) for all stable marriages A, B, C. DS Jason Juang wrote: > On Page 14 of the reading: > > "Theorem 5.8. The Mating Algorithm marries every boy to his optimal > mate and every girl to her pessimal mate." > > This is surprising. (Well, it's not _that_ surprising, and it's only > surprising before the proof is presented. But still, mildly > surprising.) Is there a practical "gender-equal" algorithm? > > Jason. From lye@MIT.EDU Sat Oct 22 10:02:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ME2hd0015759 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 10:02:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ME2fow012421 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 10:02:42 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ME2eBN002968 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 10:02:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9ME2e3R003006; Sat, 22 Oct 2005 10:02:40 -0400 Received: from RANDOM-FOUR-FORTY.MIT.EDU (RANDOM-FOUR-FORTY.MIT.EDU [18.243.6.185]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sat, 22 Oct 2005 10:02:40 -0400 Message-ID: <20051022100240.iwx7ot6u5ykosowo@webmail.mit.edu> Date: Sat, 22 Oct 2005 10:02:40 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 7 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2265 Content-Length: 430 X-Keywords: NonJunk NotJunk Section 5.5, Theorem 5.4: Everyone is married by the Mating Algorithm I found this most difficult to follow because it seems like a boy could wind up not serenading anyone, if he and another boy always happened to serenade the same girl. (Of course this isn't true because there are no ties in ranking -- there's always at least 1 girl for every boy left over.) Maybe an example of the algorithm at work would have been helpful? From ereid@MIT.EDU Sat Oct 22 16:04:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MK4Gd0001161 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 16:04:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MK4Fow018168 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 16:04:15 -0400 (EDT) Received: from [18.242.5.109] (NEXT-ONE-O-NINE.MIT.EDU [18.242.5.109]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MK48w4014517 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 16:04:08 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <6D0B2A24-A01B-46D0-8083-DAA7C23F25F5@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] reading comments Date: Sat, 22 Oct 2005 16:04:00 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2266 Content-Length: 130 X-Keywords: NotJunk I found understanding why the mating algorithm always worked the most difficult, but it makes sense if I think about it enough. From cwong08@MIT.EDU Sat Oct 22 16:29:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MKTZd0003794 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 16:29:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MKTY5Y006296 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 16:29:35 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9MKTJol017880 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 16:29:32 -0400 (EDT) Message-ID: <435AA108.2040306@mit.edu> Date: Sat, 22 Oct 2005 16:28:56 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 7 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2267 Content-Length: 275 X-Keywords: NotJunk I found the stable marriage problem a little depressing. In some cases, people were forced to marry their least preferred guy/girl, yet all the marriages were stable. It was very interesting, though, about how true it applies to the world. I really enjoyed that example. From bens@MIT.EDU Sat Oct 22 17:41:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MLfQd0018386 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 17:41:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MLfPXO005939 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 17:41:25 -0400 (EDT) Received: from [18.221.0.117] ([18.221.0.117]) (authenticated bits=0) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MLfMcD025334 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 17:41:23 -0400 (EDT) Message-ID: <435AB201.40703@mit.edu> Date: Sat, 22 Oct 2005 17:41:21 -0400 From: "Benjamin M. Schwartz" User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051014) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Stable buddy problem Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2268 Content-Length: 254 X-Keywords: NotJunk I was surprised that the stable marriage problem is always solvable. I would like to see any connection there is between it and antichains of anded partial orders (the girl's order and the boy's order, like the height and age thing in class). -Ben From scholtz@MIT.EDU Sat Oct 22 17:51:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MLpBd0018900 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 17:51:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MLpAXO010605 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 17:51:10 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MLp8Av026304 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 17:51:09 -0400 (EDT) Message-Id: <200510222151.j9MLp8Av026304@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 7 Comments Date: Sat, 22 Oct 2005 17:51:10 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0003_01C5D731.3068A1A0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXXUrb0wTDpSYL1RYi8RRJI/fJDlQ== X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2269 Content-Length: 2145 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0003_01C5D731.3068A1A0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I wasn't sure how to create the invariant predicate for Problem 2 (page 4). It would help if this example was reviewed in lecture. Eddie Scholtz ------=_NextPart_000_0003_01C5D731.3068A1A0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

I wasn’t sure how to create the = invariant predicate for Problem 2 (page 4).  It would help if this example = was reviewed in lecture.

 

Eddie Scholtz

------=_NextPart_000_0003_01C5D731.3068A1A0-- From ajshafer@MIT.EDU Sat Oct 22 18:25:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MMPud0024823 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 18:25:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MMPtXO027912 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 18:25:55 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MMPrFk000289 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 18:25:54 -0400 (EDT) Message-ID: <000201c5d757$8fbaae70$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 7 Comments Date: Sat, 22 Oct 2005 18:25:51 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2270 Content-Length: 545 X-Keywords: NotJunk For the Stable Marriage problem: This pattern showed how to make two possible lists (start with boys on balconies or girls on balconies). Are there more possible matches? Is the only way to find all posible stable marriages to do some type of deeper search, or does this algorithm generate the only two possibilities? Thanks, Andrew -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From vixen@MIT.EDU Sat Oct 22 19:02:41 2005 Return-Path: Received: from mrow.mit.edu (MROW.MIT.EDU [18.238.3.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MN2fd0028306 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 19:02:41 -0400 Received: (from vixen@localhost) by mrow.mit.edu (8.12.9) id j9MN2eeJ024582; Sat, 22 Oct 2005 19:02:40 -0400 (EDT) Date: Sat, 22 Oct 2005 19:02:39 -0400 (EDT) From: Amanda L Seybold To: 6042-probs@theory.lcs.MIT.EDU Subject: LN7 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Status: RO X-UID: 2271 Content-Length: 197 X-Keywords: NotJunk I found the section on derived variables (pg 8) confusing. I didn't understand what they were or what role they played. There also didn't seem to be much done with state machines in the notes. From hectorb@MIT.EDU Sat Oct 22 20:23:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9N0NBd0003299 for <6042-probs@theory.csail.MIT.EDU>; Sat, 22 Oct 2005 20:23:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9N0N85w019744; Sat, 22 Oct 2005 20:23:08 -0400 (EDT) Received: from besko.mit.edu (BESKO.MIT.EDU [18.246.0.69]) (authenticated bits=56) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9N0N1Bs010900 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sat, 22 Oct 2005 20:23:02 -0400 (EDT) Received: (from hectorb@localhost) by besko.mit.edu (8.12.9) id j9N0N15x020818; Sat, 22 Oct 2005 20:23:01 -0400 Subject: [Hanson] Week 7 Comments From: Hector Beltran To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Sat, 22 Oct 2005 20:23:00 -0400 Message-Id: <1130026981.20607.2.camel@besko.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2272 Content-Length: 339 X-Keywords: NotJunk "Similarly, a boy keeps serenading the girl he most prefers among those on his list until he must cross her off, at which point he serenades the next most preferred girl on his list..." p.14 I would find it useful to see a live demonstration of the mating algorithm by some 6.042 students, with boys actually serenading girls they like. From aston@MIT.EDU Sun Oct 23 01:49:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9N5n2d0018967 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 01:49:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9N5mvIJ005292 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 01:48:57 -0400 (EDT) Received: from astonlaptop (NEW-ONE-SIXTY-NINE.MIT.EDU [18.241.5.169]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9N5mtMn005481 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 01:48:55 -0400 (EDT) Message-Id: <200510230548.j9N5mtMn005481@outgoing.mit.edu> Reply-To: From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 7 Comments Date: Sun, 23 Oct 2005 01:48:54 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00AC_01C5D773.ED65F4A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXXlXQSLZg3BliaQdq0m3apvENY7Q== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2273 Content-Length: 1246 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_00AC_01C5D773.ED65F4A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Page 9's "weakly decreasing" variable is an interesting concept. I'd like to see more in lecture about the termination guarantee when it's on a variable like this. - Aston ------=_NextPart_000_00AC_01C5D773.ED65F4A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable
Page = 9's "weakly=20 decreasing" variable is an interesting concept. I'd like to see more in = lecture=20 about the termination guarantee when it's on a variable like=20 this.
 
    -=20 Aston
------=_NextPart_000_00AC_01C5D773.ED65F4A0-- From dowgun@MIT.EDU Sun Oct 23 14:09:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NI9Ld0010442 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:09:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NI9J2F029062 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:09:20 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NI9IXt024411 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:09:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9NI9IoX012999; Sun, 23 Oct 2005 14:09:18 -0400 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 14:09:18 -0400 Message-ID: <20051023140918.57wc546kxj5s0k4g@webmail.mit.edu> Date: Sun, 23 Oct 2005 14:09:18 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] state machines MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2274 Content-Length: 446 X-Keywords: NonJunk NotJunk It seems to me that proving termination is the most important use of the Well Ordered Principle. My question for the week is - are there any other ways to prove termination? Every example given here, including the Stable Marraige problem, seems to have termination proven by a strictly decreasing variable over the set of natural numbers. In fact, are there any examples of state machines that arent operating solely on the natural numbers? Neil From rnjacobs@MIT.EDU Sun Oct 23 14:17:38 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NIHcd0011468 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:38 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9NIHbDT026506 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:37 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NIHbpH028951 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:37 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9NIHUok011879 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:31 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j9NIHUeP022329; Sun, 23 Oct 2005 14:17:30 -0400 Message-Id: <200510231817.j9NIHUeP022329@white-meteo.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU Subject: [Jelani] Reading response Date: Sun, 23 Oct 2005 14:17:30 -0400 From: r n jacobs X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2275 Content-Length: 65 X-Keywords: NonJunk NotJunk Everything seemed straightforward this week. - Robert Jacobs From bilodeau@MIT.EDU Sun Oct 23 15:30:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NJUsd0021775 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 15:30:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NJUq2F008926 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 15:30:53 -0400 (EDT) Received: from pbilodeau (MACGREGOR-TWO-TWENTY-ONE.MIT.EDU [18.239.5.221]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NJUplG005476 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 15:30:51 -0400 (EDT) Message-Id: <200510231930.j9NJUplG005476@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [jelani] state machine reading comments Date: Sun, 23 Oct 2005 15:30:49 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5D7E6.BF760580" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXYCEYSmyGnqkMsRHWg5+naEpCGTQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2276 Content-Length: 2413 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5D7E6.BF760580 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit The hardest part of this reading to understand came at the end, when, in the discussion of the mating algorithm, names were replaces by variable, and similar variables at that, B B' G etc. yet the language of the mating problem was retained. (B prefers to marry G.) I guess the problem for me was there being no distinction between discussion using mathematical language, sets and functions, state machines, and the vocabulary of the simple, intuitive real world application. ------=_NextPart_000_0000_01C5D7E6.BF760580 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

The hardest part of this reading to understand came = at the end, when, in the discussion of the mating algorithm, names were = replaces by variable, and similar variables at that, B B’ G etc. yet the = language of the mating problem was retained. (B prefers to marry G…) I guess the = problem for me was there being no distinction between discussion using = mathematical language, sets and functions, state machines, and the vocabulary of the = simple, intuitive real world application.

------=_NextPart_000_0000_01C5D7E6.BF760580-- From ridell@MIT.EDU Sun Oct 23 17:01:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NL1gd0003165 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 17:01:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NL1eKA022030 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 17:01:41 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NL1YH2017754 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 17:01:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9NL1Ynr031880; Sun, 23 Oct 2005 17:01:34 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 17:01:34 -0400 Message-ID: <20051023170134.3bvzlnukri80kcc8@webmail.mit.edu> Date: Sun, 23 Oct 2005 17:01:34 -0400 From: Rebecca Idell To: 6042-probs@theory.lcs.MIT.EDU Subject: idell week 7 reading [hanson] MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2277 Content-Length: 351 X-Keywords: NonJunk NotJunk I think that the marriage problem is very interesting. In class can we go over more thoroughly how to make it into a state machine like on p12 of the reading? -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From avalys@MIT.EDU Sun Oct 23 17:07:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NL7od0003390 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:07:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NL7nKA025021 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:07:49 -0400 (EDT) Received: from [18.96.6.192] (SIMMONS-FOUR-FORTY-SEVEN.MIT.EDU [18.96.6.192]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NL7mJ5018683 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:07:48 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Reading comments Date: Sun, 23 Oct 2005 17:07:46 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2278 Content-Length: 115 X-Keywords: NonJunk NotJunk I found section 5.6 ("...And the Boys Live Especially Happily", page 14) confusing (and surprising). Alex Valys From jstritar@MIT.EDU Sun Oct 23 17:38:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NLcXd0007966 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:38:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NLcVKA010885 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:38:31 -0400 (EDT) Received: from [18.245.6.249] (BAKER-FIVE-O-FOUR.MIT.EDU [18.245.6.249]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NLcRjl023412 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:38:29 -0400 (EDT) Message-ID: <435C02D3.9020203@mit.edu> Date: Sun, 23 Oct 2005 17:38:27 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4.1 (Windows/20051006) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2279 Content-Length: 230 X-Keywords: NonJunk NotJunk 3.1 Proving Correctness I thought this part was pretty interesting because I have taken both 6.041 and 6.004 which talk about state machines, but I have never used them in this context or had to prove partial correctness. Jon From aka_kame@MIT.EDU Sun Oct 23 17:40:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NLevd0008114 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 17:40:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NLeuKA012009 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 17:40:56 -0400 (EDT) Received: from [18.187.7.125] (STRATTON-SIX-THIRTY-SIX.MIT.EDU [18.187.7.125]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NLettU023771 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 17:40:55 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <4A29788B-A474-44C9-801C-4B35EC9E12DB@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-8--674883877 From: Akari Kameyama Subject: Week 7 Comments Date: Sun, 23 Oct 2005 17:40:55 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.218 X-Spam-Level: * (1.218) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2280 Content-Length: 4611 X-Status: A X-Keywords: NonJunk NotJunk --Apple-Mail-8--674883877 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed "Acting on behalf of a consortium of major hospitals (playing the =20 role of the girls), the NRMP has, since the turn of the twentieth century, =20 assigned each year =92s pool of medical school graduates (playing the role of boys) to hospital residencies =20 (formerly called =93internships=94)" "Theorem 5.8. The Mating Algorithm marries every boy to his optimal =20 mate and every girl to her pessimal mate" So if every "girl" is married to her pessimal mate, according to the =20 Mating Algorithm, when this algorithm was applied to the intern-=20 hospital situation, every hospital was getting paired with their =20 least prefered student? I would have thought that the hospitals (who =20= are the "more important" side, i think) would have been the "boys" =20 and gotten their optimal choice. --Apple-Mail-8--674883877 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=WINDOWS-1252
"Acting on behalf of a consortium of major = hospitals (playing the role of the=A0
girls), the NRMP = has, since the turn of the twentieth century, assigned each year =92s = pool of medical=A0
school graduates (playing the role of boys) = to hospital residencies (formerly called = =93internships=94)"

"Theorem = 5.8. The Mating = Algorithm marries every boy to his optimal mate and every girl to her = pessimal=A0
mate"

So if every "girl" = is married to her pessimal mate, according to the Mating Algorithm, when = this algorithm was applied to the intern-hospital situation, every = hospital was getting paired with their least prefered student?=A0 I = would have thought that the hospitals (who are the "more important" = side, i think) would have been the "boys" and gotten their optimal = choice.

= --Apple-Mail-8--674883877-- From minilek@theory.csail.mit.edu Sun Oct 23 19:00:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NN0Cd0021529 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 19:00:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NN0BKA024655; Sun, 23 Oct 2005 19:00:11 -0400 (EDT) Received: from [18.250.5.115] (ASHDOWN-ONE-FIFTEEN.MIT.EDU [18.250.5.115]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NN03oK006173 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sun, 23 Oct 2005 19:00:06 -0400 (EDT) Message-ID: <435C15FB.3010303@theory.csail.mit.edu> Date: Sun, 23 Oct 2005 19:00:11 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051011) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda L Seybold CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: LN7 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2281 Content-Length: 864 X-Status: A X-Keywords: NonJunk NotJunk Derived variables are really functions that map states to numbers. If you recall in our proof of termination of the gcd algorithm (page 6 of the lecture notes), we argued termination by reasoning that the variable y was strictly decreasing. y is actually a derived variable, where the states are the (x,y) pairs in our execution of the Euclidean algorithm. Derived variables are useful because if we can come up with a derived variable that is a natural number and strictly decreases on state transitions, then we can invoke the well-ordering principle to prove that the state machine terminates. Does this help? -Jelani Amanda L Seybold wrote: > I found the section on derived variables (pg 8) confusing. I didn't > understand what they were or what role they played. There also didn't > seem to be much done with state machines in the notes. From meyer@imap.theory.csail.mit.edu Sun Oct 23 19:29:28 2005 Message-ID: <435C1CDD.9020803@csail.mit.edu> Date: Sun, 23 Oct 2005 19:29:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama Subject: Re: Week 7 Comments References: <4A29788B-A474-44C9-801C-4B35EC9E12DB@mit.edu> In-Reply-To: <4A29788B-A474-44C9-801C-4B35EC9E12DB@mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Content-Length: 1014 Status: RO X-UID: 2282 X-Keywords: They did originally, but there was an uproar among the interns, and the case was made that since any stable set was defensible, the interns should be given precedence. Regards, A. Akari Kameyama wrote: > "Acting on behalf of a consortium of major hospitals (playing the role > of the > girls), the NRMP has, since the turn of the twentieth century, > assigned each year s pool of medical > school graduates (playing the role of boys) to hospital residencies > (formerly called internships)" > > "Theorem 5.8. The Mating Algorithm marries every boy to his optimal > mate and every girl to her pessimal > mate" > > So if every "girl" is married to her pessimal mate, according to the > Mating Algorithm, when this algorithm was applied to the > intern-hospital situation, every hospital was getting paired with > their least prefered student? I would have thought that the hospitals > (who are the "more important" side, i think) would have been the > "boys" and gotten their optimal choice. > From kjhollen@MIT.EDU Sun Oct 23 19:47:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NNlCd0027218 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 19:47:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NNlBKA021952 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 19:47:11 -0400 (EDT) Received: from MACGREGOR-TWO-SIXTY-ONE.MIT.EDU (MACGREGOR-TWO-SIXTY-ONE.MIT.EDU [18.239.6.6]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NNl9II014252 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 19:47:10 -0400 (EDT) Subject: [Hanson] marriage state problem From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Sun, 23 Oct 2005 19:47:09 -0400 Message-Id: <1130111229.4793.9.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2283 Content-Length: 180 X-Keywords: NonJunk NotJunk I think I understand the marriage state problem/algorithm (p. 9-15) pretty well, but it might be nice to go over it tomorrow in class with more strictly mathematical terms. Kate From meyer@imap.theory.csail.mit.edu Sun Oct 23 18:55:58 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NMtvd0021128 for <6042-staff@theory.csail.mit.edu>; Sun, 23 Oct 2005 18:55:57 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NMtuBL019501; Sun, 23 Oct 2005 18:55:56 -0400 (EDT) Received: from [192.168.1.4] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9NMtmok025075; Sun, 23 Oct 2005 18:55:48 -0400 (EDT) Message-ID: <435C14E8.1050805@mit.edu> Date: Sun, 23 Oct 2005 18:55:36 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-staff@theory.csail.mit.edu Subject: Re: [David] Week 7 E-Mail Comments References: <200510232245.j9NMjkbs003903@outgoing.mit.edu> In-Reply-To: <200510232245.j9NMjkbs003903@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------060607010908010909000502" X-Spam-Score: 1.352 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,HTML_MESSAGE autolearn=ham version=2.63 Status: RO Content-Length: 4057 X-UID: 2284 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------060607010908010909000502 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit First of all, please send these emails to 6042-probs! Second, yes, choosing a good invariant is somewhat of a black art. In fact, it's funny that you specifically used the word "deterministic method" - it is possible to prove that there is NO deterministic method (i.e., "algorithm") to find an invariant capable of proving termination! More formally, given a state machine description, there is no algorithm that is able to read it in and output "HALT" exactly when the state machine terminates and output "DOESN'T HALT" exactly when it doesn't terminate. This is called the Halting Problem - we may come across this later in the course. DS Chris Yang wrote: > Selecting a correct invariant predicate for a state machine seems > really important to be able to prove a given machine is terminable. > In the examples, they seem like sort of arbitrary selections - is > there a systematic or at least deterministic method for selecting a > useful invariant predicate, or is this choice what makes state > machines sort of an art? > > > > Thanks, > > Chris Yang > --------------060607010908010909000502 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit First of all, please send these emails to 6042-probs!

Second, yes, choosing a good invariant is somewhat of a black art.  In fact, it's funny that you specifically used the word "deterministic method" - it is possible to prove that there is NO deterministic method (i.e., "algorithm") to find an invariant capable of proving termination!

More formally, given a state machine description, there is no algorithm that is able to read it in and output "HALT" exactly when the state machine terminates and output "DOESN'T HALT" exactly when it doesn't terminate.  This is called the Halting Problem - we may come across this later in the course.

DS

Chris Yang wrote:

Selecting a correct invariant predicate for a state machine seems really important to be able to prove a given machine is terminable.  In the examples, they seem like sort of arbitrary selections – is there a systematic or at least deterministic method for selecting a useful invariant predicate, or is this choice what makes state machines sort of an art?

 

Thanks,

Chris Yang

--------------060607010908010909000502-- From petek@MIT.EDU Sun Oct 23 20:56:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O0uJd0004805 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 20:56:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O0uIeQ000272 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 20:56:18 -0400 (EDT) Received: from [18.194.1.37] (SN-THIRTY-SEVEN.MIT.EDU [18.194.1.37]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O0uGMI024890 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 20:56:17 -0400 (EDT) Message-ID: <435C3130.6070305@mit.edu> Date: Sun, 23 Oct 2005 20:56:16 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Comments LN7 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.929 X-Spam-Level: * (1.929) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2285 Content-Length: 936 X-Keywords: NonJunk NotJunk
   State Machines I think are going to end up being one of the most useful topics I study in this class, especially in its relevance to artificial intelligence.  Anyway, most of the notes are fairly straight forward.  I guess I'd like to see some lecturing on derived variables, just so I have the chance to talk them over in the class' forum.  Thx

-P
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

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From mracich@MIT.EDU Sun Oct 23 21:36:56 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O1aud0011283 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:56 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9O1atkZ015101 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O1atLL011983 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:55 -0400 (EDT) Received: from MACGREGOR-TWO-SIXTY-TWO.MIT.EDU (MACGREGOR-TWO-SIXTY-TWO.MIT.EDU [18.239.6.7]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O1aqMu025813 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:53 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 7 (State Machines: Invariants and Termination) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Sun, 23 Oct 2005 21:36:51 -0400 Message-Id: <1130117811.8358.8.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2286 Content-Length: 277 X-Keywords: NotJunk I found section 5.6, "...And the Boys Live Especially Happily", (starting on page 14) to be the most surprising section of the lecture notes. I was especially surprised by Theorem 5.8. and the fact that every girl gets her worst pick in the Mating Algorithm. Moira Racich From benlu@MIT.EDU Sun Oct 23 21:38:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O1c5d0011311 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:38:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O1c4eQ023055 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:38:04 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O1bwtL001593 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:37:58 -0400 (EDT) Message-ID: <435C3B16.9050806@mit.edu> Date: Sun, 23 Oct 2005 21:38:30 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 7 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2287 Content-Length: 225 X-Keywords: NotJunk Hey Jelani, I liked the whole set up with stable marriages. It's comforting to know that even though boys do all the work and get turned down over and over again, they're the ones who end up with the optimal mates. ~Ben From mpapi@MIT.EDU Sun Oct 23 22:24:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2OLd0017895 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:24:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2OJeQ019569 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:24:19 -0400 (EDT) Received: from mpapi.MIT.EDU (MPAPI.MIT.EDU [18.239.4.219]) (authenticated bits=0) (User authenticated as mpapi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O2ODTi009009 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:24:13 -0400 (EDT) Subject: [Jelani] Week 7 comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Sun, 23 Oct 2005 22:24:16 -0400 Message-Id: <1130120656.12289.13.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2288 Content-Length: 368 X-Keywords: NonJunk NotJunk I found that Theorem 5.8 in section 5.6 on page 14, "The Mating Algorithm marries every boy to his optimal mate and every girl to her pessimal mate" was surprising. It became a little less so, though, after realizing that there were probably many cases where a girl's optimal and pessimal mates were the same, or at least the "pessimal" mate wasn't the last choice. From pgroudas@MIT.EDU Sun Oct 23 22:26:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2Q8d0018021 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 22:26:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2Q7eQ020818 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 22:26:07 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O2PujN009344 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 22:25:56 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9O2PuS6004218; Sun, 23 Oct 2005 22:25:56 -0400 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 22:25:56 -0400 Message-ID: <20051023222556.n2otxnkb15sg8sos@webmail.mit.edu> Date: Sun, 23 Oct 2005 22:25:56 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: [HANSON] Week 7 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2289 Content-Length: 213 X-Keywords: NotJunk I feel that after reading the introduction to Derived variables on page 8 that I'm not totally getting them. They seem a simple concept and I'm sure they'll probably make more sense after lecture. -Paul Groudas From lkini@MIT.EDU Sun Oct 23 22:29:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2TId0018244 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:29:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2THeQ022778 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:29:17 -0400 (EDT) Received: from [18.239.6.97] (MACGREGOR-THREE-FIFTY-TWO.MIT.EDU [18.239.6.97]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O2TEqa009905 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:29:16 -0400 (EDT) Message-ID: <435C46F5.6040204@mit.edu> Date: Sun, 23 Oct 2005 22:29:09 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 7 Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2290 Content-Length: 307 X-Keywords: NonJunk NotJunk Hi Hanson, I would like it, if possible, if the professors could go over the inductive step of Theorem 4.2 on page 8. I understand the base case and the intuitiveness behind the induction but I would like a more formal mathematical derivation. I ask because I couldn't derive it myself. Thanks, Lohith From icharny@MIT.EDU Sun Oct 23 22:54:49 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2snd0020643 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 22:54:49 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2smxN026616 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 22:54:48 -0400 (EDT) Received: from [127.0.0.1] (EASTCAMPUS-SIX-TWENTY-THREE.MIT.EDU [18.238.5.112]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O2sWMw029144 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 22:54:46 -0400 (EDT) Message-ID: <435C4CE8.9080302@mit.edu> Date: Sun, 23 Oct 2005 22:54:32 -0400 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 7 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2291 Content-Length: 107 X-Keywords: NonJunk NotJunk I don't understand derived variables (page 8, section 4). Please discuss this in lecture. ~Isaac Charny From jeffhoff@MIT.EDU Sun Oct 23 23:01:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O31Hd0021148 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 23:01:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O31Fnj011509 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 23:01:16 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O31DWe015450 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 23:01:14 -0400 (EDT) Message-Id: <5.2.1.1.2.20051023223910.05e09f48@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Sun, 23 Oct 2005 23:01:12 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2292 Content-Length: 280 X-Keywords: NonJunk NotJunk Reading section 5.6 on page 14 If in class we could explain further the proof of why boys are optimal and girls are pessimal it would be appreciated. The notes only get me to understand so far, but I feel I don't fully understand it...or maybe i'm just still a little doubtful. From ksindi@MIT.EDU Sun Oct 23 23:15:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3FJd0023599 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:15:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3FHnj020748 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:15:17 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3FGA3017873 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:15:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9O3FG8C008950; Sun, 23 Oct 2005 23:15:16 -0400 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 23:15:16 -0400 Message-ID: <20051023231516.kafh91q4locgo0wg@webmail.mit.edu> Date: Sun, 23 Oct 2005 23:15:16 -0400 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [David] Week 7 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2293 Content-Length: 143 X-Status: A X-Keywords: NotJunk I found the notes clear but still don't understand the practical use of these state concepts. It seems to me that they are mere formilizations From kevin08@MIT.EDU Sun Oct 23 23:27:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3Rmd0026267 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:27:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3Rknj028341 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:27:47 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-THREE-SIXTEEN.MIT.EDU [18.242.6.61]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3Rd0C020024 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:27:40 -0400 (EDT) Message-Id: <6.2.3.4.2.20051023232523.01e7cef8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Sun, 23 Oct 2005 23:27:33 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Week 7 Readings Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2294 Content-Length: 270 X-Keywords: NotJunk Hi Hanson, I was confused about the concept of a derived variable (Pg. 8, section 4), especially how derived functions help us to find the maximum run time of an algorithm and how it helps prove termination. An example of this would be very helpful. Thanks, Kevin From meyer@imap.theory.csail.mit.edu Sun Oct 23 23:32:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3WYd0026704 for <6042-webmaster@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:32:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3WXnj001203 for <6042-webmaster@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:32:33 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3WQP1020816 for <6042-webmaster@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:32:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9O3WQsG001185; Sun, 23 Oct 2005 23:32:26 -0400 Received: from PSK-NINETY-ONE.MIT.EDU (PSK-NINETY-ONE.MIT.EDU [18.217.1.91]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 23:32:26 -0400 Message-ID: <20051023233226.ldor0tnsxds08s8g@webmail.mit.edu> Date: Sun, 23 Oct 2005 23:32:26 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-webmaster@theory.csail.MIT.EDU Subject: [Sayan] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-3.9 required=3.7 tests=BAYES_00,FROM_ENDS_IN_NUMS autolearn=no version=2.63 Status: RO Content-Length: 317 X-UID: 2295 X-Keywords: NonJunk NotJunk I am fairly comfortable with state machines, as we just finished covering them in depth in 6.004, which I am also taking. The fuzziest parts of the reading were partial correctness (I understand what you do, just maybe the terminology in which it is described makes it confusing) and derived variables. -Ryan Young From shauni@MIT.EDU Sun Oct 23 23:42:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3g0d0028324 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:42:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3fwnj007533 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:41:58 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3fslb022437 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:41:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9O3da6j021990; Sun, 23 Oct 2005 23:39:36 -0400 Received: from SIMMONS-ONE-SEVENTY-NINE.MIT.EDU (SIMMONS-ONE-SEVENTY-NINE.MIT.EDU [18.96.5.179]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 23:39:36 -0400 Message-ID: <20051023233936.ir6twmolfdr4kw4g@webmail.mit.edu> Date: Sun, 23 Oct 2005 23:39:36 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] week 7 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2296 Content-Length: 321 X-Keywords: NonJunk NotJunk I found Theorem 5.8 (p.14) interesting: At first, I would have expected that there is only one set of stable marriages and that both girls and boys do equally well. If the algorithm is reversed the girls get their optimal mates instad of the boys, but does a stable marriage set exist where girls and boys fare equally? From rshearer@MIT.EDU Sun Oct 23 23:54:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3sAd0029931 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:54:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3s9nj015881 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:54:09 -0400 (EDT) Received: from [18.240.5.57] (MCCORMICK-FIFTY-SEVEN.MIT.EDU [18.240.5.57]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3s7fM024469 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:54:08 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <6826b5843ef05f2637fba6c0fccd784f@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: [Jelani] Week 7 Comment Date: Sun, 23 Oct 2005 23:54:06 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2297 Content-Length: 195 X-Keywords: NonJunk NotJunk I had a lot of problems with Tutor Problem 7.2 part 3. I guess I don't understand why H isn't a tree, if the edges are only marked if there isn't a marked path between the endpoints. Rachel From brevzin@MIT.EDU Mon Oct 24 00:04:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O44Zd0031387 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 00:04:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O44Ynj022820 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 00:04:34 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O44UXf026430 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 00:04:32 -0400 (EDT) Message-Id: <6.1.2.0.1.20041024000012.019ddea8@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Mon, 24 Oct 2005 00:04:29 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2298 Content-Length: 184 X-Keywords: NonJunk NotJunk I really enjoyed the Stable mating problem, and I certainly was not expected that it was so favored toward the boys. That theorem at the end was pretty cool and interesting. Barry From hsoumare@MIT.EDU Mon Oct 24 00:06:32 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O46Vd0032673 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:06:31 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O46VxN028942 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:06:31 -0400 (EDT) Received: from [18.241.5.187] (NEW-ONE-EIGHTY-SEVEN.MIT.EDU [18.241.5.187]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O46OMu002708 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:06:24 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) In-Reply-To: <20051023144946.99856.qmail@web50214.mail.yahoo.com> References: <20051023144946.99856.qmail@web50214.mail.yahoo.com> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <3190F9DA-B292-44BC-A1EC-82FD9AFBF25D@mit.edu> Content-Transfer-Encoding: 7bit From: Hamidou Soumare Subject: {SAYAN} Weekly Email Comment Date: Mon, 24 Oct 2005 00:15:16 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2299 Content-Length: 240 X-Keywords: NonJunk NotJunk I found theorem 5.8 on page 14 to be the most interesting because it was counter intuitive for me. I thought that the mating algorithm would match each girl to her optimal mate because the girls picked their favorite suitor. Hamidou From hzhou@MIT.EDU Mon Oct 24 00:21:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4LBd0003445 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:21:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4LAnj003339 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:21:10 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4L4IP029842 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:21:04 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9O4L4NA015276; Mon, 24 Oct 2005 00:21:04 -0400 Received: from BURTON-TWO-TWENTY-SIX.MIT.EDU (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (User authenticated as hzhou@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 00:21:04 -0400 Message-ID: <20051024002104.pj08nk8euo34sswo@webmail.mit.edu> Date: Mon, 24 Oct 2005 00:21:04 -0400 From: Hao Zhou To: 6042-probs@theory.lcs.MIT.EDU Subject: David TP7 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2300 Content-Length: 367 X-Keywords: NonJunk NotJunk Hi Staff, I found using the state invarient is a very clever way to solve problems. However, I find it is often times hard to find an invarient for all the states. For example, for the tiling problem we did in class, the invarient was not a very obvious choice. Other than through experience, how should we look for these invarients? Thanks - Steven (Hao) Zhou From cvnguyen@MIT.EDU Mon Oct 24 00:26:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4Qhd0004632 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:26:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4Qfnj006688 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:26:41 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-FIVE-TWENTY-EIGHT.MIT.EDU [18.238.5.17]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4Qew3000660 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:26:40 -0400 (EDT) Message-ID: <001f01c5d853$21563470$1105ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 8 comments Date: Mon, 24 Oct 2005 00:26:39 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.274 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2301 Content-Length: 143 X-Keywords: NonJunk NotJunk I found the discussion of the Mating Algorithm entertaining and interesting. I'm sure it will be covered in depth in class. --Chieu Nguyen From aeon@MIT.EDU Mon Oct 24 00:47:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4lkd0005819 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:47:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4ljnj019111 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:47:45 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4ld7t003378 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:47:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9O4ldBt029883; Mon, 24 Oct 2005 00:47:39 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 22:47:39 -0600 Message-ID: <20051023224739.neeth8xk01y8k8w8@webmail.mit.edu> Date: Sun, 23 Oct 2005 22:47:39 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 7 Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2302 Content-Length: 528 X-Keywords: NonJunk NotJunk Page 9, "The Stable Marriage Problem" Just wanted to note that this is interesting, since it's implicitly included in TONS of algorithms. I've kind of used this in many programs, never actually stating it anywhere in code, only in design. Page 14, "... and the Boys... " About the "shocking truth"... does that mean there is no possible algorithm where both sets, the-boys AND the-girls can find their optimal spouses? The proof doesn't disprove this idea, and it seems like it's pretty possible. Just a thought. John Marrero From rshroff@MIT.EDU Mon Oct 24 00:53:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4rcd0006266 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:53:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4rbnj025031 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:53:37 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4rZhh006520 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:53:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9O4rZrq009942; Mon, 24 Oct 2005 00:53:35 -0400 Received: from NEXT-THREE-FORTY-EIGHT.MIT.EDU (NEXT-THREE-FORTY-EIGHT.MIT.EDU [18.242.6.93]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 00:53:35 -0400 Message-ID: <20051024005335.w4u6tpzivq74cs0w@webmail.mit.edu> Date: Mon, 24 Oct 2005 00:53:35 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Reading Assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2303 Content-Length: 116 X-Keywords: NonJunk NotJunk I found the marriage problem extremely interesting. I hope we spend some more time over it in class. -Rahul Shroff From alisonc@MIT.EDU Mon Oct 24 01:00:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O50Fd0007230 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:00:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O50Enj004582 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:00:14 -0400 (EDT) Received: from [18.238.6.107] (EASTCAMPUS-EIGHT-SEVENTY-FOUR.MIT.EDU [18.238.6.107]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O5078p012853 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:00:08 -0400 (EDT) Message-ID: <435C6A59.4060904@mit.edu> Date: Mon, 24 Oct 2005 01:00:09 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Comments on course notes #7 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2304 Content-Length: 496 X-Keywords: NonJunk NotJunk I found the Stable Marriage problem to be the most interesting part of this week's set of notes -- as usual, the very real application and the math behind it are very neat. I'm curious about what other sorts of pairing algorithms might have insightful graphs or proofs behind them: the ones that MIT uses for housing, the readjustment lottery, the HASS lottery, the phys ed lottery, for instance? They're different and simpler in that preferences don't go both ways, but still interesting. From clintonb@MIT.EDU Mon Oct 24 01:07:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O57Yd0008464 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:07:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O57Xnj010000 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:07:33 -0400 (EDT) Received: from ccbibmx30 (BLACKBURN.MIT.EDU [18.241.3.107]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O57U74015101 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:07:31 -0400 (EDT) Message-Id: <200510240507.j9O57U74015101@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.lcs.mit.edu> Subject: Jelani: TP 7 Comments Date: Mon, 24 Oct 2005 01:06:14 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0075_01C5D837.21C867F0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXYWKflowW9hhfEQWGWgoP0m0vgOA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.668 X-Spam-Level: * (1.668) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2305 Content-Length: 4749 X-Keywords: NonJunk $Forwarded NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0075_01C5D837.21C867F0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I'm totally digging these problems with real-world relations/examples (ie. RSA, stable marriage). I learn best through these types of examples. I get the methods; although, I'm sure I need to go back over the different theorems and proofs for the stable marriage problem. Also, what happened to the pre-filled email links? I'm referring to the mailto links with the email address and subject line. I kind of liked not having to think of a subject line (the hardest part of writing emails for me). Good morning! --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0075_01C5D837.21C867F0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I’m totally digging these problems with = real-world relations/examples (ie. RSA, stable marriage). I learn best through = these types of examples. I get the methods; although, I’m sure I need to go = back over the different theorems and proofs for the stable marriage = problem.

 

Also, what happened to the pre-filled email links? = I’m referring to the mailto links with the email address and subject line. I = kind of liked not having to think of a subject line (the hardest part of = writing emails for me).

 

Good morning!

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_0075_01C5D837.21C867F0-- From hkhall@MIT.EDU Mon Oct 24 01:22:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O5Mtd0010863 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:22:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O5Mqnj025635; Mon, 24 Oct 2005 01:22:52 -0400 (EDT) Received: from [18.96.6.125] (SIMMONS-THREE-EIGHTY.MIT.EDU [18.96.6.125]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O5Mj2v023342 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Mon, 24 Oct 2005 01:22:46 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: [David] Reading Comments on LN7 Date: Mon, 24 Oct 2005 01:24:47 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2306 Content-Length: 362 X-Keywords: NonJunk NotJunk David- The coolest thing I learned is that some optimal matching occurs and that it is provable that it occurs. I tried it myself and I couldn't come up with a set that I couldn't match given the rules laid out. Is there an algorithm for matching more optimal than the the mating algorithm that can give one group their optimal spouse? Thanks. -Harrison From sergiob@MIT.EDU Mon Oct 24 01:58:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O5w3d0016199 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 01:58:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O5w1LQ013746 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 01:58:01 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O5vw3K026544 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 01:57:59 -0400 (EDT) Message-Id: <5.2.1.1.2.20051023235417.00c4e850@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Sun, 23 Oct 2005 23:57:57 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2307 Content-Length: 283 X-Keywords: NonJunk NotJunk Now, I can understand why Floyd was so excited about the invariants theorem. I would like to know more about the types of algortihms there are, and if there are any systematic ways of establishing derived variables to prove their partial correctness. Sergio Bacallado (Group 1) From scot@MIT.EDU Mon Oct 24 02:05:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O650d0016739 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 02:05:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O64xLQ017228 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 02:04:59 -0400 (EDT) Received: from [18.227.1.175] ([18.227.1.175]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O64qF0027517 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 02:04:56 -0400 (EDT) Message-ID: <435C7949.5000708@mit.edu> Date: Mon, 24 Oct 2005 02:03:53 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: comments: Jelani Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2308 Content-Length: 342 X-Keywords: NonJunk NotJunk Hi, I found the notes to be clear, interesting, and good this week. The only part I found confusing was the last, during the proof of the pessimal choice for the girls. I don't see how it can stand as a valid proof by contradition. How do we know G prefers B to her mate B' by assumption? Maybe this will be cleared up in lecture. Scot From bakster@MIT.EDU Mon Oct 24 02:14:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O6Edd0017246 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:14:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O6EcLQ022451 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:14:38 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O6EWLv028341 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:14:32 -0400 (EDT) Message-ID: <435C7BC5.10201@mit.edu> Date: Mon, 24 Oct 2005 02:14:29 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] week 7 comments Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2309 Content-Length: 193 X-Keywords: NonJunk NotJunk I found this week's reading to be particularly homophobic and sexist: "... for the boy-girl marriage problem, a stable set of marriages does always exist." and Theorem 5.8 Alexander Bakst From arup@MIT.EDU Mon Oct 24 02:37:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O6bBd0019985 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:37:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O6bALQ004065 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:37:10 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-THREE-EIGHTEEN.MIT.EDU [18.247.6.63]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O6ax6T029898 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:37:04 -0400 (EDT) Message-Id: <6.2.3.4.2.20051024023520.0352d078@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Mon, 24 Oct 2005 02:36:19 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2310 Content-Length: 172 X-Keywords: NonJunk NotJunk Section 5.6, pp 14-15, theorem 5.8 & proofs: I would like to have this discussed more fully in lecture, since I didn't really follow the proofs in the reading. |Arup| From a_lopez@MIT.EDU Mon Oct 24 03:07:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O77Wd0022457 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 03:07:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O77ULQ018850 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 03:07:30 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O77Twd001477 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 03:07:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9O77TvL025340; Mon, 24 Oct 2005 03:07:29 -0400 Received: from MDMURRAY.MIT.EDU (MDMURRAY.MIT.EDU [18.234.0.116]) (User authenticated as a_lopez@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 03:07:29 -0400 Message-ID: <20051024030729.kd5dlqyyd0kggsw4@webmail.mit.edu> Date: Mon, 24 Oct 2005 03:07:29 -0400 From: a_lopez@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [Jelani] week 7 readings MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2311 Content-Length: 522 X-Keywords: NonJunk NotJunk Reading the notes on the Marriage Algorithm, got me thinking... is this the algorithm currently used to match people to their preferences? For example, is this the algorithm that MIT uses in the freshman housing lottery? It seems like a powerful algorithm, but the fact that it favors one group over another is a bit troubling (is optimal for one group and pessimal for the other). Are there other marriage algorithms that balance the marriage choices better, so that one group is not favored over the other? Adriana From yaser@MIT.EDU Mon Oct 24 03:15:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7FTd0025119 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:15:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7FSLQ022604 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:15:28 -0400 (EDT) Received: from Charizard (NEW-ONE-THIRTY-SEVEN.MIT.EDU [18.241.5.137]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7FKJR001860 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:15:21 -0400 (EDT) Message-Id: <200510240715.j9O7FKJR001860@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: week7 reading response Date: Mon, 24 Oct 2005 03:15:16 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_002E_01C5D849.2826F0A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXYaq6TnPEQ9BHjR8CIhZ1fbTDILg== X-Spam-Score: 2.891 X-Spam-Level: ** (2.891) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2312 Content-Length: 5320 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_002E_01C5D849.2826F0A0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Hi David, Out of curiosity: does the 2-domain marriage algorithm readily scale to 3-domain? I.e., say you are matching manufactories, retailers and distributors, and each has a certain preference of the other two. does the algorithm still hold? Thanks! _Yaser ------=_NextPart_000_002E_01C5D849.2826F0A0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

Out of curiosity: does the = 2-domain marriage algorithm readily scale to 3-domain? I.e., say you are matching = manufactories, retailers and distributors, and each has a certain preference of the = other two… does the algorithm still hold?

 

Thanks!

_Yaser

------=_NextPart_000_002E_01C5D849.2826F0A0-- From mdmurray@MIT.EDU Mon Oct 24 03:24:03 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7O3d0027814 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:24:03 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7O3VH004155 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:24:03 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O7NtMu010244 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:23:55 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Michael Murray Subject: [Hanson] E-mail question Date: Mon, 24 Oct 2005 03:23:31 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2313 Content-Length: 485 X-Keywords: NonJunk NotJunk As a general comment I like this topic of state machines and wish it were more than 2 lectures long. Anyway, as regards to the reading, I like the format of the stable marriage problem and the example was very easy to follow and understand. However, I would like to have a more formal and generic description of the algorithm than the one presented in the notes. Maybe, however, the reason there is no such description is that it is not required for this class? Michael From antonk@MIT.EDU Mon Oct 24 03:24:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7OAd0027824 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:24:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7O9LQ026503 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:24:09 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7O6Sr002270 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:24:07 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] need clarification Date: Mon, 24 Oct 2005 03:24:04 -0400 Organization: Massachusetts Institute of Technology Message-ID: <003201c5d86b$ea70d2d0$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0033_01C5D84A.6360B970" X-Mailer: Microsoft Office Outlook 11 Thread-Index: AcXYa+k6R6FJwhRIQ2aAtb37KIfIjQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.214 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2314 Content-Length: 3227 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0033_01C5D84A.6360B970 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit On the first line of partial correctness of GCD I don't believe that I fully understand the definition of the proof statement. Something is not clear about the general concept of terminology when you say that if the process terminates then (the GCD(a,b)) is equal to x. what is the y then? Is it just a helper variable that we use along the way during the run of the machine? Thank you, Anton. ------=_NextPart_000_0033_01C5D84A.6360B970 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

On the first line of partial correctness of GCD I = don’t believe that I fully understand the definition of the proof = statement.

Something is not clear about the general concept of terminology when you say that if the process terminates then (the = GCD(a,b)) is equal to x. what is the y then? Is it just a helper variable that we use = along the way during the run of the machine?

 

Thank you,

 

Anton.

 

------=_NextPart_000_0033_01C5D84A.6360B970-- From vbrobbey@MIT.EDU Mon Oct 24 03:53:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7rZd0031676 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:53:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7rYLQ010285 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:53:34 -0400 (EDT) Received: from TheSlate (NEW-FOUR-TEN.MIT.EDU [18.241.6.155]) (authenticated bits=0) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7rRKe003537 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:53:28 -0400 (EDT) Message-ID: <004b01c5d870$0a4bbe40$9b06f112@TheSlate> From: "Valery Kwasi Brobbey" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 7 Comments Date: Mon, 24 Oct 2005 03:53:35 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0048_01C5D84E.820E9690" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.214 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2315 Content-Length: 2385 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0048_01C5D84E.820E9690 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable This week's material is very similar to material we've already seen. = State Machine is basically a graph but then edges exist between vertices = based on certain conditions/rules. When it comes to derived variables, = the concept of weak/strict is like the concept of weak/strict partial = orders. Halls marriage theorem is similar to the Stable marriage problem = except that the stable marriage problem takes rank into consideration. I = realized that this was a weakness in Hall's marriage theorem and = expected there to be some improved version that looks at rank/order. I = hope there's another marriage theorem that takes into account the fact = that opinion and interests change. For instance when the housing office = assigns dorms to students, it has to do a readjustment lottery to = account for change in preferences. ------=_NextPart_000_0048_01C5D84E.820E9690 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
This week's material is very similar to = material=20 we've already seen. State Machine is basically a graph but then edges = exist=20 between vertices based on certain conditions/rules. When it comes to = derived=20 variables, the concept of weak/strict is like the concept of weak/strict = partial=20 orders. Halls marriage theorem is similar to the Stable marriage problem = except=20 that the stable marriage problem takes rank into consideration. I = realized that=20 this was a weakness in Hall's marriage theorem and expected there to be = some=20 improved version that looks at rank/order. I hope there's another = marriage=20 theorem that takes into account the fact that opinion and interests=20 change. For instance when the housing office assigns dorms to = students, it=20 has to do a readjustment lottery to account for change in=20 preferences.
------=_NextPart_000_0048_01C5D84E.820E9690-- From kkdb@MIT.EDU Mon Oct 24 03:59:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7x1d0031862 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:59:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7x0LQ012773 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:59:00 -0400 (EDT) Received: from KKDB (MACLAURIN-FOURTEEN-O-FOUR.MIT.EDU [18.80.6.125]) (authenticated bits=0) (User authenticated as kkdb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7wwJb003758 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:58:58 -0400 (EDT) Message-ID: <001001c5d870$cbe00480$7d065012@KKDB> From: "Kaustuv DeBiswas" To: <6042-probs@theory.lcs.mit.edu> References: Subject: [David] Week 7 comment Date: Mon, 24 Oct 2005 03:59:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000D_01C5D84F.446C9570" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.52 X-Spam-Level: * (1.52) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2316 Content-Length: 1870 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_000D_01C5D84F.446C9570 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Hi David, I was confused by section 21./pg3 (the waterjug proof) where is it says = "proof divides into cases, according to which transition rule is used. = For example, suppose the transition followed from the "fill the little = jug" rule. This means (b, l) -> (b, 3). But P((b, l) impiles that b is = an integer multiple of 3, and of course 3 is an integer multiple of 3, = so P still holds for the new state (b, 3)." how does P(b,l) imply that b | 3? thanks, Kaustuv. ------=_NextPart_000_000D_01C5D84F.446C9570 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

Hi David,

I was confused = by section 21./pg3=20 (the waterjug proof) where is it says "proof divides into cases, = according=20 to which transition rule is used. For example, suppose the transition = followed=20 from the =93fill the little jug=94 rule. This means (b, l) -> = (b, 3). But=20 P((b, l) impiles that b is an integer multiple of 3, = and of=20 course 3 is an integer multiple of 3, so P still holds for the new state = (b,=20 3)."

how does P(b,l) imply that b = |=20 3?

thanks,=20 Kaustuv.

------=_NextPart_000_000D_01C5D84F.446C9570-- From jacques@MIT.EDU Mon Oct 24 04:02:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O82nd0032550 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:02:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O82mLQ014779 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:02:48 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O82k8L003930 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:02:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9O82kDx008910; Mon, 24 Oct 2005 04:02:46 -0400 Received: from EASTCAMPUS-SEVEN-O-SEVEN.MIT.EDU (EASTCAMPUS-SEVEN-O-SEVEN.MIT.EDU [18.238.5.196]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 04:02:46 -0400 Message-ID: <20051024040246.sx741hh16p0kkw8k@webmail.mit.edu> Date: Mon, 24 Oct 2005 04:02:46 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 7 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2317 Content-Length: 260 X-Keywords: NonJunk NotJunk On page 14, it surprised me that the Mating Algorithm actually did better for the boys than the girls. I'm glad those last two pages are there, because I probably wouldn't have given it much thought and just continued to assume the girls were the better off. From fluff@MIT.EDU Mon Oct 24 04:26:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O8QRd0003025 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:26:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O8QPLQ025320 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:26:25 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O8QJM1004891 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:26:19 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 7 reading Date: Mon, 24 Oct 2005 04:24:03 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2318 Content-Length: 423 X-Keywords: NonJunk NotJunk I thought this week's reading was way friendlier than last week's. It was nice that the state machine stuff was introduced in lecture first (the counter, jug problem, and diagonal robot), which made the reading a lot easier to understand. I also thought the "mating algorithm" was very interesting, and the fact that a stable set of marriages always exists. And also how all the girls get screwed over. ~Crystal From tonyng@MIT.EDU Mon Oct 24 04:39:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O8dId0004729 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:39:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O8dGLQ001290 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:39:17 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O8dAmk005387 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:39:11 -0400 (EDT) Message-Id: <5.2.1.1.2.20051024043123.0211eb40@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 24 Oct 2005 04:39:21 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments: Week 7 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2319 Content-Length: 482 X-Keywords: NonJunk NotJunk I found the fact that the Marriage Algorithm produces optimal "spouses" pretty interesting (Section 5.6 page 14-15). I can already see many applications of this, such as a lottery to give people their preferred choices of classes, dorms, etc (people can enter their choices as ratings, and classes can prefer seniors over juniors, etc). It can maximize the happiness of people with their results because it is unlikely people will get paired off with a worst choice. - Tony From medrano@MIT.EDU Mon Oct 24 04:55:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O8tWd0006408 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:55:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O8tNLQ008352 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:55:23 -0400 (EDT) Received: from scrubbing-bubbles.mit.edu (SCRUBBING-BUBBLES.MIT.EDU [18.7.16.68]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O8tLIY005942 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:55:22 -0400 (EDT) Received: (from medrano@localhost) by scrubbing-bubbles.mit.edu (8.12.9) id j9O8tLnf016530; Mon, 24 Oct 2005 04:55:21 -0400 (EDT) Date: Mon, 24 Oct 2005 04:55:21 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] weekly reading question Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2320 Content-Length: 267 X-Keywords: NonJunk NotJunk I was wondering if the mating ritual had any other applications than the one used in the reading? Even though it's funny to have theorems about people being happy who they are matched with, what are the actual computer applications of these theorems. Jesus Medrano From zev@MIT.EDU Mon Oct 24 05:38:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O9csd0011971 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 05:38:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O9crqJ026300 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 05:38:53 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O9cme8007565 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 05:38:51 -0400 (EDT) Message-ID: <435CABA8.5040302@mit.edu> Date: Mon, 24 Oct 2005 05:38:48 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 7 comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2321 Content-Length: 462 X-Keywords: NonJunk NotJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 Overall the lecture notes were quite straight forward. I would have liked an example of the mating algorithm, but the tutor problems took care of that. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDXKuolO3j8HLL0+4RAl5TAJ4+9I39fhy684FWZDGkFo4/DGiFoQCgmc/8 ZWAJDJwz0qcpaNXO3IZzcOk= =l0uH -----END PGP SIGNATURE----- From ozcan@MIT.EDU Mon Oct 24 08:55:37 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OCtbd0014035 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 08:55:37 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OCtbSi013766 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 08:55:37 -0400 (EDT) Received: from ebrum.mit.edu (NEW-TWO-O-FIVE.MIT.EDU [18.241.5.205]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OCtYol000938 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 08:55:35 -0400 (EDT) Message-Id: <5.2.1.1.2.20051024084808.026a4ca0@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 24 Oct 2005 08:55:32 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2322 Content-Length: 453 X-Keywords: NonJunk NotJunk I am pretty amazed with the fact that there is an entire section on marriage problem, which in fact describes a matching algorithm. I also had a class in which we tried to maximize the probability of finding the "best woman" by setting a number of dates a man should go out, and the probability we ended up with was a very high one, 1/e. These kind of approaches to math is quite fun and intriguing. Quote: Entire Section 5, The marriage problem From mukkala@MIT.EDU Mon Oct 24 09:11:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODBBd0018635 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 09:11:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODBAmj025895 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 09:11:10 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-FIFTY-TWO.MIT.EDU [18.235.1.152]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODB0YP005056 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 09:11:04 -0400 (EDT) Message-Id: <6.2.1.2.2.20051024081344.04304e20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Mon, 24 Oct 2005 09:11:01 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 7 Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2323 Content-Length: 311 X-Keywords: NonJunk NotJunk In section 3.2.1, entitled "Partial Correctness of GCD," I don't understand how partial correctness is proven. I understand that the gcd(x,y) = gcd(x1,0) = x1 = d. However, why is it that this proves only partial correctness, instead of the entire correctness of the claim? Thanks, Praveen Pamidimukkala From nancyk@MIT.EDU Mon Oct 24 09:29:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODTsd0020958 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:29:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODTpmj014254; Mon, 24 Oct 2005 09:29:51 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODTnuj012096; Mon, 24 Oct 2005 09:29:50 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9ODTnHR011619; Mon, 24 Oct 2005 09:29:49 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 09:29:49 -0400 Message-ID: <20051024092949.b7oj7majk68kgkk4@webmail.mit.edu> Date: Mon, 24 Oct 2005 09:29:49 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] week 7 reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2324 Content-Length: 330 X-Keywords: NonJunk NotJunk Hi, I thought the stable marriage problem (detailed on pages 9-12) was interesting. I don't have any specific questions about it, but it is interesting that without the gender constraint, we're not guaranteed to find a stable matching. I enjoyed reading about the concepts in terms of the marriage scenario. Thanks, Nancy Keuss From nedzel@MIT.EDU Mon Oct 24 09:43:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODhod0021834 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 09:43:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODhmmj028760 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 09:43:48 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-TWO-TWENTY-FOUR.MIT.EDU [18.96.5.224]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODhfVf017723 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 09:43:42 -0400 (EDT) Message-Id: <200510241343.j9ODhfVf017723@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 7 Comments Date: Mon, 24 Oct 2005 09:43:39 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.592 X-Spam-Level: ** (2.592) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2325 Content-Length: 111 X-Keywords: NonJunk NotJunk How are the state machines described in the reading similar or different to Markov state machines? - David From dnreshef@MIT.EDU Mon Oct 24 09:44:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODiCd0021864 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:44:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODiBmj029088 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:44:11 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODi9ah017903 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:44:09 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9ODi9Uk014692; Mon, 24 Oct 2005 09:44:09 -0400 Received: from MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU (MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU [18.239.7.33]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 09:44:09 -0400 Message-ID: <20051024094409.cm7xdsqpbu968s4k@webmail.mit.edu> Date: Mon, 24 Oct 2005 09:44:09 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] week 7 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2326 Content-Length: 183 X-Keywords: NonJunk In this week's reading on state machines, I found the topics of invariants on page 2 and derived variables on page 8 the hardest. I would appreciate gonig over these in class. -Dave From xiaoranz@MIT.EDU Mon Oct 24 09:58:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODwld0025295 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 09:58:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODwkmj014566 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 09:58:46 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODwiiO024062 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 09:58:45 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9ODwgxE023527; Mon, 24 Oct 2005 09:58:42 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 09:58:42 -0400 Message-ID: <20051024095842.um0z31procasg4g0@webmail.mit.edu> Date: Mon, 24 Oct 2005 09:58:42 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Weekly Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2327 Content-Length: 638 X-Keywords: NonJunk NotJunk I think 3.3 about Extended Euclidean Algorithm is interesting because it ties the state machine and mathematical concept to computer programming. I'm' a little confused as to how to determine the invariant for a state machine, if there's a method or a trick to determine what values are invariant or not. Sharon ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From hmzhou@blackbird.csail.mit.edu Mon Oct 24 10:12:21 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OECLd0029370; Mon, 24 Oct 2005 10:12:21 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OECLbv022762; Mon, 24 Oct 2005 10:12:21 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OECL93022759; Mon, 24 Oct 2005 10:12:21 -0400 Date: Mon, 24 Oct 2005 10:12:21 -0400 (EDT) From: Hanson Zhou To: "Xiaoran (Sharon) Zhang" cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Weekly Comments In-Reply-To: <20051024095842.um0z31procasg4g0@webmail.mit.edu> Message-ID: References: <20051024095842.um0z31procasg4g0@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2328 Content-Length: 1064 X-Keywords: NonJunk NotJunk Careful...the invariant need not be tied to any specific variable in the state machine, as the reading often demonstrates. The invariant is often a function of those variables or some other derived value. If you are asking how one comes up with an invariant, there is no good answer. That's an art, though with enough problems and examples, you get a feel for what tends to work. -Hanson > I think 3.3 about Extended Euclidean Algorithm is interesting because it ties > the state machine and mathematical concept to computer programming. I'm' a > little confused as to how to determine the invariant for a state machine, if > there's a method or a trick to determine what values are invariant or not. > > Sharon > > ************************************************** > Xiaoran (Sharon) Zhang > Class of 2008 > Department of Biology & > Department of Electrical Engineering and Computer Science > Massachusetts Institute of Technology > 320 Memorial Drive, > Cambridge, MA 02139 > E-mail: xiaoranz@mit.edu > ************************************************** > From kromer@MIT.EDU Mon Oct 24 10:16:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEGLd0029706 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:16:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEGKmj004492 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:16:20 -0400 (EDT) Received: from m12-182-9.mit.edu (M12-182-9.MIT.EDU [18.19.0.40]) (authenticated bits=56) (User authenticated as kromer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEGHD4002350 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:16:18 -0400 (EDT) Received: (from kromer@localhost) by m12-182-9.mit.edu (8.12.9) id j9OEGH3Q030056; Mon, 24 Oct 2005 10:16:17 -0400 Subject: [David] Email comments for reading From: Katherine Romer To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 24 Oct 2005 10:16:17 -0400 Message-Id: <1130163377.30018.4.camel@m12-182-9.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2329 Content-Length: 502 X-Keywords: NonJunk NotJunk p. 7: "We claim that invariant properties that can be used to prove partial correctness are: gcd(X,Y)=gcd(a,b) Sa+Tb=Y, and Ua+Vb=X" Given these invariant properties, I can verify them and see how they can be used to show the partial correctness of the extended Euclidean Algorithm, but I don't think I could come up with them on my own--it would be nice to see an example of the thinking process used to come up with the invarant properties to use in a proof of partial correctness. Katherine Romer From cbossard@MIT.EDU Mon Oct 24 10:23:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OENMd0031536 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:23:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OENLmj012064 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:23:21 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OENE6V005297 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:23:15 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9OENEu6026006; Mon, 24 Oct 2005 10:23:14 -0400 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:23:14 -0400 Message-ID: <20051024102314.baisrkhigpw0kw0s@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:23:14 -0400 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 7 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2330 Content-Length: 117 X-Keywords: NonJunk NotJunk Section 5.2 I get the general jist of the mating algorithm and I found it to be a very interesting example. Cynthia From kushan@MIT.EDU Mon Oct 24 10:37:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEbXd0003179 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:37:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEbVmj028281 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:37:32 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEbP5r011635 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:37:25 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9OEbP1d006943; Mon, 24 Oct 2005 10:37:25 -0400 Received: from RLE-13-099.MIT.EDU (RLE-13-099.MIT.EDU [18.62.13.99]) (User authenticated as kushan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:37:25 -0400 Message-ID: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:37:25 -0400 From: Kushan K Surana To: 6042-probs@theory.lcs.MIT.EDU Subject: TA - Hanson: 6.042 Week 7 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2331 Content-Length: 165 X-Status: A X-Keywords: NonJunk NotJunk - On page 6, the proof of the partial correctness of GCD is unclear. Why is the predicate P(x,y) true for every state? The GCD is found only at the very last state. From mitras@theory.csail.mit.edu Mon Oct 24 10:41:49 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEfnd0003428 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:41:49 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEfmls015818 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:41:48 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEfkZf007028; Mon, 24 Oct 2005 10:41:46 -0400 (EDT) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OEfiMu001318; Mon, 24 Oct 2005 10:41:44 -0400 (EDT) Message-ID: <435CF2A7.3090206@theory.csail.mit.edu> Date: Mon, 24 Oct 2005 10:41:43 -0400 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala , 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 7 Reading Comments References: <6.2.1.2.2.20051024081344.04304e20@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051024081344.04304e20@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2332 Content-Length: 662 X-Keywords: NotJunk Its partial correctness because it proves that if the algorithm terminates, then it gives the correct answer; however, it does not prove that the algorithm indeed terminates. This termination part is done in 3.2.2. Together these two theorems imply the (complete) correctness of the algorithm. -sayan Praveen Pamidimukkala wrote: > In section 3.2.1, entitled "Partial Correctness of GCD," I don't > understand how partial correctness is proven. I understand that the > gcd(x,y) = gcd(x1,0) = x1 = d. However, why is it that this proves > only partial correctness, instead of the entire correctness of the claim? > > Thanks, > Praveen Pamidimukkala > From shreyes19@gmail.com Mon Oct 24 10:43:41 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEhfd0003573 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:43:41 -0400 Received: by xproxy.gmail.com with SMTP id r21so192641wxc for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 07:43:40 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=j0rMZXeBFhs6Tzr72BpQ6JsTujSKfj4FcxX0z1C+R2YsGL5ql34rhGeW57vAEMMxCzcZhk4+uESkK6gCa91V0XcvciP1MCSRrrTo+WNIHlmlpckgKEMD950wV9opP7uu0Sp9K6ApJ21RDeWDeKozBq553BKHt65k25aVoEdGWmo= Received: by 10.70.73.1 with SMTP id v1mr3647223wxa; Mon, 24 Oct 2005 07:43:40 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Mon, 24 Oct 2005 07:43:40 -0700 (PDT) Message-ID: <3c33fe380510240743x51a711ei2a8ef63b3637436f@mail.gmail.com> Date: Mon, 24 Oct 2005 10:43:40 -0400 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 7 Reading comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_22743_4688988.1130165020768" Status: RO X-UID: 2333 Content-Length: 1558 X-Keywords: NonJunk NotJunk ------=_Part_22743_4688988.1130165020768 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, After reading the lecture notes for week 7, one thing I'd like to have discussed more in class is more examples of how exactly to use derived variables. I follow the definitions fine on the end of page 8 and beginning of page 9, but it's not clear to me how knowing if something is decreasing can help in an invariants problem. The only application of it that's clear to me is proving termination, but that is clear by the Well Ordering Principle. Are there other uses of knowing when state machines are weakly o= r strictly decreasing? Thanks, Shreyes Seshasai Group 7 ------=_Part_22743_4688988.1130165020768 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

After reading the lecture notes for week 7, one thing I'd like to have discussed more in class is more examples of how exactly to use derived variables.  I follow the definitions fine on the end of page 8 and beginning of page 9, but it's not clear to me how knowing if something is decreasing can help in an invariants problem.  The only application of it that's clear to me is proving termination, but that is clear by the Well Ordering Principle.  Are there other uses of knowing when state machines are weakly or strictly decreasing?

Thanks,
Shreyes Seshasai
Group 7
------=_Part_22743_4688988.1130165020768-- From crowell@MIT.EDU Mon Oct 24 10:46:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEkMd0004769 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:46:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEkLmj009400 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:46:21 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEkAF0015475 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:46:10 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9OEkA2O021354; Mon, 24 Oct 2005 10:46:10 -0400 Received: from SHYGUY.MIT.EDU (SHYGUY.MIT.EDU [18.238.4.89]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:46:10 -0400 Message-ID: <20051024104610.hvux4wbsduv48g44@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:46:10 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2334 Content-Length: 199 X-Keywords: NonJunk NotJunk I would like to have the State Machine model of the Marraige Problem explained again in class (5.3 page 12). This section of the notes was a little confusing to me when I read it. -Rob Crowell From lana@MIT.EDU Mon Oct 24 10:49:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEnFd0004964 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:49:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEnEmj012282 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:49:14 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEn5Ug016715 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:49:05 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9OEn55E021654; Mon, 24 Oct 2005 10:49:05 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:49:05 -0400 Message-ID: <20051024104905.em36hoohqzj4cgc8@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:49:05 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP7 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2335 Content-Length: 262 X-Keywords: NonJunk NotJunk The part of the reading I found particularly interesting was the marriage algorithm. I am curious if it necessarily for avery single person involved to rank all possible mates or if it can still operate in the minimal conditions of Hall's marriage theorem. Lana From zacharyozer@gmail.com Mon Oct 24 10:55:20 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.201]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEtKd0006849 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:55:20 -0400 Received: by xproxy.gmail.com with SMTP id t12so847075wxc for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 07:55:19 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=jJZl0Qn7GgebSFcepb5dFwtGSSJKz194K8mD3+C3qqAyKsMe1YiwUbXI4Zzi7AcELh535GFUs4YW1sku5pqX4HqDuEHba4R/v6+zA4CeeCBptoWfdOK8r6V0A+4BCA5cLtM0ok0sTuznuJle7CbEh1fYDBxgQ5WXf6sKLFHih3I= Received: by 10.70.7.8 with SMTP id 8mr2127843wxg; Mon, 24 Oct 2005 07:55:19 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Mon, 24 Oct 2005 07:55:19 -0700 (PDT) Message-ID: <4f2613e50510240755p13289032q8cfa2b48b4fb04af@mail.gmail.com> Date: Mon, 24 Oct 2005 10:55:19 -0400 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: 6042 comments: Request for notes and lecture MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9OEtKd0006849 Status: RO X-UID: 2336 Content-Length: 432 X-Keywords: NonJunk NotJunk First, I'm a bit confused as to how invarients are being used in the reading in general; it seems as though at one moment there's a theorem, but that they become some sort of equivalence statement. What determines the equivalence relation or that it's true? More importantly however, get rid of Die Hard, it's getting old at this point. Also, the bio type info on people is always awesome (vis Turing, Flyod, Meyer, etc). -zozer From veracarr@MIT.EDU Mon Oct 24 10:56:33 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEuXd0006934 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:56:33 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEuTSi020573 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:56:29 -0400 (EDT) Received: from [18.245.7.47] (BAKER-FIVE-FIFTY-EIGHT.MIT.EDU [18.245.7.47]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OEuMMu002578 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:56:22 -0400 (EDT) Message-ID: <435D1275.1050405@mit.edu> Date: Mon, 24 Oct 2005 10:57:25 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] State Machines Reading Assignment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2337 Content-Length: 140 X-Keywords: NonJunk NotJunk I found it difficult to understand graph algorithm works for state machines. The first problem in 7.2 with invariants was a bit confusing. From moscicki@MIT.EDU Mon Oct 24 11:30:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OFUid0011413 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 11:30:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OFUgfe025402 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 11:30:43 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OFUfnY005792 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 11:30:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9OFUe0l012954; Mon, 24 Oct 2005 11:30:40 -0400 Received: from NEW-THIRTY-TWO.MIT.EDU (NEW-THIRTY-TWO.MIT.EDU [18.241.5.32]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 11:30:40 -0400 Message-ID: <20051024113040.labrgilwhc04sg44@webmail.mit.edu> Date: Mon, 24 Oct 2005 11:30:40 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 7 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2338 Content-Length: 1195 X-Keywords: NonJunk NotJunk State Machines: The explanation of what they are is simple, but it will require a little practice to get the hang of actually coming up with invariants for them. The ususal approach is when we know what we're trying to prove, this time we must come up with what we're trying to prove and then prove it. This is the hardest part. Reachability and all that sort of thing is fine once I have some invariant to think about. Partial Correctness and Termination make sense; its just making sure your processor does the right thing and that it can't get stuck in an infinite loop. Derived Variables seem just to be whatever variable your state machine is using; all the registers in the processor. I'm not entirely sure what this f states -> R means. The Mating Algorighm: at first I did not want to believe that one set gets matched with their pessimal mate, but thinking about it, it makes sense that if one set is matched with their optimal mates and those mates prefer them better than the rest of the bunch (so the losing set gets their mate in order of least-pessimal), it will at least form a stable system. Other than that, it reminds me of non-linear system stability from 18.03. From meyer@imap.theory.csail.mit.edu Mon Oct 24 10:33:06 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEX6d0001551 for <6042-staff@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:33:06 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEX4Si018787; Mon, 24 Oct 2005 10:33:04 -0400 (EDT) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OEX1Mu000612; Mon, 24 Oct 2005 10:33:02 -0400 (EDT) Message-ID: <435CF09C.3060509@theory.csail.mit.edu> Date: Mon, 24 Oct 2005 10:33:00 -0400 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "David A. Nedzel" CC: 6042-staff@theory.csail.mit.edu Subject: Re: [Sayan] Week 7 Comments References: <200510241343.j9ODhfVf017723@outgoing.mit.edu> In-Reply-To: <200510241343.j9ODhfVf017723@outgoing.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 817 X-UID: 2339 X-Keywords: NotJunk They are quite similar actually. As you have noticed their descriptions---states, starting states, and transitions---are same as Markov chains, modulo the probabilities associated with the transitions in the case of MCs. In fact, all the concepts that we have seen in class for state machines (finite automata) have their probabilistic analogues in the MC world. For example: Invariant property of the states ----- Invariant distribution over states Reachability of a state ----- Hitting time or expected time to reach a state Traditionally, MCs have been used for different type of analysis than what finite automata have been used for. -Sayan David A. Nedzel wrote: >How are the state machines described in the reading similar or different to >Markov state machines? > > > >- David > > > From meyer@imap.theory.csail.mit.edu Mon Oct 24 12:14:27 2005 Message-ID: <435D0866.8010009@csail.mit.edu> Date: Mon, 24 Oct 2005 12:14:30 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: David Shin CC: clintonb@MIT.EDU Subject: [Fwd: Jelani: TP 7 Comments] Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Content-Length: 830 Status: RO X-UID: 2340 X-Keywords: Let's keep these links updated. regards, A. -------- Original Message -------- Subject: Jelani: TP 7 Comments Date: Mon, 24 Oct 2005 01:06:14 -0400 From: Clinton Blackburn To: <6042-probs@theory.lcs.mit.edu> Im totally digging these problems with real-world relations/examples (ie. RSA, stable marriage). I learn best through these types of examples. I get the methods; although, Im sure I need to go back over the different theorems and proofs for the stable marriage problem. Also, what happened to the pre-filled email links? Im referring to the mailto links with the email address and subject line. I kind of liked not having to think of a subject line (the hardest part of writing emails for me). Good morning! --- Clinton Blackburn DLP Have you seen it? From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:15:03 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGF2d0023070; Mon, 24 Oct 2005 12:15:02 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGF2bL023870; Mon, 24 Oct 2005 12:15:02 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGF2QX023867; Mon, 24 Oct 2005 12:15:02 -0400 Date: Mon, 24 Oct 2005 12:15:02 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 7 Readings In-Reply-To: <6.2.3.4.2.20051023232523.01e7cef8@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051023232523.01e7cef8@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2341 Content-Length: 600 X-Keywords: NonJunk NotJunk For example, if you have a derived variable X of the algorithm at each step, with termination corresponding to a value of X=0, and each iteration lowers the value of X by at least 1, then you know that the algorithm can take no longer than the initial value of X. -Hanson On Sun, 23 Oct 2005, Kevin Wang wrote: > Hi Hanson, > > I was confused about the concept of a derived variable (Pg. 8, > section 4), especially how derived functions help us to find the > maximum run time of an algorithm and how it helps prove termination. > An example of this would be very helpful. > > Thanks, > Kevin > > From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:25:41 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGPfd0026281; Mon, 24 Oct 2005 12:25:41 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGPfiE023968; Mon, 24 Oct 2005 12:25:41 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGPfTH023965; Mon, 24 Oct 2005 12:25:41 -0400 Date: Mon, 24 Oct 2005 12:25:41 -0400 (EDT) From: Hanson Zhou To: Kushan K Surana cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: TA - Hanson: 6.042 Week 7 comments In-Reply-To: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> Message-ID: References: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2342 Content-Length: 322 X-Keywords: NonJunk NotJunk Review the number theory notes: gcd(x,y) = gcd(y, remainder(x,y)). This gives the invariant. -Hanson On Mon, 24 Oct 2005, Kushan K Surana wrote: > - On page 6, the proof of the partial correctness of GCD is unclear. Why is the > predicate P(x,y) true for every state? The GCD is found only at the very last > state. > From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:26:29 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGQTd0026321; Mon, 24 Oct 2005 12:26:29 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGQTP5023981; Mon, 24 Oct 2005 12:26:29 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGQTSm023978; Mon, 24 Oct 2005 12:26:29 -0400 Date: Mon, 24 Oct 2005 12:26:29 -0400 (EDT) From: Hanson Zhou To: Zachary Adam Ozer cc: 6042-probs@theory.lcs.mit.edu Subject: Re: 6042 comments: Request for notes and lecture In-Reply-To: <4f2613e50510240755p13289032q8cfa2b48b4fb04af@mail.gmail.com> Message-ID: References: <4f2613e50510240755p13289032q8cfa2b48b4fb04af@mail.gmail.com> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2343 Content-Length: 556 X-Keywords: NonJunk NotJunk I am thoroughly confused by this question :). -Hanson On Mon, 24 Oct 2005, Zachary Adam Ozer wrote: > First, I'm a bit confused as to how invarients are being used in the > reading in general; it seems as though at one moment there's a > theorem, but that they become some sort of equivalence statement. What > determines the equivalence relation or that it's true? > > More importantly however, get rid of Die Hard, it's getting old at > this point. Also, the bio type info on people is always awesome (vis > Turing, Flyod, Meyer, etc). > > -zozer > > From tylerw@MIT.EDU Mon Oct 24 12:28:13 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGSDd0026797 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:28:13 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OGSCgE025416 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:28:12 -0400 (EDT) Received: from [18.246.1.160] (TYLERW.MIT.EDU [18.246.1.160]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OGS9ok015200 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:28:09 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <11F6B62F-0919-428E-A71F-368926B5FDC4@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Tyler Williams Subject: [SAYAN] Week 7 Comments Date: Mon, 24 Oct 2005 12:28:08 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2344 Content-Length: 164 X-Keywords: NonJunk NotJunk The part of this weeks reading that I didn't really understand the best was the stuff about derived variables. I hope we see these in lecture. Tyler Williams From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:34:29 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGYTd0028213; Mon, 24 Oct 2005 12:34:29 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGYSmZ024056; Mon, 24 Oct 2005 12:34:28 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGYS0v024053; Mon, 24 Oct 2005 12:34:28 -0400 Date: Mon, 24 Oct 2005 12:34:28 -0400 (EDT) From: Hanson Zhou To: Angelique E Moscicki cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Week 7 comments In-Reply-To: <20051024113040.labrgilwhc04sg44@webmail.mit.edu> Message-ID: References: <20051024113040.labrgilwhc04sg44@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2345 Content-Length: 1817 X-Keywords: NonJunk NotJunk Careful...derived variables need not be the actual variables specified in the problem. They can be functions of them, as the notes demonstrate. For example, your invariant may well be A+B rather than either A or B. The notation f: states->R simply says that a derived variable function maps a state of the state machine to a real number value. For example, if the derived variable is being used as a counter towards termination, then the associated value of a state indicates how close the state is to a finishing one. -Hanson On Mon, 24 Oct 2005, Angelique E Moscicki wrote: > State Machines: The explanation of what they are is simple, but it will require > a little practice to get the hang of actually coming up with invariants for > them. The ususal approach is when we know what we're trying to prove, this > time we must come up with what we're trying to prove and then prove it. This > is the hardest part. Reachability and all that sort of thing is fine once I > have some invariant to think about. Partial Correctness and Termination make > sense; its just making sure your processor does the right thing and that it > can't get stuck in an infinite loop. > > Derived Variables seem just to be whatever variable your state machine is using; > all the registers in the processor. I'm not entirely sure what this f states -> > R means. > > The Mating Algorighm: at first I did not want to believe that one set gets > matched with their pessimal mate, but thinking about it, it makes sense that if > one set is matched with their optimal mates and those mates prefer them better > than the rest of the bunch (so the losing set gets their mate in order of > least-pessimal), it will at least form a stable system. Other than that, it > reminds me of non-linear system stability from 18.03. > > > > From jehan@MIT.EDU Mon Oct 24 12:52:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGqvd0032551 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:52:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OGqtfe024755 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:52:56 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OGqmQx011697 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:52:49 -0400 (EDT) Message-Id: <5.2.1.1.2.20051024124927.00c0cd50@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 24 Oct 2005 12:50:50 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2346 Content-Length: 90 X-Keywords: NonJunk NotJunk I'm confused as to why state machines are useful at all. What benefits do they provide? From kktyan@MIT.EDU Mon Oct 24 12:55:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGtEd0032702 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 12:55:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OGtDfe026987 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 12:55:13 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OGtBIO012513 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 12:55:11 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9OGtB1w020642; Mon, 24 Oct 2005 12:55:11 -0400 Received: from BURTON-FOUR-SIXTY-SEVEN.MIT.EDU (BURTON-FOUR-SIXTY-SEVEN.MIT.EDU [18.247.6.212]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 12:55:11 -0400 Message-ID: <20051024125511.vpt9an0gshs0gggw@webmail.mit.edu> Date: Mon, 24 Oct 2005 12:55:11 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2347 Content-Length: 280 X-Keywords: NonJunk NotJunk I had little trouble understanding the reading in general. I just have a point that I'd like to clarify a little: a derived variable (page 8) is a function, essentially? Like, a stored function? Not a value? - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From rian@MIT.EDU Mon Oct 24 13:24:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OHOUd0007793 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 13:24:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OHOSfe028871 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 13:24:29 -0400 (EDT) Received: from m56-129-21.mit.edu (M56-129-21.MIT.EDU [18.56.0.50]) (authenticated bits=56) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OHOQ6H025235 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 13:24:27 -0400 (EDT) Received: (from rian@localhost) by m56-129-21.mit.edu (8.12.9) id j9OHOQdu004259; Mon, 24 Oct 2005 13:24:26 -0400 Subject: (jelani) required reading From: rian To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 24 Oct 2005 13:24:26 -0400 Message-Id: <1130174666.4226.1.camel@m56-129-21.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2348 Content-Length: 75 X-Status: A X-Keywords: NonJunk NotJunk I honestly found no difficulty with this weeks lecture notes. Rian Hunter From meyer@imap.theory.csail.mit.edu Mon Oct 24 13:25:06 2005 Message-ID: <435D18F4.8030704@csail.mit.edu> Date: Mon, 24 Oct 2005 13:25:08 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kushan K Surana CC: Hanson M Zhou Subject: Re: TA - Hanson: 6.042 Week 7 comments References: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> In-Reply-To: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 867 Status: RO X-UID: 2349 X-Keywords: I'm guessing you're confused by the fact that the invariant mentions the unknown quantity gcd(a,b), which is exactly what the algorithm is trying to find. But there's nothing wrong with reasoning about an unknown quantity based on the ([possibly limited) info you may have about it. For example, suppose I let Q(d) be the predicate "d is a nonnegative integer". Then I know that Q(gcd(a,b,)) is true, even though I haven't found gcd(a,b) yet. Similarly, if at the start of a procedure I set x:=a and y:=b, then I know immediately that "gcd(x,y)=gcd(a,b)" holds in the start state, even though I haven't calculated anything yet. Does that help? regards, A. Kushan K Surana wrote: > - On page 6, the proof of the partial correctness of GCD is unclear. Why is the > predicate P(x,y) true for every state? The GCD is found only at the very last > state. From meyer@csail.mit.edu Mon Oct 24 13:33:47 2005 Return-Path: Received: from [128.30.49.31] (theory-dhcp-31.csail.mit.edu [128.30.49.31]) (authenticated bits=0) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OHXUd0008368; Mon, 24 Oct 2005 13:33:46 -0400 Message-ID: <435D1AEA.2080104@csail.mit.edu> Date: Mon, 24 Oct 2005 13:33:30 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: rian CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: (jelani) required reading References: <1130174666.4226.1.camel@m56-129-21.mit.edu> In-Reply-To: <1130174666.4226.1.camel@m56-129-21.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 2350 Content-Length: 423 X-Keywords: I believe you, but your email doesn't have to identify parts you found hard: you could indicate what you found most (or least) interesting, surprising, poorly (or specially well) explained, relevant to your professional interests (or not),.... I'll look forward to more specific & responsive comments next week. regards, A. rian wrote: > I honestly found no difficulty with this weeks lecture notes. > > Rian Hunter From meyer@imap.theory.csail.mit.edu Mon Oct 24 14:34:29 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OIYSd0024112 for ; Mon, 24 Oct 2005 14:34:28 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EU79Q-0007r9-KK for meyer@csail.MIT.EDU; Mon, 24 Oct 2005 14:34:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OIYQp4012779; Mon, 24 Oct 2005 14:34:26 -0400 (EDT) Received: from ccbibmx30 (BLACKBURN.MIT.EDU [18.241.3.107]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OIYHdh024826 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT); Mon, 24 Oct 2005 14:34:17 -0400 (EDT) Message-Id: <200510241834.j9OIYHdh024826@outgoing.mit.edu> From: "Clinton Blackburn" To: "'David Shin'" , "'Albert R. Meyer'" Subject: RE: [Fwd: Jelani: TP 7 Comments] Date: Mon, 24 Oct 2005 14:32:14 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 In-Reply-To: <20051024125045.s69nn6imizkg4kow@webmail.mit.edu> Thread-Index: AcXYuzUGgdrPXXw+QISUKvJ+Tw273QADerKg X-Spam-Score: 2.037 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1478 X-UID: 2351 X-Keywords: NotJunk Oops. I must have been clicking and scrolling too fast. My apologies. --- Clinton Blackburn DLP - Have you seen it? http://www.dlp.com -----Original Message----- From: David Shin [mailto:dshin@MIT.EDU] Sent: Monday, October 24, 2005 12:51 PM To: Albert R. Meyer Cc: clintonb@mit.edu Subject: Re: [Fwd: Jelani: TP 7 Comments] Clinton, When I log on to the tutor and select TP7, the first page I come across has a prefilled email link ("subject=[TA-name] Week 7 Comments"). Is there another link somewhere? Please let me know. DS Quoting "Albert R. Meyer" : > Let's keep these links updated. > regards, A. > > -------- Original Message -------- > Subject: Jelani: TP 7 Comments > Date: Mon, 24 Oct 2005 01:06:14 -0400 > From: Clinton Blackburn > To: <6042-probs@theory.lcs.mit.edu> > > > > I'm totally digging these problems with real-world relations/examples > (ie. RSA, stable marriage). I learn best through these types of > examples. I get the methods; although, I'm sure I need to go back over > the different theorems and proofs for the stable marriage problem. > > > > Also, what happened to the pre-filled email links? I'm referring to the > mailto links with the email address and subject line. I kind of liked > not having to think of a subject line (the hardest part of writing > emails for me). > > > > Good morning! > > --- > > Clinton Blackburn > > DLP - Have you seen it? > > > > > > From mitras@theory.csail.mit.edu Mon Oct 24 18:27:38 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OMRcd0008261 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 18:27:38 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OMRaXA012523; Mon, 24 Oct 2005 18:27:37 -0400 (EDT) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OMRTok007535; Mon, 24 Oct 2005 18:27:29 -0400 (EDT) Message-ID: <435D5FD1.2050807@theory.csail.mit.edu> Date: Mon, 24 Oct 2005 18:27:29 -0400 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jehan deFonseka CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 7 Comments References: <5.2.1.1.2.20051024124927.00c0cd50@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20051024124927.00c0cd50@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2352 Content-Length: 1613 X-Keywords: NotJunk Finite state machines (FSM) serve as a natural and often tractable model for systems. For example (from Wikipedia): In a digital circuit , a FSM may be built using a programmable logic device , a programmable logic controller , logic gates and flip flops or relays . More specifically, a hardware implementation requires a register to store state variables, a block of combinational logic which determines the state transition, and a second block of combinational logic that determines the output of a FSM. There are innumerable other applications both in hardware and software. The state machine model is attractive because it comes with a bunch of proof methods, like the Invariant Theorem that we have studied in class. These mathematical proof methods can often be incorporated into analysis tools for verifying correctness of systems. For example, there are Model checking tools that can automatically check if a given state machine model with more than 10^80 states satisfies a certain invariant property! These tools are useful for catching bugs that would be hard to find otherwise. The Pentium FDIV bug was found using such tools. -Sayan Jehan deFonseka wrote: > I'm confused as to why state machines are useful at all. What benefits > do they provide? > From dshin@MIT.EDU Mon Oct 24 20:11:09 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9P0B9d0019063 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 20:11:09 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9P0B5sh016012; Mon, 24 Oct 2005 20:11:05 -0400 (EDT) Received: from [192.168.1.3] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9P0AvMu005438; Mon, 24 Oct 2005 20:10:57 -0400 (EDT) Message-ID: <435D7810.8010806@mit.edu> Date: Mon, 24 Oct 2005 20:10:56 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hao Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: David TP7 Reading Comment References: <20051024002104.pj08nk8euo34sswo@webmail.mit.edu> In-Reply-To: <20051024002104.pj08nk8euo34sswo@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2353 Content-Length: 754 X-Keywords: NotJunk Unfortunately, experience and intuition is the best answer we can really give. One can actually prove that there is no preset method that will find the right invariant! When you are designing your own state machines (i.e., "algorithms"), I think you will find that finding that invariant often goes hand-in-hand with designing the algorithm. DS Hao Zhou wrote: >Hi Staff, > >I found using the state invarient is a very clever way to solve problems. >However, I find it is often times hard to find an invarient for all the states. > For example, for the tiling problem we did in class, the invarient was not a >very obvious choice. Other than through experience, how should we look for >these invarients? Thanks > >- Steven (Hao) Zhou > > From meyer@imap.theory.csail.mit.edu Tue Oct 25 10:01:11 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9PCqqd0002059 for ; Tue, 25 Oct 2005 08:52:52 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EUOIO-0007p4-Cl for meyer@csail.MIT.EDU; Tue, 25 Oct 2005 08:52:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9PCqnwN008390; Tue, 25 Oct 2005 08:52:50 -0400 (EDT) Received: from cs37.mit.edu (CS37.MIT.EDU [18.62.4.237]) (authenticated bits=56) (User authenticated as kushan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9PCqgG6015504 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 25 Oct 2005 08:52:43 -0400 (EDT) Received: (from kushan@localhost) by cs37.mit.edu (8.12.9) id j9PCqgC9028746; Tue, 25 Oct 2005 08:52:42 -0400 Date: Tue, 25 Oct 2005 08:52:42 -0400 (EDT) From: Kushan K Surana To: "Albert R. Meyer" cc: Hanson M Zhou Subject: Re: TA - Hanson: 6.042 Week 7 comments In-Reply-To: <435D18F4.8030704@csail.mit.edu> Message-ID: References: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> <435D18F4.8030704@csail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1043 X-UID: 2354 X-Keywords: NonJunk I think so. I have to review the notes again and clear it in my mind. Thanks, Kushan On Mon, 24 Oct 2005, Albert R. Meyer wrote: > I'm guessing you're confused by the fact that the invariant mentions the > unknown quantity gcd(a,b), which is exactly what the algorithm is trying to > find. > > But there's nothing wrong with reasoning about an unknown quantity based on > the ([possibly limited) info you may have about it. For example, suppose I > let Q(d) be the predicate "d is a nonnegative integer". Then I know that > Q(gcd(a,b,)) is true, even though I haven't found gcd(a,b) yet. Similarly, > if at the start of a procedure I set x:=a and y:=b, then I know immediately > that "gcd(x,y)=gcd(a,b)" holds in the start state, even though I haven't > calculated anything yet. > > Does that help? > > regards, A. > > Kushan K Surana wrote: >> - On page 6, the proof of the partial correctness of GCD is unclear. Why is >> the >> predicate P(x,y) true for every state? The GCD is found only at the very >> last >> state. > From meyer@imap.theory.csail.mit.edu Mon Oct 24 13:33:47 2005 Message-ID: <435D1AEA.2080104@csail.mit.edu> Date: Mon, 24 Oct 2005 13:33:30 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: rian CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: (jelani) required reading References: <1130174666.4226.1.camel@m56-129-21.mit.edu> In-Reply-To: <1130174666.4226.1.camel@m56-129-21.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 424 X-UID: 2355 X-Keywords: I believe you, but your email doesn't have to identify parts you found hard: you could indicate what you found most (or least) interesting, surprising, poorly (or specially well) explained, relevant to your professional interests (or not),.... I'll look forward to more specific & responsive comments next week. regards, A. rian wrote: > I honestly found no difficulty with this weeks lecture notes. > > Rian Hunter From meyer@imap.theory.csail.mit.edu Mon Oct 24 13:47:05 2005 Message-ID: <435D1E19.4070902@csail.mit.edu> Date: Mon, 24 Oct 2005 13:47:05 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kamil Y Sindi Subject: Re: [David] Week 7 Comments References: <20051023231516.kafh91q4locgo0wg@webmail.mit.edu> In-Reply-To: <20051023231516.kafh91q4locgo0wg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 332 X-UID: 2356 X-Keywords: Didn't you think verification of the extended GCD was a practical application? Or did you feel it really didn't use the state machine formalism? Regards, A. Kamil Y Sindi wrote: > I found the notes clear but still don't understand the practical use of these > state concepts. It seems to me that they are mere formilizations From rnjacobs@MIT.EDU Tue Oct 25 14:33:33 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9PIXX5o012775 for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 14:33:33 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9PIXWsj028560 for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 14:33:33 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9PIXWsF013077 for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 14:33:32 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9PIXUok010149 for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 14:33:30 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j9PIXRNR015326; Tue, 25 Oct 2005 14:33:27 -0400 Message-Id: <200510251833.j9PIXRNR015326@white-meteo.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Reading response Date: Tue, 25 Oct 2005 14:33:27 -0400 From: r n jacobs X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2357 Content-Length: 259 X-Status: A X-Keywords: NonJunk NotJunk The material made sense. But I had a stupid problem on the tutor problems where i misread the integral a^b x^r as "the integral of a to the b times x to the r" instead of that it should have been "the integral from a to b of x to the r". - robert jacobs From meyer@imap.theory.csail.mit.edu Tue Oct 25 20:11:15 2005 Message-ID: <435EC9A3.7000506@csail.mit.edu> Date: Tue, 25 Oct 2005 20:11:15 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: r n jacobs CC: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Jelani] Reading response References: <200510251833.j9PIXRNR015326@white-meteo.mit.edu> In-Reply-To: <200510251833.j9PIXRNR015326@white-meteo.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 709 X-UID: 2358 X-Keywords: r n jacobs wrote: >The material made sense. > > > I believe you, but your email doesn't have to identify parts you found hard: you could indicate what you found most (or least) interesting, surprising, poorly (or specially well) explained, relevant to your professional interests (or not),.... I'll look forward to more specific & responsive comments next week. regards, A. >But I had a stupid problem on the tutor problems where i misread the >integral a^b x^r as "the integral of a to the b times x to the r" >instead of that it should have been "the integral from a to b of x to >the r". > > - robert jacobs > > have been "the integral from a to b of x to >the r". > > - robert jacobs > > From vixen@MIT.EDU Tue Oct 25 20:41:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q0fV5o016404 for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 20:41:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q0fUQk013622 for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 20:41:30 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q0fQ4o004814 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 20:41:28 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Tue, 25 Oct 2005 20:39:41 -0400 To: 6042-probs@theory.csail.mit.edu From: Amanda Seybold Subject: [Jelani] LN8 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2359 Content-Length: 61 X-Status: A X-Keywords: NotJunk I don't understand the difference between O(n) and omega(n). From juang@MIT.EDU Tue Oct 25 20:44:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q0iY5o016634 for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 20:44:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q0iWQk015780 for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 20:44:32 -0400 (EDT) Received: from [18.238.6.214] (EASTCAMPUS-NINE-EIGHTY-ONE.MIT.EDU [18.238.6.214]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q0iUju005306 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 20:44:31 -0400 (EDT) Message-ID: <435ED16D.6060206@mit.edu> Date: Tue, 25 Oct 2005 20:44:29 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 8 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2360 Content-Length: 210 X-Keywords: NonJunk NotJunk I was surprised by the demonstration, beginning on page 7, that a stack of books can extend arbitrarily far past the edge of the table. It's very much counterintuitive; the proof is very interesting. Jason. From hectorb@MIT.EDU Tue Oct 25 21:49:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q1n95o025578 for <6042-probs@theory.lcs.mit.edu>; Tue, 25 Oct 2005 21:49:09 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q1n7Qk006956; Tue, 25 Oct 2005 21:49:07 -0400 (EDT) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q1n0v4016639 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 25 Oct 2005 21:49:01 -0400 (EDT) Message-ID: <435EE08C.8060808@mit.edu> Date: Tue, 25 Oct 2005 21:49:00 -0400 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 8 Comment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2361 Content-Length: 339 X-Keywords: NonJunk NotJunk "As a second best, however, we can find closed forms for very good approximations to Hn using the Integral Method. The idea of the Integral Method is to bound terms of the sum above and below by simple functions as suggested in Figure 5." p. 10 Would like to see more examples of integral method, how to find lower and upper bounds. From cwong08@MIT.EDU Tue Oct 25 22:26:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q2QC5o030568 for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 22:26:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q2QBQk013194 for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 22:26:11 -0400 (EDT) Received: from w20-575-62.mit.edu (W20-575-62.MIT.EDU [18.187.0.81]) (authenticated bits=56) (User authenticated as cwong08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q2Q9MI026356 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Tue, 25 Oct 2005 22:26:10 -0400 (EDT) Received: (from cwong08@localhost) by w20-575-62.mit.edu (8.12.9) id j9Q2Q9Lk031640; Tue, 25 Oct 2005 22:26:09 -0400 Subject: [Sayan] Week 8 Comments From: Christopher Wong To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 25 Oct 2005 22:26:08 -0400 Message-Id: <1130293568.31582.2.camel@w20-575-62.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2362 Content-Length: 186 X-Keywords: NotJunk I thought this week's notes were the most confusing notes so far. I would like better examples with the integral method (pg. 10) and Stirling's approximations (pg 13 and 14) -Chris Won From ajshafer@MIT.EDU Tue Oct 25 23:34:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q3Yr5o005437 for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 23:34:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q3YqXK007294 for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 23:34:52 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q3YjTM009235 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 25 Oct 2005 23:34:46 -0400 (EDT) Message-ID: <000201c5d9de$35d2c620$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 8 Comments Date: Tue, 25 Oct 2005 23:34:45 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2363 Content-Length: 374 X-Keywords: NonJunk NotJunk I'm not sure how harmonic numbers are used beyond the given problem. Are there more examples? Also, I would like more time spent work on "order of" problems. Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From ksindi@MIT.EDU Wed Oct 26 00:21:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q4L05o015182 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 00:21:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q4KxXK011707 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 00:20:59 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q4Kwts017072 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 00:20:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9Q4Kw7D017622; Wed, 26 Oct 2005 00:20:58 -0400 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 26 Oct 2005 00:20:58 -0400 Message-ID: <20051026002058.grh3sgatdu68kcgo@webmail.mit.edu> Date: Wed, 26 Oct 2005 00:20:58 -0400 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [David] Week 8 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2364 Content-Length: 327 X-Keywords: NotJunk I found this weeks notes to be great! I liked the book stacking and appreciated the pitfalls of the Big Oh. Maybe the Professor can mention how much this notation facilitates concepts and techniques in analytic number theory and that if we can pay homage to Edmund Landau for introducing it in his glorious book, 'Primzahlen'. From bens@MIT.EDU Wed Oct 26 01:14:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9Q5Er5o022236 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 01:14:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9Q5EqXK019608 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 01:14:52 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9Q5Ep1V023791 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 01:14:51 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9Q5EpYB013671; Wed, 26 Oct 2005 01:14:51 -0400 Received: from SHTREASURY.MIT.EDU (SHTREASURY.MIT.EDU [18.221.0.63]) (User authenticated as bens@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 26 Oct 2005 01:14:51 -0400 Message-ID: <20051026011451.izrjgyz1x1wkoooo@webmail.mit.edu> Date: Wed, 26 Oct 2005 01:14:51 -0400 From: Benjamin M Schwartz To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] O-notation MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2365 Content-Length: 129 X-Status: A X-Keywords: NotJunk How can we write f = O(g) or f(n) = O(g(n))? What's the equivalence? I think we should write "f is O(g)" or "f \in O(g)". -Ben From dowgun@MIT.EDU Wed Oct 26 14:49:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9QInk5o000940 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 26 Oct 2005 14:49:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9QInjh1020935 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 26 Oct 2005 14:49:45 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9QIncXv023553 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 26 Oct 2005 14:49:38 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9QInbaT011452; Wed, 26 Oct 2005 14:49:37 -0400 Received: from W20-575-53.MIT.EDU (W20-575-53.MIT.EDU [18.187.0.72]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 26 Oct 2005 14:49:37 -0400 Message-ID: <20051026144937.ss9y06ihebwokksg@webmail.mit.edu> Date: Wed, 26 Oct 2005 14:49:37 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] ln8 questions MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2366 Content-Length: 836 X-Status: A X-Keywords: NonJunk NotJunk Although I have a little trouble following some of the arithmatic used these lecture notes, Im sure you guys will be going over the important concepts, which I think I get. However there are little things that are bothering me, such 1) what is the difference between lim sup and lim? Can we just use lHopital's method to find big Oh or are these two things fundamentally different? 2) Figure 3 does not seem to show the right way to stack the books - each book should overhang the one beneath it more than the one beneath it overhung the one one beneath that one. Like in Figure 4, basically. 3) what happens if g is negative in f = O(g) ? for instance, if you're comparing two functions to see if f = (theta(g)), then you would not really be allowed to choose g, as you would if you wanted to find an upper asymtotic limit for f. Neil From icharny@MIT.EDU Wed Oct 26 18:47:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9QMlN5o010129 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 18:47:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9QMlMnM000261 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 18:47:22 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9QMlGqj018113 for <6042-probs@theory.csail.MIT.EDU>; Wed, 26 Oct 2005 18:47:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9QMlG8Y017466; Wed, 26 Oct 2005 18:47:16 -0400 Received: from ab6.draper.com (ab6.draper.com [192.80.95.243]) (User authenticated as icharny@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 26 Oct 2005 18:47:16 -0400 Message-ID: <20051026184716.3g2nf247gxn4c088@webmail.mit.edu> Date: Wed, 26 Oct 2005 18:47:16 -0400 From: Isaac Charny To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Week 8 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2367 Content-Length: 162 X-Keywords: NonJunk NotJunk I don't understand what little oh/big oh are and what they are used for (section 7 starting on page 15). Please cover these thoroughly in lecture. ~Isaac Charny From aston@MIT.EDU Thu Oct 27 00:07:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9R47e5o016235 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 00:07:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9R47dZc002854 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 00:07:39 -0400 (EDT) Received: from aston (ASTON.MIT.EDU [18.241.3.49]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9R47Wmt014825 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 00:07:33 -0400 (EDT) Message-Id: <200510270407.j9R47Wmt014825@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 8 Comments Date: Thu, 27 Oct 2005 00:07:30 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00BB_01C5DA8A.6C5307D0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXaq/LrmXsHDyliTdCg0fkAjRz4ew== X-Spam-Score: 2.593 X-Spam-Level: ** (2.593) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2368 Content-Length: 2233 X-Status: A X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_00BB_01C5DA8A.6C5307D0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The book stacking problem is ridiculous. Definitely not a conclusion I expected. Even after reading over the explanation a few times, I still was getting lost right around the first full paragraph of page 8. - Aston ------=_NextPart_000_00BB_01C5DA8A.6C5307D0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

The book stacking problem is ridiculous. Definitely = not a conclusion I expected. Even after reading over the explanation a few = times, I still was getting lost right around the first full paragraph of page = 8.

 

         =    - Aston

------=_NextPart_000_00BB_01C5DA8A.6C5307D0-- From dnreshef@MIT.EDU Thu Oct 27 00:50:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9R4os5o020921 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 00:50:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9R4orZc002240 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 00:50:53 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9R4op9X020469 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 00:50:51 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9R4opIX028385; Thu, 27 Oct 2005 00:50:51 -0400 Received: from PLP-ONE-FORTY-EIGHT.MIT.EDU (PLP-ONE-FORTY-EIGHT.MIT.EDU [18.218.1.148]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 27 Oct 2005 00:50:51 -0400 Message-ID: <20051027005051.fx974far8fswc0ko@webmail.mit.edu> Date: Thu, 27 Oct 2005 00:50:51 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: [HANSON] T8 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2369 Content-Length: 208 X-Keywords: NonJunk In this week's reading, I found the sections on Infintie Geometric Sums on page 4 and on Finding Summation Formulas on page 11 especially confusing. We should definitely cover these in lecture please. -Dave From lye@MIT.EDU Thu Oct 27 05:50:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9R9oi5o028597 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 05:50:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9R9ohZc020986 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 05:50:43 -0400 (EDT) Received: from w20-575-56.mit.edu (W20-575-56.MIT.EDU [18.187.0.75]) (authenticated bits=56) (User authenticated as lye@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9R9oeGI007954 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 05:50:40 -0400 (EDT) Received: (from lye@localhost) by w20-575-56.mit.edu (8.12.9) id j9R9oe51020024; Thu, 27 Oct 2005 05:50:40 -0400 Subject: [Sayan] Week 8 Comments From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 27 Oct 2005 05:50:40 -0400 Message-Id: <1130406640.19953.5.camel@w20-575-56.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2370 Content-Length: 425 X-Keywords: NonJunk NotJunk 2.6 Related Sum: "But suppose that we differentiate that formula..." Is there an analogous trick of integrating? Say, you have a formula for summation of f(x) over some range. In what instances can you integrate your closed form to get the formula for summation of F(x) [where dF(x)/dx = f(x)]? Because F(x) = G(x) implies dF/dx = dG/dx, but dF/dx = dG/dx does not necessarily imply F(x) = G(x). Lunduo/Linda Ye From meyer@imap.theory.csail.mit.edu Thu Oct 27 08:34:46 2005 Message-ID: <4360C968.3040706@csail.mit.edu> Date: Thu, 27 Oct 2005 08:34:48 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Aston Motes Subject: Re: [Hanson] Week 8 Comments References: <200510270407.j9R47Wmt014825@outgoing.mit.edu> In-Reply-To: <200510270407.j9R47Wmt014825@outgoing.mit.edu> Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Length: 1815 Status: RO X-UID: 2371 X-Keywords: "ridiculous" means "silly": is that what you mean to say?
Regards, A.

Aston Motes wrote:

The book stacking problem is ridiculous. Definitely not a conclusion I expected. Even after reading over the explanation a few times, I still was getting lost right around the first full paragraph of page 8.

 

            - Aston

From meyer@imap.theory.csail.mit.edu Thu Oct 27 11:03:48 2005 Message-ID: <4360EC60.8030804@csail.mit.edu> Date: Thu, 27 Oct 2005 11:04:00 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Aston Motes Subject: Re: [Hanson] Week 8 Comments References: <200510271430.j9REU9VY006741@outgoing.mit.edu> In-Reply-To: <200510271430.j9REU9VY006741@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 1081 X-UID: 2372 X-Keywords: comes from "ridicule" meaning "to mock." i can see using it your way with proper voice intonation to indicate astonishment rather than skepticism, but intonation doesn't come across well in email :-) regards, A. Aston Motes wrote: > Hehe. When I use ridiculous, I almost always mean extremely surprising > and somewhat hard to believe. A remark of astonishment. > > > > I guess most people dont use it that way > > > > > > - Aston > > > > ------------------------------------------------------------------------ > > *From:* Prof. Albert R. Meyer [mailto:meyer@csail.mit.edu] > *Sent:* Thursday, October 27, 2005 8:35 AM > *To:* Aston Motes > *Subject:* Re: [Hanson] Week 8 Comments > > > > "ridiculous" means "silly": is that what you mean to say? > Regards, A. > > Aston Motes wrote: > > The book stacking problem is ridiculous. Definitely not a conclusion I > expected. Even after reading over the explanation a few times, I still > was getting lost right around the first full paragraph of page 8. > > > > - Aston > From meyer@imap.theory.csail.mit.edu Thu Oct 27 10:30:13 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9REUC5o005230 for ; Thu, 27 Oct 2005 10:30:12 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EV8lg-0006Nw-Og for meyer@csail.mit.edu; Thu, 27 Oct 2005 10:30:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9REUBZc010124 for ; Thu, 27 Oct 2005 10:30:11 -0400 (EDT) Received: from aston (ASTON.MIT.EDU [18.241.3.49]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9REU9VY006741 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for ; Thu, 27 Oct 2005 10:30:09 -0400 (EDT) Message-Id: <200510271430.j9REU9VY006741@outgoing.mit.edu> From: "Aston Motes" To: "'Prof. Albert R. Meyer'" Subject: RE: [Hanson] Week 8 Comments Date: Thu, 27 Oct 2005 10:30:06 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00F8_01C5DAE1.664D0EB0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 In-Reply-To: <4360C968.3040706@csail.mit.edu> X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXa8tT5W4EJW5DWTLy7zOhYFcAoEgAD95ew X-Spam-Score: 0.179 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.6 required=3.7 tests=AWL,BAYES_00,HTML_70_80, HTML_FONTCOLOR_UNKNOWN,HTML_MESSAGE autolearn=no version=2.63 Status: RO X-Status: A Content-Length: 6212 X-UID: 2373 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_00F8_01C5DAE1.664D0EB0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hehe. When I use ridiculous, I almost always mean extremely surprising and somewhat hard to believe. A remark of astonishment. I guess most people don't use it that way. - Aston _____ From: Prof. Albert R. Meyer [mailto:meyer@csail.mit.edu] Sent: Thursday, October 27, 2005 8:35 AM To: Aston Motes Subject: Re: [Hanson] Week 8 Comments "ridiculous" means "silly": is that what you mean to say? Regards, A. Aston Motes wrote: The book stacking problem is ridiculous. Definitely not a conclusion I expected. Even after reading over the explanation a few times, I still was getting lost right around the first full paragraph of page 8. - Aston ------=_NextPart_000_00F8_01C5DAE1.664D0EB0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hehe. When I use ridiculous, I = almost always mean extremely surprising and somewhat hard to believe. A remark = of astonishment.

 

I guess most people don’t use = it that way…

 

 

      =       - Aston

 

From: Prof. Albert R. Meyer [mailto:meyer@csail.mit.edu] =
Sent: Thursday, October = 27, 2005 8:35 AM
To: Aston Motes
Subject: Re: [Hanson] = Week 8 Comments

 

"ridiculous" means = "silly": is that what you mean to say?
Regards, A.

Aston Motes wrote:

The book stacking problem is ridiculous. = Definitely not a conclusion I expected. Even after reading over the explanation a = few times, I still was getting lost right around the first full paragraph of = page 8.

 

        = ;    - Aston

------=_NextPart_000_00F8_01C5DAE1.664D0EB0-- From meyer@imap.theory.csail.mit.edu Thu Oct 27 11:08:39 2005 Message-ID: <4360ED83.80005@csail.mit.edu> Date: Thu, 27 Oct 2005 11:08:51 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin M Schwartz CC: Ronitt Rubinfeld Subject: Re: [Jelani] O-notation References: <20051026011451.izrjgyz1x1wkoooo@webmail.mit.edu> In-Reply-To: <20051026011451.izrjgyz1x1wkoooo@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 328 X-UID: 2374 X-Keywords: I agree, but the regrettable notation using equality has proved too widely accepted to change (despite serious past efforts in the Math community). regards, A Benjamin M Schwartz wrote: > How can we write f = O(g) or f(n) = O(g(n))? What's the equivalence? > I think we should write "f is O(g)" or "f \in O(g)". > > -Ben From meyer@imap.theory.csail.mit.edu Thu Oct 27 11:10:40 2005 Message-ID: <4360EDFC.9080205@csail.mit.edu> Date: Thu, 27 Oct 2005 11:10:52 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: Re: [Jelani] LN8 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 188 X-UID: 2375 X-Keywords: There shouldn't be any omega in the notes, unless we slipped up. Maybe you mean Theta? regards, A Amanda Seybold wrote: > I don't understand the difference between O(n) and omega(n). From meyer@imap.theory.csail.mit.edu Thu Oct 27 11:12:03 2005 Message-ID: <4360EE4F.40904@csail.mit.edu> Date: Thu, 27 Oct 2005 11:12:15 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Neil M Dowgun Subject: Re: [Hanson] ln8 questions References: <20051026144937.ss9y06ihebwokksg@webmail.mit.edu> In-Reply-To: <20051026144937.ss9y06ihebwokksg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1857 X-UID: 2376 X-Keywords: Neil M Dowgun wrote: > Although I have a little trouble following some of the arithmatic used these > lecture notes, Im sure you guys will be going over the important concepts, > which I think I get. However there are little things that are bothering me, > such 1) what is the difference between lim sup and lim? defs and example are in the notes, but here's another simple one: the sequence 0,1,0,1,0,1,... has no limit, but has a limsup of 1. Another way to think of it is, if you have a sequence of real numbers that does not have a limit, then try to find a subsequence that does have a limit. The largest limit you can obtain this way is the limsup. Can we just use > lHopital's method to find big Oh or are these two things fundamentally > different? l'Hopital is good for limits, not directly suited for limsup or BigOh. 2) Figure 3 does not seem to show the right way to stack the books - > each book should overhang the one beneath it more than the one beneath it > overhung the one one beneath that one. Like in Figure 4, basically. In figs 2 & 3, we are defining overhang and deriving the increment n overhang from adding the nth book. We mustn't assume the shape of the optimal stack, since that what we're trying to derive. The figs, especially fig 2, were drawn that way to emphasize that the definitions made sense for any stack, and so could be used w/o circularity in deducing what the optimal stack would be. 3) what > happens if g is negative in f = O(g) ? for instance, if you're comparing two > functions to see if f = (theta(g)), then you would not really be allowed to > choose g, as you would if you wanted to find an upper asymtotic limit for f. There's no consensus on how to define BigOh for negative g. We could specify one, but there's no point, since there's never been a need. > Neil Good questions. Regards, A. From yangc@MIT.EDU Thu Oct 27 15:58:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9RJwd5o010840 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 15:58:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9RJwcge024947 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 15:58:38 -0400 (EDT) Received: from christope8f0c6 (MACGREGOR-ONE-TWENTY-FOUR.MIT.EDU [18.239.5.124]) (authenticated bits=0) (User authenticated as yangc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9RJwZ9a027373 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 15:58:36 -0400 (EDT) Message-Id: <200510271958.j9RJwZ9a027373@outgoing.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 8 E-mail Comments Date: Thu, 27 Oct 2005 16:00:35 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0033_01C5DB0F.91751910" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 thread-index: AcXbMRfu3EyGNUkSTl2jw5LNnW8Dng== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2377 Content-Length: 2671 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0033_01C5DB0F.91751910 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I did not really understand the derivation of Stirling's approximation of the factorial - and I guess, to some extent, I did not really understand the Integral Method. I assume both of these things will be covered in greater detail in class. Thanks, Chris Yang ------=_NextPart_000_0033_01C5DB0F.91751910 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I did not really understand the derivation of = Stirling’s approximation of the factorial = – and I guess, to some extent, I did not really understand the Integral = Method.  I assume both of these things will be covered in greater detail in = class.

 

Thanks,

Chris Yang

------=_NextPart_000_0033_01C5DB0F.91751910-- From rian@MIT.EDU Thu Oct 27 16:05:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9RK5v5o012197 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 16:05:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9RK5tge002979 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 16:05:56 -0400 (EDT) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9RK5r7U000295 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 16:05:53 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <3ACD480C-B9BF-4D19-B2A9-E703999FA202@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rian Hunter Subject: (jelani) required reading comments Date: Thu, 27 Oct 2005 16:05:53 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2378 Content-Length: 1395 X-Keywords: NotJunk These notes were more confusing than the past notes have been. Particularly in the proofs (sometimes i was not able to intuitively follow on the steps). --- Firstly in page 10, where the integral method is described when H(sub) n is less than 1+ln(n) it doesn't say where the 1 comes from (it can be inferred that 1 is the area under the first bar: 1 * the-first- term, but it was confusing since it wasn't explicitly stated). Also if the integral method means: (integral from 0->n) term * dx <= actual sum <= (1 * first-term) + (intergral from 1->n) this is different from how it is used at the bottom of page 11. I understand its different because where you use x+1 depends on if that equation upper bounds or lower bounds the sum terms. This isn't clear from the notes, especially since a general equation is given earlier. This isn't necessarily bad, but math students usually have an affinity for digesting general equations more than general graphs. --- Also the derivation of the distance of the tip of the bottom book from the edge of the table in n+1 books on page 8 is confusing simply because the definition of center of mass is expected to be known. Also usually the center of mass equation is a weighted average by mass, in the notes it is weighted by amount of books which may not intuitively correlate to mass for some people. Thanks. Rian From shauni@MIT.EDU Thu Oct 27 17:09:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9RL9P5o024210 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 17:09:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9RL1LPK028791 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 17:09:24 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9RKw0Tf020702 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 16:58:00 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9RKw03g021166; Thu, 27 Oct 2005 16:58:00 -0400 Received: from SIMMONS-ONE-SEVENTY-NINE.MIT.EDU (SIMMONS-ONE-SEVENTY-NINE.MIT.EDU [18.96.5.179]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 27 Oct 2005 16:58:00 -0400 Message-ID: <20051027165800.hged0dmbeke840w4@webmail.mit.edu> Date: Thu, 27 Oct 2005 16:58:00 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] week 8 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2379 Content-Length: 98 X-Keywords: NotJunk I am confused as to the differences between o, O, and theta; i.e. which one is used when and why. From moscicki@MIT.EDU Thu Oct 27 18:52:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9RMql5o010781 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 18:52:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9RMqjNg001789 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 18:52:45 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9RMqd4I020112 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 18:52:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9RMqd36021928; Thu, 27 Oct 2005 18:52:39 -0400 Received: from NEW-FOUR-NINETY-ONE.MIT.EDU (NEW-FOUR-NINETY-ONE.MIT.EDU [18.241.6.236]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 27 Oct 2005 18:52:39 -0400 Message-ID: <20051027185239.5y7gn1zapt7o0k40@webmail.mit.edu> Date: Thu, 27 Oct 2005 18:52:39 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] week 8 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2380 Content-Length: 1131 X-Keywords: NotJunk The value of money: The formulas make sense, if the financial practices are somewhat confusing :) After that, I am a little shaky with the formulas about summations. It'll be good to see them presented in class. Harmonic numbers: Cool, seen these before, but not presented this way. Stirling's Approximation: Where did he come up with the sqrt(2n) term? The notes don't really explain that; they explain about taking the sum of ln, which is followable, but then that other term just seems to appear. Book Stacking... i am here stacking up cd's tryint to test... interesting. Little o: The proof of Lemma 7.3 is mystifying. Once I just accepted Lemma 7.3, the rest was just some juggling of exponents. I remember seeing the proofs they mentioned (L'Hopital's rule and the MacLaurian Series ones) in my Calculus class but don't remember them now. Big O: Necessarily more familiar from CS and math classes. errrrm... lim sup? I know lim... and I thought I knew big O pretty well too... this lim sup confuses the matter I think. All the other stuff about Big O is familar. Theta is just an extension of the above. From ridell@MIT.EDU Thu Oct 27 19:16:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9RNGU5o013554 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 19:16:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9RNGSNg017320 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 19:16:28 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9RNGRdW023851 for <6042-probs@theory.csail.MIT.EDU>; Thu, 27 Oct 2005 19:16:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9RNGRQd003503; Thu, 27 Oct 2005 19:16:27 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 27 Oct 2005 19:16:27 -0400 Message-ID: <20051027191627.81cahawyea5ck0co@webmail.mit.edu> Date: Thu, 27 Oct 2005 19:16:27 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: tp 8 reading comments [Hanson] MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2381 Content-Length: 308 X-Keywords: NonJunk NotJunk Can we gover Stirling's formula from page 15 of the reading? Maybe go into more about how it can be used in proofs. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From cvnguyen@MIT.EDU Thu Oct 27 19:18:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9RNIq5o013669 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 19:18:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9RNIoNg018915 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 19:18:50 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-FIVE-TWENTY-EIGHT.MIT.EDU [18.238.5.17]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9RNImnN024268 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 19:18:49 -0400 (EDT) Message-ID: <004f01c5db4c$ca7f3960$1105ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 8 comments Date: Thu, 27 Oct 2005 19:18:49 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.644 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2382 Content-Length: 126 X-Keywords: NotJunk I found the book-stacking problem interesting and insightful. I've been thinking about it for a long time. --Chieu Nguyen From avalys@MIT.EDU Thu Oct 27 20:06:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S06U5o020321 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 20:06:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S06SSu018349 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 20:06:28 -0400 (EDT) Received: from [18.96.7.30] (SIMMONS-FIVE-FOURTY-ONE.MIT.EDU [18.96.7.30]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S06RPk000937 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 20:06:27 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <2ECD1ABF-249A-45EF-80BA-D94FCEE82346@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Week 8 Required Comments Date: Thu, 27 Oct 2005 20:06:29 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2383 Content-Length: 231 X-Keywords: NonJunk NotJunk I am not clear on the specifics of how to apply the integral method. I understand the example given on page 10, but I don't know how to apply it generally. Having it stated formally, in general terms, would be nice. Alex From bakster@MIT.EDU Thu Oct 27 20:12:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S0CZ5o021031 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 20:12:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S0CYSu022474 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 20:12:34 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S0CVCE001808 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 20:12:32 -0400 (EDT) Message-ID: <43616CEB.6030809@mit.edu> Date: Thu, 27 Oct 2005 20:12:27 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] Week 8 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2384 Content-Length: 136 X-Keywords: NonJunk NotJunk I'd like it if we could go over what it means to 'use the integral method' to find the bounds on a summation. Thanks, Alexander Bakst From jeffhoff@MIT.EDU Thu Oct 27 20:34:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S0Ye5o024313 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 20:34:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S0YdSu007403 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 20:34:39 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S0YVll005181 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 20:34:32 -0400 (EDT) Message-Id: <6.2.3.4.2.20051027203022.01e22678@hesiod> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 27 Oct 2005 20:32:27 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 8 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2385 Content-Length: 250 X-Keywords: NonJunk NotJunk Page 1 Section 2 / 2.1 Annuities Apparently I'm confused on how they work because I'm having a tough time with the first TP problem. I think i'm confused mostly with the future values and annuities and summations. can we do an example in class? From hzhou@MIT.EDU Thu Oct 27 20:37:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S0bv5o024511 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 20:37:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S0buSu009850 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 20:37:56 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-SEVENTY-ONE.MIT.EDU [18.247.5.71]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S0brXt005811 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 20:37:54 -0400 (EDT) Message-Id: <5.2.1.1.2.20051027203636.02aba728@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 27 Oct 2005 20:37:54 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: David-tp8 reading comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2386 Content-Length: 225 X-Keywords: NonJunk NotJunk Hi, I simply found the different methods of determining convergence to be very interesting. The one concerning comparing a series with another known convergent or divergent series is a very clever method. - Steven Zhou From pgroudas@MIT.EDU Thu Oct 27 21:17:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S1Ho5o029974 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 21:17:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S1HnSu005624 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 21:17:49 -0400 (EDT) Received: from sicp-31.mit.edu (SICP-31.MIT.EDU [18.62.1.234]) (authenticated bits=56) (User authenticated as pgroudas@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S1Hl1r011494 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 21:17:47 -0400 (EDT) Received: (from pgroudas@localhost) by sicp-31.mit.edu (8.12.9) id j9S1HlWj011499; Thu, 27 Oct 2005 21:17:47 -0400 Subject: [Hanson] Week 8 comments From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 27 Oct 2005 21:17:46 -0400 Message-Id: <1130462267.11210.7.camel@sicp-31.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2387 Content-Length: 283 X-Keywords: NonJunk NotJunk I found that overall, i grasped most of the comments, but working through some of the math was very confusing. For instance, the double sum example on page 13 would have been easier to follow with perhaps less steps and more explanation, or simply more explanation. -Paul Groudas From brevzin@MIT.EDU Thu Oct 27 21:42:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S1g05o003342 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 21:42:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S1fxSu021607 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 21:41:59 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S1fqUu015860 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 21:41:53 -0400 (EDT) Message-Id: <6.1.2.0.1.20051027213721.019ec6c8@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Thu, 27 Oct 2005 21:41:55 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2388 Content-Length: 731 X-Keywords: NotJunk So, I already showed Myers this, but the reading says that the source of the sum of squares formula is inexplicable. But it's not... Let Sn = 1^2 + 2^2 + ... + n^2, so that S(0) = 0, S(1) = 1, S(2) = 5, ... You know S(n) can be modeled by a polynomial function with rational coefficients, and since S(0), the constant term is 0, so there is a constant c such that n | cS(n) for all n. So n+1 | cS(n+1), but cS(n+1) = cS(n) + c(n+1)^2... so n+1 | cS(n+1) iff n+1 | cS(n). cS(2n+1) = 2cS(n) (mod 2n+1) and since 2n+1 | cS(2n+1), 2n+1 | 2cS(n), so 2n+1 | cS(n). So cS(n) has n, n+1, and 2n+1 as divisors. These are all relatively prime. So n(n+1)(2n+1) | cS(n). Plug in n = 1, 6 | c. Thus, S(n) = n(n+1)(2n+1) / 6. Barry From zacharyozer@gmail.com Thu Oct 27 22:03:22 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.197]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S23L5o005576 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:03:21 -0400 Received: by xproxy.gmail.com with SMTP id s8so135200wxc for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 19:03:21 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=gkseT77SH6Eoix0WATXVzKQxLvPmSaAWF3q7Au73GAtiNX3KY4/ojABVRilLEoe5C/CPOquz/WMFM+HCsqdroRPjQZqgKCpN+CWmmXQQXexSbK+NaeU9vLUd7PLBlIFQtPEn7Z43TyEGOSx4ImthrYTlG28pEDYyYo/p9z6eQFY= Received: by 10.65.137.14 with SMTP id p14mr1616633qbn; Thu, 27 Oct 2005 19:03:21 -0700 (PDT) Received: by 10.64.27.9 with HTTP; Thu, 27 Oct 2005 19:03:20 -0700 (PDT) Message-ID: <4f2613e50510271903h1aedf5e6scdfb037c6b14a231@mail.gmail.com> Date: Thu, 27 Oct 2005 22:03:21 -0400 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Question regarding Little O vs Big O MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9S23L5o005576 Status: RO X-UID: 2389 Content-Length: 236 X-Keywords: NotJunk What's the difference between them? It seems as though they're defined as practically the same thing. Also, Theta appears to be the same relation, only in a slightly different form. Is Theta derivable from Big Oh or Little Oh? -zozer From petek@MIT.EDU Thu Oct 27 22:24:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S2OX5o008631 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:24:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S2OVSu020024 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:24:32 -0400 (EDT) Received: from [18.194.1.37] (SN-THIRTY-SEVEN.MIT.EDU [18.194.1.37]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S2OTU6022488 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:24:29 -0400 (EDT) Message-ID: <43618BEA.1070106@mit.edu> Date: Thu, 27 Oct 2005 22:24:42 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] LN8 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.929 X-Spam-Level: * (1.929) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2390 Content-Length: 752 X-Keywords: NonJunk NotJunk I think I'd like to see a good lecture on Big O / Little O / Theta, just to help me get a real sense on how to differentiate between them.....often I've used Theta and Big O interchangeably.....

-P
 
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Pete Kruskall
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From ozcan@MIT.EDU Thu Oct 27 22:37:22 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S2bM5o011199 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:37:22 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9S2bKh8011872 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:37:21 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S2bIHG013785 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:37:18 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9S2bBol010618 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:37:11 -0400 (EDT) Message-Id: <5.2.1.1.2.20051027223300.02000d68@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 27 Oct 2005 22:37:08 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2391 Content-Length: 228 X-Keywords: NonJunk NotJunk section 3.2: evaluating the sum, the integral method: the entire section This method is so amazing, I would have never thought of approximating the sum by bounding it from up and down, and it also looks quite useful. yasin From bilodeau@MIT.EDU Thu Oct 27 22:42:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S2gx5o012220 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:42:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S2gvSu001939 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:42:58 -0400 (EDT) Received: from pbilodeau (MACGREGOR-TWO-TWENTY-ONE.MIT.EDU [18.239.5.221]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S2gpTR025509 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 22:42:51 -0400 (EDT) Message-Id: <200510280242.j9S2gpTR025509@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [jelani] Reading Comments Date: Thu, 27 Oct 2005 22:42:43 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0003_01C5DB47.BEBDFC10" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXbaUUydtnMXQPFQkOaqhkpCEvSMg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.593 X-Spam-Level: ** (2.593) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2392 Content-Length: 2347 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0003_01C5DB47.BEBDFC10 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit For this reading I thought that the notation f = O(g) to be particularly confusing. It seems to be another example of when "=" does not mean equal, like modular arithmetic. Also I find it not quite intuitive that the degree of f = degree of g if f = O(g) yet f ~ g is not true. Peter Bilodeau ------=_NextPart_000_0003_01C5DB47.BEBDFC10 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

For this reading I thought that the notation f =3D = O(g) to be particularly confusing.  It seems to be another example of when = “=3D” does not mean equal, like modular arithmetic.  Also I find it not = quite intuitive that the degree of f =3D degree of g if f =3D O(g) yet  f = ~ g is not true.

 

Peter Bilodeau

------=_NextPart_000_0003_01C5DB47.BEBDFC10-- From mwangi@MIT.EDU Thu Oct 27 22:47:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S2li5o012574 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 22:47:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S2lhSu005005 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 22:47:43 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-ONE-NINETY.MIT.EDU [18.241.5.190]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S2laPE026276 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 22:47:36 -0400 (EDT) Message-Id: <6.2.1.2.2.20051027224518.039f89e0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Thu, 27 Oct 2005 22:47:31 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 8 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2393 Content-Length: 261 X-Keywords: NonJunk NotJunk Passage: 3.1 Formalizing the problem Page: 7 I found this passage most surprising because, intuitively, I did not think that it was possible to make a stable stack whose overhang was more than 1/2 the length of a book. Sincerely, Timothy M. Mwangi From arup@MIT.EDU Thu Oct 27 23:34:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S3Yb5o017979 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 23:34:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S3YZWI004229 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 23:34:35 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-THREE-EIGHTEEN.MIT.EDU [18.247.6.63]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S3YTqN003386 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 23:34:29 -0400 (EDT) Message-Id: <6.2.3.4.2.20051027232904.035ebf40@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 27 Oct 2005 23:33:19 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2394 Content-Length: 174 X-Keywords: NonJunk NotJunk Section 3, pp 7-9: I just got kindof lost reading the explanation of how we get to the expression for B_n, so it would be helpful if we went over this in lecture. |Arup| From fluff@MIT.EDU Thu Oct 27 23:34:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S3Yq5o017984 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 23:34:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S3YoWI004414 for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 23:34:50 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S3Yn85003451 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 27 Oct 2005 23:34:49 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 8 reading Date: Thu, 27 Oct 2005 23:32:50 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2395 Content-Length: 383 X-Keywords: NonJunk NotJunk I thought the sudden appearance of sqrt(2*pi*n) in Stirling's Formula on pg.14 was like, whoa, where the hell did that come from? The whole part before that only talked about getting (n/e)^n. But I'm assuming the justification is long and nasty and that I wouldn't want to see it anyway. Also, the differences between the various oh's are way too subtle for me. ~Crystal From jjmonzon@MIT.EDU Thu Oct 27 23:38:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S3ck5o018365 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 23:38:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S3cjWI006869 for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 23:38:45 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-TWO-TWELVE.MIT.EDU [18.247.5.212]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S3chZA004049 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 27 Oct 2005 23:38:44 -0400 (EDT) Message-Id: <6.2.3.4.2.20051027232558.01cd1ae8@po9.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 27 Oct 2005 23:38:55 -0400 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: [David] - Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2396 Content-Length: 359 X-Keywords: NotJunk This topic (sums, products) is very interesting. I partically was amazed by the book stacking problem. I can't believe that this is possible. I hope that the big and small o formulas be discussed in the lecture. Josh Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From lana@MIT.EDU Thu Oct 27 23:57:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S3vr5o021901 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 23:57:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S3vqWI019115 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 23:57:52 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S3vphf006688 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 27 Oct 2005 23:57:51 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9S3vpH9025223; Thu, 27 Oct 2005 23:57:51 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 27 Oct 2005 23:57:51 -0400 Message-ID: <20051027235751.l8pj0cps6vr0gsw0@webmail.mit.edu> Date: Thu, 27 Oct 2005 23:57:51 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]-TP8 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2397 Content-Length: 219 X-Keywords: NonJunk NotJunk In my opinion, the most confusing part of the notes was the part about the sum bounded by integrals. I would like to see an efficient and rigorous method for finding these bounds for a general function in lecture. Lana From ryan786@MIT.EDU Fri Oct 28 00:01:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S41o5o022717 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 00:01:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S41nWI021541 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 00:01:49 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S41lER007241 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 00:01:47 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9S41lF8032734; Fri, 28 Oct 2005 00:01:47 -0400 Received: from PSK-NINETEEN.MIT.EDU (PSK-NINETEEN.MIT.EDU [18.217.1.19]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 00:01:47 -0400 Message-ID: <20051028000147.23zrxw3jzles0kcs@webmail.mit.edu> Date: Fri, 28 Oct 2005 00:01:47 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [sayan] tp8 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2398 Content-Length: 258 X-Keywords: NonJunk NotJunk I don't understand big O notation. I had a very hard time with the TPs dealing with it, even after going through the reading section multiple times. Hopefully this will be covered in MUCH detail in lecture, or I will come in for office hours. -Ryan Young From clintonb@MIT.EDU Fri Oct 28 00:04:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S44u5o023530 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:04:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S44tWI023334 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:04:55 -0400 (EDT) Received: from ccbibmx30 (NEW-FIFTY-EIGHT.MIT.EDU [18.241.5.58]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S44qm1007633 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:04:53 -0400 (EDT) Message-Id: <200510280404.j9S44qm1007633@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Week 8 Comments Date: Fri, 28 Oct 2005 00:03:38 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0098_01C5DB53.0D3DCE50" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXbdJLPm2wFbCnfQSmRwhzWl7vUOQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.781 X-Spam-Level: * (1.781) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2399 Content-Length: 3712 X-Status: A X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0098_01C5DB53.0D3DCE50 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit IRRELEVANT: That new logo scares me; take it down. Could we go over the Integral Method in class? I think a few more examples would be helpful. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0098_01C5DB53.0D3DCE50 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

IRRELEVANT: That new logo scares me; take it down. =

 

Could we go over the Integral Method in class? I = think a few more examples would be helpful.

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_0098_01C5DB53.0D3DCE50-- From nedzel@MIT.EDU Fri Oct 28 00:14:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S4EQ5o026153 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:14:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S4EPWI029032 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:14:25 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-TWO-TWENTY-FOUR.MIT.EDU [18.96.5.224]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S4EN2S009337 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:14:24 -0400 (EDT) Message-Id: <200510280414.j9S4EN2S009337@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 8 Comments Date: Fri, 28 Oct 2005 00:14:18 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 thread-index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: 2.037 X-Spam-Level: ** (2.037) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2400 Content-Length: 108 X-Keywords: NonJunk NotJunk I'm a bit confused by the presentation of asymptotic notation: "little oh" vs. "big oh" vs. theta - David From meyer@csail.mit.edu Fri Oct 28 00:19:08 2005 Return-Path: Received: from [192.168.1.101] (pool-71-248-160-242.bstnma.fios.verizon.net [71.248.160.242]) (authenticated bits=0) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S4J75o026424; Fri, 28 Oct 2005 00:19:07 -0400 Message-ID: <4361A6BB.7080702@csail.mit.edu> Date: Fri, 28 Oct 2005 00:19:07 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Clinton Blackburn CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 8 Comments References: <200510280404.j9S44qm1007633@outgoing.mit.edu> In-Reply-To: <200510280404.j9S44qm1007633@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Status: RO X-UID: 2401 Content-Length: 382 X-Keywords: it's gone. was was only there for a few minutes (preparing for Halloween :-) ) regards, A. Clinton Blackburn wrote: > IRRELEVANT: That new logo scares me; take it down. > > > > Could we go over the Integral Method in class? I think a few more > examples would be helpful. > > --- > > Clinton Blackburn > > DLP Have you seen it? > > > > > From meyer@imap.theory.csail.mit.edu Fri Oct 28 00:19:08 2005 Message-ID: <4361A6BB.7080702@csail.mit.edu> Date: Fri, 28 Oct 2005 00:19:07 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Clinton Blackburn CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 8 Comments References: <200510280404.j9S44qm1007633@outgoing.mit.edu> In-Reply-To: <200510280404.j9S44qm1007633@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 383 X-UID: 2402 X-Keywords: it's gone. was was only there for a few minutes (preparing for Halloween :-) ) regards, A. Clinton Blackburn wrote: > IRRELEVANT: That new logo scares me; take it down. > > > > Could we go over the Integral Method in class? I think a few more > examples would be helpful. > > --- > > Clinton Blackburn > > DLP Have you seen it? > > > > > From benlu@MIT.EDU Fri Oct 28 00:24:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S4O15o026810 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 00:24:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S4O0WI004549 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 00:24:00 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S4NrMR010534 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 00:23:53 -0400 (EDT) Message-ID: <4361A7DD.6000202@mit.edu> Date: Fri, 28 Oct 2005 00:23:57 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051013) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 8 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2403 Content-Length: 494 X-Keywords: NotJunk I had a very good understanding of Big Oh, Little Oh, and Theta notation in 6.001 and 6.170. For some reason, seeing the formal definitions just totally confused me. I want to think about them in terms of orders of growth in time or space, not in terms of limits and partial orders. Also, the Integral Method was never really well described. Is this just something that we have to infer by looking at a graph? By inspection? Is there an actual method? Does it have to be exact? ~Ben Lu From tonyng@MIT.EDU Fri Oct 28 00:58:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S4wZ5o029535 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:58:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S4wYWI024674 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:58:34 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S4wVx4014633 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 00:58:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20051028004725.02101c20@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 28 Oct 2005 00:58:44 -0400 To: 6042-probs@theory.csail.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments Week 8 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2404 Content-Length: 404 X-Keywords: NonJunk NotJunk I am a little uncertain about the Theta notation (Section 7.3, starting in pg 18). Does theta imply both an upper and a lower bound? So if I have something such as n^2, it can be bounded by 2n^2 and 0.5n^2 so it would be theta(n^2). And if we have an unusual function that has a different asymptotic upper and lower bound, would that mean that there is no theta notation for that function? - Tony From hsoumare@MIT.EDU Fri Oct 28 01:03:10 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S53A5o030737 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:03:10 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S539Rb000801 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:03:10 -0400 (EDT) Received: from [18.241.5.187] (NEW-ONE-EIGHTY-SEVEN.MIT.EDU [18.241.5.187]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9S534ok017603 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:03:08 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8752D710-4F95-4B94-9DFB-4CD9C7AD06AB@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: {Sayan} Weekly Email Comment Date: Fri, 28 Oct 2005 01:12:00 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2405 Content-Length: 190 X-Keywords: NonJunk NotJunk I found the most interesting passage to be the one on page 6-7 about how $1 Million today is worth more than an annuity that increases as time goes on paid out forever. Hamidou Soumare From rshroff@MIT.EDU Fri Oct 28 01:12:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S5C65o032262 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 01:12:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S5C5WI001649 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 01:12:05 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S5BwL6015982 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 01:11:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9S5Bwt3017358; Fri, 28 Oct 2005 01:11:58 -0400 Received: from NEXT-FOUR-NINETY-SIX.MIT.EDU (NEXT-FOUR-NINETY-SIX.MIT.EDU [18.242.6.241]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 01:11:58 -0400 Message-ID: <20051028011158.4o7xzvp6fu4e8ok4@webmail.mit.edu> Date: Fri, 28 Oct 2005 01:11:58 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David]Reading Assigment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2406 Content-Length: 193 X-Keywords: NonJunk NotJunk Reference: Section 7, Asymptotic Notation. I found this part of the reading extremely interesting but difficult as well. Hopefully we will go over it in lecture in some detail. -Rahul Shroff From jstritar@MIT.EDU Fri Oct 28 01:12:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S5CN5o032267 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:12:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S5CMWI001828 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:12:22 -0400 (EDT) Received: from JSTRITAR.MIT.EDU (JSTRITAR.MIT.EDU [18.245.4.29]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S5CK5I016013 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:12:20 -0400 (EDT) Subject: [Sayan] Reading comments From: Jon Stritar To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 28 Oct 2005 01:16:16 -0400 Message-Id: <1130476576.18698.1.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2407 Content-Length: 162 X-Keywords: NonJunk NotJunk 3. Book Stacking I thought this was an interesting application of sums and Harmonic numbers. It seemed like a very clever approach to the problem. Jon Stritar From lkini@MIT.EDU Fri Oct 28 01:19:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S5JP5o001275 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:19:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S5JMWI005407; Fri, 28 Oct 2005 01:19:23 -0400 (EDT) Received: from [18.239.6.100] (MACGREGOR-THREE-FIFTY-FIVE.MIT.EDU [18.239.6.100]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S5JJho016724 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Fri, 28 Oct 2005 01:19:21 -0400 (EDT) Message-ID: <4361B4D7.10008@mit.edu> Date: Fri, 28 Oct 2005 01:19:19 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu CC: lkini@MIT.EDU Subject: [Hanson] Week 8 Lecture Notes Question Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2408 Content-Length: 330 X-Keywords: NonJunk NotJunk Hi Hanson, The thing I found most difficult to understand (but I sorta get it) is the use of the integral method in 3.2 Evaluating the Sum page 10. I had, at first, difficulty understanding why the integral was 1 + S (1/x) and the lower bound was S (1/[x+1]). I now realize the 1 is to offset the stairstep integral. Lohith From ereid@MIT.EDU Fri Oct 28 01:44:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S5iM5o004428 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:44:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S5iLWI018390 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:44:21 -0400 (EDT) Received: from [18.242.7.22] (NEXT-FIVE-THIRTY-THREE.MIT.EDU [18.242.7.22]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S5iA4f018802 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:44:14 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <5612830C-7A43-478F-9CC4-A55B8FBF53CF@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Reading comments Date: Fri, 28 Oct 2005 01:43:58 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2409 Content-Length: 168 X-Keywords: NonJunk NotJunk I'm still a little confused as to what the methodology for the integral method entails - what is the standard method used to figure out which integrals to look at? From hkhall@MIT.EDU Fri Oct 28 01:50:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S5ot5o004768 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:50:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S5ogWI021536; Fri, 28 Oct 2005 01:50:42 -0400 (EDT) Received: from [18.96.6.79] (SIMMONS-THREE-THIRTY-FOUR.MIT.EDU [18.96.6.79]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S5od4l019367 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Fri, 28 Oct 2005 01:50:40 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: [David] Reading Comments 8, Harrison Hall Date: Fri, 28 Oct 2005 01:52:43 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2410 Content-Length: 717 X-Keywords: NonJunk NotJunk David- Two things intrigued me this week with the reading. I understand the book stacking algorithm, but can it be done on objects where the center of mass is not centralized or the objects are considered linearly symmetric, i feel sure that there can, but is there an algorithm for such? I find it cool that they(CS people) finally gave a formal definition of Big O notation, little O, and Theta. I was always able to do them but never understood the workings behind them or from where they derived other that intuition. And where does this constant C applied to Big O come from? Is it assumed that 10x^2=O(x^2)-->10x^2=O(c*x^2)? Thanks, this was my favorite reading so far, cool topic. -Harrison From mracich@MIT.EDU Fri Oct 28 01:57:19 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S5vJ5o007386 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:57:19 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S5vI6I002116 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:57:18 -0400 (EDT) Received: from MACGREGOR-TWO-NINETY-TWO.MIT.EDU (MACGREGOR-TWO-NINETY-TWO.MIT.EDU [18.239.6.37]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9S5vBok019362 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 01:57:11 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 8 (Sums, Products, & Asymptotics) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 28 Oct 2005 01:57:10 -0400 Message-Id: <1130479030.8370.23.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2411 Content-Length: 205 X-Keywords: NotJunk I found section 6, "Stirling's Approximation", (starting on page 13) interesting, but also kind of confusing. I would appreciate it if this was reviewed in greater depth during lecture. Moira Racich From lmccart@MIT.EDU Fri Oct 28 02:06:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S6605o007952 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 02:06:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S61NAk025585 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 02:05:59 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S5wK3K019887 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 01:58:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9S5wKxp008423; Fri, 28 Oct 2005 01:58:20 -0400 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 01:58:20 -0400 Message-ID: <20051028015820.0k9lnqxy75tkw80g@webmail.mit.edu> Date: Fri, 28 Oct 2005 01:58:20 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 8 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2412 Content-Length: 428 X-Status: A X-Keywords: NotJunk The part I found most confusing in this week's reading was the stuff about Stirling's Approximation. If we could go over this in class that would be awesome. About my group, it has gotten somewhat better. If you would prefer me to stay with my current group, I guess this late in the semester it doesn't really matter that much. I appreciate yours and the professors efforts to help our group work better together. -lauren From kevin08@MIT.EDU Fri Oct 28 02:15:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S6Fr5o008486 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 02:15:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S6FpAa002637 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 02:15:52 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-FORTY-SEVEN.MIT.EDU [18.242.5.47]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S6FjmU021012 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 02:15:45 -0400 (EDT) Message-Id: <6.2.3.4.2.20051028021423.02828608@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 28 Oct 2005 02:15:44 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson]@mit.edu 6.042 Reading 8 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2413 Content-Length: 228 X-Keywords: NotJunk Hi Hanson, I was confused about the double summation in section 5, page 14. I don't understand how many of the equalities are rationalized. If we could go over that step by step in class, that would be great. Thanks, Kevin From kevin08@MIT.EDU Fri Oct 28 02:16:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S6Gr5o008809 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 02:16:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S6GpAa003218 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 02:16:52 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-FORTY-SEVEN.MIT.EDU [18.242.5.47]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S6Gnro021085 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 02:16:49 -0400 (EDT) Message-Id: <6.2.3.4.2.20051028021605.0283f570@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 28 Oct 2005 02:16:48 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: small typo Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2414 Content-Length: 53 X-Keywords: NotJunk on page 15, 1+10^-6 should be 1*10^-6 (I think ...) From jehan@MIT.EDU Fri Oct 28 02:33:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S6Xc5o013210 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 02:33:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S6XbAa011004 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 02:33:37 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S6XZ3H022109 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 02:33:36 -0400 (EDT) Message-Id: <5.2.1.1.2.20051028023133.00bc2b08@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 28 Oct 2005 02:33:35 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 8 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2415 Content-Length: 38 X-Keywords: NonJunk NotJunk I don't understand harmonic numbers. From antonk@MIT.EDU Fri Oct 28 03:05:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S75l5o018928 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:05:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S75kAa025630 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:05:46 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S75hGN023783 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:05:43 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] need a clarification Date: Fri, 28 Oct 2005 03:05:28 -0400 Organization: Massachusetts Institute of Technology Message-ID: <000001c5db8d$fa855180$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0001_01C5DB6C.7373B180" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXbjfm9uUZcQtbUSxmuzNgqeTA+Pw== X-Spam-Score: 0.645 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2416 Content-Length: 3513 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0001_01C5DB6C.7373B180 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi, In the chapter about the Big Oh I have some question regarding the constant use of the limit functions. If it is possible I would like to hear more about the different lims that are used there and why are they used. I understand that the Big Oh the upper bound, sometimes treated as the worse case scenario in programs, but I don't understand the different notations of the lims. Thank you, Anton. ------=_NextPart_000_0001_01C5DB6C.7373B180 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi,

In the chapter about the Big Oh I have some question regarding the constant use of the limit = functions.

If it is possible I would like to hear more about the different lims that are used there and why are they = used.

I understand that the Big Oh the upper bound, = sometimes treated as the worse case scenario in programs, but I don’t = understand the different notations of the lims.

 

 

Thank you,

 

Anton.

------=_NextPart_000_0001_01C5DB6C.7373B180-- From mpapi@MIT.EDU Fri Oct 28 03:41:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S7fd5o025906 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:41:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S7fcAa011806 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:41:38 -0400 (EDT) Received: from mpapi.MIT.EDU (MPAPI.MIT.EDU [18.239.4.219]) (authenticated bits=0) (User authenticated as mpapi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S7faJi025352 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:41:37 -0400 (EDT) Subject: [Jelani] Week 8 comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 28 Oct 2005 03:41:42 -0400 Message-Id: <1130485302.13339.11.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2417 Content-Length: 294 X-Keywords: NonJunk NotJunk Page 17, section 7.2 on Big Oh - it's pretty straightforward but it's something that I tend to mess up anyway. (It doesn't help that other CS courses sort of blur the distinction between big oh and theta.) I'd definitely like to see more examples, for sure, especially ones that use log. -M From ctsims@MIT.EDU Fri Oct 28 03:41:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S7fm5o025934 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:41:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S7fkAa011858 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:41:47 -0400 (EDT) Received: from HowardRoark.mit.edu (EASTCAMPUS-SEVEN-O-EIGHT.MIT.EDU [18.238.5.197]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S7fhm6025357 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 03:41:44 -0400 (EDT) Message-Id: <5.2.1.1.2.20051028023443.01ddbd18@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 28 Oct 2005 03:39:33 -0400 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: [6.042] Reading Comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2418 Content-Length: 1365 X-Keywords: NonJunk NotJunk 1) f(x) = log(x) f(1) = 0 g(x) = x g(1) = 1 lim g(x)/f(x) = x/log(x) = (By L'Hopital's) 1/(1/(x*ln(10))) = lim x*ln(10) = oo x->oo x->oo x = log(x) has no intersections. g(x) starts higher than f(x), ends higher than f(x), and the two graphs do not intersect, therefore g(x) > f(x) for all x > 1. 2) Transitivity lim f(x) = 0 lim g(x) = 0 f(x) < g(x) < h(x) so lim f(x) = 0 x-oo g(x) x->oo h(x) h(x) fRg gRh fRh asymetric lim f(x) = 0 lim g(x) = oo x->oo g(x) x->oo f(x) if fRg, not gRf 3) f ~g iff f and g are asymptotically equal, f and g are asymptotically equal iff at x ->oo df/dx = dg/dx At x ->oo df/dx = dg/dx iff f = g + j(function that relates non x->oo, to add factors that were ignored due to x being infinite) j = h iff h = o(g), otherwise f != O(g + h) which must be true for the ~ relation. Thus f ~g iff f = g + h I found the passage on page 12 regarding Exchanging the order of summation to be somewhat confusing. I would like to have the sections about double sums to be explained moreso in lecture. In particular, how sums which are not cleanly and directly related are handled. From mdmurray@MIT.EDU Fri Oct 28 03:51:21 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S7pL5o026677 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:21 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9S7pK4j011434 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:20 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S7pKec022037 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:20 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9S7pHok022978 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:17 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <43ECA1EA-E5E4-4415-AFCA-2814AF5AE195@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Michael Murray Subject: [Hanson] E-mail question Date: Fri, 28 Oct 2005 03:50:44 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2419 Content-Length: 505 X-Keywords: NonJunk NotJunk I don't really understand the section on Double Summing. I don't understand the "triangle chart" and also some lines of the derivation at the end of the section. Maybe it would help if there was some explanation of how to get from one line to the next in the derivation. Also in general I feel the readings are getting very long. We had a 15 page reading last friday for this monday, and now we have a 20 page reading from wednesday. I think it is alot to take in in one week. Thanks, Michael From a_lopez@MIT.EDU Fri Oct 28 03:51:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S7pX5o026707 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S7pVAa016103 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:32 -0400 (EDT) Received: from [18.234.1.117] (DP-ONE-SEVENTEEN.MIT.EDU [18.234.1.117]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S7pUWN025759 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 03:51:30 -0400 (EDT) Message-ID: <4361D886.9080509@mit.edu> Date: Fri, 28 Oct 2005 03:51:34 -0400 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] week 8 reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2420 Content-Length: 449 X-Keywords: NonJunk NotJunk I didn't really understand the book stacking problem. I specially didn't understand how the formula B_(n+1) = B_n + 1/(2(n+1)) was derived. And I don't know how to actually stack up the books, that is, if someone gave me n books and told me to stack them up, I know I can do it, but I don't know how. Also, if I already have n books stacked and I want to add one more, do I have to move the n books that are already stacked? Adriana Lpez From kjhollen@MIT.EDU Fri Oct 28 04:17:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9S8H75o001871 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 04:17:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9S8H6Aa027520 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 04:17:06 -0400 (EDT) Received: from MACGREGOR-FOUR-SIXTY-TWO.MIT.EDU (MACGREGOR-FOUR-SIXTY-TWO.MIT.EDU [18.239.6.207]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9S8H4xg026806 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 04:17:04 -0400 (EDT) Subject: [Hanson] interesting From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 28 Oct 2005 04:17:03 -0400 Message-Id: <1130487424.21837.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2421 Content-Length: 120 X-Keywords: NonJunk NotJunk The section on the integral method (p. 10) was interesting. Hope to get some of the math down in class tomorrow. Kate From jacques@MIT.EDU Fri Oct 28 06:25:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SAPH5o013210 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 06:25:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SAPG1p022221 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 06:25:16 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SAPEUa001613 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 06:25:14 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9SAPEsp018548; Fri, 28 Oct 2005 06:25:14 -0400 Received: from EASTCAMPUS-SEVEN-O-SEVEN.MIT.EDU (EASTCAMPUS-SEVEN-O-SEVEN.MIT.EDU [18.238.5.196]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 06:25:14 -0400 Message-ID: <20051028062514.qcdu1bgib9s80s00@webmail.mit.edu> Date: Fri, 28 Oct 2005 06:25:14 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 8 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2422 Content-Length: 509 X-Status: A X-Keywords: NonJunk NotJunk For Week 8, page 14-15, I found the sections on Stirling's Formula and Approximation sort of hand-wavy (technical term). It does say for Stirling's Approximation that the reason there is no proof of it's legitimacy is that "the details are nasty", so I can understand why it was omitted (although it probably would have been cool). But I don't get where the root 2*pi*n comes from in Stirling's Formula; it just seems to come out of thin air without any reason but "more careful analysis" (what analysis?). From cbossard@MIT.EDU Fri Oct 28 06:25:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SAPn5o013237 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 06:25:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SAPl1p022450 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 06:25:48 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SAPf94001627 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 06:25:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9SAPfWN030769; Fri, 28 Oct 2005 06:25:41 -0400 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 06:25:41 -0400 Message-ID: <20051028062541.6zt0zhuyl1hc4ogo@webmail.mit.edu> Date: Fri, 28 Oct 2005 06:25:41 -0400 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 8 Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2423 Content-Length: 184 X-Keywords: NonJunk NotJunk I find big/little O and Theta very confusing. Also can you do a short recap on annuities b/c I thought I understood it and then I was unable to solve the first tutor problem. Cynthia From xiaoranz@MIT.EDU Fri Oct 28 08:30:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SCUi5o024541 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 08:30:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SCUh2g024302 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 08:30:43 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SCUfKa013696 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 08:30:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9SCUfTq026610; Fri, 28 Oct 2005 08:30:41 -0400 Received: from MCCORMICK-SEVEN-SIXTY.MIT.EDU (MCCORMICK-SEVEN-SIXTY.MIT.EDU [18.240.7.249]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 08:30:41 -0400 Message-ID: <20051028083041.a6m8npvwc80k00g0@webmail.mit.edu> Date: Fri, 28 Oct 2005 08:30:41 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Weekly Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2424 Content-Length: 556 X-Keywords: NonJunk NotJunk I thought finding annuity and big/little O are interesting topics. But the rules for mannipulating more than one summations are not clear, is there a list of rules we can apply to summations like those we can apply to mod? Sharon ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From meyer@imap.theory.csail.mit.edu Fri Oct 28 08:54:43 2005 Message-ID: <43621F96.80406@csail.mit.edu> Date: Fri, 28 Oct 2005 08:54:46 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Lauren McCarthy CC: David Shin , Ronitt Rubinfeld Subject: Re: [david] week 8 References: <20051028015820.0k9lnqxy75tkw80g@webmail.mit.edu> In-Reply-To: <20051028015820.0k9lnqxy75tkw80g@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 1036 Status: RO X-UID: 2425 X-Keywords: Glad to hear we've been able to help. I think it's wiser to stick with your group for now, but the choice remains yours -- feel free to request a switch at any time, and let me know of any other problems that may come up. Re: Stirling: it's a bit of elementary, but ingenious, 18th century calculus reasoning that leads to the final formula. It could be understood by 6.042 students, but would cost a lecture to spell out fully and would dilute our focus on discrete Math, so we omit the details. Proofs are available in many calculus texts. regards, A. Lauren McCarthy wrote: >The part I found most confusing in this week's reading was the stuff about >Stirling's Approximation. If we could go over this in class that would be >awesome. > >About my group, it has gotten somewhat better. If you would prefer me to stay >with my current group, I guess this late in the semester it doesn't really >matter that much. I appreciate yours and the professors efforts to help our >group work better together. > >-lauren > > From alisonc@MIT.EDU Fri Oct 28 09:06:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SD6C5o030245 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 09:06:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SD6B2g019693 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 09:06:11 -0400 (EDT) Received: from [18.238.6.107] (EASTCAMPUS-EIGHT-SEVENTY-FOUR.MIT.EDU [18.238.6.107]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SD68wY021725 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 09:06:09 -0400 (EDT) Message-ID: <43622240.5040006@mit.edu> Date: Fri, 28 Oct 2005 09:06:08 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Comments on course notes #8 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2426 Content-Length: 255 X-Keywords: NonJunk NotJunk It would be nice to hear more in class on the summation and approximation formulae. I'm having a little trouble getting a firm grasp on the Integral Method particularly... how much of the given example is general and how much is specific to that case. From meyer@imap.theory.csail.mit.edu Fri Oct 28 09:08:06 2005 Message-ID: <436222B9.4030402@csail.mit.edu> Date: Fri, 28 Oct 2005 09:08:09 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: jacques Subject: Re: [David] Week 8 Reading Comment References: <20051028062514.qcdu1bgib9s80s00@webmail.mit.edu> In-Reply-To: <20051028062514.qcdu1bgib9s80s00@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 969 Status: RO X-UID: 2427 X-Keywords: Some elementary, but ingenious, 18th century calculus reasoning leads to Stirling's formula. It could be understood by 6.042 students, but would cost a lecture to spell out fully and would dilute our focus on discrete Math. So we settle for proving a rough bound using the integral method, and then just state the exact bound w/o proof. You'll find proofs in many calculus texts if you want to pursue this. Regards, A. jacques wrote: >For Week 8, page 14-15, I found the sections on Stirling's Formula and >Approximation sort of hand-wavy (technical term). It does say for Stirling's >Approximation that the reason there is no proof of it's legitimacy is that "the >details are nasty", so I can understand why it was omitted (although it probably >would have been cool). But I don't get where the root 2*pi*n comes from in >Stirling's Formula; it just seems to come out of thin air without any reason >but "more careful analysis" (what analysis?). > > From aeon@MIT.EDU Fri Oct 28 09:17:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SDHc5o031200 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:17:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SDHb2g027975 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:17:37 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SDHV0P024927 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:17:31 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9SDHVOC028659; Fri, 28 Oct 2005 09:17:31 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 07:17:31 -0600 Message-ID: <20051028071731.ow8q4btmblkw04gs@webmail.mit.edu> Date: Fri, 28 Oct 2005 07:17:31 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 8 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2428 Content-Length: 375 X-Keywords: NonJunk NotJunk Page 15, "Asymptotic Notation" Not much to say other than there are very few examples for the three sections. They're not bad examples, but they're not particularly elucidating and they're only given for special cases. Also, if there are some visuals, like graphs, they'd be perfect to help demonstrate these concepts (particularly with respect to algorithms). John Marrero From vbrobbey@MIT.EDU Fri Oct 28 09:21:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SDL25o032640 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 09:21:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SDL12g000582 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 09:21:01 -0400 (EDT) Received: from TheSlate (NEW-FIVE-SEVENTEEN.MIT.EDU [18.241.7.6]) (authenticated bits=0) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SDKdGR025911 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 09:20:49 -0400 (EDT) Message-ID: <002701c5dbc2$715c9b40$8906f112@TheSlate> From: "Valery Kwasi Brobbey" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson]Week 8 Comments Date: Fri, 28 Oct 2005 09:20:47 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0024_01C5DBA0.E1BBAAB0" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.84 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2429 Content-Length: 1911 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0024_01C5DBA0.E1BBAAB0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable There were a couple of things in this week's notes that I have seen = before. I saw the order of growth function in 6.001. For some reason the = explanation I got in 6.001 made more intuitive sense that the one in the = notes. It took me quite a while to figure out that they're the same = thing. In high school I learnt a way to find closed formulas for series. If you = have a series F(n), write out the first and las few terms of F(n), = -F(n-1) and -F(n+1). You would find a pattern and be able to cross out = all the terms besides a first one or 2 and the last few terms. ------=_NextPart_000_0024_01C5DBA0.E1BBAAB0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
There were a couple of things in this = week's notes=20 that I have seen before. I saw the order of growth function in 6.001. = For some=20 reason the explanation I got in 6.001 made more intuitive sense that the = one in=20 the notes. It took me quite a while to figure out that they're the same=20 thing.
In high school I learnt a way to find = closed=20 formulas for series. If you have a series F(n), write out the first and = las few=20 terms of F(n), -F(n-1) and -F(n+1). You would find a pattern and be able = to=20 cross out all the terms besides a first one or 2 and the last few=20 terms.
------=_NextPart_000_0024_01C5DBA0.E1BBAAB0-- From miki_tnd@MIT.EDU Fri Oct 28 09:23:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SDNu5o000320 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:23:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SDNs2g003031 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:23:55 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SDNmpV026931 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:23:48 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9SDNmsD025832; Fri, 28 Oct 2005 09:23:48 -0400 Received: from 18.214.1.233 ([18.214.1.233]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 09:23:48 -0400 Message-ID: <20051028092348.q1o25mccnge8owgk@webmail.mit.edu> Date: Fri, 28 Oct 2005 09:23:48 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: (Sayan) TP8 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2430 Content-Length: 101 X-Keywords: NotJunk I'm not understanding the integral method described on pg.10. Can you go over this in class? ~Thu From kromer@MIT.EDU Fri Oct 28 09:24:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SDOZ5o000358 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:24:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SDOY2g003735 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:24:34 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SDOSvo027192 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:24:28 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9SDOSkH030754; Fri, 28 Oct 2005 09:24:28 -0400 Received: from BURTON-EIGHTY-FIVE.MIT.EDU (BURTON-EIGHTY-FIVE.MIT.EDU [18.247.5.85]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 09:24:28 -0400 Message-ID: <20051028092428.710ergh40n40kwog@webmail.mit.edu> Date: Fri, 28 Oct 2005 09:24:28 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2431 Content-Length: 497 X-Keywords: NotJunk p. 8 "So the maximum overhang, Bn+1, of a stack of n+1 books is obtained by placing a maximum overhang stable stack of n books on top of the bottom book. And we get the biggest overhang for the stack of n+1 books by placing the center of mass of the n books right over the edge of the bottom book..." This sounds reasonable, but I'm not sure how I'd prove that its true (in particular, how you know that you do best by placing the center of mass of the n books over the edge of the bottom book). From fgreen@MIT.EDU Fri Oct 28 09:42:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SDgg5o003574 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 09:42:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SDge2g020677 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 09:42:41 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SDgdAY003739 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 09:42:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9SDgdpW028035; Fri, 28 Oct 2005 09:42:39 -0400 Received: from EASTCAMPUS-EIGHT-FIFTY.MIT.EDU (EASTCAMPUS-EIGHT-FIFTY.MIT.EDU [18.238.6.83]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 09:42:39 -0400 Message-ID: <20051028094239.39wuze8ep9fok8kg@webmail.mit.edu> Date: Fri, 28 Oct 2005 09:42:39 -0400 From: Forrest O Green To: 6042-probs@theory.csail.MIT.EDU Subject: Julani Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2432 Content-Length: 180 X-Keywords: NonJunk NotJunk Section 2, Pages 3 and 4 Is there any particular significance of the annuity type problems to computer science, or is it just a good example of geometric series? -Forrest Green From sil_03@MIT.EDU Fri Oct 28 09:51:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SDpV5o005232 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:51:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SDpU2g028675 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:51:30 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SDpS8I006943 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 09:51:28 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9SDpS48029085; Fri, 28 Oct 2005 09:51:28 -0400 Received: from NEW-TWO-NINETY-EIGHT.MIT.EDU (NEW-TWO-NINETY-EIGHT.MIT.EDU [18.241.6.43]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 09:51:28 -0400 Message-ID: <20051028095128.79eg9xds4iuscsco@webmail.mit.edu> Date: Fri, 28 Oct 2005 09:51:28 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] week 8 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2433 Content-Length: 251 X-Keywords: NonJunk NotJunk I found this week's notes very hard to understand. An explanation of what was done in each step to simplify the math problem would be nice, like on page 8 and 13. Also, more examples for big Oh and theta would help me understand it better. ~Silvia From rshearer@MIT.EDU Fri Oct 28 10:26:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SEQ95o011368 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 10:26:09 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SEQ82g000705 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 10:26:08 -0400 (EDT) Received: from [18.240.7.234] (MCCORMICK-SEVEN-FORTY-FIVE.MIT.EDU [18.240.7.234]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SEQ4EU020453 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 10:26:05 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) In-Reply-To: <6826b5843ef05f2637fba6c0fccd784f@mit.edu> References: <6826b5843ef05f2637fba6c0fccd784f@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Rachel Shearer Subject: [Jelani] TP8 Comment Date: Fri, 28 Oct 2005 10:26:03 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2434 Content-Length: 118 X-Keywords: NonJunk NotJunk I don't think I understand Big Oh very well. I got the tutor problems right but I don't really know why :) Rachel From nancyk@MIT.EDU Fri Oct 28 10:36:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SEam5o012726 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 10:36:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SEaj2g011021; Fri, 28 Oct 2005 10:36:45 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SEacrr024711; Fri, 28 Oct 2005 10:36:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9SEacb4006279; Fri, 28 Oct 2005 10:36:38 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 10:36:38 -0400 Message-ID: <20051028103638.83jw0n83zrc4g44s@webmail.mit.edu> Date: Fri, 28 Oct 2005 10:36:38 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 8 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2435 Content-Length: 394 X-Keywords: NonJunk NotJunk Hi, I think that the asymptotic notation is explained well, but some more examples might be helpful in the notes. Also, for one of the online tutorial asymptotic notation problems, the answer was "none." I realize that this was because we were limited to choosing choosing n in the form O(x^n), but in general does there ever not exist asymptotic notation for a function? Thanks, Nancy Keuss From mukkala@MIT.EDU Fri Oct 28 10:47:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SEl65o014286 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 10:47:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SEl52g021765 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 10:47:05 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-SEVENTY-NINE.MIT.EDU [18.235.1.79]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SEkula029000 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 10:46:59 -0400 (EDT) Message-Id: <6.2.1.2.2.20051028104035.01e76150@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 28 Oct 2005 10:47:02 -0400 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 8 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2436 Content-Length: 658 X-Keywords: NotJunk In section 3.2, the notes describe evaluating the Sum using the Integral Method. In their example, in order to create upper and lower bounds for the summation of 1/i from i = 1 to n, they say that "the function 1/x is everywhere greater than or equal to the stairstep and so the integral of 1/x over this interval is an upper bound on the sum." However, in their calculation of the integral, they use Hn <= 1 + (integral of dx/x from 1 to n) for the upper bound calculation. Did they need to add an extra 1? In general, how does one pick these functions for the upper bound and lower bound? They seem fairly arbitrarily based on the summation. From harelw@gmail.com Fri Oct 28 10:56:28 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.204]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SEuS5o015088 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 10:56:28 -0400 Received: by xproxy.gmail.com with SMTP id s6so881075wxc for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 07:56:27 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=AjbLfZxpiMPHrRRCND6oSWPhYrqWUL8aDrcql+1rr2VC4c4b7oPndFdRwTTifCOAAQCV/QgATf49JkpYPRkBr4ZVbsXZP1YgdaDsHcy56eVV5tp7YpLBXbdxBz3KaZIE3mbjJkz7nf0r/j2HtR+2ZVLBZnnHkYDCv+kS/4oyx5A= Received: by 10.70.48.19 with SMTP id v19mr222525wxv; Fri, 28 Oct 2005 07:56:27 -0700 (PDT) Received: by 10.70.13.17 with HTTP; Fri, 28 Oct 2005 07:56:27 -0700 (PDT) Message-ID: <8c5248a80510280756hfe84489yb81ab59bc94e880f@mail.gmail.com> Date: Fri, 28 Oct 2005 10:56:27 -0400 From: "Harel M. Williams" Sender: harelw@gmail.com To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Week 8 Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_3283_5285812.1130511387700" Status: RO X-UID: 2437 Content-Length: 1082 X-Keywords: NonJunk NotJunk ------=_Part_3283_5285812.1130511387700 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hey Jelani, I would like to go over n-time stuff as well as the integral method; it is not really clear to me how to do them by the notes. ~H -- Harel M. Williams '06 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science cell: 516.610.3393 email: harelw@mit.edu ------=_Part_3283_5285812.1130511387700 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hey Jelani,

I would like to go over n-time stuff as well as the integral method; it is = not really clear to me how to do them by the notes.

~H

--
Harel M. Williams '06
Massachusetts Insti= tute of Technology
Department of Electrical Engineering and Computer Sc= ience

cell: 516.610.3393
email: harelw@mit.edu
------=_Part_3283_5285812.1130511387700-- From yaser@MIT.EDU Fri Oct 28 10:58:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SEwa5o015216 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 10:58:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SEwZ2g003650 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 10:58:35 -0400 (EDT) Received: from Charizard (NEW-THREE-FIFTY-THREE.MIT.EDU [18.241.6.98]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SEwXn9003803 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 10:58:33 -0400 (EDT) Message-Id: <200510281458.j9SEwXn9003803@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: week8 reading response Date: Fri, 28 Oct 2005 10:58:31 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0048_01C5DBAE.89340F50" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXb0A+cLtdj57P2QXSGe5Tn3e6cJA== X-Spam-Score: 1.781 X-Spam-Level: * (1.781) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2438 Content-Length: 5441 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0048_01C5DBAE.89340F50 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi David, I would like it if in lecture you guys can make clear the distinctions between the various notations in the reading. I realize the basic differences between little Oh, big Oh, theta, etc., but perhaps having a bit of explicit explanation regarding when to use them would be helpful. Thanks! _Yaser ------=_NextPart_000_0048_01C5DBAE.89340F50 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

I would like it if in lecture you = guys can make clear the distinctions between the various notations in the = reading. I realize the basic differences between little Oh, big Oh, theta, etc., = but perhaps having a bit of explicit explanation regarding when to use them = would be helpful.

 

Thanks!

 

_Yaser

------=_NextPart_000_0048_01C5DBAE.89340F50-- From crowell@MIT.EDU Fri Oct 28 11:03:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SF3O5o016219 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 11:03:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SF2x2g008221 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 11:03:00 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SF2wiH005628 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 11:02:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9SF2wsQ030262; Fri, 28 Oct 2005 11:02:58 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 11:02:58 -0400 Message-ID: <20051028110258.v5nh7ewrkh6xcc04@webmail.mit.edu> Date: Fri, 28 Oct 2005 11:02:58 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2439 Content-Length: 85 X-Keywords: NotJunk I don't understand the section on adding up the double series, page 12-13 section 5. From hmzhou@blackbird.csail.mit.edu Fri Oct 28 11:46:13 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SFkD5o023827; Fri, 28 Oct 2005 11:46:13 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9SFkDmb020514; Fri, 28 Oct 2005 11:46:13 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9SFkDl1020511; Fri, 28 Oct 2005 11:46:13 -0400 Date: Fri, 28 Oct 2005 11:46:13 -0400 (EDT) From: Hanson Zhou To: Angelique E Moscicki cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] week 8 comments In-Reply-To: <20051027185239.5y7gn1zapt7o0k40@webmail.mit.edu> Message-ID: References: <20051027185239.5y7gn1zapt7o0k40@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2440 Content-Length: 1075 X-Keywords: NotJunk > Harmonic numbers: Cool, seen these before, but not presented this way. > Stirling's Approximation: Where did he come up with the sqrt(2n) term? The You are not meant to know where the sqrt term comes from. Not necessary for our purposes here. > The proof of Lemma 7.3 is mystifying. Once I just accepted Lemma 7.3, the rest > was just some juggling of exponents. I remember seeing the proofs they Don't worry too much about it. Your understanding of this is most likely good enough. Lemma 7.3 simply proves what you already know by reducing it to something even easier to see: that x^a = o(x^b) for a Big O: > Necessarily more familiar from CS and math classes. > errrrm... lim sup? I know lim... and I thought I knew big O pretty well too... Sometimes sequences don't have limits, ie, 1, 0, 1, 0, 1, 0, 1... lim sup is the largest limit obtainable from any subsequence(in this case, 1). -Hanson From hmzhou@blackbird.csail.mit.edu Fri Oct 28 11:51:17 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SFpH5o025301; Fri, 28 Oct 2005 11:51:17 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9SFpHiP020566; Fri, 28 Oct 2005 11:51:17 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9SFpHw2020563; Fri, 28 Oct 2005 11:51:17 -0400 Date: Fri, 28 Oct 2005 11:51:17 -0400 (EDT) From: Hanson Zhou To: Elizabeth Reid cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Reading comments In-Reply-To: <5612830C-7A43-478F-9CC4-A55B8FBF53CF@MIT.EDU> Message-ID: References: <5612830C-7A43-478F-9CC4-A55B8FBF53CF@MIT.EDU> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2441 Content-Length: 563 X-Keywords: NotJunk Well...you usually simply integrate the function in the sum. For example, \sum x^2 can be approximated by looking at the integral of x^2 because summing is just a discrete version of integrating. When you integrate, you take the area under the curve. When you sum, you add up the areas of the rectangles under the curve. Make sense? -Hanson On Fri, 28 Oct 2005, Elizabeth Reid wrote: > I'm still a little confused as to what the methodology for the > integral method entails - what is the standard method used to figure > out which integrals to look at? > From hmzhou@blackbird.csail.mit.edu Fri Oct 28 11:52:12 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SFqC5o025401; Fri, 28 Oct 2005 11:52:12 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9SFqCUX020579; Fri, 28 Oct 2005 11:52:12 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9SFqCgT020576; Fri, 28 Oct 2005 11:52:12 -0400 Date: Fri, 28 Oct 2005 11:52:12 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson]@mit.edu 6.042 Reading 8 In-Reply-To: <6.2.3.4.2.20051028021423.02828608@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051028021423.02828608@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2442 Content-Length: 324 X-Keywords: NotJunk See me if you remain confused. -Hanson On Fri, 28 Oct 2005, Kevin Wang wrote: > Hi Hanson, > > I was confused about the double summation in section 5, page 14. I > don't understand how many of the equalities are rationalized. If we > could go over that step by step in class, that would be great. > > Thanks, > Kevin > > From hmzhou@blackbird.csail.mit.edu Fri Oct 28 11:55:07 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SFt75o025597; Fri, 28 Oct 2005 11:55:07 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9SFt6rH020610; Fri, 28 Oct 2005 11:55:06 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9SFt6X8020607; Fri, 28 Oct 2005 11:55:06 -0400 Date: Fri, 28 Oct 2005 11:55:06 -0400 (EDT) From: Hanson Zhou To: Michael Murray cc: 6042-probs@theory.csail.mit.edu Subject: Re: [Hanson] E-mail question In-Reply-To: <43ECA1EA-E5E4-4415-AFCA-2814AF5AE195@mit.edu> Message-ID: References: <43ECA1EA-E5E4-4415-AFCA-2814AF5AE195@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2443 Content-Length: 605 X-Keywords: NotJunk See me if this remains confusing. -Hanson On Fri, 28 Oct 2005, Michael Murray wrote: > I don't really understand the section on Double Summing. I don't > understand the "triangle chart" and also some lines of the derivation > at the end of the section. Maybe it would help if there was some > explanation of how to get from one line to the next in the derivation. > > Also in general I feel the readings are getting very long. We had a > 15 page reading last friday for this monday, and now we have a 20 > page reading from wednesday. I think it is alot to take in in one week. > > Thanks, > Michael > From hmzhou@blackbird.csail.mit.edu Fri Oct 28 11:58:40 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SFwe5o026436; Fri, 28 Oct 2005 11:58:40 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9SFweca020645; Fri, 28 Oct 2005 11:58:40 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9SFwdMH020642; Fri, 28 Oct 2005 11:58:40 -0400 Date: Fri, 28 Oct 2005 11:58:39 -0400 (EDT) From: Hanson Zhou To: "Xiaoran (Sharon) Zhang" cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Weekly Comments In-Reply-To: <20051028083041.a6m8npvwc80k00g0@webmail.mit.edu> Message-ID: References: <20051028083041.a6m8npvwc80k00g0@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2444 Content-Length: 787 X-Keywords: NotJunk Which summation tricks are not clear? There is no "big list" but you should understand the various tricks that we do use in the course. -Hanson On Fri, 28 Oct 2005, Xiaoran (Sharon) Zhang wrote: > I thought finding annuity and big/little O are interesting topics. But the rules > for mannipulating more than one summations are not clear, is there a list of > rules we can apply to summations like those we can apply to mod? > > Sharon > > ************************************************** > Xiaoran (Sharon) Zhang > Class of 2008 > Department of Biology & > Department of Electrical Engineering and Computer Science > Massachusetts Institute of Technology > 320 Memorial Drive, > Cambridge, MA 02139 > E-mail: xiaoranz@mit.edu > ************************************************** > From hmzhou@blackbird.csail.mit.edu Fri Oct 28 12:07:16 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SG7G5o027173; Fri, 28 Oct 2005 12:07:16 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9SG7G4J020745; Fri, 28 Oct 2005 12:07:16 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9SG7GI5020742; Fri, 28 Oct 2005 12:07:16 -0400 Date: Fri, 28 Oct 2005 12:07:15 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 8 reading comments In-Reply-To: <20051028103638.83jw0n83zrc4g44s@webmail.mit.edu> Message-ID: References: <20051028103638.83jw0n83zrc4g44s@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2445 Content-Length: 585 X-Keywords: NotJunk Well...nothing reasonable that I can think of. Given f(x), one can always construct the asymptotically larger xf(x). -Hanson On Fri, 28 Oct 2005, Nancy L Keuss wrote: > Hi, > > I think that the asymptotic notation is explained well, but some more examples > might be helpful in the notes. Also, for one of the online tutorial asymptotic > notation problems, the answer was "none." I realize that this was because we > were limited to choosing choosing n in the form O(x^n), but in general does > there ever not exist asymptotic notation for a function? > > Thanks, > Nancy Keuss > From scholtz@MIT.EDU Fri Oct 28 12:45:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SGjB5o002075 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 12:45:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SGjAFF015079 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 12:45:10 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SGj8Uv015939 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 12:45:08 -0400 (EDT) Message-Id: <200510281645.j9SGj8Uv015939@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 8 Comments Date: Fri, 28 Oct 2005 12:45:09 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5DBBD.6E83F210" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXb3vTn4b87RnIoSre7nSwkmnDIHA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.781 X-Spam-Level: * (1.781) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2446 Content-Length: 1680 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5DBBD.6E83F210 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I would like methods for finding summation formulas discussed in lecture. ------=_NextPart_000_0000_01C5DBBD.6E83F210 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I would like methods for finding summation = formulas discussed in lecture.

------=_NextPart_000_0000_01C5DBBD.6E83F210-- From kushan@MIT.EDU Fri Oct 28 13:12:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SHCf5o007544 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 13:12:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SHCeFF010573 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 13:12:40 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SHCcXl025041 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 28 Oct 2005 13:12:38 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9SHCc7U022455; Fri, 28 Oct 2005 13:12:38 -0400 Received: from RLE-13-099.MIT.EDU (RLE-13-099.MIT.EDU [18.62.13.99]) (User authenticated as kushan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 13:12:38 -0400 Message-ID: <20051028131238.316n32hafdsk4kso@webmail.mit.edu> Date: Fri, 28 Oct 2005 13:12:38 -0400 From: Kushan K Surana To: 6042-probs@theory.lcs.MIT.EDU Subject: TA: Hanson (Week 8 comments) MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2447 Content-Length: 675 X-Keywords: NonJunk NotJunk The notes are easy to follow until the book stacking problem. It would be helpful if the professor could go over the idea step by step. I understand that one book can overhang half a book length. But then generalizing that idea to (n+1) books is unclear. The use of equality for asymptotic notation was unclear until the last paragraph which explained it Also, from last week's notes, the last two explanations of why every boy should be matched with a girl and why every marriage is stable seem circular. It's hard to follow the argument in so many words. I would prefer a step-by-step explanation where one step follows from another, instead of a paragraph form. Kushan From dangut@MIT.EDU Fri Oct 28 13:35:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SHZD5o013746 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 13:35:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SHZCFF029097 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 13:35:12 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9SHZ6YV002711 for <6042-probs@theory.csail.MIT.EDU>; Fri, 28 Oct 2005 13:35:06 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9SHZ6R1030486; Fri, 28 Oct 2005 13:35:06 -0400 Received: from MACLAURIN-FIFTY-SIX.MIT.EDU (MACLAURIN-FIFTY-SIX.MIT.EDU [18.90.5.56]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 28 Oct 2005 13:35:06 -0400 Message-ID: <20051028133506.veo6mbacxkeo808k@webmail.mit.edu> Date: Fri, 28 Oct 2005 13:35:06 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: [Hansen] TP8 Reading Response MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2448 Content-Length: 258 X-Keywords: NonJunk NotJunk I just have a question concerning Big O, one of the problems discussed at the end is 4^x is not O(2^x), but it doesn't say what the Big-oh for it actually is. Is the big-oh for 4^x just a number arbitrarily larger than 4 to the power of x? Daniel Gutierrez From dshin@MIT.EDU Fri Oct 28 14:22:28 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SIMS5o019038 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 14:22:28 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SIMPh3029282; Fri, 28 Oct 2005 14:22:25 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-72.hsd1.ma.comcast.net [71.192.58.72]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9SIMIok026420; Fri, 28 Oct 2005 14:22:19 -0400 (EDT) Message-ID: <43626C43.8040306@mit.edu> Date: Fri, 28 Oct 2005 14:21:55 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Email comments for reading References: <20051028092428.710ergh40n40kwog@webmail.mit.edu> In-Reply-To: <20051028092428.710ergh40n40kwog@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2449 Content-Length: 598 X-Keywords: NotJunk Well, if it was any further out, it would fall over, right? Katherine A Romer wrote: >p. 8 "So the maximum overhang, Bn+1, of a stack of n+1 books is obtained by >placing a maximum overhang stable stack of n books on top of the bottom book. >And we get the biggest overhang for the stack of n+1 books by placing the >center of mass of the n books right over the edge of the bottom book..." > >This sounds reasonable, but I'm not sure how I'd prove that its true (in >particular, how you know that you do best by placing the center of mass of the >n books over the edge of the bottom book). > > From dshin@MIT.EDU Fri Oct 28 14:30:10 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SIUA5o022518 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 14:30:10 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9SIU8CM005416 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 14:30:09 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SIU8BJ016420; Fri, 28 Oct 2005 14:30:08 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-72.hsd1.ma.comcast.net [71.192.58.72]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9SIU5ok027253; Fri, 28 Oct 2005 14:30:05 -0400 (EDT) Message-ID: <43626E1D.5080306@mit.edu> Date: Fri, 28 Oct 2005 14:29:49 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Reading Comments 8, Harrison Hall References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2450 Content-Length: 1079 X-Keywords: NotJunk Yes, it is true that c*x^2 = O(x^2). As for book stacking with non-centralized centers of mass, the problem actually just gets easier the further your center of mass gets from the center. You can see this by placing unevenly balanced books so that their centers of mass are closer towards the table. Harrison King Hall wrote: > David- > Two things intrigued me this week with the reading. I understand the > book stacking algorithm, but can it be done on objects where the > center of mass is not centralized or the objects are considered > linearly symmetric, i feel sure that there can, but is there an > algorithm for such? I find it cool that they(CS people) finally gave > a formal definition of Big O notation, little O, and Theta. I was > always able to do them but never understood the workings behind them > or from where they derived other that intuition. And where does this > constant C applied to Big O come from? Is it assumed that > 10x^2=O(x^2)-->10x^2=O(c*x^2)? Thanks, this was my favorite reading > so far, cool topic. > -Harrison > From dshin@MIT.EDU Fri Oct 28 14:35:03 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SIZ35o023638 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 14:35:03 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9SIZ2CM011639 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 14:35:02 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9SIZ1BJ016688; Fri, 28 Oct 2005 14:35:02 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-72.hsd1.ma.comcast.net [71.192.58.72]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9SIYsok027689; Fri, 28 Oct 2005 14:34:54 -0400 (EDT) Message-ID: <43626F3E.6080101@mit.edu> Date: Fri, 28 Oct 2005 14:34:38 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 8 E-mail Comments References: <200510271958.j9RJwZ9a027373@outgoing.mit.edu> In-Reply-To: <200510271958.j9RJwZ9a027373@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------010607060602010104030705" X-Spam-Score: 1.218 X-Spam-Level: * (1.218) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2451 Content-Length: 3267 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------010607060602010104030705 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit The derivation of Stirling's approximation is not really important. What's important is that you know that log(n!) = theta(n log n). That is an extremely important relationship in computer science. We will get to see the Integral Method in more detail. DS Chris Yang wrote: > I did not really understand the derivation of Stirling's approximation > of the factorial - and I guess, to some extent, I did not really > understand the Integral Method. I assume both of these things will be > covered in greater detail in class. > > > > Thanks, > > Chris Yang > --------------010607060602010104030705 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit The derivation of Stirling's approximation is not really important.  What's important is that you know that log(n!) = theta(n log n).  That is an extremely important relationship in computer science.

We will get to see the Integral Method in more detail.

DS

Chris Yang wrote:

I did not really understand the derivation of Stirling’s approximation of the factorial – and I guess, to some extent, I did not really understand the Integral Method.  I assume both of these things will be covered in greater detail in class.

 

Thanks,

Chris Yang

--------------010607060602010104030705-- From shreyes19@gmail.com Fri Oct 28 15:20:13 2005 Return-Path: Received: from nproxy.gmail.com (nproxy.gmail.com [64.233.182.204]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9SJKC5o029581 for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 15:20:12 -0400 Received: by nproxy.gmail.com with SMTP id d4so175171nfe for <6042-probs@theory.lcs.mit.edu>; Fri, 28 Oct 2005 12:20:11 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=MCfTwxW+gwbB9EVa1FnRlik2Yu25NQNvJMozoUqAEoU90yHhsoQ04vSCUK23KQb8mLehIdROGKAt+oxOgiQQvSMxLuHcCxwrBqmpafxE9jx+KDSc/Dk+9c+MQvTJVCR+JywYmstxhhZ+0itCEHj4KR0vu6MYlFwzVQGZpLhQZeM= Received: by 10.48.31.7 with SMTP id e7mr271092nfe; Fri, 28 Oct 2005 12:20:11 -0700 (PDT) Received: by 10.48.161.13 with HTTP; Fri, 28 Oct 2005 12:20:11 -0700 (PDT) Message-ID: <3c33fe380510281220w209d2f7aw3f4cf068477f7523@mail.gmail.com> Date: Fri, 28 Oct 2005 15:20:11 -0400 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 8 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_23774_32243744.1130527211568" Status: RO X-UID: 2452 Content-Length: 1537 X-Keywords: NonJunk NotJunk ------=_Part_23774_32243744.1130527211568 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hey, Sorry again for the lateness. The part I'd like to see more discussed in lecture is the section on little oh and big oh notation. From other classes I've learned the basics of big O notation, but I'm confused about the difference between the two notations. >From the wording on page 17, it seems as if big O notation is equivalent to little oh , as Lemma 7.6 says If f =3D o(g) or f =18 g, then f =3D O(g). Basically, my questions are when are the different notations used, and whic= h is preferred when discussing an asymptotic relation. Thanks, Shreyes ------=_Part_23774_32243744.1130527211568 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hey,

Sorry again for the lateness.

The part I'd like to see more discussed in lecture is the section on little oh and big oh notation.  From other classes I've learned the basics of big O notation, but I'm confused about the difference between the two notations.  From the wording on page 17, it seems as if big O notation is equivalent to little oh , as Lemma 7.6 says If f =3D o(g) or f =18 g, then f =3D O(g).
Basically, my questions are when are the different notations used, and which is preferred when discussing an asymptotic relation.

Thanks,
Shreyes
------=_Part_23774_32243744.1130527211568-- From mwangi@MIT.EDU Fri Oct 28 23:11:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9T3BF5o008323 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:11:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9T3BE2e005948 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:11:14 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-TWO-EIGHTY-NINE.MIT.EDU [18.241.6.34]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9T3B7xt007468 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:11:08 -0400 (EDT) Message-Id: <6.2.1.2.2.20051028230113.039fbeb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 28 Oct 2005 23:11:06 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2453 Content-Length: 262 X-Keywords: NonJunk NotJunk Passage: 5.6 ....And the Boys Live Especially Happily Page: 14 I found this passage most surprising because it had seemed to me that the mating algorithm favored the girls since they were the ones choosing their suitors. Sincerely, Timothy M. Mwangi From mwangi@MIT.EDU Fri Oct 28 23:20:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9T3KU5o008763 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:20:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9T3KT2e010222 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:20:29 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-TWO-EIGHTY-NINE.MIT.EDU [18.241.6.34]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9T3KRFt008617 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:20:28 -0400 (EDT) Message-Id: <6.2.1.2.2.20051028231157.03980550@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 28 Oct 2005 23:20:27 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2454 Content-Length: 181 X-Status: A X-Keywords: NonJunk NotJunk Passage: 8.3 Fermat's Theorem Page: 21 I found this passage most difficult because I did not understand the proof given of Fermat's Theorem. Sincerely, Timothy M. Mwangi From mwangi@MIT.EDU Fri Oct 28 23:37:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9T3bl5o010272 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:37:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9T3bk2e017417 for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:37:46 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-TWO-EIGHTY-NINE.MIT.EDU [18.241.6.34]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9T3bdud009819 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 28 Oct 2005 23:37:40 -0400 (EDT) Message-Id: <6.2.1.2.2.20051028232037.039eaa40@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 28 Oct 2005 23:37:39 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2455 Content-Length: 238 X-Status: A X-Keywords: NonJunk NotJunk Passage: 5.1 Euler's Formula Page: 12 I found this passage most difficult because, although Euler's Formula is shown to be true by induction, I have no intuitive understanding of why it is true. Sincerely, Timothy M. Mwangi From meyer@imap.theory.csail.mit.edu Sat Oct 29 11:05:03 2005 Message-ID: <43638FA3.3020405@csail.mit.edu> Date: Sat, 29 Oct 2005 11:05:07 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Timothy Mwangi Subject: Re: [Sayan] Week 6 Comments References: <6.2.1.2.2.20051028231157.03980550@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051028231157.03980550@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 368 X-UID: 2456 X-Keywords: it's important to get this proof and the similar proof of Euler's thm straight. I suggest you take it up in office hours with your TA. regards, A. Timothy Mwangi wrote: > Passage: 8.3 Fermat's Theorem > Page: 21 > I found this passage most difficult because I did not understand the > proof given of Fermat's Theorem. > > Sincerely, > Timothy M. Mwangi From meyer@imap.theory.csail.mit.edu Sat Oct 29 11:10:16 2005 Message-ID: <436390DC.6060701@csail.mit.edu> Date: Sat, 29 Oct 2005 11:10:20 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Timothy Mwangi Subject: Re: [Sayan] Week 5 Comments References: <6.2.1.2.2.20051028232037.039eaa40@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051028232037.039eaa40@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 759 Status: RO X-UID: 2457 X-Keywords: I'm not sure what further intuition you're looking for; the only proof I've ever seen is essentially the one by induction in the notes. Sometimes seeing generalizations can help refine intuition, and Euler's formula does generalize to higher "genus" shapes beyond the plane -- like a torus. But this would take you well beyond 6.042, and I would not recommend it unless you think you're independently interested in this field called "combinatorial topology." regards, A. Timothy Mwangi wrote: > Passage: 5.1 Euler's Formula > Page: 12 > I found this passage most difficult because, although Euler's Formula > is shown to be true by induction, I have no intuitive understanding > of why it is true. > > > Sincerely, > Timothy M. Mwangi From hmzhou@blackbird.csail.mit.edu Mon Oct 31 09:51:56 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9VEpu5o000599; Mon, 31 Oct 2005 09:51:56 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9VEpuAd005913; Mon, 31 Oct 2005 09:51:56 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9VEpu5h005910; Mon, 31 Oct 2005 09:51:56 -0500 Date: Mon, 31 Oct 2005 09:51:55 -0500 (EST) From: Hanson Zhou To: Daniel A Gutierrez cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hansen] TP8 Reading Response In-Reply-To: <20051028133506.veo6mbacxkeo808k@webmail.mit.edu> Message-ID: References: <20051028133506.veo6mbacxkeo808k@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2458 Content-Length: 541 X-Keywords: NotJunk Well, you should look at the definition of big-oh and try to understand it. See me if it remains confusing. 4^x is O(4^x), O(4^x-2x), O(7^x), O(2^(x log x)), but certainly not O(2^x) or O(x^c) for any constant c. -Hanson On Fri, 28 Oct 2005, Daniel A Gutierrez wrote: > I just have a question concerning Big O, one of the problems discussed > at the end is 4^x is not O(2^x), but it doesn't say what the Big-oh for > it actually is. Is the big-oh for 4^x just a number arbitrarily larger > than 4 to the power of x? Daniel Gutierrez > From meyer@imap.theory.csail.mit.edu Wed Nov 23 07:44:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANCif5o026557 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 07:44:41 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANCieKh013580 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 07:44:40 -0500 (EST) Received: from ryan.mit.edu (PSK-SIXTEEN.MIT.EDU [18.217.1.16]) (authenticated bits=0) (User authenticated as ryan786@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANCibsV022642 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 07:44:38 -0500 (EST) Message-Id: <6.2.3.4.2.20051123073853.02901450@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 23 Nov 2005 07:44:39 -0500 To: 6042-probs@theory.lcs.mit.edu From: ryan Subject: [sayan] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 545 X-UID: 2459 X-Keywords: NonJunk NotJunk I found the part about mutual independence tricky, although it seems relatively simple, I found myself getting it wrong on the Tutorial Problems as well as in the notes. Also, I have never heard of a posteriori problems before, but am getting used to them. I discovered that my last email comments never sent, so I will append them here. This entire notion is very new to me, so I feel behind already. It is difficult for me to look at a sequence and instantly find the generating functions, even on some of the trivial ones. -Ryan From meyer@imap.theory.csail.mit.edu Tue Nov 22 10:03:59 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMF3x5o007180 for ; Tue, 22 Nov 2005 10:03:59 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EeZgc-0007G0-TK for meyer@csail.MIT.EDU; Tue, 22 Nov 2005 10:03:58 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAMF3vKG001895 for ; Tue, 22 Nov 2005 10:03:57 -0500 (EST) Received: from no-knife.mit.edu (NO-KNIFE.MIT.EDU [18.7.16.64]) (authenticated bits=56) (User authenticated as hmzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAMF3mQT010247 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for ; Tue, 22 Nov 2005 10:03:51 -0500 (EST) Received: (from hmzhou@localhost) by no-knife.mit.edu (8.12.9) id jAMF3mQe004898; Tue, 22 Nov 2005 10:03:48 -0500 (EST) Date: Tue, 22 Nov 2005 10:03:48 -0500 (EST) From: Hanson M Zhou To: meyer@csail.MIT.EDU Subject: reading comment (fwd) Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 543 X-UID: 2460 X-Status: A X-Keywords: $Forwarded Elizabeth Reid's comments below. Might just be something with email in general today. -Hanson ---------- Forwarded message ---------- Date: Tue, 22 Nov 2005 04:19:34 -0500 From: Elizabeth Reid To: Hanson Zhou Subject: reading comment I sent a reading comment in via the 6042-probs mailing list, but I got a weird error message, so I figured I'd send it directly to you, just in case. Anyway, the a posteriori problems were a bit odd, but I think I get the idea behind them now. Elizabeth From shauni@MIT.EDU Tue Nov 22 14:19:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMJJI5o014065 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 14:19:18 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAMJJHLH027143 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 14:19:17 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAMJJFvL029203 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 14:19:15 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jAMJJFEU011572; Tue, 22 Nov 2005 14:19:15 -0500 Received: from WOMENS-STUDIES.MIT.EDU (WOMENS-STUDIES.MIT.EDU [18.51.0.243]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 22 Nov 2005 14:19:15 -0500 Message-ID: <20051122141915.qvcizm6yhtgkwook@webmail.mit.edu> Date: Tue, 22 Nov 2005 14:19:15 -0500 From: "I. Shauni Deshmukh" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] week 12 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2461 Content-Length: 75 X-Keywords: NotJunk I would like to see more explanation/examples with infinite sample spaces. From pgroudas@MIT.EDU Tue Nov 22 14:53:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMJrc5o031991 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 14:53:38 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAMJraLH010115 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 14:53:37 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAMJrUQF015047 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 14:53:30 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAMJrU9E011751; Tue, 22 Nov 2005 14:53:30 -0500 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 22 Nov 2005 14:53:30 -0500 Message-ID: <20051122145330.b2djqwd3utgk8w0c@webmail.mit.edu> Date: Tue, 22 Nov 2005 14:53:30 -0500 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: [HANSON] Week 12 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2462 Content-Length: 245 X-Keywords: NotJunk I found the reading fairly straightforward, although the last example about mutual vs. pairwise independence could have been explained further. I'm just not exactly grasping how the calculation drops from 1 in 170 million to 1 in 17000. -Paul From bakster@MIT.EDU Tue Nov 22 15:50:22 2005 Return-Path: Received: from bexxxley.mit.edu (BEXXXLEY.MIT.EDU [18.246.1.238]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMKoM5o024709 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 15:50:22 -0500 Received: (from bakster@localhost) by bexxxley.mit.edu (8.12.9) id jAMKoM2Q014134; Tue, 22 Nov 2005 15:50:22 -0500 Date: Tue, 22 Nov 2005 15:50:22 -0500 (EST) From: Alexander G Bakst X-X-Sender: bakster@bexxxley To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Lecture Notes 12 Comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2463 Content-Length: 140 X-Keywords: NotJunk The coin problem example from the lecture notes confused me. Could you explain it a little? Or can we cover it in class? Alexander Bakst From ajshafer@MIT.EDU Tue Nov 22 15:58:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMKwD5o029233 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 15:58:13 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAMKwCLH022858 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 15:58:12 -0500 (EST) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAMKw4e8014406 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 15:58:04 -0500 (EST) Message-ID: <004601c5efa7$715b4170$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 12 Comments Date: Tue, 22 Nov 2005 15:58:08 -0500 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2464 Content-Length: 262 X-Keywords: NotJunk Will many of our problems be dealing with infinite trees? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From dangut@MIT.EDU Tue Nov 22 17:33:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMMX35o031688 for <6042-probs@theory.csail.MIT.EDU>; Tue, 22 Nov 2005 17:33:03 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAMMX1L7026270 for <6042-probs@theory.csail.MIT.EDU>; Tue, 22 Nov 2005 17:33:02 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAMMWx18019923 for <6042-probs@theory.csail.MIT.EDU>; Tue, 22 Nov 2005 17:33:00 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAMMWxBw003186; Tue, 22 Nov 2005 17:32:59 -0500 Received: from MAJI.MIT.EDU (MAJI.MIT.EDU [18.239.1.17]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 22 Nov 2005 17:32:59 -0500 Message-ID: <20051122173259.dw4rq0gjg9c8cg0k@webmail.mit.edu> Date: Tue, 22 Nov 2005 17:32:59 -0500 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Probability reading response MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2465 Content-Length: 408 X-Keywords: NotJunk This is all fairly straight forward. I guess the only thing I haven't really seen before and would like to see in class is probability as variables appproach infinity (I have seen some funny proofs before that sound illogical,but become logical when carried to infinite, such as the odds out of any number possible of picking a number divisible by 11 is zero? or maybe it was just a false proof...) -Daniel From kjhollen@mit.edu Tue Nov 22 18:27:50 2005 Return-Path: Received: from takanawa.isao.net (fi01-84CBc5.tokyo.flets.isao.net [211.132.203.197]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMNRn5o026574 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 18:27:50 -0500 Received: from [172.16.0.19] (unknown [172.16.0.19]) by takanawa.isao.net (Postfix) with ESMTP id 819023798 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 08:27:43 +0900 (JST) Subject: [Hanson] a posteriori probabilities From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Tue, 22 Nov 2005 18:27:42 -0500 Message-Id: <1132702062.7874.1.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit Status: RO X-UID: 2466 Content-Length: 114 X-Keywords: NotJunk i found this section in the reading (p. 18) to be a bit confusing. it would be helpful to go over it in lecture. From vixen@MIT.EDU Tue Nov 22 18:40:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMNeX5o003166 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 18:40:33 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAMNeWL7015735 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 18:40:32 -0500 (EST) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAMNePHj005681 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 18:40:26 -0500 (EST) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Tue, 22 Nov 2005 18:38:18 -0500 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Subject: LN12 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2467 Content-Length: 56 X-Keywords: NotJunk I was confused by the section on pairwise independence. From hzhou@MIT.EDU Tue Nov 22 19:21:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN0LT5o028829 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 19:21:29 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN0LRL7013609 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 19:21:27 -0500 (EST) Received: from hzhou.mit.edu (BURTON-SEVENTY-ONE.MIT.EDU [18.247.5.71]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN0LKid012409 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 19:21:21 -0500 (EST) Message-Id: <5.2.1.1.2.20051122192047.02b05dd8@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 22 Nov 2005 19:21:25 -0500 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: David-tp12 reading comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2468 Content-Length: 160 X-Keywords: NonJunk NotJunk Hey, Nothing too confusing. Independence is a bit tricky, but I guess its easy to be determined if you just do the conditional probability. - steven zhou From hectorb@MIT.EDU Tue Nov 22 19:26:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN0Qi5o029322 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 19:26:44 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN0QfL7017154; Tue, 22 Nov 2005 19:26:42 -0500 (EST) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN0QYF1013200 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 22 Nov 2005 19:26:35 -0500 (EST) Message-ID: <4383B739.1060300@mit.edu> Date: Tue, 22 Nov 2005 19:26:33 -0500 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 12 Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2469 Content-Length: 278 X-Keywords: NonJunk NotJunk "The three events A1, A2, and A3 are not mutually independent, even though all pairs of events are independent!" p.26 I found independence to be the hardest topic to understand. Distinguishing different types of independence is tricky. Probability itself is tricky. -Hector From rshroff@MIT.EDU Tue Nov 22 19:42:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN0gd5o008829 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 19:42:39 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN0gcL7027754 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 19:42:38 -0500 (EST) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN0gV8E015625 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 19:42:31 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id jAN0gVot019137; Tue, 22 Nov 2005 19:42:31 -0500 Received: from NEXT-THREE-EIGHTY-TWO.MIT.EDU (NEXT-THREE-EIGHTY-TWO.MIT.EDU [18.242.6.127]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 22 Nov 2005 19:42:31 -0500 Message-ID: <20051122194231.tbchecdfkkhwoc8w@webmail.mit.edu> Date: Tue, 22 Nov 2005 19:42:31 -0500 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Reading Assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2470 Content-Length: 164 X-Keywords: NonJunk NotJunk I realy enjoyeed the reading. I am currently also taking 6.041 so it was intersting to see the material taught in a somewhat different perspective. -Rahul Shroff From yangc@MIT.EDU Tue Nov 22 20:17:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN1HD5o021808 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:17:14 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN1HCBV021210 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:17:12 -0500 (EST) Received: from christope8f0c6 (MACGREGOR-ONE-TWENTY-FOUR.MIT.EDU [18.239.5.124]) (authenticated bits=0) (User authenticated as yangc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN1H4mA021113 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:17:05 -0500 (EST) Message-Id: <200511230117.jAN1H4mA021113@outgoing.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 12 Comments Date: Tue, 22 Nov 2005 20:18:54 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00EE_01C5EFA1.F6B94FC0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXvy97Os6BNr2WpSvSo3mINGkUcnQ== X-Spam-Score: 1.668 X-Spam-Level: * (1.668) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2471 Content-Length: 2913 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_00EE_01C5EFA1.F6B94FC0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Quick question regarding conditional probability: is asking the probability of something given another event the same as the probability of that something happening using all the probabilities on the subtree after that "another" event? Hm, that is a difficult question to ask in e-mail form. Assuming an event is given - is that equivalent to, say, traversing down one path on the tree of that given event? Have a good break! -Chris Yang ------=_NextPart_000_00EE_01C5EFA1.F6B94FC0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Quick question regarding conditional = probability:  is asking the probability of something given another event the same as the probability of that something happening using all the probabilities on = the subtree after that “another” event?  Hm, that is a = difficult question to ask in e-mail form.  Assuming an event is given – = is that equivalent to, say, traversing down one path on the tree of that given = event?

 

Have a good break!

 

-Chris Yang

------=_NextPart_000_00EE_01C5EFA1.F6B94FC0-- From xiaoranz@MIT.EDU Tue Nov 22 20:24:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN1OP5o026687 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 20:24:25 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN1ONBV026108 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 20:24:24 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN1OH5O022232 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 20:24:17 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jAN1OH9e026508; Tue, 22 Nov 2005 20:24:17 -0500 Received: from MCCORMICK-SEVEN-SIXTY.MIT.EDU (MCCORMICK-SEVEN-SIXTY.MIT.EDU [18.240.7.249]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 22 Nov 2005 20:24:17 -0500 Message-ID: <20051122202417.3xeq2oaicm2o4oso@webmail.mit.edu> Date: Tue, 22 Nov 2005 20:24:17 -0500 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 12 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2472 Content-Length: 517 X-Keywords: NonJunk NotJunk THe Monty Hall problem was most interesting because it demonstrated how common sense might be deceiving in calculating probability. The 4 step method is also demonstrated to be very useful. ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From dowgun@MIT.EDU Tue Nov 22 21:22:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN2MA5o026655 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 21:22:10 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN2M9BV005389 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 21:22:09 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN2M3j4001062 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 22 Nov 2005 21:22:03 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jAN2M3AU031886; Tue, 22 Nov 2005 21:22:03 -0500 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 22 Nov 2005 21:22:03 -0500 Message-ID: <20051122212203.rxvlztqlteaoo0sk@webmail.mit.edu> Date: Tue, 22 Nov 2005 21:22:03 -0500 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] ln 12 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2473 Content-Length: 748 X-Keywords: NotJunk I think I understand everything here, but I do have one question - how could the OJ DNA be pairwise independent but not mutually independent? I assume that two markers would only be pairwise independent if they were on different chromosones, otherwise there would be some recombination frequency. Thus, all five markers would be on different chromosones - how could any combination of markers/lack of markers affect the appearance/lack of a marker on a different chromosone? The example with the coins seems to not be mutually independent because there is a pigeonhole problem - there are three coins and only two outcomes! But here there should be 46 chromosones, and only five are being used. Was this really an argument used in this trial? Neil From hsoumare@MIT.EDU Tue Nov 22 22:02:42 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN32g5o010061 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 22:02:42 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN32fA1023206 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 22:02:42 -0500 (EST) Received: from [18.241.7.23] (NEW-FIVE-THIRTY-FOUR.MIT.EDU [18.241.7.23]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAN32Tok002012 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 22:02:29 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <9CF52F48-4496-4C7A-825B-B3AF949B96A7@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: {Sayan} Weekly Email Comment Date: Tue, 22 Nov 2005 22:11:57 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2474 Content-Length: 120 X-Keywords: NonJunk NotJunk Hi, Could you please go over where the Law of Total Probability on pg 17 comes from more in detail? Thanks. Hamidou From hkhall@MIT.EDU Tue Nov 22 22:13:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN3Dc5o020682 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 22:13:38 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN3DaBV010502; Tue, 22 Nov 2005 22:13:36 -0500 (EST) Received: from [18.96.5.125] (SIMMONS-ONE-TWENTY-FIVE.MIT.EDU [18.96.5.125]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN3DTBJ008773 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Tue, 22 Nov 2005 22:13:29 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-1--210413645 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 12 Comments From: Harrison King Hall Date: Tue, 22 Nov 2005 22:13:28 -0500 X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2475 Content-Length: 1711 X-Status: A X-Keywords: NonJunk NotJunk --Apple-Mail-1--210413645 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=ISO-2022-JP; format=flowed David- Wow, I know nothing about probability. Well not absolutely nothing, but close enough for the rigor required by this course. By far the most interesting case in probability that I had not even considered was Posteriori probabilities. I had always heard and made phrases like that, but now I understand where the mathematical basis is from. I realize it is only a special case of conditional probability, but I think it is awesome. The most difficult thing for me for a long time was independence since I didn't find a rigorous definition of Pr{A $B"A(B B} in the notes. Maybe it was there but I didn't catch it for a while and I am still a little fuzzy on it. -Harrison --Apple-Mail-1--210413645 Content-Transfer-Encoding: 7bit Content-Type: text/enriched; charset=ISO-2022-JP David- Wow, I know nothing about probability. Well not absolutely nothing, but close enough for the rigor required by this course. By far the most interesting case in probability that I had not even considered was Posteriori probabilities. I had always heard and made phrases like that, but now I understand where the mathematical basis is from. I realize it is only a special case of conditional probability, but I think it is awesome. The most difficult thing for me for a long time was independence since I didn't find a rigorous definition of Pr{A Hiragino Kaku Gothic Pro$B"A(B B} in the notes. Maybe it was there but I didn't catch it for a while and I am still a little fuzzy on it. -Harrison --Apple-Mail-1--210413645-- From ozcan@MIT.EDU Tue Nov 22 22:26:20 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN3QK5o002359 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 22:26:20 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN3QJA1024013 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 22:26:20 -0500 (EST) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAN3QGMv012154 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 22:26:16 -0500 (EST) Message-Id: <5.2.1.1.2.20051122221658.020e5e50@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 22 Nov 2005 22:26:12 -0500 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [Hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2476 Content-Length: 209 X-Keywords: NotJunk I found the conditional probability equation very interesting and helpful. It is at the start of Section 3. Prob(A intersection B ) = Prob (A given B) / Prob (B) = Prob (B given A) / Prob (A) Yasin Ozcan From aston@MIT.EDU Tue Nov 22 22:29:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN3Te5o003663 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 22:29:40 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN3TdBV021921 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 22:29:39 -0500 (EST) Received: from astonlaptop (NEW-SEVENTY-TWO.MIT.EDU [18.241.5.72]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN3Ta3Q011463 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 22:29:37 -0500 (EST) Message-Id: <200511230329.jAN3Ta3Q011463@outgoing.mit.edu> Reply-To: From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 12 Comments Date: Tue, 22 Nov 2005 22:29:34 -0500 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_008F_01C5EFB4.371BFCE0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXv3h9LzzwOav+fR5amhjEmiRaPIQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.313 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2477 Content-Length: 1644 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_008F_01C5EFB4.371BFCE0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The notes claim that two events are independent iff the probabilities multiply to their intersection with no proof of the statement (page 22). I'm pretty sure I lost points on a 6.041 test for using this proposition and have, ever since, been sort of wary of using that reasoning to prove independence. Is it provable or is it a definition for this course? - Aston ------=_NextPart_000_008F_01C5EFB4.371BFCE0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable
The = notes claim that=20 two events are independent iff the probabilities multiply to their = intersection=20 with no proof of the statement (page 22). I'm pretty sure I lost points = on a=20 6.041 test for using this proposition and have, ever since, been sort of = wary of=20 using that reasoning to prove independence. Is it provable or is it a = definition=20 for this course?
 
    -=20 Aston
------=_NextPart_000_008F_01C5EFB4.371BFCE0-- From meyer@imap.theory.csail.mit.edu Tue Nov 22 20:13:59 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN1Dx5o018225 for <6042-webmaster@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:13:59 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAN1Dv26026535 for <6042-webmaster@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:13:58 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN1Dv4N021229 for <6042-webmaster@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:13:57 -0500 (EST) Received: from [18.246.1.160] (TYLERW.MIT.EDU [18.246.1.160]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAN1DtMu006043 for <6042-webmaster@theory.csail.mit.edu>; Tue, 22 Nov 2005 20:13:55 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-webmaster@theory.csail.mit.edu From: Tyler Williams Subject: (Sayan) email comment Date: Tue, 22 Nov 2005 20:13:45 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 181 X-UID: 2478 X-Keywords: NonJunk NotJunk It would be cool if, in class, we could go over a little bit more of drawing the probability trees. Also, I thought the game show thing was cool because it was so unintuitive. From meyer@imap.theory.csail.mit.edu Tue Nov 22 20:58:17 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN1wG5o004887 for <6042-staff@theory.csail.MIT.EDU>; Tue, 22 Nov 2005 20:58:17 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jAN1wF26000739 for <6042-staff@theory.csail.MIT.EDU>; Tue, 22 Nov 2005 20:58:16 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN1wF4N022729; Tue, 22 Nov 2005 20:58:15 -0500 (EST) Received: from [192.168.1.104] (c-66-30-24-123.hsd1.ma.comcast.net [66.30.24.123]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAN1w7Mu007837; Tue, 22 Nov 2005 20:58:07 -0500 (EST) Message-ID: <4383CCAE.5030709@theory.lcs.mit.edu> Date: Tue, 22 Nov 2005 20:58:06 -0500 From: sayan mitra User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jesus Medrano , 6042-staff@theory.csail.MIT.EDU Subject: Re: [Sayan] indepence with conditional reading assignment References: <1132690759.17310.1.camel@localhost> In-Reply-To: <1132690759.17310.1.camel@localhost> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 419 X-UID: 2479 X-Keywords: NonJunk NotJunk Jesus Medrano wrote: >What is suppose to be done when the events in conditional probability >are independent. > Well, if A and B are independent, then as shown in the notes, P(A|B) = P(A /\ B) / P(B) = P(A). > Also what is suppose to be done when the conditional >probability has more than one dependent event. > > I don't quite understand this question, can you rephrase it ? Best, Sayan >Jesus Medrano > > From meyer@imap.theory.csail.mit.edu Tue Nov 22 16:39:54 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAMLdr5o009561 for ; Tue, 22 Nov 2005 16:39:53 -0500 Received: from smtp104.mail.sc5.yahoo.com ([66.163.169.223]) by rozz.csail.mit.edu with smtp (Exim 4.50) id 1Eefrl-0004S5-85 for meyer@csail.mit.edu; Tue, 22 Nov 2005 16:39:53 -0500 Received: (qmail 69679 invoked from network); 22 Nov 2005 21:39:52 -0000 Received: from unknown (HELO ?192.168.1.103?) (armeyer10@71.248.165.112 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 22 Nov 2005 21:39:52 -0000 Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <48e5d16a79172eeeef6f1335b36137bd@csail.mit.edu> Content-Transfer-Encoding: 7bit From: "Albert R. Meyer" Subject: Fwd: [Hanson] Reading Comments 12 (fwd) Date: Tue, 22 Nov 2005 16:39:51 -0500 To: dnreshef@MIT.EDU X-Mailer: Apple Mail (2.623) X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1502 X-UID: 2480 X-Keywords: Begin forwarded message: > From: "Albert R. Meyer" > Date: November 22, 2005 4:37:56 PM EST > To: Hanson M Zhou > Subject: Re: [Hanson] Reading Comments 12 (fwd) > > email ink should work now... > regards, A. > On Nov 22, 2005, at 3:24 PM, Hanson M Zhou wrote: > >> One more... >> >> ---------- Forwarded message ---------- >> Date: Tue, 22 Nov 2005 15:18:59 -0500 >> From: David N Reshef >> To: hmzhou@mit.edu >> Subject: Fwd: [Hanson] Reading Comments 12 >> >> Hanson, >> I've tried emailing my comments twice now to >> 6042-probs@theory.lcs.MIT.EDU, >> and both times, it said that it was an inccorrect adress. If that is >> an >> incorrect adress, could you please send me the real one. In any >> case, my >> reading comments for tomorrow are forwarded below. >> -Dave >> >> >> >> ----- Forwarded message from dnreshef@MIT.EDU ----- >> Date: Tue, 22 Nov 2005 13:11:49 -0500 >> From: David N Reshef >> Reply-To: David N Reshef >> Subject: [Hanson] Reading Comments 12 >> To: 6042-probs@theory.lcs.MIT.EDU >> >> This week's reading was particularly easy, and I think I understood >> just about >> everything. I did, however, find the sections on infinite sample >> spaces (like >> yesterday's in class problem) and conditional probability (we have >> done a lot >> with this in my genetics class) particularly interesting. >> -Dave >> >> >> ----- End forwarded message ----- From fluff@MIT.EDU Tue Nov 22 23:07:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN47h5o017652 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 23:07:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN47gMV015530 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 23:07:42 -0500 (EST) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN47es4017228 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 23:07:40 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <24438ECC-3BFD-4D89-A4C6-57C3D7560165@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 12 reading Date: Tue, 22 Nov 2005 23:05:58 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2481 Content-Length: 152 X-Keywords: NonJunk NotJunk I thought the reading very effectively conveyed the fact that probability is a completely unintuitive subject. It was really depressing. ~Crystal From benlu@MIT.EDU Tue Nov 22 23:31:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN4VF5o028090 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 23:31:15 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN4VEMV000960 for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 23:31:14 -0500 (EST) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN4UwUc020393 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 22 Nov 2005 23:30:58 -0500 (EST) Message-ID: <4383F089.5000002@mit.edu> Date: Tue, 22 Nov 2005 23:31:05 -0500 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051013) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 12 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2482 Content-Length: 425 X-Status: A X-Keywords: NotJunk No real comments besides counter-intuitive things are cool, and I think I found two errors in the notes: * The first is the lower branch of the tree in the fair VS trick coin tree (top of page 19). I think it should read 1^100 and not (1/2)^100. * The second is the inclusion-exclusion formula (page 22) "Pr{A|BuC} = Pr{A|C} + Pr{A|C} - Pr{AnB|C}" which should read "Pr{A|BuC} = Pr{A|C} + Pr{B|C} - Pr{AnB|C}" ~Ben Lu From tonyng@MIT.EDU Tue Nov 22 23:36:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN4aX5o028417 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 23:36:33 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN4aWMV004251 for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 23:36:32 -0500 (EST) Received: from TNG.mit.edu (BURTON-SIX-EIGHTY-NINE.MIT.EDU [18.247.7.178]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN4aFU2021088 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 22 Nov 2005 23:36:16 -0500 (EST) Message-Id: <5.2.1.1.2.20051122232734.00bb1688@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 22 Nov 2005 23:36:30 -0500 To: 6042-probs@theory.csail.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments Week 12 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2483 Content-Length: 389 X-Keywords: NotJunk I found the "Medical Testing" example (Section 3.6 on pg20) very interesting. It made me more aware about how easy it is to just listen to some numbers and be fooled by what it actually means. I think it is a great example to show that it is important to calculate probabilities carefully because initially intuition led me to think that this medical test wasn't too bad. - Tony Ng From sergiob@MIT.EDU Wed Nov 23 00:14:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN5Ex5o003595 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 00:14:59 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN5EwMV027269 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 00:14:58 -0500 (EST) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN5Eq8r026041 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 00:14:53 -0500 (EST) Message-Id: <5.2.1.1.2.20051122220644.00c3e5e8@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 22 Nov 2005 22:14:51 -0700 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 12 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2484 Content-Length: 316 X-Keywords: NotJunk I found the topic of independence very interesting. I wonder if there is a simpler way to determine independence between events, because evaluating the probabilities of intersections for every possible subset of outcomes must become cumbersome as the outcome space becomes very large. Sergio Bacallado Group A. From yaser@MIT.EDU Wed Nov 23 00:24:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN5OV5o004267 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 00:24:31 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN5OUMV002656 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 00:24:30 -0500 (EST) Received: from Charizard (NEW-THREE-SEVENTY-FOUR.MIT.EDU [18.241.6.119]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN5OJF6027049 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 00:24:28 -0500 (EST) Message-Id: <200511230524.jAN5OJF6027049@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David- tp12 reading comments Date: Wed, 23 Nov 2005 00:24:12 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00BB_01C5EFC4.41F8CCA0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXv7iM3r6FldEKuT/ahzJnaWuCXjA== X-Spam-Score: 5.851 X-Spam-Level: ***** (5.851) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2485 Content-Length: 5675 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_00BB_01C5EFC4.41F8CCA0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Hi David, At first read, the prob stuff seemed pretty similar to that in 6.041, with just a few more rigorous definitions. I was wondering, what are some of the other caveats/subtleties with the pairwise/mutually independent definition? (The one given is that it's possibly for three events to not be mutually independent, even though their pairs are.) Thanks! _Yaser ------=_NextPart_000_00BB_01C5EFC4.41F8CCA0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

At first read, the prob stuff seemed pretty similar to that in 6.041, with just a few more = rigorous definitions. I was wondering, what are some of the other = caveats/subtleties with the pairwise/mutually independent = definition?  (The one given is that = it’s possibly for three events to not be mutually independent, even though their pairs = are.)

 

Thanks!

 

_Yaser =

------=_NextPart_000_00BB_01C5EFC4.41F8CCA0-- From a_lopez@MIT.EDU Wed Nov 23 01:16:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN6GG5o020805 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 01:16:16 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN6GEMV002363 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 01:16:15 -0500 (EST) Received: from [18.234.1.61] (DP-SIXTY-ONE.MIT.EDU [18.234.1.61]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN6GCGo002238 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 01:16:13 -0500 (EST) Message-ID: <43840932.3040503@mit.edu> Date: Wed, 23 Nov 2005 01:16:18 -0500 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] week 12 readings Content-Type: multipart/mixed; boundary="------------030201030901050307070104" X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2486 Content-Length: 1145 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. --------------030201030901050307070104 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit --------------030201030901050307070104 Content-Type: message/rfc822; name="[Jelani] week 12 readings" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="[Jelani] week 12 readings" Message-ID: <4382933A.1080904@mit.edu> Date: Mon, 21 Nov 2005 22:40:42 -0500 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] week 12 readings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit I didn't really understand section 3.7 ("other identities"). How is the formula Pr{A U B | C} = Pr{A | C} + Pr{A | C} - Pr{A n B | C} derived? Why not write it Pr{A U B | C} = 2Pr{A | C} - Pr{A n B | C}. Why is the other formula not valid? And why is B n C empty? I also thought the DNA example was specially confusing. Adriana --------------030201030901050307070104-- From mracich@MIT.EDU Wed Nov 23 01:20:38 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN6Kc5o021089 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 01:20:38 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN6Kctd029422 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 01:20:38 -0500 (EST) Received: from MACGREGOR-THREE-FORTY-FIVE.MIT.EDU (MACGREGOR-THREE-FORTY-FIVE.MIT.EDU [18.239.6.90]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAN6KZMu019508 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 01:20:35 -0500 (EST) Subject: [Sayan] Comments for Course Notes, Week 12 (Introduction to Probability) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 23 Nov 2005 01:20:34 -0500 Message-Id: <1132726835.8136.6.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2487 Content-Length: 235 X-Keywords: NotJunk I found section 5. "Mutual Independence", (starting on page 24) confusing. I would appreciate it if this was explained further. Also, in general, I am confused about how to convert from set notation to probabilities. Moira Racich From brevzin@MIT.EDU Wed Nov 23 01:21:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN6Lr5o021213 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 01:21:53 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN6LqMV005371 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 01:21:52 -0500 (EST) Received: from BARRY.mit.edu (PLP-ONE-TWENTY-FOUR.MIT.EDU [18.218.1.124]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN6LjYL002768 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 01:21:45 -0500 (EST) Message-Id: <6.1.2.0.1.20051123012009.019fe210@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Wed, 23 Nov 2005 01:21:50 -0500 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2488 Content-Length: 204 X-Keywords: NotJunk Um, yeah, I'm learning all this stuff in 18.103 (Measure Theory). Um. Yeah. Union bound is called subadditivity, I thing it makes more sense conceptually. Otherwise... Have a good thanksgiving? Barry From kktyan@MIT.EDU Wed Nov 23 01:45:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN6jB5o002222 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 01:45:11 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN6jAMV017269 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 01:45:10 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN6j45K004515 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 01:45:04 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jAN6j3qg015626; Wed, 23 Nov 2005 01:45:03 -0500 Received: from BURTON-THREE-THIRTY-SIX.MIT.EDU (BURTON-THREE-THIRTY-SIX.MIT.EDU [18.247.6.81]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 01:45:03 -0500 Message-ID: <20051123014503.5lactfb6xr4kowcs@webmail.mit.edu> Date: Wed, 23 Nov 2005 01:45:03 -0500 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments 12 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2489 Content-Length: 764 X-Keywords: NotJunk Just for clarification (this isn't my main question), two events that are independent are *also* mutually independent, by the definition of independence and mutual independence, correct? (pages 23-25) I am confused about the a posteriori probability covered in pages 17-18. In the tutor problem 12.2, it is said that Pr{A} = 0.8 and Pr{L} = 0.4; in a case like this, how would you compute Pr{L|A} without being given that it is 0.3? I mean, I know that Pr{L|A} = Pr{A n L}/Pr{L}, but A n L is dependent upon the value of Pr{A|L}, which is in turn dependent upon the value of Pr{L|A} in that Pr{A|L} = Pr{A n L}/Pr{L}. In general, I'm just a little confused over how to find A n B or Pr{A|B}. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From dnreshef@MIT.EDU Wed Nov 23 02:25:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN7Ph5o029046 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 02:25:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN7PgLX003989 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 02:25:42 -0500 (EST) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN7PZlq006925 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 02:25:35 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id jAN7PYC5019523; Wed, 23 Nov 2005 02:25:34 -0500 Received: from pcp09558804pcs.verona01.nj.comcast.net (pcp09558804pcs.verona01.nj.comcast.net [68.36.248.152]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 02:25:34 -0500 Message-ID: <20051123022534.pcraj4vpmx440c4g@webmail.mit.edu> Date: Wed, 23 Nov 2005 02:25:34 -0500 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments 12 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2490 Content-Length: 300 X-Keywords: This week's reading was particularly easy, and I think I understood just about everything. I did, however, find the sections on infinite sample spaces (like yesterday's in class problem) and conditional probability (we have done a lot with this in my genetics class) particularly interesting. -Dave From mdmurray@MIT.EDU Wed Nov 23 02:37:56 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN7bu5o001810 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 02:37:56 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN7bu0h001639 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 02:37:56 -0500 (EST) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jAN7bhok013295 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 02:37:44 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Michael Murray Subject: [Hanson] ln12 reading Date: Wed, 23 Nov 2005 02:36:28 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2491 Content-Length: 445 X-Status: A X-Keywords: NonJunk NotJunk I found it surprising that the independence equation was Pr(A n B) = Pr(A).Pr(B). My original thought was that the A and B would be independent if Pr(A u B) = Pr(A) + Pr(B), this would imply that there is no overlap in the sets A and B and so they would be independent. I am not sure where this intuition fails and why we instead look at the intersection. What is the intuition behind multiplying two probabilities? Michael Murray From bilodeau@MIT.EDU Wed Nov 23 02:52:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN7q95o013295 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 02:52:09 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN7q8LX015134 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 02:52:08 -0500 (EST) Received: from pbilodeau (MACGREGOR-TWO-TWENTY-ONE.MIT.EDU [18.239.5.221]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN7q1Qv008371 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 02:52:02 -0500 (EST) Message-Id: <200511230752.jAN7q1Qv008371@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [jelani] reading comments Date: Wed, 23 Nov 2005 02:51:48 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5EFD8.D992D420" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXwAsHw61iyGuDqR7+6BKtt20MGhw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2492 Content-Length: 2255 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5EFD8.D992D420 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit This reading was, I think, made unduly confusing, especially in the "four step method" section. The method was explained by means of an example, but an overall summary of the process is hard to extract from amidst the rest of the explanation. Peter Bilodeau ------=_NextPart_000_0000_01C5EFD8.D992D420 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

This reading was, I think, made unduly confusing, = especially in the “four step method” section.  The method was = explained by means of an example, but an overall summary of the process is hard to = extract from amidst the rest of the explanation.

 

Peter Bilodeau

------=_NextPart_000_0000_01C5EFD8.D992D420-- From ksindi@MIT.EDU Wed Nov 23 02:55:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN7tP5o020751 for <6042-probs@theory.csail.MIT.EDU>; Wed, 23 Nov 2005 02:55:25 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN7tNLX016540 for <6042-probs@theory.csail.MIT.EDU>; Wed, 23 Nov 2005 02:55:24 -0500 (EST) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN7tF9k008561 for <6042-probs@theory.csail.MIT.EDU>; Wed, 23 Nov 2005 02:55:16 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id jAN7tFYH021237; Wed, 23 Nov 2005 02:55:15 -0500 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 02:55:15 -0500 Message-ID: <20051123025515.8qd9e0hgiie8oso4@webmail.mit.edu> Date: Wed, 23 Nov 2005 02:55:15 -0500 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [David] Week 12 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2493 Content-Length: 170 X-Keywords: NotJunk I found the many detailed examples very helpful. To me, the touphest part is getting around in modeling conditional probability in to a formula as I always get mixed up. From lmccart@MIT.EDU Wed Nov 23 03:48:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN8m85o007607 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 03:48:08 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN8m6LX008498 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 03:48:07 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN8m2sT011102 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 03:48:02 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jAN8m2qK025958; Wed, 23 Nov 2005 03:48:02 -0500 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 03:48:02 -0500 Message-ID: <20051123034802.pu7i84sxi1hicoo4@webmail.mit.edu> Date: Wed, 23 Nov 2005 03:48:02 -0500 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2494 Content-Length: 141 X-Keywords: NotJunk I understood the reading but a review of the section about posteriori probabilities like on the tutor problems would be appreciated. -lauren From shreyes19@gmail.com Wed Nov 23 03:48:47 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN8mk5o009512 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 03:48:46 -0500 Received: from nproxy.gmail.com ([64.233.182.192]) by rozz.csail.mit.edu with esmtp (Exim 4.50) id 1EeqJ4-0006Pa-G8 for 6042-probs@theory.lcs.mit.edu; Wed, 23 Nov 2005 03:48:46 -0500 Received: by nproxy.gmail.com with SMTP id l36so227711nfa for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 00:47:45 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=SYz8Bcsn2N2wkCxsBMZPzTybtBky1W+ZO1U6DyBxK2VivssTchCoBUiRBXryPyf4TpWTceBcf0K1ZiYftQcJlzwNDZysniLwxDxiMOmMI/BnQIPhnH3Kdi1DNpMRwIirEwA4RsKkpxzLxxg+kYE4aBnG0th7kHSP1VrrxeXV6AA= Received: by 10.48.225.2 with SMTP id x2mr1760nfg; Tue, 22 Nov 2005 23:44:18 -0800 (PST) Received: by 10.48.161.8 with HTTP; Tue, 22 Nov 2005 23:44:18 -0800 (PST) Message-ID: <3c33fe380511222344j1453ec0em8813e669c3946e51@mail.gmail.com> Date: Wed, 23 Nov 2005 02:44:18 -0500 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 12 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_2798_21161657.1132731858295" Status: RO X-UID: 2495 Content-Length: 1623 X-Keywords: NonJunk NotJunk ------=_Part_2798_21161657.1132731858295 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, The part of the notes that most interested me was section 3.6 - Medical Testing (pg 20-21). I was able to follow the derivation of the conditional probability of 1/4 that you have the disease if tested positively. I think it would be good to see more examples in class of cases where drawing out the probability tree and calculating leads to non-intuitive answers, especially when working with conditional probabilities. These tend to be the more interesting ones, at least compared with regular probabilities wit= h discrete random variables (like questions with coin flips). Thanks, Shreyes Group 7 ------=_Part_2798_21161657.1132731858295 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

The part of the notes that most interested me was section 3.6 - Medical Testing (pg 20-21).  I was able to follow the derivation of the conditional probability of 1/4 that you have the disease if tested positively.  I think it would be good to see more examples in class of cases where drawing out the probability tree and calculating leads to non-intuitive answers, especially when working with conditional probabilities.  These tend to be the more interesting ones, at least compared with regular probabilities with discrete random variables (like questions with coin flips).

Thanks,
Shreyes
Group 7
------=_Part_2798_21161657.1132731858295-- From ctsims@MIT.EDU Wed Nov 23 03:56:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN8um5o011980 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 03:56:48 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN8ulLX011889 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 03:56:47 -0500 (EST) Received: from HowardRoark.mit.edu (EASTCAMPUS-NINE-TWENTY-TWO.MIT.EDU [18.238.6.155]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN8ueOb011489 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 03:56:40 -0500 (EST) Message-Id: <5.2.1.1.2.20051123035151.10020518@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 23 Nov 2005 03:54:39 -0500 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: [6.042] Reading problem - Clayton Sims Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2496 Content-Length: 284 X-Keywords: NonJunk NotJunk I think that the passage and formula on pg. 13 about conditional probability could require most explanation. Primarily the introduction and use of the A (intersection) B probability, and how it is derived. I think this would be a good thing to talk about in class. -Clayton Sims From mpapi@MIT.EDU Wed Nov 23 04:36:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAN9aK5o027159 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 04:36:20 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAN9aJLX027280 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 04:36:19 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN9aDFo013027 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 04:36:13 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jAN9aDpk029142; Wed, 23 Nov 2005 04:36:13 -0500 Received: from MACGREGOR-THREE-FORTY-FIVE.MIT.EDU (MACGREGOR-THREE-FORTY-FIVE.MIT.EDU [18.239.6.90]) (User authenticated as mpapi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 04:36:13 -0500 Message-ID: <20051123043613.vuyh0wis0jykockk@webmail.mit.edu> Date: Wed, 23 Nov 2005 04:36:13 -0500 From: Matthew M Papi To: 6042-probs@theory.lcs.MIT.EDU Subject: [Jelani] week 12 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2497 Content-Length: 145 X-Keywords: NonJunk NotJunk Section 5 on Mutual Independence, pages 24ff. This stuff was tricky the first time around in 6.041, so the more explanation the better. - Matt From kevin08@MIT.EDU Wed Nov 23 05:05:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANA5I5o003197 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 05:05:18 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANA5GKh007810 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 05:05:16 -0500 (EST) Received: from kevlar.mit.edu (NEXT-THREE-FIFTY-SEVEN.MIT.EDU [18.242.6.102]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAN9vqUB013841 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 04:57:53 -0500 (EST) Message-Id: <6.2.3.4.2.20051123044514.02a12490@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 23 Nov 2005 04:57:39 -0500 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Reading Comments Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="=====================_68461522==_" X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2498 Content-Length: 481920 X-Keywords: NotJunk --=====================_68461522==_ Content-Type: text/plain; charset="us-ascii"; format=flowed Hi, I have no questions about this week's reading since I'm taking 6.041 at the same time and we covered this material already. However, I think it might be helpful to students if you explain Total Probability Theorem (Section 3.3) to them with a diagram like the one I attached. I found that picture very helpful when learning it. 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////////AAAA AAAAAAAA//////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////4eHhAAAAAAAAjIyM//////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////jIyMAAAAAAAA4eHh//////////////////////////// ////////////////////////////////////////////////////////////4eHhAAAAAAAAjIyM //////////////////////////////////////////////////////////////////////////// ////AAAAAAAAAAAA//////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////////jIyM AAAAAAAA4eHh//////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////jIyMAAAAAAAA4eHh//////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////4eHhAAAAAAAAjIyM//////////////////// ////////////////////////////////////////////////////////////////////jIyMAAAA AAAA4eHh//////////////////////////////////////////////////////////////////// ////////////AAAAAAAAAAAA//////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////4eHhAAAAAAAAjIyM//////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////6enpAAAAAAAAfHx8//////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////jIyMAAAAAAAA6enp//////// ////////////////////////////////////////////////////////////////////////6enp AAAAAAAAfHx8//////////////////////////////////////////////////////////////// ////////////////////AAAAAAAAAAAA//////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////jIyMAAAAAAAA6enp//////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////p6enAAAAAAAA0NDQ //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////0NDQAAAAAAAAp6en //////////////////////////////////////////////////////////////////////////// ////p6enAAAAAAAA0NDQ//////////////////////////////////////////////////////// ////////////////////////////AAAAAAAAAAAA//////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////0NDQAAAAAAAAp6en//////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////TU1NAAAA AAAA//////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////////AAAA AAAATU1N//////////////////////////////////////////////////////////////////// ////////////TU1NAAAAAAAA//////////////////////////////////////////////////// ////////////////////////////////////AAAAAAAAAAAA//////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////AAAAAAAATU1N//////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////AAAAAAAAAAAA////////////////////////////////////////////////4eHh AAAAAAAAjIyM////////////fHx8AAAAAAAA0NDQ//////////////////////////////////// 0NDQAAAAAAAAfHx8////////////8PDwp6enTU1NAAAATU1Np6en8PDw//////////////////// ////jIyMAAAAAAAA4eHh////////////AAAAAAAAAAAA//////////////////////////////// ////////////////4eHhAAAAAAAAjIyM////////////////////AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAjIyM0NDQ////////////////////AAAAAAAAAAAA////////fHx8AAAAAAAA 0NDQ////////////////////////////////////0NDQAAAAAAAAfHx8////////////8PDwp6en TU1NAAAATU1Np6en8PDw////////////////////////jIyMAAAAAAAA4eHh//////////////// 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//////////////////////////////////////////////////////////////////////////// AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA4eHh////////aGhoAAAAAAAA6enp ////////////////////////jIyMAAAAAAAA0NDQ////////////0NDQAAAAAAAAjIyM//////// ////////////////////////AAAAAAAAAAAAAAAAAAAAjIyM////////////////////6enpAAAA AAAAaGho////////////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA4eHh//// ////aGhoAAAAAAAA6enp////////////////////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAjIyM8PDw////////////////AAAAAAAAAAAA////////////////////jIyMAAAAAAAA 0NDQ////////////0NDQAAAAAAAAjIyM////////////////////////////////AAAAAAAAAAAA AAAAAAAAjIyM////////////////////6enpAAAAAAAAaGho//////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////AAAAAAAAAAAA////////////////////0NDQTU1NAAAAAAAAmpqa////////jIyMAAAA AAAA4eHh////////////////////////0NDQAAAAAAAAfHx8////////////jIyMAAAAAAAA0NDQ ////////////////////////////////jIyMAAAAAAAAAAAA0NDQ//////////////////////// 4eHhAAAAAAAAjIyM////////////AAAAAAAAAAAA////////////////////0NDQTU1NAAAAAAAA mpqa////////jIyMAAAAAAAA4eHh////////////////////AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAATU1N8PDw////////////////AAAAAAAAAAAA////////////////////0NDQ AAAAAAAAfHx8////////////jIyMAAAAAAAA0NDQ////////////////////////////////jIyM AAAAAAAAAAAA0NDQ////////////////////////4eHhAAAAAAAAjIyM//////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////AAAAAAAAAAAA////////////////////////4eHhAAAAAAAATU1N//////// vb29AAAAAAAAvb29////////////////////////////TU1NAAAAAAAA8PDw////8PDwAAAAAAAA TU1N////////////////////////////////////jIyMAAAAAAAAAAAATU1N8PDw//////////// ////////vb29AAAAAAAAvb29////////////AAAAAAAAAAAA////////////////////////4eHh AAAAAAAATU1N////////vb29AAAAAAAAvb29////////////////////AAAAAAAAAAAA//////// ////////////6enpmpqaAAAAAAAAp6en////////////////AAAAAAAAAAAA//////////////// ////////TU1NAAAAAAAA8PDw////8PDwAAAAAAAATU1N//////////////////////////////// ////jIyMAAAAAAAAAAAATU1N8PDw////////////////////vb29AAAAAAAAvb29//////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// 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--=====================_68461522==_-- From ryoung@MIT.EDU Wed Nov 23 07:44:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANCif5o026557 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 07:44:41 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANCieKh013580 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 07:44:40 -0500 (EST) Received: from ryan.mit.edu (PSK-SIXTEEN.MIT.EDU [18.217.1.16]) (authenticated bits=0) (User authenticated as ryan786@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANCibsV022642 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 07:44:38 -0500 (EST) Message-Id: <6.2.3.4.2.20051123073853.02901450@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 23 Nov 2005 07:44:39 -0500 To: 6042-probs@theory.lcs.mit.edu From: ryan Subject: [sayan] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2499 Content-Length: 545 X-Keywords: NonJunk NotJunk I found the part about mutual independence tricky, although it seems relatively simple, I found myself getting it wrong on the Tutorial Problems as well as in the notes. Also, I have never heard of a posteriori problems before, but am getting used to them. I discovered that my last email comments never sent, so I will append them here. This entire notion is very new to me, so I feel behind already. It is difficult for me to look at a sequence and instantly find the generating functions, even on some of the trivial ones. -Ryan From mwangi@MIT.EDU Wed Nov 23 08:28:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANDSP5o014595 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 08:28:25 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANDSOLG006060; Wed, 23 Nov 2005 08:28:24 -0500 (EST) Received: from IBM-1E80C73FC15.mit.edu (NEW-THREE-NINETY-NINE.MIT.EDU [18.241.6.144]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANDSDPD029046 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 23 Nov 2005 08:28:17 -0500 (EST) Message-Id: <6.2.1.2.2.20051123082057.03a06eb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 23 Nov 2005 08:28:11 -0500 To: 6042-probs@theory.lcs.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 12 Comments Cc: mitras@MIT.EDU Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2500 Content-Length: 258 X-Keywords: NotJunk Passage: 3.4 A Posteriori Probabilities Page: 17 I found this passage most surprising because of the notion that we could use probability to say not only what will happen in the future but what happened in the past. Sincerely, Timothy M. Mwangi From nancyk@MIT.EDU Wed Nov 23 09:32:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANEW45o004569 for <6042-probs@theory.csail.MIT.EDU>; Wed, 23 Nov 2005 09:32:04 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANEW2LG025301; Wed, 23 Nov 2005 09:32:02 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANEVxpN015976; Wed, 23 Nov 2005 09:31:59 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jANEVxlD019946; Wed, 23 Nov 2005 09:31:59 -0500 Received: from WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU (WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU [18.139.6.237]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 09:31:59 -0500 Message-ID: <20051123093159.isipff4n878k8g48@webmail.mit.edu> Date: Wed, 23 Nov 2005 09:31:59 -0500 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 12 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2501 Content-Length: 306 X-Keywords: NonJunk NotJunk Hi, I thought these notes were pretty straightforward and well-written, and I can't think of any difficulties to bring up. The applications of probability that the author brings up (tests for diseases, DNA testing) are interesting, and the way they are explained is very clear and logical. Thanks, Nancy From nedzel@MIT.EDU Wed Nov 23 09:39:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANEdl5o006073 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 09:39:47 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANEdkLG002429 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 09:39:46 -0500 (EST) Received: from ThinkPadT43 (SIMMONS-ONE-NINETY-NINE.MIT.EDU [18.96.5.199]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANEddut018520 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 09:39:40 -0500 (EST) Message-Id: <200511231439.jANEddut018520@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 12 Comments Date: Wed, 23 Nov 2005 09:39:35 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: 2.777 X-Spam-Level: ** (2.777) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2502 Content-Length: 161 X-Keywords: NonJunk NotJunk I've taken 6.041, so the notes this week seemed very straight-forward. Will we be covering much over the rest of the term that 6.041 doesn't touch on? - David From jehan@MIT.EDU Wed Nov 23 09:46:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANEkv5o007101 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 09:46:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANEktLG008626 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 09:46:55 -0500 (EST) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANEkruw021009 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 09:46:53 -0500 (EST) Message-Id: <5.2.1.1.2.20051123094356.00bc5c50@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 23 Nov 2005 09:44:58 -0500 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 12 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2503 Content-Length: 91 X-Keywords: NonJunk NotJunk This stuff is pretty easy. You should give the class the family boy/girl problem. jehan From arup@MIT.EDU Wed Nov 23 09:52:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANEqB5o011425 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 09:52:12 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANEqALG013433 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 09:52:10 -0500 (EST) Received: from [18.247.7.36] (BURTON-FIVE-FORTY-SEVEN.MIT.EDU [18.247.7.36]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANEpsiV022711 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 09:51:54 -0500 (EST) Message-ID: <438481DC.9010407@mit.edu> Date: Wed, 23 Nov 2005 09:51:08 -0500 From: Arup Sarma User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Week 12 Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2504 Content-Length: 194 X-Keywords: NotJunk Most of the probability reading seemed pretty straightforward, I guess the only thing is I was confused about independence (esp. mutual independence) section 5 p. 24 for a little bit. |Arup| From kromer@MIT.EDU Wed Nov 23 09:56:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANEuZ5o012000 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 09:56:35 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANEuYLG017749 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 09:56:34 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANEuWav024379 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 09:56:32 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jANEuWoQ012372; Wed, 23 Nov 2005 09:56:32 -0500 Received: from wibr-pub-7-101.wi.mit.edu (wibr-pub-7-101.wi.mit.edu [18.157.7.101]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 09:56:32 -0500 Message-ID: <20051123095632.2gu2rfengjo84c8o@webmail.mit.edu> Date: Wed, 23 Nov 2005 09:56:32 -0500 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2505 Content-Length: 260 X-Keywords: NotJunk p. 24: "Here's a better mental picture of what independent events look like. The sample space is the whole rectangle. Event A is the vertical stripe, and event B is the horizontal stripe." I was confused by this diagram--what is it trying to show? Katherine From jstritar@MIT.EDU Wed Nov 23 10:12:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANFCo5o018040 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 10:12:50 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANFCnLG005175 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 10:12:49 -0500 (EST) Received: from BAKER-FOUR-EIGHTY-NINE.MIT.EDU (BAKER-FOUR-EIGHTY-NINE.MIT.EDU [18.245.6.234]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANFCfrT001093 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 10:12:42 -0500 (EST) Subject: [Sayan] Reading comments From: Jon Stritar To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 23 Nov 2005 10:12:40 -0500 Message-Id: <1132758760.19347.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2506 Content-Length: 294 X-Keywords: NonJunk NotJunk 1.3 Step 1: Find the Sample Space I always think its amazing how important this step is for me to write down. Once I have the sample space down, and in the correct format for the problem, (like a tree, table, or all the events written out), the solution emerges quite naturally. Jon Stritar From vbrobbey@MIT.EDU Wed Nov 23 10:52:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANFqk5o008258 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 10:52:46 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANFqjLG014734 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 10:52:45 -0500 (EST) Received: from napoleon (NAPOLEON-BONAPARTE.MIT.EDU [18.18.3.59]) (authenticated bits=0) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANFqbod016551 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 10:52:38 -0500 (EST) Message-ID: <001301c5f045$ed049f80$3b031212@napoleon> From: "Valery Kwasi Brobbey" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 12 Comments Date: Wed, 23 Nov 2005 10:52:35 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0010_01C5F01C.032ABC10" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -0.718 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2507 Content-Length: 1335 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0010_01C5F01C.032ABC10 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Probability is so far the topic I'm liking the most in the course. I = took some courses in high school that dealt probability but I've been = surprised by the analysis of the coin problem in the notes and the game = we looked at in class on Monday. It was cool to see applications f = probability in Medicin and Law. ------=_NextPart_000_0010_01C5F01C.032ABC10 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Probability is so far the topic I'm = liking the most=20 in the course. I took some courses in high school that dealt probability = but=20 I've been surprised by the analysis of the coin problem in the notes and = the=20 game we looked at in class on Monday. It was cool to see applications f=20 probability in Medicin and Law.
------=_NextPart_000_0010_01C5F01C.032ABC10-- From petek@MIT.EDU Wed Nov 23 10:53:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANFrD5o009123 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 10:53:13 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANFrCLG015218 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 10:53:12 -0500 (EST) Received: from [128.30.5.10] (30-5-10.wireless.csail.mit.edu [128.30.5.10]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANFr92B016770 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 10:53:10 -0500 (EST) Message-ID: <4384906B.2020106@mit.edu> Date: Wed, 23 Nov 2005 10:53:15 -0500 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Probability Notes (This Week's Comments) Content-Type: multipart/alternative; boundary="------------040008090904050709070408" X-Spam-Score: 1.928 X-Spam-Level: * (1.928) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2508 Content-Length: 1550 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. --------------040008090904050709070408 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Probability is probably, as Ronnitt said, REALLY the most important thing we'll learn in this class. I'm looking forward to the conditional probability lecture today to help get a good picture of how to solve those kinds of problems. -P -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------040008090904050709070408 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Probability is probably, as Ronnitt said, REALLY the most important thing we'll learn in this class.  I'm looking forward to the conditional probability lecture today to help get a good picture of how to solve those kinds of problems.

-P
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------040008090904050709070408-- From meyer@imap.theory.csail.mit.edu Wed Nov 23 10:55:00 2005 Message-ID: <438490D4.4050201@csail.mit.edu> Date: Wed, 23 Nov 2005 10:55:00 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [Jelani][6.042] comments on reading 12 References: <4383F089.5000002@mit.edu> In-Reply-To: <4383F089.5000002@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 643 Status: RO X-UID: 2509 X-Keywords: Will fix 1/2^100. Thx for catching this. I think the text on p22 was correct, but I added a clarification (I hope it is anyway :-) in response to your concern. regards, A. Benjamin Lu wrote: > No real comments besides counter-intuitive things are cool, and I > think I found two errors in the notes: > * The first is the lower branch of the tree in the fair VS trick coin > tree (top of page 19). I think it should read 1^100 and not (1/2)^100. > * The second is the inclusion-exclusion formula (page 22) "Pr{A|BuC} = > Pr{A|C} + Pr{A|C} - Pr{AnB|C}" which should read "Pr{A|BuC} = Pr{A|C} > + Pr{B|C} - Pr{AnB|C}" > > ~Ben Lu > > From juang@MIT.EDU Wed Nov 23 10:58:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANFwD5o010901 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 10:58:13 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANFvlLG020517 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 10:57:47 -0500 (EST) Received: from [18.238.5.154] (EASTCAMPUS-SIX-SIXTY-FIVE.MIT.EDU [18.238.5.154]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANFvi7P018690 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 10:57:45 -0500 (EST) Message-ID: <4384916A.207@mit.edu> Date: Wed, 23 Nov 2005 10:57:30 -0500 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 12 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2510 Content-Length: 268 X-Keywords: NonJunk NotJunk I found it surprising (on page 27) that a set of pairwise independent events is not necessarily mutually independent. So much for intuition! I am actually working on that chessboard-tiling generating functions problem. ... Slowly. Have a good Thanksgiving, Jason. From zacharyozer@gmail.com Wed Nov 23 11:00:42 2005 Return-Path: Received: from wproxy.gmail.com (wproxy.gmail.com [64.233.184.193]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANG0g5o011315 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:00:42 -0500 Received: by wproxy.gmail.com with SMTP id i30so870396wra for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 08:00:42 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=PAXBZEC73KMl2iSnL3BQYAr3QbZ7g4jLa4IVxzxSHlvx473fWlez2eRd+RDmkdgoaFx93B8sb897N3nlbVcvfJAhXjdaDetiT/a7MYUZQ9EPmSzT1rZC583aVx5OweEszem9XEwSlszTVXHfKOUQPwygVXgja1t51Zbhd5JWcik= Received: by 10.65.103.12 with SMTP id f12mr6233395qbm; Wed, 23 Nov 2005 08:00:41 -0800 (PST) Received: by 10.65.152.13 with HTTP; Wed, 23 Nov 2005 08:00:41 -0800 (PST) Message-ID: <4f2613e50511230800t26b1610cm711c8049438076ca@mail.gmail.com> Date: Wed, 23 Nov 2005 11:00:41 -0500 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Question regarding Posteriori Probabilities MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jANG0g5o011315 Status: RO X-UID: 2511 Content-Length: 318 X-Keywords: NonJunk NotJunk So, why is it that the probability of B given A is the interesction of A and B divided by the probability of A? This seems to give a circular definition if you substitute the intersection for the probability of A given B * probability of B and repeating for the probability of A given B. Best regards, Zachary Ozer From meyer@imap.theory.csail.mit.edu Wed Nov 23 11:04:45 2005 Message-ID: <4384931E.8060306@csail.mit.edu> Date: Wed, 23 Nov 2005 11:04:46 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Michael Murray Subject: Re: [Hanson] ln12 reading References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 933 Status: RO X-UID: 2512 X-Keywords: we thought the rationale for multiplying was pretty well motivated in section 4.2: that is, if we took the better motivated version of independence given in 4.2, then we instantly derive the initial, multiplication version. The multiply version is technically preferable, because it works even for zero-probability events. Take another look at sect 4.2 and let me know if you're still not satisfied with the explanation. regards, A. Michael Murray wrote: > I found it surprising that the independence equation was Pr(A n B) = > Pr(A).Pr(B). > > My original thought was that the A and B would be independent if Pr(A > u B) = Pr(A) + Pr(B), this would imply that there is no overlap in > the sets A and B and so they would be independent. I am not sure > where this intuition fails and why we instead look at the > intersection. What is the intuition behind multiplying two > probabilities? > > Michael Murray From crowell@MIT.EDU Wed Nov 23 11:15:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANGFB5o014070 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:15:11 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANGFA7H006796 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:15:10 -0500 (EST) Received: from [18.238.6.73] (EASTCAMPUS-EIGHT-FORTY.MIT.EDU [18.238.6.73]) (authenticated bits=0) (User authenticated as crowell@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANGEjqt025708 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:15:03 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <17A1589D-CDD3-4659-90A7-884E8DE12846@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Robert Crowell Subject: [Hanson] Reading comments Date: Wed, 23 Nov 2005 11:14:42 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2513 Content-Length: 246 X-Keywords: NotJunk Hi, I'd like to see a little more about A Posteriori probabilities in class. I'm having some trouble understanding how to work backwards and arrive at some answer, like the Halting Problem example on page 18, section 3.4 --Rob Crowell From meyer@imap.theory.csail.mit.edu Wed Nov 23 11:17:51 2005 Message-ID: <43849631.8050307@csail.mit.edu> Date: Wed, 23 Nov 2005 11:17:53 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall Subject: Re: [David] Week 12 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-2022-JP Content-Transfer-Encoding: 7bit Content-Length: 968 Status: RO X-UID: 2514 X-Keywords: events are SETS (of outcomes), so A$B"A(BB is the intersection of events A and B. It consists of those outcomes in both A and B, so A$B"A(BB is typically referred to as "the event that both A and B occur". make sense? regards, A. Harrison King Hall wrote: > David- > Wow, I know nothing about probability. Well not absolutely nothing, > but close enough for the rigor required by this course. By far the > most interesting case in probability that I had not even considered > was Posteriori probabilities. I had always heard and made phrases like > that, but now I understand where the mathematical basis is from. I > realize it is only a special case of conditional probability, but I > think it is awesome. The most difficult thing for me for a long time > was independence since I didn't find a rigorous definition of Pr{A $B"A(B > B} in the notes. Maybe it was there but I didn't catch it for a while > and I am still a little fuzzy on it. > -Harrison From meyer@imap.theory.csail.mit.edu Wed Nov 23 11:29:50 2005 Message-ID: <438498FF.3020100@csail.mit.edu> Date: Wed, 23 Nov 2005 11:29:51 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall Subject: Re: [David] Week 12 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-2022-JP Content-Transfer-Encoding: 7bit Status: RO Content-Length: 969 X-UID: 2515 X-Keywords: events are SETS (of outcomes), so A$B"A(BB is the intersection of events A and B. It consists of those outcomes in both A and B, so A$B"A(BB is typically referred to as "the event that both A and B occur". make sense? regards, A. Harrison King Hall wrote: > David- > Wow, I know nothing about probability. Well not absolutely nothing, > but close enough for the rigor required by this course. By far the > most interesting case in probability that I had not even considered > was Posteriori probabilities. I had always heard and made phrases like > that, but now I understand where the mathematical basis is from. I > realize it is only a special case of conditional probability, but I > think it is awesome. The most difficult thing for me for a long time > was independence since I didn't find a rigorous definition of Pr{A $B"A(B > B} in the notes. Maybe it was there but I didn't catch it for a while > and I am still a little fuzzy on it. > -Harrison From cvnguyen@MIT.EDU Wed Nov 23 11:32:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANGWf5o019961 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:32:41 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANGW185025423 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:32:40 -0500 (EST) Received: from MULTIVAC (EASTCAMPUS-SIX-SIXTY.MIT.EDU [18.238.5.149]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANG3xjJ021224 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 11:04:01 -0500 (EST) Message-ID: <039201c5f047$859f21b0$9505ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 12 comments Date: Wed, 23 Nov 2005 11:03:54 -0500 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2516 Content-Length: 181 X-Keywords: NotJunk I found the Monty Hall example interesting, amusing, and entertaining. It's a good example of how often people rely on intuition rather than reasoning out the problem. --Chieu From cbossard@MIT.EDU Wed Nov 23 11:53:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANGrf5o006408 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 11:53:41 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANGrd7H021125 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 11:53:40 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANGrbEZ011052 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 23 Nov 2005 11:53:37 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jANGrbtd030995; Wed, 23 Nov 2005 11:53:37 -0500 Received: from pool-141-156-39-164.res.east.verizon.net (pool-141-156-39-164.res.east.verizon.net [141.156.39.164]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 23 Nov 2005 11:53:37 -0500 Message-ID: <20051123115337.v75luc9tqym8488g@webmail.mit.edu> Date: Wed, 23 Nov 2005 11:53:37 -0500 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 12 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2517 Content-Length: 266 X-Keywords: NonJunk NotJunk I found this to be a very interesting reading because probability is an interesting topic. I don't have any specific questions on any one section in particular because I have been at least somewhat introduced to all of these topics in my probability class. Cynthia From hmzhou@blackbird.csail.mit.edu Wed Nov 23 12:12:26 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHCQ5o015845; Wed, 23 Nov 2005 12:12:26 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jANHCP5L002915; Wed, 23 Nov 2005 12:12:25 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jANHCPoK002912; Wed, 23 Nov 2005 12:12:25 -0500 Date: Wed, 23 Nov 2005 12:12:25 -0500 (EST) From: Hanson Zhou To: Michael Murray cc: 6042-probs@theory.csail.mit.edu Subject: Re: [Hanson] ln12 reading In-Reply-To: Message-ID: References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2518 Content-Length: 850 X-Keywords: NotJunk Addition would mean that the events are mutually exclusive. Independence is different. Intuitively, event A and event B are independent means that A occurring affects B occurring not at all, and vice versa. Thus, we can compute Pr[A and B] by simply multiplying Pr[A]Pr[B], ie consider the probability of each event occurring separately. -Hanson On Wed, 23 Nov 2005, Michael Murray wrote: > I found it surprising that the independence equation was Pr(A n B) = > Pr(A).Pr(B). > > My original thought was that the A and B would be independent if Pr(A > u B) = Pr(A) + Pr(B), this would imply that there is no overlap in > the sets A and B and so they would be independent. I am not sure > where this intuition fails and why we instead look at the > intersection. What is the intuition behind multiplying two > probabilities? > > Michael Murray > From hmzhou@blackbird.csail.mit.edu Wed Nov 23 12:29:18 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHTH5o025703; Wed, 23 Nov 2005 12:29:17 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jANHTH7m003064; Wed, 23 Nov 2005 12:29:17 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jANHTHSZ003061; Wed, 23 Nov 2005 12:29:17 -0500 Date: Wed, 23 Nov 2005 12:29:17 -0500 (EST) From: Hanson Zhou To: Karena Tyan cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Reading Comments 12 In-Reply-To: <20051123014503.5lactfb6xr4kowcs@webmail.mit.edu> Message-ID: References: <20051123014503.5lactfb6xr4kowcs@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2519 Content-Length: 1285 X-Keywords: NotJunk Yes...independence for two variables is also mutual independence for two variables. It is true that the two probabilities are related...in fact they are related by precisely the formula for conditional probability that you wrote. The point is that often it is easy to compute one and not the other. For example, to compute Pr[A|B], you can compute Pr[A n B] and Pr[B] using a tree diagram and then use the formula to find Pr[A|B]. -Hanson On Wed, 23 Nov 2005, Karena Tyan wrote: > Just for clarification (this isn't my main question), two events that are > independent are *also* mutually independent, by the definition of independence > and mutual independence, correct? (pages 23-25) > > I am confused about the a posteriori probability covered in pages 17-18. In the > tutor problem 12.2, it is said that Pr{A} = 0.8 and Pr{L} = 0.4; in a case like > this, how would you compute Pr{L|A} without being given that it is 0.3? I > mean, I know that Pr{L|A} = Pr{A n L}/Pr{L}, but A n L is dependent upon the > value of Pr{A|L}, which is in turn dependent upon the value of Pr{L|A} in that > Pr{A|L} = Pr{A n L}/Pr{L}. In general, I'm just a little confused over how to > find A n B or Pr{A|B}. > > - Karena > > -- > 410 Memorial Drive > Cambridge, MA 02139 > (585)957-5923 > From dshin@MIT.EDU Wed Nov 23 12:54:51 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHsp5o030095 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:54:51 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANHsnSK026591; Wed, 23 Nov 2005 12:54:49 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jANHsjMu023547; Wed, 23 Nov 2005 12:54:46 -0500 (EST) Message-ID: <4384ACE2.2030105@mit.edu> Date: Wed, 23 Nov 2005 12:54:42 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 12 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-2022-JP Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2520 Content-Length: 960 X-Keywords: NotJunk A $B"A(B B is just an event. For example, if we are considering the roll of a die, and A is the event "roll a 1, 3 or 5", and B is the event "roll a 2, 3, or 4", then A $B"A(B B is the event "roll a 3". Hope that helps DS Harrison King Hall wrote: > David- > Wow, I know nothing about probability. Well not absolutely nothing, > but close enough for the rigor required by this course. By far the > most interesting case in probability that I had not even considered > was Posteriori probabilities. I had always heard and made phrases like > that, but now I understand where the mathematical basis is from. I > realize it is only a special case of conditional probability, but I > think it is awesome. The most difficult thing for me for a long time > was independence since I didn't find a rigorous definition of Pr{A $B"A(B > B} in the notes. Maybe it was there but I didn't catch it for a while > and I am still a little fuzzy on it. > -Harrison From mitras@theory.csail.mit.edu Wed Nov 23 12:56:19 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHuJ5o030250 for <6042-probs@theory.lcs.mit.edu>; Wed, 23 Nov 2005 12:56:19 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANHuISK026651; Wed, 23 Nov 2005 12:56:18 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jANHuAok013859; Wed, 23 Nov 2005 12:56:11 -0500 (EST) Message-ID: <4384AD3A.40406@theory.csail.mit.edu> Date: Wed, 23 Nov 2005 12:56:10 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Timothy Mwangi CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 12 Comments References: <6.2.1.2.2.20051123082057.03a06eb0@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051123082057.03a06eb0@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.74 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2521 Content-Length: 808 X-Keywords: NotJunk If you think of probabilities as functions attaching values in [0,1] to sets in the sample space (events), then this is not so surprising. The notion of time, (and therefore the notions of past and future) is not a part of the probability model. It just so happens that sometimes we attach more meaning to the events, like order them in time, for our convenience. The probability is just a function on the sets, does not matter if the sets "correspond" to events in the past or the future. Best, Sayan Timothy Mwangi wrote: > > Passage: 3.4 A Posteriori Probabilities > Page: 17 > I found this passage most surprising because of the notion that we > could use probability to say not only what will happen in the future > but what happened in the past. > > Sincerely, > Timothy M. Mwangi From dshin@MIT.EDU Wed Nov 23 12:56:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHuQ5o030256 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:56:26 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jANHuPb2019868 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:56:25 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANHuOgS028600; Wed, 23 Nov 2005 12:56:24 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jANHuCok013860; Wed, 23 Nov 2005 12:56:13 -0500 (EST) Message-ID: <4384AD39.5090206@mit.edu> Date: Wed, 23 Nov 2005 12:56:09 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sergio Bacallado CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 12 Comments References: <5.2.1.1.2.20051122220644.00c3e5e8@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20051122220644.00c3e5e8@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2522 Content-Length: 520 X-Keywords: NotJunk In fact, the outcome space can be infinite! In subsequent lectures, we should be seeing ways to avoid such a brute-force "identify all outcomes" strategy. DS Sergio Bacallado wrote: > I found the topic of independence very interesting. I wonder if there > is a simpler way to determine independence between events, because > evaluating the probabilities of intersections for every possible > subset of outcomes must become cumbersome as the outcome space becomes > very large. > > Sergio Bacallado > Group A. > From dshin@MIT.EDU Wed Nov 23 12:57:25 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHvP5o030406 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:57:25 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jANHvOb2021234 for <6042-probs@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:57:24 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANHrOgS028498; Wed, 23 Nov 2005 12:53:25 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jANHrGok013724; Wed, 23 Nov 2005 12:53:17 -0500 (EST) Message-ID: <4384AC89.7070802@mit.edu> Date: Wed, 23 Nov 2005 12:53:13 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 12 Comments References: <200511230117.jAN1H4mA021113@outgoing.mit.edu> In-Reply-To: <200511230117.jAN1H4mA021113@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2523 Content-Length: 613 X-Keywords: NotJunk Yes...assuming that the "given" event corresponds to a single nice sub-path down the tree (that's not always the case). DS Chris Yang wrote: > Quick question regarding conditional probability: is asking the > probability of something given another event the same as the > probability of that something happening using all the probabilities on > the subtree after that another event? Hm, that is a difficult > question to ask in e-mail form. Assuming an event is given is that > equivalent to, say, traversing down one path on the tree of that given > event? > > Have a good break! > > -Chris Yang > From meyer@imap.theory.csail.mit.edu Wed Nov 23 12:47:45 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANHlh5o028515 for <6042-staff@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:47:43 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jANHlgb2007245 for <6042-staff@theory.csail.mit.edu>; Wed, 23 Nov 2005 12:47:42 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANHlggS028178; Wed, 23 Nov 2005 12:47:42 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jANHlcMu023093; Wed, 23 Nov 2005 12:47:38 -0500 (EST) Message-ID: <4384AB39.3000602@theory.csail.mit.edu> Date: Wed, 23 Nov 2005 12:47:37 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jon Stritar , 6042 staff <6042-staff@theory.csail.mit.edu> Subject: Re: [Sayan] Reading comments References: <1132758760.19347.2.camel@localhost.localdomain> In-Reply-To: <1132758760.19347.2.camel@localhost.localdomain> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 383 X-UID: 2524 X-Keywords: NotJunk Glad you appreciate this systematic approach. -sayan Jon Stritar wrote: >1.3 Step 1: Find the Sample Space > >I always think its amazing how important this step is for me to write >down. Once I have the sample space down, and in the correct format for >the problem, (like a tree, table, or all the events written out), the >solution emerges quite naturally. > >Jon Stritar > > > From meyer@imap.theory.csail.mit.edu Wed Nov 23 14:16:50 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jANJGn5o012875 for ; Wed, 23 Nov 2005 14:16:49 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1Ef06r-0007tL-NP for meyer@csail.mit.edu; Wed, 23 Nov 2005 14:16:49 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jANJGmW4004056; Wed, 23 Nov 2005 14:16:48 -0500 (EST) Received: from [18.96.6.116] (SIMMONS-THREE-SEVENTY-ONE.MIT.EDU [18.96.6.116]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jANJGfQk027542 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Wed, 23 Nov 2005 14:16:41 -0500 (EST) In-Reply-To: <438498FF.3020100@csail.mit.edu> References: <438498FF.3020100@csail.mit.edu> Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=ISO-2022-JP; format=flowed Message-Id: <8c3ccf380da4c59ae306e8babc47e1c1@mit.edu> Cc: David D Shin From: Harrison King Hall Subject: Re: [David] Week 12 Comments Date: Wed, 23 Nov 2005 14:16:40 -0500 To: "Prof. Albert R. Meyer" X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1333 X-UID: 2525 X-Keywords: NotJunk I finally got it on the tutor problems after doing it 4 or 5 different ways and getting different answers each time, then I reverse engineered the meaning from that. David emailed me as well and I think I got the formal definition now. Thanks for all of the help. See you in class today. -Harrison On Nov 23, 2005, at 11:29 AM, Prof. Albert R. Meyer wrote: events are SETS (of outcomes), so A$B"A(BB is the intersection of events A and B. It consists of those outcomes in both A and B, so A$B"A(BB is typically referred to as "the event that both A and B occur". make sense? regards, A. Harrison King Hall wrote: > David- > Wow, I know nothing about probability. Well not absolutely nothing, > but close enough for the rigor required by this course. By far the > most interesting case in probability that I had not even considered > was Posteriori probabilities. I had always heard and made phrases like > that, but now I understand where the mathematical basis is from. I > realize it is only a special case of conditional probability, but I > think it is awesome. The most difficult thing for me for a long time > was independence since I didn't find a rigorous definition of Pr{A $B"A(B > B} in the notes. Maybe it was there but I didn't catch it for a while > and I am still a little fuzzy on it. > -Harrison From jeffhoff@MIT.EDU Mon Sep 26 21:22:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R1McaJ012361 for <6042-probs@theory.lcs.mit.edu>; Mon, 26 Sep 2005 21:22:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R1Mamx019819 for <6042-probs@theory.lcs.mit.edu>; Mon, 26 Sep 2005 21:22:36 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R1MW6W009583 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 26 Sep 2005 21:22:33 -0400 (EDT) Message-Id: <5.2.1.1.2.20050926211914.05405898@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 26 Sep 2005 21:22:29 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.701 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2526 Content-Length: 192 X-Keywords: NonJunk NotJunk Section: 1.3 (Surjective and the like) Is Surjective and total the same? also, what is the difference between RB = A and BR = A ??? i had a problem with that on the online tutor questions From hectorb@MIT.EDU Mon Sep 26 21:52:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R1qraJ017701 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 21:52:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R1qomx006603; Mon, 26 Sep 2005 21:52:51 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R1qnWu015391; Mon, 26 Sep 2005 21:52:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8R1qnQR000481; Mon, 26 Sep 2005 21:52:49 -0400 Received: from BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (User authenticated as hectorb@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 26 Sep 2005 21:52:49 -0400 Message-ID: <20050926215249.yrdmzari4l4w4k4c@webmail.mit.edu> Date: Mon, 26 Sep 2005 21:52:49 -0400 From: Hector Beltran To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 4 Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.714 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2527 Content-Length: 477 X-Keywords: NonJunk NotJunk "Definition 4.6. A topological sort of a partial order, , on a set, A, is a total ordering, @, on A such that a  b implies a @ b." pg. 13 (the mathematical symbols do not show) I might be wrong, but I think I remember from 6.034 that there's an easy, systematic way to find the topological sort of a DAG. Something about writing all the relations as a pair and them crossing them out or something like that. If I'm not mistaken can you please show this procedure. -Hector From lye@MIT.EDU Mon Sep 26 22:09:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R293aJ018733 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 22:09:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R292mx015906 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 22:09:02 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R28xZq019216 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 22:09:00 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8R28xuQ002975; Mon, 26 Sep 2005 22:08:59 -0400 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 26 Sep 2005 22:08:59 -0400 Message-ID: <20050926220859.7fxq62sdjf480k8k@webmail.mit.edu> Date: Mon, 26 Sep 2005 22:08:59 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 4 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2528 Content-Length: 304 X-Keywords: NonJunk NotJunk Page 3, Equivalence Relations, Def. 2.1 I don't understand what that symbol means in: "Given any total function, f, with domain A, define the binary relation f on A by the rule..." If the "=f" the relation itself? Is it a relation between A and the function f in the definition? Lunduo (Linda) Ye From ksindi@MIT.EDU Mon Sep 26 22:18:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R2I3aJ019365 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 22:18:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R2I2mx021038 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 22:18:02 -0400 (EDT) Received: from KAMIL.mit.edu (SENIOR-FOUR-NINETY-THREE.MIT.EDU [18.244.6.238]) (authenticated bits=0) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R2Hxot021157 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 22:18:00 -0400 (EDT) Message-Id: <6.2.3.4.2.20050926215317.039b3eb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Mon, 26 Sep 2005 22:17:57 -0400 To: 6042-probs@theory.csail.mit.edu From: Kamil Sindi Subject: [David] Week 4 Comments Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="=====================_724570327==.ALT" X-Spam-Score: 1.637 X-Spam-Level: * (1.637) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2529 Content-Length: 3485 X-Keywords: NonJunk NotJunk --=====================_724570327==.ALT Content-Type: text/plain; charset="us-ascii"; format=flowed 6042- Reading Comments 4 Student: Kamil Sindi Tutor: David "3 Partial Orders Partial orders are another class of binary relations that are particularly important in Computer Science, with applications that include task scheduling, database concurrency control, and proving that computations terminate. A general example of a partial order is the subset relation, , on sets. In fact, we will define partial orders via the subset relation in much the same way we defined equivalence relations. Namely, for any element, a, we think of a function, g, such that g(a) is the set of properties that a has. Then we relate different elements according to how their properties compare. All partial order will arise in this way." I found the material in this reading, on he whole to be pretty straight forward. I'd like to see some applications and examples of partial orders and digraphs. Also if the fact that for total relations AR=B is not true is emphasized because I thought that it was both AR=B and RB=A (I learnt my lesson from the TPs). --=====================_724570327==.ALT Content-Type: text/html; charset="us-ascii" 6042- Reading Comments 4
Student: Kamil Sindi
Tutor: David

"3 Partial Orders
Partial orders are another class of binary relations that are particularly important in Computer Science,
with applications that include task scheduling, database concurrency control, and proving
that computations terminate.
A general example of a partial order is the subset relation, , on sets. In fact, we will define partial
orders via the subset relation in much the same way we defined equivalence relations. Namely,
for any element, a , we think of a function, g , such that g ( a ) is the set of properties that a has. Then
we relate different elements according to how their properties compare. All partial order will arise
in this way."

I found the material in this reading, on he whole to be pretty straight forward. I'd like to see some applications and examples of partial orders and digraphs. Also if the fact that for total relations AR=B is not true is emphasized because I thought that it was both AR=B and RB=A (I learnt my lesson from the TPs).
 --=====================_724570327==.ALT-- From a_lopez@MIT.EDU Mon Sep 26 23:09:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R392aJ027248 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 23:09:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R391mx021463 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 23:09:01 -0400 (EDT) Received: from [18.234.1.10] (DP-TEN.MIT.EDU [18.234.1.10]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R38lu2002213 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 23:08:53 -0400 (EDT) Message-ID: <4338B7C2.3080906@mit.edu> Date: Mon, 26 Sep 2005 23:08:50 -0400 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Week 4 Readings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.811 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2530 Content-Length: 240 X-Keywords: NonJunk NotJunk The part of the readings that I found most confusing was Dilworth's Lemma and the Corollaries that followed it (pgs. 16-17). I still don't understand example 4.19: R is never defined. I'll have to wait for the in-class demo. Adriana From mdmurray@MIT.EDU Tue Sep 27 00:24:16 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R4OFaJ007314 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:16 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8R4OEF7028819 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:14 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R4OEOW023472 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:14 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8R4OB0v018175 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:11 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <19A13BDF-BC24-467B-AD72-833AE3C2EBC5@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Michael Murray Subject: [Hanson] E-mail comment on notes Date: Tue, 27 Sep 2005 00:23:55 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.671 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2531 Content-Length: 292 X-Keywords: NotJunk Hi, Overall I felt that the notes were very dense and difficult to read in many areas. In particular, i thought the section on partial orders was difficult to understand. Definition 3.1 is unnecessarily complicated and hard to understand despite several reads. Thanks, Michael Murray From iyzhang@MIT.EDU Tue Sep 27 01:35:47 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R5ZlaJ015354 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 01:35:47 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R5ZkZw007181 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 01:35:46 -0400 (EDT) Received: from [18.239.7.4] (MACGREGOR-FIVE-FIFTEEN.MIT.EDU [18.239.7.4]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8R5Zh0v020327 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 01:35:43 -0400 (EDT) Message-ID: <4338DA2B.5050003@mit.edu> Date: Tue, 27 Sep 2005 01:35:39 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 4 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2532 Content-Length: 99 X-Keywords: NotJunk I would be interested in hearing the graph theories about determining isomorphism. Thanks, Irene From hmzhou@blackbird.csail.mit.edu Tue Sep 27 12:00:09 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RG09aJ021657; Tue, 27 Sep 2005 12:00:09 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8RG09rd019857; Tue, 27 Sep 2005 12:00:09 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8RG09VM019854; Tue, 27 Sep 2005 12:00:09 -0400 Date: Tue, 27 Sep 2005 12:00:09 -0400 (EDT) From: Hanson Zhou To: "Jeffrey D. Hoff" cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 4 Comments In-Reply-To: <5.2.1.1.2.20050926211914.05405898@hesiod> Message-ID: References: <5.2.1.1.2.20050926211914.05405898@hesiod> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2533 Content-Length: 586 X-Keywords: NonJunk NotJunk Section 1.2 clarifies the notation that you're asking about. Surjective and total are NOT the same. For example, a relation that maps everything in A to one element in B would be total, but clearly not surjective. Again section 1.2 will help you to understand section 1.3. Go to office hours if this is still not clear. -Hanson On Mon, 26 Sep 2005, Jeffrey D. Hoff wrote: > > Section: 1.3 (Surjective and the like) > > Is Surjective and total the same? > also, what is the difference between > RB = A and BR = A ??? > > i had a problem with that on the online tutor questions > > From rnjacobs@MIT.EDU Tue Sep 27 14:20:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RIKRaJ019820 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:20:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8RIKPB2008091 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:20:26 -0400 (EDT) Received: from m2-225-9.mit.edu (M2-225-9.MIT.EDU [18.21.0.91]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8RIKLDj023330 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:20:22 -0400 (EDT) Received: (from rnjacobs@localhost) by m2-225-9.mit.edu (8.12.9) id j8RIKNmn016435; Tue, 27 Sep 2005 14:20:23 -0400 Message-Id: <200509271820.j8RIKNmn016435@m2-225-9.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] reading response Date: Tue, 27 Sep 2005 14:20:23 -0400 From: r n jacobs X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2534 Content-Length: 243 X-Keywords: NonJunk NotJunk I found all the discussion on graphs familiar and entertaining. I think I've reviewed the paper enough by now that, in the process of trying to figure out something to comment on, I understand what I was fuzzy on at first. - Robert Jacobs From dangut@MIT.EDU Tue Sep 27 14:24:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RIOHaJ020175 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:24:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8RIOFB2011539 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:24:15 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8RIODnG025422 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:24:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8RIODkp004499; Tue, 27 Sep 2005 14:24:13 -0400 Received: from MACLAURIN-TWO-SEVENTY-EIGHT.MIT.EDU (MACLAURIN-TWO-SEVENTY-EIGHT.MIT.EDU [18.90.6.23]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 27 Sep 2005 14:24:13 -0400 Message-ID: <20050927142413.aa9nkz9v9elckk0g@webmail.mit.edu> Date: Tue, 27 Sep 2005 14:24:13 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: Reading assignment #4 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.666 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2535 Content-Length: 221 X-Keywords: NotJunk I didn't really have much of a background in set relations, but I feel fairly confident in them. They're pretty natural feeling once I made the short step in my head and thinking of them like I do arrays in programming. From dshin@MIT.EDU Tue Sep 27 14:54:16 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RIsGaJ028307 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 14:54:16 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8RIsEBl003570; Tue, 27 Sep 2005 14:54:14 -0400 (EDT) Received: from [18.54.6.60] (NONE-THREE-FIFTEEN.MIT.EDU [18.54.6.60]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8RIrQ0v029657; Tue, 27 Sep 2005 14:53:26 -0400 (EDT) Message-ID: <43399522.70804@mit.edu> Date: Tue, 27 Sep 2005 14:53:22 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez CC: 6042-probs@theory.csail.mit.edu Subject: Re: Reading assignment #4 References: <20050927142413.aa9nkz9v9elckk0g@webmail.mit.edu> In-Reply-To: <20050927142413.aa9nkz9v9elckk0g@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2536 Content-Length: 441 X-Keywords: NotJunk If this helps you to understand sets better, that's fine, but just be aware of one difference between sets and arrays: An array can have the same element twice. A set cannot. DS Daniel A Gutierrez wrote: >I didn't really have much of a background in set relations, but I feel fairly >confident in them. They're pretty natural feeling once I made the short step >in my head and thinking of them like I do arrays in programming. > > From kevin08@MIT.EDU Tue Sep 27 14:56:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RIu6aJ028845 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 14:56:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8RIu5B2012304 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 14:56:05 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8RIu0sF013553 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 14:56:01 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8RIu0OS009664; Tue, 27 Sep 2005 14:56:00 -0400 Received: from SCI-12.MIT.EDU (SCI-12.MIT.EDU [18.51.0.197]) (User authenticated as kevin08@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 27 Sep 2005 14:56:00 -0400 Message-ID: <20050927145600.exez20leipkw8oks@webmail.mit.edu> Date: Tue, 27 Sep 2005 14:56:00 -0400 From: Kevin Y Wang To: 6042-probs@theory.lcs.MIT.EDU Subject: Friday Reading Comments: MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2537 Content-Length: 335 X-Keywords: NonJunk NotJunk Dear Prof. Meyer and Rubinfeld, I found section 5.2 (Pg. 20-21) on this weeks reading to be slightly confusing, especially in terms of a bijection. If you could provide one or two more examples in the notes and explicitly state the bijection functions along with how you arrived at them, that would be very helpful. Thank you, Kevin From jehan@MIT.EDU Tue Sep 27 19:41:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RNfaaJ030314 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 19:41:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8RNfZxw008762 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 19:41:35 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8RNfRMc022808 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 19:41:28 -0400 (EDT) Message-Id: <5.2.1.1.2.20050927193417.00befb90@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 27 Sep 2005 19:39:26 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2538 Content-Length: 108 X-Keywords: NonJunk NotJunk I'm confused as to how to formally show a partial order, in terms of the process and the notation. Jehan From dowgun@MIT.EDU Tue Sep 27 23:45:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8S3jwaJ009057 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 23:45:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8S3jv2l007973 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 23:45:57 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8S3jsjX010016 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 23:45:55 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8S3js8m026304; Tue, 27 Sep 2005 23:45:54 -0400 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 27 Sep 2005 23:45:54 -0400 Message-ID: <20050927234554.dta9x43ao1kw8gg0@webmail.mit.edu> Date: Tue, 27 Sep 2005 23:45:54 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] ln4 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.023 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2539 Content-Length: 576 X-Keywords: NonJunk NotJunk After reading this week's notes, I would greatly appreciated it if we were to be given a chart of all the symbol that we use in this course. Now on top of 'less than' and 'subset of' we have 'less then in partial order' and 'less than in total order' (see page 13). And those are just the comparison ones! Anyway, I found Dilworth's Lemma the most interesting part of teh notes, because it is not an immediately obvious fact about partially ordered sets, and it could potentially used among laymen to make absurd sounding predictions that mathematically have to be true. Neil From shauni@MIT.EDU Wed Sep 28 04:08:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8S884aJ004172 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 04:08:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8S882uO008535 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 04:08:02 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8S881to003949 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 04:08:01 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8S8814b013422; Wed, 28 Sep 2005 04:08:01 -0400 Received: from SIMMONS-ONE-SEVENTY-FIVE.MIT.EDU (SIMMONS-ONE-SEVENTY-FIVE.MIT.EDU [18.96.5.175]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 28 Sep 2005 04:08:01 -0400 Message-ID: <20050928040801.shdcyqt45pogcwow@webmail.mit.edu> Date: Wed, 28 Sep 2005 04:08:01 -0400 From: shauni@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.683 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2540 Content-Length: 361 X-Keywords: NonJunk NotJunk (Digraphs p.11) I found it interesting that digraphs are formally the same as binary relations, since I seemed to better understand digraphs, while the concept of binary relations is still a bit confusing to me. I think the visual nature of the topic makes it easier to grasp. I think it would be helpful to use more visual examples for other concepts as well. From lmccart@MIT.EDU Wed Sep 28 08:57:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8SCvVaJ007717 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 08:57:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8SCvUaP018053 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 08:57:30 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8SCvRj7000198 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 08:57:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8SCvR40029972; Wed, 28 Sep 2005 08:57:27 -0400 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 28 Sep 2005 08:57:27 -0400 Message-ID: <20050928085727.svxcu9nbxwpw48w0@webmail.mit.edu> Date: Wed, 28 Sep 2005 08:57:27 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 4 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.618 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2541 Content-Length: 236 X-Keywords: NonJunk NotJunk I was confused about the digraphs and the question on the tutor problems. The one that asked how many bijections there were of the star graph. I understand the things in the reading but I was unsure how to solve that problem. -lauren From minilek@theory.csail.mit.edu Wed Sep 28 09:14:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8SDE3w0009940 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:14:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8SDE1aP029949; Wed, 28 Sep 2005 09:14:01 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8SDDku2005432 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 28 Sep 2005 09:13:54 -0400 (EDT) Message-ID: <433A9724.1040201@theory.csail.mit.edu> Date: Wed, 28 Sep 2005 09:14:12 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Adriana Lopez Subject: Re: [Jelani] Week 4 Readings References: <4338B7C2.3080906@mit.edu> In-Reply-To: <4338B7C2.3080906@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2542 Content-Length: 544 X-Keywords: NonJunk NotJunk Adriana, In example 4.19, aRb iff b is both older and taller than a (note that this relation is not total since if b is older but not taller than a, then neither aRb nor bRa hold). If you are still confused after today's lecture, please ask more questions. -Jelani Adriana Lopez wrote: > The part of the readings that I found most confusing was Dilworth's > Lemma and the Corollaries that followed it (pgs. 16-17). I still > don't understand example 4.19: R is never defined. I'll have to wait > for the in-class demo. > Adriana From minilek@theory.csail.mit.edu Wed Sep 28 09:15:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8SDFuw0010055 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:15:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8SDFsaP001422; Wed, 28 Sep 2005 09:15:54 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8SDFhOl006155 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 28 Sep 2005 09:15:48 -0400 (EDT) Message-ID: <433A9798.9040207@theory.csail.mit.edu> Date: Wed, 28 Sep 2005 09:16:08 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: r n jacobs Subject: Re: [Jelani] reading response References: <200509271820.j8RIKNmn016435@m2-225-9.mit.edu> In-Reply-To: <200509271820.j8RIKNmn016435@m2-225-9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.413 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2543 Content-Length: 504 X-Keywords: NonJunk NotJunk Were you able to solve the "Not So Quick Exercise" on page 17 of the notes? "Devise an efficient procedure for finding the longest increasing and the longest decreasing subsequence in any given sequence of integers." -Jelani r n jacobs wrote: >I found all the discussion on graphs familiar and entertaining. > >I think I've reviewed the paper enough by now that, in the process of >trying to figure out something to comment on, I understand what I was >fuzzy on at first. > > - Robert Jacobs > > From ajshafer@MIT.EDU Wed Sep 28 09:31:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8SDVDw0011175 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:31:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8SDVBaP014982 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:31:12 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8SDV5Jj012033 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:31:06 -0400 (EDT) Message-ID: <000201c5c430$e100fda0$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 4 Comments Date: Wed, 28 Sep 2005 09:31:05 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2544 Content-Length: 368 X-Keywords: NonJunk NotJunk Page 16, Dilworth's Lemma. I'm not sure I understand what a chain or anti-chain are yet. Can you clarify those? A graph may be helpful to visualize it. Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From cwong08@MIT.EDU Wed Sep 28 19:14:42 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8SNEgw0021188 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:42 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8SNEfmi014956 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:41 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8SNEeUM013791 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:40 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8SNET0w018916 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:37 -0400 (EDT) Message-ID: <433B23C5.4000306@mit.edu> Date: Wed, 28 Sep 2005 19:14:13 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 4 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2545 Content-Length: 120 X-Keywords: NonJunk NotJunk I found the explanation of isomorphism on pages 20-21 to be unclear. I'm still confused about what it actually means. From vixen@MIT.EDU Wed Sep 28 20:20:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T0K7w0030769 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 20:20:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8T0K6fc019460 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 20:20:06 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8T0K2um012391 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 20:20:03 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 28 Sep 2005 20:19:05 -0400 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Subject: [Jelani] LN4 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2546 Content-Length: 239 X-Keywords: NonJunk NotJunk I was confused while reading the notes until pictures started showing up! Pictures--even little cirlces + arrows, with captions like, "The graph on the left is partial order, and the graph on the right is total order" would be useful. From benlu@MIT.EDU Wed Sep 28 23:14:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T3EGw0002792 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 23:14:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8T3EEk4002575 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 23:14:14 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8T3EBHW013342 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 23:14:11 -0400 (EDT) Message-ID: <433B5C07.7020905@mit.edu> Date: Wed, 28 Sep 2005 23:14:15 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [jelani][6.042] comments on reading 4 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.871 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2547 Content-Length: 277 X-Keywords: NotJunk Gah! Strange symbols! What's up with that union looking thingy on the bottom of page 3? And what's the deal with the bent < signs? Are those supposed to be a new symbol used to denote partial order, binary relationships, or just a fun thing to throw around in Latex? ~Ben From hejing85@MIT.EDU Thu Sep 29 01:10:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T5A1w0015428 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 01:10:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8T5A0k4006894 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 01:10:00 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8T59rsD001158 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 01:09:53 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8T59rT4007413; Thu, 29 Sep 2005 01:09:53 -0400 Received: from NEW-THREE-FORTY-NINE.MIT.EDU (NEW-THREE-FORTY-NINE.MIT.EDU [18.241.6.94]) (User authenticated as hejing85@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 01:09:53 -0400 Message-ID: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> Date: Thu, 29 Sep 2005 01:09:53 -0400 From: Jing He To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] relations and graph theory reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=GB2312 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2548 Content-Length: 355 X-Status: A X-Keywords: NonJunk NotJunk The most difficult part of this section for me to understand was what a total order was, since it wasn't given the formal definitions that the partial order came with in the reading. I also didn't understand the reasoning for the "proof" of Corollary 4.17 on page 16. The last section on graphs (and applications to computing) was the most interesting. From dnreshef@MIT.EDU Thu Sep 29 01:28:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T5Slw0016444 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 01:28:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8T5Skk4016361 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 01:28:46 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8T5SguX003000 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 01:28:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8T5SgFB018830; Thu, 29 Sep 2005 01:28:42 -0400 Received: from FOURTHFRONT.MIT.EDU (FOURTHFRONT.MIT.EDU [18.206.0.66]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 01:28:42 -0400 Message-ID: <20050929012842.vwjqga46c3wo0ko0@webmail.mit.edu> Date: Thu, 29 Sep 2005 01:28:42 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] READING ASSIGNMENT FOR 9/30 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2549 Content-Length: 236 X-Keywords: NonJunk I thought that this week's reading was easier than the last few have been. I did however find the sections on relations and images (especially about properties of relations) on pages 1 and 2 the most confusing. Hope that helps. -Dave From icharny@mit.edu Thu Sep 29 03:28:14 2005 Return-Path: Received: from auathena (EASTCAMPUS-SIX-NINETY-EIGHT.MIT.EDU [18.238.5.187]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T7SEw0028321 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 03:28:14 -0400 Received: (from icharny@localhost) by auathena (8.12.9) id j8T7SCC7021740; Thu, 29 Sep 2005 03:28:13 -0400 Subject: Week 4 Comments [Sayan] From: Isaac E Charny To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 29 Sep 2005 03:28:05 -0400 Message-Id: <1127978886.12799.1.camel@auathena> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 Status: RO X-UID: 2550 Content-Length: 141 X-Keywords: NonJunk NotJunk I'm still very hazy about the idea of partial order (section 3, page 7). Please add more real-life examples of partial orders. Isaac Charny From minilek@theory.csail.mit.edu Thu Sep 29 09:39:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TDdiw0019001 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 09:39:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TDdgRw014988; Thu, 29 Sep 2005 09:39:42 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TDdSat023690 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 29 Sep 2005 09:39:32 -0400 (EDT) Message-ID: <433BEEA7.1020607@theory.csail.mit.edu> Date: Thu, 29 Sep 2005 09:39:51 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [jelani][6.042] comments on reading 4 References: <433B5C07.7020905@mit.edu> In-Reply-To: <433B5C07.7020905@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2551 Content-Length: 697 X-Keywords: NonJunk NotJunk >> What's up with that union looking thingy on the bottom of page 3? It is a union symbol. It's saying A can be written as the union of elements of curvy-A (curvy-A is a set whose elements are sets). For the curvy-<, I'm not sure whether it's standard for all binary relations or just partial orders, or if it is even standard at all. I'll ask and get back to you. -Jelani Benjamin Lu wrote: > Gah! Strange symbols! > > What's up with that union looking thingy on the bottom of page 3? > > And what's the deal with the bent < signs? Are those supposed to be a > new symbol used to denote partial order, binary relationships, or just > a fun thing to throw around in Latex? > > ~Ben From minilek@theory.csail.mit.edu Thu Sep 29 09:48:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TDmLw0019756 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 09:48:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TDmIRw023312; Thu, 29 Sep 2005 09:48:18 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TDmC5F027533 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 29 Sep 2005 09:48:15 -0400 (EDT) Message-ID: <433BF0B9.3010308@theory.csail.mit.edu> Date: Thu, 29 Sep 2005 09:48:41 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [jelani][6.042] comments on reading 4 References: <433B5C07.7020905@mit.edu> In-Reply-To: <433B5C07.7020905@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2552 Content-Length: 538 X-Keywords: NonJunk NotJunk It seems that the curvy-< (or \prec in LaTeX), is for partial orders. A previous year's version of the 6.042 lecture notes makes this more explicit: http://theory.lcs.mit.edu/classes/6.042/fall04/relations.pdf. -Jelani Benjamin Lu wrote: > Gah! Strange symbols! > > What's up with that union looking thingy on the bottom of page 3? > > And what's the deal with the bent < signs? Are those supposed to be a > new symbol used to denote partial order, binary relationships, or just > a fun thing to throw around in Latex? > > ~Ben From meyer@csail.mit.edu Thu Sep 29 12:02:31 2005 Return-Path: Received: from smtp106.mail.sc5.yahoo.com (smtp106.mail.sc5.yahoo.com [66.163.169.226]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8TG2Uw0018858 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 12:02:31 -0400 Received: (qmail 37751 invoked from network); 29 Sep 2005 16:02:30 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp106.mail.sc5.yahoo.com with SMTP; 29 Sep 2005 16:02:29 -0000 Message-ID: <433C1019.6070903@csail.mit.edu> Date: Thu, 29 Sep 2005 12:02:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jing He CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] relations and graph theory reading comments References: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> In-Reply-To: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> Content-Type: text/plain; charset=GB2312 Content-Transfer-Encoding: 7bit Status: RO X-UID: 2553 Content-Length: 692 X-Keywords: 4.16 says there is a chain of size t, or an antichain of size n/t, for ANY t. So let t=sqrt(n) , and you get a chain of size sqrt(n) or an antichain of size n/sqrt(n), which also equals sqrt(n). So you 're sure to have a sqrt(n) size chain or antichain, which is what 4.17 says. Does that help? regards, A. Jing He wrote: >The most difficult part of this section for me to understand was what a total >order was, since it wasn't given the formal definitions that the partial order >came with in the reading. I also didn't understand the reasoning for the >"proof" of Corollary 4.17 on page 16. The last section on graphs (and >applications to computing) was the most interesting. > > From aeon@MIT.EDU Thu Sep 29 14:29:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TIT9w0011586 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 14:29:09 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TIT7fP013780 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 14:29:07 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TIT5JO024338 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 14:29:05 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8TIT5N9001636; Thu, 29 Sep 2005 14:29:05 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 12:29:05 -0600 Message-ID: <20050929122905.o3evhbb71k0gwg00@webmail.mit.edu> Date: Thu, 29 Sep 2005 12:29:05 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments LN4 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.704 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2554 Content-Length: 624 X-Keywords: NonJunk NotJunk Page 17, "Undirected Graphs" This piqued my interest, because it relates to things such as data network topology or pathfinding, both important topics in computer science. It was surprising to find out that it could be expressed mathematically almost exactly as it is expressed in program code. Just a personal note. =) Page 20, "Isomorphism" The bit about isophomorphism existing iff there is a bijective function that relates G1 and G2 seems very important, but there wasn't really an example to illustrate that point.. I think it'd be great to use one (or several simple ones) in lecture, if possible. - John Marrero From yangc@MIT.EDU Thu Sep 29 14:55:51 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TItpw0015337 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 14:55:51 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TItpBA028085 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 14:55:51 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8TIth0v011087 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 14:55:43 -0400 (EDT) Message-Id: <200509291855.j8TIth0v011087@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 4 Comments Date: Thu, 29 Sep 2005 14:55:31 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0036_01C5C505.D691B9B0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXFJ11GjcYVM07RT0GcIoKDDwuBqw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 0.419 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2555 Content-Length: 2207 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0036_01C5C505.D691B9B0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Is the problem of a general procedure for finding isomorphic graph related to P/N-P computability theory? It's pretty interesting how isomorphism is related to internet security - I look forward to hearing about it tomorrow. Chris Yang ------=_NextPart_000_0036_01C5C505.D691B9B0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Is the problem of a general procedure for finding = isomorphic graph related to P/N-P computability theory?  It’s pretty interesting how isomorphism is related to internet security – I = look forward to hearing about it tomorrow.

 

Chris Yang

------=_NextPart_000_0036_01C5C505.D691B9B0-- From cbossard@MIT.EDU Thu Sep 29 17:03:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TL3Qw0007235 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 17:03:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TL3Pek023883 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 17:03:25 -0400 (EDT) Received: from m12-182-21.mit.edu (M12-182-21.MIT.EDU [18.19.0.52]) (authenticated bits=56) (User authenticated as cbossard@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TL3IqU029138 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 17:03:18 -0400 (EDT) Received: (from cbossard@localhost) by m12-182-21.mit.edu (8.12.9) id j8TL3HEO014430; Thu, 29 Sep 2005 17:03:17 -0400 (EDT) Subject: [David] Week 4 Comments From: Cynthia C Bossard To: 6042-probs@theory.lcs.MIT.EDU Content-Type: multipart/alternative; boundary="=-557AlRUN6Crp9U/9OLU5" Date: Thu, 29 Sep 2005 17:03:17 -0400 Message-Id: <1128027797.14379.3.camel@m12-182-21.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.077 X-Spam-Level: * (1.077) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2556 Content-Length: 1094 X-Keywords: NonJunk NotJunk --=-557AlRUN6Crp9U/9OLU5 Content-Type: text/plain Content-Transfer-Encoding: 7bit Passage: mainly passage 2 Pages: 2-6 I believe that I have a general handle on most of the concepts, but I find that when trying to apply say partial order to something I do it improperly. My main mistakes are in testing for the equivalence relations. Can there be some extra real life type of problems given out on these? Cynthia --=-557AlRUN6Crp9U/9OLU5 Content-Type: text/html; charset=utf-8 Passage: mainly passage 2
Pages: 2-6
I believe that I have a general handle on most of the concepts, but I find that when trying to apply say partial order to something I do it improperly.  My main mistakes are in testing for the equivalence relations.  Can there be some extra real life type of problems given out on these?
Cynthia --=-557AlRUN6Crp9U/9OLU5-- From jstritar@MIT.EDU Thu Sep 29 17:13:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TLD0w0008098 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:13:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TLCrek003260 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:12:54 -0400 (EDT) Received: from JSTRITAR.MIT.EDU (JSTRITAR.MIT.EDU [18.245.4.29]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TLCk19002123 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:12:48 -0400 (EDT) Subject: [Sayan] Reading comments From: Jon Stritar To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Thu, 29 Sep 2005 17:12:47 -0400 Message-Id: <1128028367.7329.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2557 Content-Length: 198 X-Keywords: NonJunk NotJunk 4.4 Parallel Task Scheduling I thought it was really cool when we formalized this concept. It seems like it could have some pretty interesting applications if it were written into a program. Jon From mukkala@MIT.EDU Thu Sep 29 17:46:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TLkfw0012762 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:46:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TLkeek002675 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:46:40 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TLkWN5012048 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:46:33 -0400 (EDT) Message-Id: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Thu, 29 Sep 2005 17:46:34 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2558 Content-Length: 505 X-Status: A X-Keywords: NonJunk NotJunk In Section 3.2, the last 2 sentences read, "So < and < are total orders on [the set of all Real Numbers]. On the other hand, the subset relation is generally not total: any two finite sets of the same size will be incomparable under [the contained-in-or-equal subset relation]." Don't they mean just the [proper subset relation]? If the two finite sets contain the exact same elements, then aren't they comparable under [the contain-in-or-equal subset relation]? --Praveen Pamidimukkala Team 7 From avalys@MIT.EDU Thu Sep 29 18:10:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TMAow0016866 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TMAmek019487 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:48 -0400 (EDT) Received: from [18.96.5.102] (SIMMONS-ONE-O-TWO.MIT.EDU [18.96.5.102]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TMAYxG017937 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:42 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Required reading comments Date: Thu, 29 Sep 2005 18:10:26 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2559 Content-Length: 301 X-Keywords: NonJunk NotJunk I was "surprised" to read in section 5.0 (page 18) about the varied applications of ugraphs: solving routing problems, scheduling, etc. I'm curious about how such applications work. Meaning, I understand digraphs and ugraphs in theory, but what are we supposed to do with them? Alex Valys From cvnguyen@MIT.EDU Thu Sep 29 18:10:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TMAsw0016879 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TMAqek019557 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:53 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-ONE-O-FOUR-HUNDRED-TWENTY.MIT.EDU [18.238.7.153]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TMAiX2017995 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:46 -0400 (EDT) Message-ID: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 4 comments Date: Thu, 29 Sep 2005 18:10:47 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2560 Content-Length: 40 X-Status: A X-Keywords: NotJunk (Pages 11-21) Graphs are cool. --Chieu From aka_kame@MIT.EDU Thu Sep 29 18:12:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TMCKw0016952 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 18:12:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TMCIek020758 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 18:12:19 -0400 (EDT) Received: from [18.234.1.130] (DP-ONE-THIRTY.MIT.EDU [18.234.1.130]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TMC9xY018330 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 18:12:10 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-6--599128064 From: Akari Kameyama Subject: Week 4 Comments Date: Thu, 29 Sep 2005 18:12:07 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.819 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2561 Content-Length: 8217 X-Keywords: NonJunk NotJunk --Apple-Mail-6--599128064 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed In more formal terms, if G1 is a graph with vertices, V1 , and edges, =20= E1 , and likewise for G2 , then G1 is isomorphic to G2 iff there exists a bijective function f : V1 =20 =E2=86=92 V2 such that for every pair of vertices u, v =E2=88=88 V1 : u=E2=80=94v =E2=88=88 E1 iff f (u)=E2=80=94f = (v) =E2=88=88 E2 . I understand the overall gist of what this implies but I am finding =20 it difficult to interpret the terminology used above. So, as in =20 TP4.5 I can see how what is relevant is the number of edges to =20 vertices but in what specific way are they to be related?= --Apple-Mail-6--599128064 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
In more formal terms, if G1 is a graph with = vertices, V1 = , and = edges, E1 = , and likewise = for G2 = , = then=C2=A0
G1 is = isomorphic to G2 = iff there exists = a bijective function f : = V1=C2=A0=E2=86=92 V2 = such that for = every pair of=C2=A0
vertices u, v=C2=A0=E2=88=88 V1 :=C2=A0u=E2=80=94v=C2=A0=E2=88=88 E1 = iff = f = (u)=E2=80=94f (v) =E2=88=88 E2 = .=C2=A0

I understand = the overall gist of what this implies but I am finding it difficult to = interpret the terminology used above.=C2=A0 So, as in TP4.5 I can see = how what is relevant is the number of edges to vertices but in what = specific way are they to be related?
= --Apple-Mail-6--599128064-- From meyer@csail.mit.edu Thu Sep 29 18:14:33 2005 Return-Path: Received: from smtp106.mail.sc5.yahoo.com (smtp106.mail.sc5.yahoo.com [66.163.169.226]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8TMEXw0018292 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:14:33 -0400 Received: (qmail 43163 invoked from network); 29 Sep 2005 22:14:32 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp106.mail.sc5.yahoo.com with SMTP; 29 Sep 2005 22:14:31 -0000 Message-ID: <433C674B.7010806@csail.mit.edu> Date: Thu, 29 Sep 2005 18:14:35 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 4 Comments References: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 2562 Content-Length: 788 X-Keywords: If they had the same elements, they'd be the same set, not two sets :-) But I grant you it would have been clearer to say "two distinct sets," and I'm editing the notes to say that right now. Thanks for the comment. Regards, A. Praveen Pamidimukkala wrote: > In Section 3.2, the last 2 sentences read, "So < and < are total > orders on [the set of all Real Numbers]. On the other hand, the subset > relation is generally not total: any two finite sets of the same size > will be incomparable under [the contained-in-or-equal subset > relation]." Don't they mean just the [proper subset relation]? If > the two finite sets contain the exact same elements, then aren't they > comparable under [the contain-in-or-equal subset relation]? > > --Praveen Pamidimukkala > Team 7 From meyer@csail.mit.edu Thu Sep 29 18:17:11 2005 Return-Path: Received: from smtp102.mail.sc5.yahoo.com (smtp102.mail.sc5.yahoo.com [216.136.174.140]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8TMHBw0019156 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:17:11 -0400 Received: (qmail 57507 invoked from network); 29 Sep 2005 22:17:09 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp102.mail.sc5.yahoo.com with SMTP; 29 Sep 2005 22:17:09 -0000 Message-ID: <433C67E9.1050704@csail.mit.edu> Date: Thu, 29 Sep 2005 18:17:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chieu Nguyen CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Jelani] Week 4 comments References: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> In-Reply-To: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 2563 Content-Length: 253 X-Keywords: Glad you think so, but we'd like more specific reactions to particular sections of the notes that caught your attention. I'll look forward to fuller comments next week. regards, A. Chieu Nguyen wrote: > (Pages 11-21) Graphs are cool. > > --Chieu From fluff@MIT.EDU Thu Sep 29 19:38:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TNcZw0031657 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:38:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TNcXek017057 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:38:33 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TNcQEu004076 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:38:26 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 4 reading Date: Thu, 29 Sep 2005 19:38:00 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2564 Content-Length: 248 X-Keywords: NonJunk NotJunk I pretty much sort of got everything except page 2, 1.2 and 1.3. You know, the notation "we'll get a lot of mileage out of" (why?). I didn't see it being used anywhere else. I have no idea what those sections are trying to tell me. ~Crystal From sil_03@MIT.EDU Thu Sep 29 19:49:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TNn4w0000760 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 19:49:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TNmpek023955 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 19:48:52 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TNmjGD005640 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 19:48:45 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8TNmjNw030238; Thu, 29 Sep 2005 19:48:45 -0400 Received: from NEW-FOUR-O-NINE.MIT.EDU (NEW-FOUR-O-NINE.MIT.EDU [18.241.6.154]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 19:48:45 -0400 Message-ID: <20050929194845.vfu5rlutx0kk0ck8@webmail.mit.edu> Date: Thu, 29 Sep 2005 19:48:45 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2565 Content-Length: 261 X-Keywords: NonJunk NotJunk Total orders: I dont understand the definition given on page 8 section 3.2. I also don't understand what comparable means. g(a) : I dont understand this function and the weird less than sign thing on page 7 section 3.1. How do you read it? Silvia Baptista From sriaz@MIT.EDU Thu Sep 29 19:56:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TNukw0001998 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:56:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TNtfPM027606 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:56:45 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TNsa59006512 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:54:36 -0400 (EDT) Message-ID: <433C7EBE.5090601@mit.edu> Date: Thu, 29 Sep 2005 19:54:38 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2566 Content-Length: 200 X-Keywords: NonJunk NotJunk There was one thing that confused me about this week's reading. I would like more explanation about partial orders (Section 3, Page 7), as I did not fully understand the defintion Thanks Sameer From sheldons@MIT.EDU Thu Sep 29 20:07:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U07ow0003304 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:07:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U07mNS004283 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:07:48 -0400 (EDT) Received: from [10.0.0.2] (209-6-159-26.c3-0.smr-ubr3.sbo-smr.ma.cable.rcn.com [209.6.159.26]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U07eiu008469 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:07:41 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Sheldon Chan Subject: [David] Week 4 Comments Date: Thu, 29 Sep 2005 20:07:39 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2567 Content-Length: 232 X-Keywords: NonJunk NotJunk I found the concepts of minimal and maximal the hardest to grasp, on page 14. I know we discussed it in class, but simply the idea of how specific elements within a set can be both minimal and maximal is interesting. Sheldon From hzhou@MIT.EDU Thu Sep 29 20:20:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U0Kaw0004949 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:20:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U0KZNS012074 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:20:35 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-ONE-O-ONE.MIT.EDU [18.247.5.101]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0KJLB009051 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:20:29 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929200541.02a7a2e0@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 20:11:17 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2568 Content-Length: 267 X-Keywords: NonJunk NotJunk I am still really confused with equivalence relation. Is the relation the function itself, or a description of how a set maps to the other? I was wondering if possible, in class, you show some more examples of how to write a equivalence relation. - Steve Zhou From tonyng@MIT.EDU Thu Sep 29 20:51:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U0pBw0010840 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:51:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U0pANS003735 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:51:10 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0p7JB013783 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:51:08 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929204720.02105a00@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 20:51:15 -0400 To: 6042-probs@theory.csail.mit.edu From: Tony Ng Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2569 Content-Length: 353 X-Keywords: NotJunk I am confused by Images and Reverse Images (Section 1.2 page 2). I don't understand why the answers are what they are on the tutorial problems and the tutorial problems refer me to section 1.4 of the lecture notes for the explanation but section 1.4 does not exist. I find the notation confusing and I don't get why a relation R is total iff RB=A. From bilodeau@MIT.EDU Thu Sep 29 20:54:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U0s4w0011120 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:54:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U0s3NS005608 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:54:03 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THIRTY.MIT.EDU [18.239.5.30]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0s12M014170 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:54:01 -0400 (EDT) Message-Id: <200509300054.j8U0s12M014170@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] reading comments 9/30 Date: Thu, 29 Sep 2005 20:53:51 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C537.E5E488C0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXFWWxqWceFAiHpQ7qkkgg+RRv+Kg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.274 X-Spam-Level: * (1.274) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2570 Content-Length: 2129 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C537.E5E488C0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Peter Bilodeau The most difficult part of the reading was dilworth's dilemma and the ordered subset thing. While the demo with height and age in class was entertaining, the concept got lost somehow. ------=_NextPart_000_0000_01C5C537.E5E488C0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Peter Bilodeau

 

The most difficult part of the reading was = dilworth’s dilemma and the ordered subset thing.  While the demo with height and age = in class was entertaining, the concept got lost somehow.

------=_NextPart_000_0000_01C5C537.E5E488C0-- From brevzin@MIT.EDU Thu Sep 29 20:58:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U0wLw0011396 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:58:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U0wKNS008484 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:58:20 -0400 (EDT) Received: from BARRY.mit.edu ([18.218.1.202]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0w5ua014767 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:58:11 -0400 (EDT) Message-Id: <6.1.2.0.1.20050929205341.019c6650@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Thu, 29 Sep 2005 20:58:07 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2571 Content-Length: 860 X-Keywords: NotJunk So my question is actually from the in-class 3d: Describe a set with a unique minimal element and no mimumum. I don't think this set exists. So we have a set with n elemetns x1... xn, for every xi there exists a xj such that xj < xi EXCEPT for x1. Now if any xi is unrelated, it would be minimal, so we can say that this set is arranged in some order, so that xi !> xi+1 for all i. But then, by transivity, it is easy to see that x1 < xi for all i, but then x1 would be a minimum! But there can't be one ? The only way this would work is if you have a group of elements related circularly with a disjoint element. i.e. Some element x, with three other elements a,b,c such that a < b and b < c and c < a. Thus, x is minimal but there is no minimum. But this is ridiculous because partial orders are transitive. So, Does 3d have a solution? Barry From jjmonzon@MIT.EDU Thu Sep 29 21:10:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U1ASw0012907 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:10:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U1ANNS016568 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:10:24 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-ONE.MIT.EDU [18.247.5.1]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U1AGVi016527 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:10:17 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929210803.0214fd68@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 21:10:30 -0400 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2572 Content-Length: 407 X-Keywords: NotJunk I found the CR RC AR BR notation for relations quite confusing especially in the tutorial problems. I wish you could go over that part in class. When I answered the question, the tutorial also said to refer to part 1.4 of the notes which doesn't exist. Sincerely, Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From dshin@MIT.EDU Thu Sep 29 21:27:34 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U1RYw0015049 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:27:34 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U1RWuE010765; Thu, 29 Sep 2005 21:27:33 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U1RU3a004901; Thu, 29 Sep 2005 21:27:30 -0400 (EDT) Message-ID: <433C947C.80104@mit.edu> Date: Thu, 29 Sep 2005 21:27:24 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-probs@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050929210803.0214fd68@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050929210803.0214fd68@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2573 Content-Length: 673 X-Keywords: NotJunk Sorry, Joshua, the tutorial should have read part 1.3, not 1.4. Please take a look at that section, as well as section 1.2, where the notation is defined. Let me know if you still are confused after that. DS Joshua Jen C. Monzon wrote: > I found the CR RC AR BR notation for relations quite confusing > especially in the tutorial problems. I wish you could go over that > part in class. When I answered the question, the tutorial also said to > refer to part 1.4 of the notes which doesn't exist. > > Sincerely, > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From dshin@MIT.EDU Thu Sep 29 21:32:25 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U1WPw0015281 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:32:25 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U1WOuE010872; Thu, 29 Sep 2005 21:32:24 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U1WK3a005084; Thu, 29 Sep 2005 21:32:21 -0400 (EDT) Message-ID: <433C959F.2010304@mit.edu> Date: Thu, 29 Sep 2005 21:32:15 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Tony Ng CC: 6042-probs@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050929204720.02105a00@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050929204720.02105a00@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2574 Content-Length: 768 X-Keywords: NotJunk Sorry, the TP should read section 1.3, not section 1.4 (it has been modified accordingly). The set RB denotes the set of all a in A such that aRb for some b in B (definition given in section 1.2). A relation is total if domain element is related to some codomain element. In other words, R is total if for every a in A, aRb for some b in B. Do you see, then, why RB=A? DS Tony Ng wrote: > I am confused by Images and Reverse Images (Section 1.2 page 2). I > don't understand why the answers are what they are on the tutorial > problems and the tutorial problems refer me to section 1.4 of the > lecture notes for the explanation but section 1.4 does not exist. I > find the notation confusing and I don't get why a relation R is total > iff RB=A. > From ridell@MIT.EDU Thu Sep 29 21:52:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U1qLw0016864 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 21:52:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U1qJNS016707 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 21:52:19 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U1qCYq023797 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 21:52:12 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8U1qCSx017231; Thu, 29 Sep 2005 21:52:12 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 21:52:12 -0400 Message-ID: <20050929215212.8u1na8vqmsw8c4k8@webmail.mit.edu> Date: Thu, 29 Sep 2005 21:52:12 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: week 4 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2575 Content-Length: 329 X-Keywords: NonJunk NotJunk I am having trouble understanding partial orders. In class, it would help if we could do more problems similar to example 3.4 on page 8. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From dshin@MIT.EDU Thu Sep 29 21:56:13 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U1uDw0017035 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 21:56:13 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8U1uCe4008218 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 21:56:13 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U1uCB9029980; Thu, 29 Sep 2005 21:56:12 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U1uA3a005949; Thu, 29 Sep 2005 21:56:10 -0400 (EDT) Message-ID: <433C9B34.4020006@mit.edu> Date: Thu, 29 Sep 2005 21:56:04 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Silvia F. Baptista" CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] comments References: <20050929194845.vfu5rlutx0kk0ck8@webmail.mit.edu> In-Reply-To: <20050929194845.vfu5rlutx0kk0ck8@webmail.mit.edu> Content-Type: multipart/alternative; boundary="------------040202050508030303070709" X-Spam-Score: -1.658 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2576 Content-Length: 4197 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------040202050508030303070709 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sorry, the definition on page 8 section 3.2 had a typo. We have corrected it. It now reads: "...given any two numbers, one will be bigger *than* the other..." (before, it read "that" instead of "than") Basically, an order is total if whenever you take two distinct elements x and y in the set, one of the following will be true: "x is bigger than y" or "y is bigger than x" This may seem obvious if the example you have in mind is something like the natural numbers or the real numbers. However, think about the divisors-relation (call it "div-rel") we used in the class problems, where x is less than y iff x divides y. In this setting, neither "4 is-less-than-with-respect-to-div-rel 5" nor "5 is-less-than-with-respect-to-div-rel 4" are true. So this order is not total. Since you can't tell me which of the two (4 or 5) are bigger, they're not comparable. Otherwise, they would be comparable. And, wouldn't it be convenient to have a simple symbol we could use instead of this cumbersome "is-less-than-with-respect-to-div-rel"? Well, that's why we have the "weird less than sign thing"! Just convenient notation, when it's clear from the contex what the order relation is. The g(a) function is a little harder to explain over email. I'd suggest trying to read over section 3.1 carefully - if you're still stuck, stop by office hours. Hope that helps. DS Silvia F. Baptista wrote: >Total orders: I dont understand the definition given on page 8 section 3.2. I >also don't understand what comparable means. > >g(a) : I dont understand this function and the weird less than sign thing on >page 7 section 3.1. How do you read it? > >Silvia Baptista > > --------------040202050508030303070709 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Sorry, the definition on page 8 section 3.2 had a typo.  We have corrected it.  It now reads:

"...given any two numbers, one will be bigger than the other..."

(before, it read "that" instead of "than")

Basically, an order is total if whenever you take two distinct elements x and y in the set, one of the following will be true:

"x is bigger than y"

or

"y is bigger than x"

This may seem obvious if the example you have in mind is something like the natural numbers or the real numbers.  However, think about the divisors-relation (call it "div-rel") we used in the class problems, where x is less than y iff x divides y.  In this setting, neither "4 is-less-than-with-respect-to-div-rel 5" nor "5 is-less-than-with-respect-to-div-rel 4" are true.  So this order is not total.

Since you can't tell me which of the two (4 or 5) are bigger, they're not comparable.  Otherwise, they would be comparable.

And, wouldn't it be convenient to have a simple symbol we could use instead of this cumbersome "is-less-than-with-respect-to-div-rel"?  Well, that's why we have the "weird less than sign thing"!  Just convenient notation, when it's clear from the contex what the order relation is.

The g(a) function is a little harder to explain over email.  I'd suggest trying to read over section 3.1 carefully - if you're still stuck, stop by office hours.

Hope that helps.

DS

Silvia F. Baptista wrote: 
Total orders: I dont understand the definition given on page 8 section 3.2.  I
also don't understand what comparable means.

g(a) : I dont understand this function and the weird less than sign thing on
page 7 section 3.1.   How do you read it?

Silvia Baptista

--------------040202050508030303070709-- From dshin@MIT.EDU Thu Sep 29 22:18:02 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U2I1w0018334 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 22:18:02 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8U2I0e4001604 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 22:18:00 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U2I0B9000478; Thu, 29 Sep 2005 22:18:00 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U2Hu0v002992; Thu, 29 Sep 2005 22:17:56 -0400 (EDT) Message-ID: <433CA04E.7020802@mit.edu> Date: Thu, 29 Sep 2005 22:17:50 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Steven Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: comment References: <5.2.1.1.2.20050929200541.02a7a2e0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050929200541.02a7a2e0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.779 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2577 Content-Length: 785 X-Keywords: NotJunk A (binary) relation R on a set S is a subset of SxS. We write "aRb" if (a,b) is in R. For example, if R is "less than" on the set N (natural numbers), R contains (0,1), (0,2), (0,3), ..., (1,2), (1,3), ... (a whole bunch of more pairs). We write "0R6" to mean "(0,6) is in R". In this particular example, we typically use the symbol '<' instead of 'R'. An equivalence relation is a type of relation that obeys reflexivity, symmetry, and transitivity. Hope that helps DS Steven Zhou wrote: > I am still really confused with equivalence relation. Is the relation > the function itself, or a description of how a set maps to the other? > I was wondering if possible, in class, you show some more examples of > how to write a equivalence relation. > > - Steve Zhou > From sergiob@MIT.EDU Thu Sep 29 23:28:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3S4w0032671 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:28:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3S2pT024036 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:28:02 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U3Ro18009582 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:27:55 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929212427.00c40e38@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 21:27:46 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2578 Content-Length: 292 X-Keywords: NonJunk NotJunk I think the most interesting part of the reading was the section on proofs using Dilworth's lemma. You can prove very useful and non-trivial theorems using this lemma. I would like to know more about devising methods for search of fast paths of scheduling, etc. Sergio Bacallado Group 1 From aston@MIT.EDU Thu Sep 29 23:31:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3VPw0000522 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:31:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3VNpT026399 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:31:23 -0400 (EDT) Received: from aston (BOOKX.MIT.EDU [18.241.0.191]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U3VGQX010136 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:31:17 -0400 (EDT) Message-Id: <200509300331.j8U3VGQX010136@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: Week 4 Comments Date: Thu, 29 Sep 2005 23:33:15 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_006D_01C5C54E.2AB23450" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXFb7E3iFSJ01CzRnixYzBsGk8qUg== X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2579 Content-Length: 2345 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_006D_01C5C54E.2AB23450 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I was extremely excited by the mention of the possible ramifications of proving/disproving the isomorphism between graphs (short of the brute force approach). I'm looking forward to hearing about it in the lecture tomorrow. - Aston ------=_NextPart_000_006D_01C5C54E.2AB23450 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I was extremely excited by the mention of the = possible ramifications of proving/disproving the isomorphism between graphs = (short of the brute force approach). I’m looking forward to hearing about it = in the lecture tomorrow.

         =   

         =    - Aston

------=_NextPart_000_006D_01C5C54E.2AB23450-- From dshin@MIT.EDU Thu Sep 29 23:40:48 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3emw0001592 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:40:48 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3ekhP013864; Thu, 29 Sep 2005 23:40:46 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3ed3a009264; Thu, 29 Sep 2005 23:40:39 -0400 (EDT) Message-ID: <433CB3AF.5060405@mit.edu> Date: Thu, 29 Sep 2005 23:40:31 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 4 Comments References: <200509291855.j8TIth0v011087@outgoing-legacy.mit.edu> In-Reply-To: <200509291855.j8TIth0v011087@outgoing-legacy.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2580 Content-Length: 1028 X-Keywords: NotJunk An excellent question! Graph isomorphism represents a huge gap in our current understanding of complexity theory. We don't know whether it is easy to solve (i.e., we don't know whether graph-isomorphism is in P), and we also don't have strong evidence that it is hard to solve (i.e., we don't know whether graph-isomorphism is NP-complete). There aren't many other computational problems like this - for all other problems, we usually either have proven that it's easy, or have provided strong evidence that it is hard. Sorry for being a bit vague, but we don't quite have the machinery built up to talk precisely about notions of "easiness" or "hardness" or "evidence" - I can point you to further resources if you find this interesting. DS Chris Yang wrote: > Is the problem of a general procedure for finding isomorphic graph > related to P/N-P computability theory? Its pretty interesting how > isomorphism is related to internet security I look forward to > hearing about it tomorrow. > > Chris Yang > From dshin@MIT.EDU Thu Sep 29 23:44:22 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3iLw0001697 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 23:44:21 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3iKhP013932; Thu, 29 Sep 2005 23:44:20 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3iE0v005945; Thu, 29 Sep 2005 23:44:15 -0400 (EDT) Message-ID: <433CB486.3090302@mit.edu> Date: Thu, 29 Sep 2005 23:44:06 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Cynthia C Bossard CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 4 Comments References: <1128027797.14379.3.camel@m12-182-21.mit.edu> In-Reply-To: <1128027797.14379.3.camel@m12-182-21.mit.edu> Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.606 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2581 Content-Length: 516 X-Keywords: NotJunk The next pset has a few problems on partial orders/equivalence relations. Hopefully, this will give you the practice you need. DS Cynthia C Bossard wrote: > Passage: mainly passage 2 > Pages: 2-6 > I believe that I have a general handle on most of the concepts, but I > find that when trying to apply say partial order to something I do it > improperly. My main mistakes are in testing for the equivalence > relations. Can there be some extra real life type of problems given > out on these? > Cynthia From dshin@MIT.EDU Thu Sep 29 23:55:05 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3t4w0002277 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:55:04 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3t2hP014218; Thu, 29 Sep 2005 23:55:03 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3ss0v006342; Thu, 29 Sep 2005 23:54:55 -0400 (EDT) Message-ID: <433CB706.7080909@mit.edu> Date: Thu, 29 Sep 2005 23:54:46 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sergio Bacallado CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 4 Comments References: <5.2.1.1.2.20050929212427.00c40e38@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20050929212427.00c40e38@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2582 Content-Length: 1066 X-Keywords: NotJunk Scheduling is a fascinating problem. There are all sorts of variants on the scheduling problem (single processor/multiple processor, set-up times, penalties associated with uncompleted tasks, etc.). Some of these variants are easy to solve, but we have very strong evidence that others are hard to solve. Here's a sample of some of the variants for which we have strong evidence of their hardness: http://www.nada.kth.se/~viggo/wwwcompendium/node173.html (The concept of "easy", "hard", and "evidence" might seem subjective to you, but they are actually mathematically rigorous terms - to give their exact definitions is beyond the scope of this class). DS Sergio Bacallado wrote: > I think the most interesting part of the reading was the section on > proofs using Dilworth's lemma. You can prove very useful and > non-trivial theorems using this lemma. I would like to know more about > devising methods for search of fast paths of scheduling, etc. > > Sergio Bacallado > Group 1 > From dshin@MIT.EDU Thu Sep 29 23:56:38 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3ucw0003207 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 23:56:38 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3uahP014246; Thu, 29 Sep 2005 23:56:36 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3uS0v006411; Thu, 29 Sep 2005 23:56:28 -0400 (EDT) Message-ID: <433CB764.3070902@mit.edu> Date: Thu, 29 Sep 2005 23:56:20 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Lauren McCarthy CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [david] week 4 reading References: <20050928085727.svxcu9nbxwpw48w0@webmail.mit.edu> In-Reply-To: <20050928085727.svxcu9nbxwpw48w0@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2583 Content-Length: 351 X-Keywords: NotJunk Did you submit an answer to check the rationale given in the TP solution? DS Lauren McCarthy wrote: >I was confused about the digraphs and the question on the tutor problems. The >one that asked how many bijections there were of the star graph. I understand >the things in the reading but I was unsure how to solve that problem. >-lauren > > From bens@MIT.EDU Fri Sep 30 00:07:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U47Zw0006637 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:07:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U47YpT021239 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:07:34 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U47Rr4015592 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:07:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8U47RkF018421; Fri, 30 Sep 2005 00:07:27 -0400 Received: from LSLATER.MIT.EDU (LSLATER.MIT.EDU [18.221.0.171]) (User authenticated as bens@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 00:07:27 -0400 Message-ID: <20050930000727.7g9rx2spwprksswo@webmail.mit.edu> Date: Fri, 30 Sep 2005 00:07:27 -0400 From: Benjamin M Schwartz To: 6042-probs@theory.lcs.MIT.EDU Subject: Relations II MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2584 Content-Length: 135 X-Keywords: NonJunk NotJunk P. 11: Directed graphs are equivalent to relations? That's so awesome! That totally explains how to store graphs in a data structure. From dshin@MIT.EDU Fri Sep 30 00:08:58 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U48ww0006867 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 00:08:58 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U48uhP014538; Fri, 30 Sep 2005 00:08:56 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U48l0v006840; Fri, 30 Sep 2005 00:08:48 -0400 (EDT) Message-ID: <433CBA47.5010209@mit.edu> Date: Fri, 30 Sep 2005 00:08:39 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Wong CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 4 Comments References: <433B23C5.4000306@mit.edu> In-Reply-To: <433B23C5.4000306@mit.edu> Content-Type: multipart/alternative; boundary="------------090402000509070000020202" X-Spam-Score: -1.872 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2585 Content-Length: 3184 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------090402000509070000020202 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Formally, a graph G is a set of vertices V, and a set of edges E, which is a subset of VxV. G=(V,E) and H'=(V',E') are isomorphic if there exists a bijection f: V->V' such that (u,v) is in E iff (f(u),f(v)) is in E'. Informally... Think about the graph your team drew for the last in-class problem on Wednesday (involving 6.042 TA's trying to defeat Microsoft). Your team probably drew the graph differently from other teams around you - maybe your "path" went from left to right, while theirs went top to bottom. Maybe they had a pair of edges cross each other, while you arranged your nodes so that no edges needed to cross each other. Even despite all these differences, what would you say if a TA came by and said, "Nope, your graph is wrong - look at that team's graph - their graph is the correct one." You'd say, "No, but they're actually the same graph! They encode the same information! *We just drew them differently*!" Isomorphism tries to capture this notion, that the two graphs actually are equivalent in some sense. Hope that helps, DS Chris Wong wrote: > I found the explanation of isomorphism on pages 20-21 to be unclear. > I'm still confused about what it actually means. > --------------090402000509070000020202 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Formally, a graph G is a set of vertices V, and a set of edges E, which is a subset of VxV.  G=(V,E) and H'=(V',E') are isomorphic if there exists a bijection f: V->V' such that (u,v) is in E iff (f(u),f(v)) is in E'. 

Informally...

Think about the graph your team drew for the last in-class problem on Wednesday (involving 6.042 TA's trying to defeat Microsoft).  Your team probably drew the graph differently from other teams around you - maybe your "path" went from left to right, while theirs went top to bottom.  Maybe they had a pair of edges cross each other, while you arranged your nodes so that no edges needed to cross each other. 

Even despite all these differences, what would you say if a TA came by and said, "Nope, your graph is wrong - look at that team's graph - their graph is the correct one."  You'd say, "No, but they're actually the same graph!  They encode the same information!  We just drew them differently!"

Isomorphism tries to capture this notion, that the two graphs actually are equivalent in some sense. 

Hope that helps,

DS

Chris Wong wrote:
I found the explanation of isomorphism on pages 20-21 to be unclear. I'm still confused about what it actually means.


--------------090402000509070000020202-- From dshin@MIT.EDU Fri Sep 30 00:15:23 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U4FNw0007264 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 00:15:23 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4FLhP014667; Fri, 30 Sep 2005 00:15:21 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U4F70v007071; Fri, 30 Sep 2005 00:15:08 -0400 (EDT) Message-ID: <433CBBC3.4010707@mit.edu> Date: Fri, 30 Sep 2005 00:14:59 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Barry Revzin CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Hanson] Comments References: <6.1.2.0.1.20050929205341.019c6650@po14.mit.edu> In-Reply-To: <6.1.2.0.1.20050929205341.019c6650@po14.mit.edu> Content-Type: multipart/alternative; boundary="------------060500010706000409040001" X-Spam-Score: -1.872 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2586 Content-Length: 2935 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------060500010706000409040001 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit You have successfully proven that any *finite* set with a unique minimal element must have a minimum. Now, can you describe a set with a unique minimal element and no minimum? DS Barry Revzin wrote: > So my question is actually from the in-class 3d: Describe a set with a > unique minimal element and no mimumum. I don't think this set exists. > > So we have a set with n elemetns x1... xn, for every xi there exists a > xj such that xj < xi EXCEPT for x1. Now if any xi is unrelated, it > would be minimal, so we can say that this set is arranged in some > order, so that xi !> xi+1 for all i. But then, by transivity, it is > easy to see that x1 < xi for all i, but then x1 would be a minimum! > But there can't be one ? > > The only way this would work is if you have a group of elements > related circularly with a disjoint element. i.e. Some element x, with > three other elements a,b,c such that a < b and b < c and c < a. Thus, > x is minimal but there is no minimum. But this is ridiculous because > partial orders are transitive. > > So, > > Does 3d have a solution? > > Barry > --------------060500010706000409040001 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit You have successfully proven that any finite set with a unique minimal element must have a minimum.

Now, can you describe a set with a unique minimal element and no minimum?

DS

Barry Revzin wrote:
So my question is actually from the in-class 3d: Describe a set with a unique minimal element and no mimumum. I don't think this set exists.

So we have a set with n elemetns x1... xn, for every xi there exists a xj such that xj < xi EXCEPT for x1. Now if any xi is unrelated, it would be minimal, so we can say that this set is arranged in some order, so that xi !> xi+1 for all i. But then, by transivity, it is easy to see that x1 < xi for all i, but then x1 would be a minimum! But there can't be one ?

The only way this would work is if you have a group of elements related circularly with a disjoint element. i.e. Some element x, with three other elements a,b,c such that a < b and b < c and c < a. Thus, x is minimal but there is no minimum. But this is ridiculous because partial orders are transitive.

So,

Does 3d have a solution?

Barry


--------------060500010706000409040001-- From ozcan@MIT.EDU Fri Sep 30 00:19:11 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U4JAw0007464 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:10 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4J99x025940 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:10 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4J9MZ003306 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:09 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U4J13b010587 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:02 -0400 (EDT) Message-Id: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 30 Sep 2005 00:18:59 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2587 Content-Length: 254 X-Status: A X-Keywords: NonJunk NotJunk Hi, The only comments I have about this week's readings is that it was too long. Also, I am a little bit confused by the notation in the True/False questions in the Tutor problems, i.e. Question 4.4. Can we go over it in the lecture thanks yasin From xiaoranz@MIT.EDU Fri Sep 30 00:34:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U4Ygw0008988 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:34:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4YepT009373 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:34:41 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U4YY1w018862 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:34:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8U4YYtg030629; Fri, 30 Sep 2005 00:34:34 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 00:34:34 -0400 Message-ID: <20050930003434.peberlpwka0osskk@webmail.mit.edu> Date: Fri, 30 Sep 2005 00:34:34 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 4 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2588 Content-Length: 593 X-Keywords: NonJunk NotJunk I was confused about isomorphism for undirected graphs. Page 17, section 5. I am still unsure which characteristics to evaluate when determining whether two graphs are isomorphic or not. The correspondence between the vertex of the graphs are not clear to me. Sharon ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From lkini@MIT.EDU Fri Sep 30 01:05:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U55Uw0010930 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:05:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U55TpT028874 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:05:29 -0400 (EDT) Received: from [18.239.5.240] (MACGREGOR-TWO-FORTY.MIT.EDU [18.239.5.240]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U55O64022100 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:05:26 -0400 (EDT) Message-ID: <433CC791.7090107@mit.edu> Date: Fri, 30 Sep 2005 01:05:21 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 4 Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.164 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2589 Content-Length: 218 X-Keywords: NonJunk NotJunk Hi, I was totally confused about parallel task scheduling and theorem 4.11. I didn't understand the corollaries either (because of this). I would like it if you could go over that a little bit more. Thanks, Lohith From clintonb@MIT.EDU Fri Sep 30 01:09:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U59gw0012656 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:09:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U59epT001589 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:09:41 -0400 (EDT) Received: from ccbibmx30 (NEW-EIGHTY-TWO.MIT.EDU [18.241.5.82]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U59WYL022496 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:09:33 -0400 (EDT) Message-Id: <200509300509.j8U59WYL022496@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.lcs.mit.edu> Subject: Jelani: Reading #4 Date: Fri, 30 Sep 2005 01:08:26 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000C_01C5C55B.771A3830" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXFfP0id5mRPCReRxCm/RaUpus+WA== X-Spam-Score: 1.15 X-Spam-Level: * (1.15) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2590 Content-Length: 3146 X-Status: A X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_000C_01C5C55B.771A3830 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I'm just totally confused by this subject. I don't know. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_000C_01C5C55B.771A3830 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I’m just totally confused by this subject. I = don’t know…

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_000C_01C5C55B.771A3830-- From zev@MIT.EDU Fri Sep 30 01:42:26 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U5gQw0015480 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 01:42:26 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U5gLhP016462 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 01:42:21 -0400 (EDT) Received: from [18.243.2.60] (PRINTER-ON-FIRE.MIT.EDU [18.243.2.60]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U5gE0v010315 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 01:42:14 -0400 (EDT) Message-ID: <433CD035.4010005@mit.edu> Date: Fri, 30 Sep 2005 01:42:13 -0400 From: Zev Benjamin User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050726) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 4 comments Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2591 Content-Length: 370 X-Keywords: NonJunk NotJunk The only sections I found confusing were 1.2 and 1.3 on page 2. The rest of the notes were really cool. It was a lot of information, but the connections drawn between them was interesting. The fact that set theory can be examined from so many angles is fascinating. In addition, I enjoyed the practical examples (here's how we can do parallel processing). Zev From ereid@MIT.EDU Fri Sep 30 01:55:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U5tnw0016425 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U5tmuA029863 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:48 -0400 (EDT) Received: from [18.242.6.66] (NEXT-THREE-TWENTY-ONE.MIT.EDU [18.242.6.66]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U5tJPa026454 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:19 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <348E403C-2681-4A5D-BE9B-0F623B985801@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Reading Comments Date: Fri, 30 Sep 2005 01:55:15 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.078 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2592 Content-Length: 195 X-Keywords: NonJunk NotJunk If they could talk a bit about parallel task scheduling in lecture that would be nice. I was wondering if there was any mathematical way to maximize it, or if it's more of a guess and check? From mpapi@MIT.EDU Fri Sep 30 01:55:58 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U5tww0016442 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:58 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8U5tvgG027672 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:57 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U5tvGR005298 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:57 -0400 (EDT) Received: from MACGREGOR-THREE-FORTY-SIX.MIT.EDU (MACGREGOR-THREE-FORTY-SIX.MIT.EDU [18.239.6.91]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U5tl3a014044 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:47 -0400 (EDT) Subject: [Jelani] Week 4 Comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 30 Sep 2005 01:55:45 -0400 Message-Id: <1128059745.12411.40.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2593 Content-Length: 390 X-Keywords: NonJunk NotJunk The section on images, following "The set C R is called the image of C under R"; the notes mention that it's notation "that we'll get a lot of mileage out of". It might be helpful to see some examples or have the topic appear somehow in an in-class problem. While the concept itself isn't that hard to understand, I could imagine situations where it could be used in tricky ways. -- Matt From crowell@MIT.EDU Fri Sep 30 02:26:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U6Qgw0020313 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:26:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U6QeuA017073 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:26:40 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U6QcIw028433 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:26:38 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8U6QckL029923; Fri, 30 Sep 2005 02:26:38 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 02:26:38 -0400 Message-ID: <20050930022638.fxnit9klgug0kk8k@webmail.mit.edu> Date: Fri, 30 Sep 2005 02:26:38 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2594 Content-Length: 180 X-Keywords: NonJunk NotJunk I'm a little unclear on the section involving the AR=B notation found on page 2 in section 1.2 It's a bit confusing as to what is meant by having AR = something. -Rob Crowell From mracich@MIT.EDU Fri Sep 30 02:48:47 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U6mlw0022002 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:48:47 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U6mkMf017924 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:48:46 -0400 (EDT) Received: from MACGREGOR-THREE-SIXTY-TWO.MIT.EDU (MACGREGOR-THREE-SIXTY-TWO.MIT.EDU [18.239.6.107]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U6mh3a015432 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:48:44 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 4 (Binary Relations) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 30 Sep 2005 02:48:40 -0400 Message-Id: <1128062920.30191.6.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 0.999 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2595 Content-Length: 181 X-Keywords: NonJunk NotJunk I found section 3.5, Products and Restrictions of Relations, (located on page 10) kind of confusing. I would find it very helpful if this was reviewed in class. Moira Racich From ryan786@MIT.EDU Fri Sep 30 02:49:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U6n1w0022046 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:49:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U6mxuA000082 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:48:59 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U6mwYu029662 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:48:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8U6mwiR013101; Fri, 30 Sep 2005 02:48:58 -0400 Received: from PSK-SIX.MIT.EDU (PSK-SIX.MIT.EDU [18.217.1.6]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 02:48:58 -0400 Message-ID: <20050930024858.plht40d3x3lcssg0@webmail.mit.edu> Date: Fri, 30 Sep 2005 02:48:58 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [sayan] email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.071 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2596 Content-Length: 209 X-Keywords: NonJunk NotJunk This weeks reading was alright, since I have had some experience with graphs of this nature. Partial orders will probably give me the most trouble, but overall I think I understand the material. -Ryan Young From kjhollen@MIT.EDU Fri Sep 30 02:57:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U6vGw0023955 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:57:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U6vFuA004912 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:57:15 -0400 (EDT) Received: from MACGREGOR-FOUR-EIGHTY.MIT.EDU (MACGREGOR-FOUR-EIGHTY.MIT.EDU [18.239.6.225]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U6utP6000037 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:56:59 -0400 (EDT) Subject: [Hanson] confused about isomorphism From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 30 Sep 2005 02:56:55 -0400 Message-Id: <1128063415.13619.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.227 X-Spam-Level: *** (3.227) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2597 Content-Length: 273 X-Keywords: NonJunk NotJunk I think that I've almost got this concept (graphs and isomorphism) but the explanations in the reading (p. 20-22) didn't really give me a good feel for how to solve problems with isomorphism. It might be nice to have a few more examples of how to use/apply this property. From lana@MIT.EDU Fri Sep 30 03:04:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U74aw0024273 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:04:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U74YuA008996 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:04:34 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U74RN8000360 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:04:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8U74RrL013710; Fri, 30 Sep 2005 03:04:27 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 03:04:27 -0400 Message-ID: <20050930030427.q2wiidvufjls00o8@webmail.mit.edu> Date: Fri, 30 Sep 2005 03:04:27 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP4 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2598 Content-Length: 258 X-Keywords: NonJunk NotJunk The part of the reading that intrigued me the most was parallel task scheduling, particularly the question of existence of an efficient way of doing it. So I would like to know if there is a purely computational (i.e. non-intuitive) way of schceduling. Lana From adnaan@MIT.EDU Fri Sep 30 03:26:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U7QKw0029048 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:26:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U7QJuA021127 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:26:19 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U7QFJA001313 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:26:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8U7QFfA009788; Fri, 30 Sep 2005 03:26:15 -0400 Received: from NEW-FOUR-SIXTY-SEVEN.MIT.EDU (NEW-FOUR-SIXTY-SEVEN.MIT.EDU [18.241.6.212]) (User authenticated as adnaan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 03:26:15 -0400 Message-ID: <20050930032615.lhnye9xspwoo4ck8@webmail.mit.edu> Date: Fri, 30 Sep 2005 03:26:15 -0400 From: Adnaan N Jiwaji To: 6042-probs@theory.lcs.MIT.EDU Subject: relation rwading - hanson MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.043 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2599 Content-Length: 73 X-Keywords: NonJunk NotJunk I found the distintion between partial and total order confusing Adnaan From juang@MIT.EDU Fri Sep 30 03:28:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U7SYw0029158 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 03:28:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U7SXuA022312 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 03:28:33 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U7SLLg001364 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 03:28:26 -0400 (EDT) Message-ID: <433CE8F5.9030605@mit.edu> Date: Fri, 30 Sep 2005 03:27:49 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 4 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2600 Content-Length: 238 X-Keywords: NonJunk NotJunk "So with an unlimited number of processors, the time to complete all the tasks is the size of the largest chain." Is there a general way to determine the fastest way of scheduling tasks if number of processors is finite? Jason Juang. From medrano@MIT.EDU Fri Sep 30 03:35:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U7Z5w0029675 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:35:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U7Z4uA025933 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:35:04 -0400 (EDT) Received: from scrubbing-bubbles.mit.edu (SCRUBBING-BUBBLES.MIT.EDU [18.7.16.68]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U7YvDS001626 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:34:57 -0400 (EDT) Received: (from medrano@localhost) by scrubbing-bubbles.mit.edu (8.12.9) id j8U7Yvr8016362; Fri, 30 Sep 2005 03:34:57 -0400 (EDT) Date: Fri, 30 Sep 2005 03:34:56 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Isomorphism Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2601 Content-Length: 353 X-Keywords: NonJunk NotJunk The examples in the text about isomorphism are simplistic. Is there something more complicated we can work with. Some isomorphism that will not be as obvious just so we can understand the concept a bit better. Changing the name of the vertices for A,B,C,D to 1,2,3,4 was not difficult to determine that those two graphs are isomorphic. Jesus Medrano From rshroff@MIT.EDU Fri Sep 30 03:58:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U7wWw0030762 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:58:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U7wUuA009301 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:58:31 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U7wToG002494 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:58:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8U7wTDA001647; Fri, 30 Sep 2005 03:58:29 -0400 Received: from NEXT-SIXTY-SEVEN.MIT.EDU (NEXT-SIXTY-SEVEN.MIT.EDU [18.242.5.67]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 03:58:29 -0400 Message-ID: <20050930035829.7jemghp8sj4s8sg8@webmail.mit.edu> Date: Fri, 30 Sep 2005 03:58:29 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David]E-mail Comments for assigned reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.694 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2602 Content-Length: 166 X-Keywords: NonJunk NotJunk Reference: Theorem 4.11 I didn't really understand parallel task scheduling or its realted corollaries and would like it if we went over it in class. -Rahul Shroff From jacques@MIT.EDU Fri Sep 30 04:07:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U87dw0031479 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 04:07:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U87cuA014246 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 04:07:38 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U87aMx002803 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 04:07:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8U87aT4002295; Fri, 30 Sep 2005 04:07:36 -0400 Received: from EASTCAMPUS-ONE-O-THREE-HUNDRED-FORTY.MIT.EDU (EASTCAMPUS-ONE-O-THREE-HUNDRED-FORTY.MIT.EDU [18.238.7.73]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 04:07:36 -0400 Message-ID: <20050930040736.fk9hd1m30p8oowwc@webmail.mit.edu> Date: Fri, 30 Sep 2005 04:07:36 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 4 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.917 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2603 Content-Length: 232 X-Keywords: NonJunk NotJunk In the lecture notes, page 8, there is a very good definition of the property of antisymmetry. However, asymmetry is mentioned in the lecture slides but never really defined in the lecture notes. That's something I'd like to see. From scot@MIT.EDU Fri Sep 30 04:11:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U8B2w0032679 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 04:11:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U8B1uA016199 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 04:11:01 -0400 (EDT) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U8Ar0Q002931 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 04:10:55 -0400 (EDT) Message-ID: <433CF299.5040004@mit.edu> Date: Fri, 30 Sep 2005 04:08:57 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 4 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2604 Content-Length: 253 X-Keywords: NonJunk NotJunk I found partial orders (from page 7) to be confusing during lecture. Perhaps it was gone over too quickly for me. Also, I had never heard of modulo before. I did not find it in the appendix or explained, so I wlll need to look it up. Thanks, Scot From knightfu@MIT.EDU Fri Sep 30 05:29:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U9TTw0008718 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 05:29:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U9TSkI028204 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 05:29:28 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U9TMVY005553 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 05:29:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8U9TLWS016208; Fri, 30 Sep 2005 05:29:21 -0400 Received: from EASTCAMPUS-ONE-FORTY-FOUR.MIT.EDU (EASTCAMPUS-ONE-FORTY-FOUR.MIT.EDU [18.248.5.144]) (User authenticated as knightfu@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 05:29:21 -0400 Message-ID: <20050930052921.ls56zys1yk8wsgkc@webmail.mit.edu> Date: Fri, 30 Sep 2005 05:29:21 -0400 From: Knight W Fu To: 6042-probs@theory.csail.MIT.EDU Subject: Email Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2605 Content-Length: 1124 X-Keywords: NonJunk NotJunk Definition 2.1. Given any total function, f, with domain A, define the binary relation :=f on A by the rule: a :=f b iff f(a) = f(b) (1) for all a, b 2 A. A binary relation is an equivalence relation iff it equals :=f for some f. (the above was taken from Relations II, page 3.) 2.3 Properties of Equivalence Relations Equivalence relations have some obvious properties that occur so frequently they merit names: Definition 2.7. A binary relation R on a set A is: ? reflexive iff for every a 2 A, aRa, Course Notes, Week 4: Binary Relations 5 ? symmetric iff for every a, a0 2 A, aRa0 implies a0 Ra, ? transitive iff for every a, b, c 2 A, [aRb and bRc] implies aRc. (the above was taken from Relations II, page 4 The excerpts were most interesting because in Algebra (18.701), equivalent relationships are defined to be a relationship that is reflexive, symmetric, and transitive. The definition offered here is actually a consequence of the fact that the inverse image of a function partitions the domain (and the partition defines an equivalent relationship under the inverse image and creates equivalent classes.) From shreyes19@gmail.com Fri Sep 30 06:07:42 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.193]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UA7gw0014053 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 06:07:42 -0400 Received: by xproxy.gmail.com with SMTP id t11so373023wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 03:07:37 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=IU5OwGbsTjTvFI3JXlP+viVoNI7szFgWq+xTNH3EJE88sxukfV5bhdG6nO/+w+k5Ln0g0QrsM6jt4QjfJmwz1xPj+rmoxstTjMD+uVJIhbFaOYR6mQdBc3eHk6BG1AFpIAGMjwshtZA73c8FFJEvQXPJl9uDjDlvIeYTdp0nC1Q= Received: by 10.70.105.11 with SMTP id d11mr840976wxc; Fri, 30 Sep 2005 03:07:37 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Fri, 30 Sep 2005 03:07:37 -0700 (PDT) Message-ID: <3c33fe380509300307j2d1cb405ye12ab51ede8fb8e9@mail.gmail.com> Date: Fri, 30 Sep 2005 06:07:37 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 4 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_1915_2104057.1128074857232" Status: RO X-UID: 2606 Content-Length: 1974 X-Keywords: NonJunk NotJunk ------=_Part_1915_2104057.1128074857232 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Good morning, Here are my comments on the week 4 reading: The part I found most difficult in this week's reading was the definitions on page 10. Specifically, the definition of restrictions. "Definition 3.13. Let R be a relation on a set, A, and let B be a subset of A. The restrictio= n of R to B is the relation on B whose graph is graph (R) \ (B =D7 B). " (pag= e 10). I understand the concept of a restriction, but the specific definition that's provided is a little confusing to me, especially the graph of R [int= ] (B x B) part. I like how it goes through the properties of relations for th= e restriction on a set, but perhaps a couple examples of how restrictions are used will be helping in getting past the technical definition. Thanks, Shreyes Seshasai Group 7 ------=_Part_1915_2104057.1128074857232 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Good morning,

Here are my comments on the week 4 reading:

The part I found most difficult in this week's reading was the definitions on page 10.  Specifically, the definition of restrictions.  "Definition 3.13. Let R be a relation on a set, A, and let B be a subset of A. The restriction of R to B is the relation on B whose graph is graph (R) \ (B =D7 B). " (page 10).  I understand the concept of a restriction, but the specific definition that's provided is a little confusing to me, especially the graph of R [int] (B x B) part.  I like how it goes through the properties of relations for the restriction on a set, but perhaps a couple examples of how restrictions are used will be helping in getting past the technical definition.

Thanks,
Shreyes Seshasai
Group 7
------=_Part_1915_2104057.1128074857232-- From miki_tnd@MIT.EDU Fri Sep 30 06:33:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UAXUw0017034 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 06:33:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UAXTkI004709 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 06:33:29 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UAXMfR008073 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 06:33:23 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UAXMu1020980; Fri, 30 Sep 2005 06:33:22 -0400 Received: from PIKA-ONE-O-NINE.MIT.EDU (PIKA-ONE-O-NINE.MIT.EDU [18.214.1.109]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 06:33:22 -0400 Message-ID: <20050930063322.bnc5qws54xz40gw8@webmail.mit.edu> Date: Fri, 30 Sep 2005 06:33:22 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: (Sayan) Reading Assignment 4 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2607 Content-Length: 245 X-Keywords: NonJunk NotJunk I found the section on Parellel Task Scheduling on pg. 14 pretty interesting. If there is time in class, it would be great if more strategies for parellel task scheduling (or related operations research applications) were covered. ~Thu Duong From natalia3@MIT.EDU Fri Sep 30 06:33:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UAXWw0017043 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 06:33:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UAXUkI004716 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 06:33:31 -0400 (EDT) Received: from Drache (NEW-ONE-EIGHTY-NINE.MIT.EDU [18.241.5.189]) (authenticated bits=0) (User authenticated as natalia3@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UAXMSo008071 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 06:33:24 -0400 (EDT) Message-Id: <200509301033.j8UAXMSo008071@outgoing.mit.edu> From: "Natalia Chernenko" To: <6042-probs@theory.csail.mit.edu> Subject: Week 4 Comments Date: Fri, 30 Sep 2005 06:33:19 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0006_01C5C588.DA0286F0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXFql+umw1OudKiQgSO9LCsteBOjA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.15 X-Spam-Level: * (1.15) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2608 Content-Length: 1964 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0006_01C5C588.DA0286F0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I found images and inverse images confusing (p. 2; 1-2). Could you please explain them more? Thank you. Natalia Chernenko ------=_NextPart_000_0006_01C5C588.DA0286F0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

I found images and inverse images confusing (p. 2; = 1-2). Could you please explain them more? Thank = you.

 

Natalia Chernenko

------=_NextPart_000_0006_01C5C588.DA0286F0-- From pgroudas@MIT.EDU Fri Sep 30 07:02:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UB2Gw0020442 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 07:02:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UB2EkI021065 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 07:02:14 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UB2Ds0009624 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 07:02:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UB2D5N022489; Fri, 30 Sep 2005 07:02:13 -0400 Received: from DKE-TWENTY-SEVEN.MIT.EDU (DKE-TWENTY-SEVEN.MIT.EDU [18.233.1.27]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 07:02:13 -0400 Message-ID: <20050930070213.8f5443mr1sz9c8ow@webmail.mit.edu> Date: Fri, 30 Sep 2005 07:02:13 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2609 Content-Length: 223 X-Keywords: NonJunk NotJunk I found the pages on isomorphic graphs were interesting (18-20). After doing the tutor problems, I thought that it would have been useful if it gave any strategys on determining the number of isomorphisms. -Paul Groudas From fgreen@MIT.EDU Fri Sep 30 07:15:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UBFcw0022525 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 07:15:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UBFbkI029092 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 07:15:37 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UBFSAP010660 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 07:15:28 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UBFSln023216; Fri, 30 Sep 2005 07:15:28 -0400 Received: from GREENONE.MIT.EDU (GREENONE.MIT.EDU [18.238.2.183]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 07:15:28 -0400 Message-ID: <20050930071528.6whrjgkfn0g0koos@webmail.mit.edu> Date: Fri, 30 Sep 2005 07:15:28 -0400 From: Forrest O Green To: 6042-probs@theory.csail.MIT.EDU Subject: Required Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2610 Content-Length: 139 X-Keywords: NonJunk NotJunk I found the isomorphism test to be very interesting. I would be interested in knowing if it is NP complete or not. -Forrest Green From petek@MIT.EDU Fri Sep 30 07:54:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UBstw0028935 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:54:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UBsnkI023074 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:54:49 -0400 (EDT) Received: from [18.194.1.144] (SN-ONE-FORTY-FOUR.MIT.EDU [18.194.1.144]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UBsjho014254 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:54:48 -0400 (EDT) Message-ID: <433D2792.5080900@mit.edu> Date: Fri, 30 Sep 2005 07:54:58 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Partial Orders ? Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -1.892 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2611 Content-Length: 649 X-Keywords: NonJunk NotJunk [3.1]

I think I've having most difficulty with the vocabulary for partial orders....


 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

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From zacharyozer@gmail.com Fri Sep 30 08:47:56 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.193]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UCltw0002444 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 08:47:56 -0400 Received: by xproxy.gmail.com with SMTP id h27so84558wxd for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 05:47:50 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=k2RP7SRfouHC6qRS6gg3Qopk7sn1eXIKO6tqA+r8gkzqyJt7Bv+q+TgkuynC7oCaUD6IGxe8JKjA6TJdTb1+KqBVGrjdbwPmbNEUyRBySqMzgPzxkKBgrdd8/W9zxYQ6lrYrLO8vcSbu6WEn5RNMe3lM0WG3nDcxfInonmGBd/8= Received: by 10.70.87.13 with SMTP id k13mr876067wxb; Fri, 30 Sep 2005 05:47:49 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Fri, 30 Sep 2005 05:47:49 -0700 (PDT) Message-ID: <4f2613e50509300547k7cde16f7le8d435085afeb24f@mail.gmail.com> Date: Fri, 30 Sep 2005 08:47:49 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8UCltw0002444 Status: RO X-UID: 2612 Content-Length: 592 X-Keywords: NonJunk NotJunk While the reading does give a mathematical definition for terms such as maximal and minimal, it does not give the reader an intuitive understanding of these terms, nor do the examples on pages 13 or 14 really help the reader at all. What would be better would be several examples involving numbers and properties, ie maximum of the property multiplication over some set of integers. I only understood the terms for the property division after a lengthy explination, and I am still uncertain as to whether I understand them for other properties. Examples would help me, I do believe. -zozer From yuhsin@MIT.EDU Fri Sep 30 09:25:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UDP7w0007759 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:25:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UDP61O002941 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:25:06 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UDOxfw004661 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:24:59 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8UDOx0x022283; Fri, 30 Sep 2005 09:24:59 -0400 Received: from NONE-FOUR-NINETY-SIX.MIT.EDU (NONE-FOUR-NINETY-SIX.MIT.EDU [18.54.6.241]) (User authenticated as yuhsin@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 09:24:59 -0400 Message-ID: <20050930092459.1zplgv7pb94w0o4k@webmail.mit.edu> Date: Fri, 30 Sep 2005 09:24:59 -0400 From: Yuhsin Chen To: 6042-probs@theory.lcs.MIT.EDU Subject: questions MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2613 Content-Length: 992 X-Keywords: NonJunk NotJunk About Partial Orders: On page 7, the formal definition of a partial order is given as "A binary relation, R, on a set, A, is a partial order iff there is a g such that R agrees with *squiggly*g for every pair of distinct elements". Since last lecture, I've seen examples of partial orders, and if given a relation I can probably identify it as a partial order or not. However, I'm not convinced I understand how to define what a partial order is if someone asked me. (this is more or less an "afterthought" question) About Isomorphism: Most surprising sentence--"having an efficient isomorphism finding/testing procedure would, for example, make it easy to search for a particular molecule in a database given the molecular bonds". I know this is going to be explained further today, but the implications of isomorphism are very surprising, and I'd like to know more about fields that are deeply related. (The fields could serve as examples to strengthen my concept of isomorphism) Joyce From arup@MIT.EDU Fri Sep 30 09:25:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UDPNw0007778 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:25:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UDPM1O003197 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:25:22 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-TWO-TWENTY-SEVEN.MIT.EDU [18.247.5.227]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UDP5s2004704 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:25:06 -0400 (EDT) Message-Id: <6.2.3.4.2.20050930012730.04696c10@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 30 Sep 2005 09:25:03 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2614 Content-Length: 315 X-Keywords: NonJunk NotJunk Section 3.1, page 7: I had trouble understanding partial orders because I wasn't sure what "g(a) is the set of properties that a has" meant, since I confused it with f(a) like in equivalence functions. In general, the notation in this reading confused me for some reason, but I'm getting used to it. |Arup| From veracarr@MIT.EDU Fri Sep 30 09:37:17 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UDbGw0009857 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:17 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8UDbF6K013278 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:15 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UDbF2G015607 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:15 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UDbB0v028298 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:12 -0400 (EDT) Message-ID: <433D5BE5.7010309@mit.edu> Date: Fri, 30 Sep 2005 09:38:13 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Binary Relations Reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2615 Content-Length: 288 X-Keywords: NonJunk NotJunk The section that I found most difficult to understand was the one on Partial Orders (pg 7). I had difficulty recognizing them in the online tutor. Also, when they are defining weak and strict partial orders for both they say iff it is reflexive. For strict did they mean irreflexive? From mwangi@MIT.EDU Fri Sep 30 09:42:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UDg7w0010132 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 09:42:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UDg51O018735 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 09:42:05 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-THREE-FOURTEEN.MIT.EDU [18.241.6.59]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UDfwtY010481 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 09:41:59 -0400 (EDT) Message-Id: <6.2.1.2.2.20050930092937.04434400@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 30 Sep 2005 09:41:55 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2616 Content-Length: 281 X-Keywords: NonJunk NotJunk Passage: 1.2 Images and Inverse Images Page: 2 I found this passage most difficult because the definition of the image of C under R and the definition of the inverse image of C under R was unclear. An example would definitely have helped. Sincerely, Timothy M. Mwangi From kromer@MIT.EDU Fri Sep 30 09:47:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UDlaw0010805 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:47:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UDlZ1O024661 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:47:35 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UDlX6L012790 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:47:33 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8UDlXdh014719; Fri, 30 Sep 2005 09:47:33 -0400 Received: from BURTON-THREE-TEN.MIT.EDU (BURTON-THREE-TEN.MIT.EDU [18.247.6.55]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 09:47:33 -0400 Message-ID: <20050930094733.2gko3s8aj808kw8s@webmail.mit.edu> Date: Fri, 30 Sep 2005 09:47:33 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.468 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2617 Content-Length: 345 X-Keywords: NonJunk NotJunk p. 9: "Theorem: A binary relation is a partial order iff it is transitive and antisymmetric.... To prove the theorem in the right to left direction, assume R is transitive and antisymmetric. Define g(a) ::=R{a} union {a}" I was confused by the notation R{a} union {a}: does this mean: "the set containing a and all the elements that map to a"? From hkhall@MIT.EDU Fri Sep 30 10:19:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEJ6w0019655 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:19:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEJ41O028324; Fri, 30 Sep 2005 10:19:04 -0400 (EDT) Received: from [18.96.6.143] (SIMMONS-THREE-NINETY-EIGHT.MIT.EDU [18.96.6.143]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEIvmo025815 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Fri, 30 Sep 2005 10:18:58 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <884bfc7b6180f5205264705c4878ea84@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: Reading Comments on LN4 [David]- Harrison Hall Date: Fri, 30 Sep 2005 10:20:57 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: -1.036 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2618 Content-Length: 552 X-Keywords: NonJunk NotJunk While I understand the axioms that define a partial order(e.g. transitivity and antisymmetry) I could not explain to you in a way coherent way what one is. While it is a type of regression I don't own the material until I can explain it to someone else or even myself. Right now I cant do that with partial orders. As such I am not really able to understand all of the proofs in the lecture notes that use partial orders or their properties to prove some property. Some additional explanation of a partial order would be awesome. -Harrison From nedzel@MIT.EDU Fri Sep 30 10:26:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEQrw0020339 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:26:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEQq1O006800 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:26:52 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SEVENTY-SEVEN.MIT.EDU [18.96.5.77]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEQos4029247 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:26:50 -0400 (EDT) Message-Id: <200509301426.j8UEQos4029247@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 4 Comments Date: Fri, 30 Sep 2005 10:26:40 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: 2.224 X-Spam-Level: ** (2.224) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2619 Content-Length: 107 X-Keywords: NonJunk NotJunk In section 4.5, I am a little confused by Dilworth's Lemma and its implications to subsequences. - David From mike_a@MIT.EDU Fri Sep 30 10:27:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UERvw0020429 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:27:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UERt1O007777 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:27:56 -0400 (EDT) Received: from [18.238.6.106] (EASTCAMPUS-EIGHT-SEVENTY-THREE.MIT.EDU [18.238.6.106]) (authenticated bits=0) (User authenticated as mike_a@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UERpiE029672 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:27:51 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.lcs.mit.edu Message-Id: <5B7D6F99-9F9E-4306-B7CA-326B336C68BD@mit.edu> Content-Type: multipart/mixed; boundary=Apple-Mail-1--540584741 From: mike anderson Subject: [David] weekly comment Date: Fri, 30 Sep 2005 10:27:50 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.844 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2620 Content-Length: 161029 X-Keywords: NonJunk NotJunk --Apple-Mail-1--540584741 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Especially the "there is a g such that R agrees w/ ...". What does agrees w/ mean in this context? -mike anderson --Apple-Mail-1--540584741-- From moscicki@MIT.EDU Fri Sep 30 10:28:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UESCw0020444 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 10:28:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UESB1O007909 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 10:28:11 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UES9dt029783 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 10:28:09 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8UES9ZS020773; Fri, 30 Sep 2005 10:28:09 -0400 Received: from NEW-ONE-SIXTY-EIGHT.MIT.EDU (NEW-ONE-SIXTY-EIGHT.MIT.EDU [18.241.5.168]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 10:28:09 -0400 Message-ID: <20050930102809.v8g0gnfit0cgko8k@webmail.mit.edu> Date: Fri, 30 Sep 2005 10:28:09 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: week4comment MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_1yozm3wntfnk" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.064 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2621 Content-Length: 4121 X-Keywords: NonJunk NotJunk This message is in MIME format. --=_1yozm3wntfnk Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit attached. --=_1yozm3wntfnk Content-Type: text/plain; charset=ISO-8859-1; name="week4comment.txt" Content-Disposition: attachment; filename="week4comment.txt" Content-Transfer-Encoding: 7bit Get ready for a big substantial comment... did you write me that letter knowing that the next week would be so much more challenging? Relations and Functions: Its a concept we've all been dealing with for many years, but for some reason the idea is hard for me to catch abstractly enough. The notation is confusing. I understand the doman, codomain, and graph, but somehow the ideas do not have concrete meaning for me. This is why all the rest of the reading is so hard. "So a function is a special case of binary relation" is the most helpful sentence of them all I think. It is also hard for me to grasp that the codomain of the function is not defined by the domain and the mapping; that there can be elements that are not used, so there is a certain arbitrariness about the codomain of a relation. Images and Inverse Images: Completely confusing. The notation especially. If CR is a set made from cRb, where B is in the codomain of the relation between A and B, where did A go and why is it relevant? CR is cRb so C acts as the domain RC is aRc C acts as the codomain If I think of it this way, then the idea of images makes sense. Surjective... Again the definitions: surjective -> every element in codomain used at least once. ah! it makes senes! AR= the set of little bs = B -> surjective because the set of bs = the total domain! Sorry for the flame. Equivalence: Symetric, Reflexive, and Transitive. Equivalence by function: Two equal objects give the same hashCode() Partitions: If a set is partitioned, the set of the partitions contains all the elements of the original set and the partitions do not overlap. This one makes sense more than others. Interesting that being in the same block is equivalent. So, an equivalence class is the block the element belongs to. This was very unclear from the lecture. Now it makes a bit more sense. A bit on notation: Saying that reflexivity is defined for R on A by aRa is extremely confusing and I didn't get it all week until now. I think its because there are too many as. Call a q or p or anything but little a and it makes it more clear. Likewise for symmetry: aRa' is not particularly clear I think. The examples are helpful. Partial Orders: The definition give in the first two paragraphs of page 6 is unclear to me. Perhaps it will become clearer as I read on... It seems to be a total order that is not true in all cases, i.e. it does not specify whether elements are related to themselves. weak-> reflexive strict-> irreflexive all are transitive all are antisymmetric Total Orders: pick any element of the domain and any element of the codomain and either the relation or its converse will be true. Note: My summarization of the key ideas I guess is just to show that I understand it, or maybe how the reader might take it. Digraphs: Networking. Searching algorithms, therefore interesting. Noticed the connection that this really is the definition of the relation, just like the equation of a line is only an expression for the actual thing. Path length is the important thing here, as in class problems. Minimal: No edge points to this thing. Its the very bottom, the least element Maximal: The most edges point to this thing. It is farthest up the tree. Topological Sort: Pick the minimal elements first. Many possible orders depending on the set and relation. Critical Path= longest chain= shortest possible number of steps to complete everything The part with chains and antichains is refreshingly clear, and was clear in lecture as well. Ugraphs: Fairly stright forward. The part about isomorphism was not clear at first, but someone explained it to me, so I do understand it... but not from the reading. --=_1yozm3wntfnk-- From yaser@MIT.EDU Fri Sep 30 10:44:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEigw0025095 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:44:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEif1O026131 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:44:41 -0400 (EDT) Received: from Charizard (NEW-TWO-EIGHTY-NINE.MIT.EDU [18.241.6.34]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEidEa007018 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:44:39 -0400 (EDT) Message-Id: <200509301444.j8UEidEa007018@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] Week 4 comments Date: Fri, 30 Sep 2005 10:44:36 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C5AB.F43A5430" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXFzXntEats+8hXS2KwkwzUZmB+HQ== X-Spam-Score: 1.015 X-Spam-Level: * (1.015) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2622 Content-Length: 6220 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C5AB.F43A5430 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Hi David, On the topic of isomorphism, I read that there is no generalizable short-cut method to determine if two graphs are isomorphic, short of exhaustively examining each possible bijection. During lecture, could you please elaborate on whether there are some short cuts that hold in certain cases, or what some of the other properties of isomorphic bjiection are? Also, I still don't understand what the take-away message of the height- age demonstration was. Thanks! _Yaser ------=_NextPart_000_0000_01C5C5AB.F43A5430 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

On the topic of isomorphism, I = read that there is no generalizable short-cut method = to determine if two graphs are isomorphic, short of exhaustively examining = each possible bijection. During lecture, could = you please elaborate on whether there are some short cuts that hold in certain = cases, or what some of the other properties of isomorphic bjiection are?

 

Also, I still don’t = understand what the take-away message of the height- age demonstration = was…

 

Thanks!

 

_Yaser

------=_NextPart_000_0000_01C5C5AB.F43A5430-- From harelw@gmail.com Fri Sep 30 10:49:27 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.207]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEnQw0027010 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:49:26 -0400 Received: by xproxy.gmail.com with SMTP id h27so104199wxd for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:49:21 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type; b=blJvvddhnLhjfbOI7NWmMlg078svkv603AQcz5QIEFL8eFWsV3ihrv4TVoXbG6HfduyFZKvKNxrBRDIefbXILL6KmpV5+E5Z7u3GDGDv8DWTIGMMc8v7wn3qmfhfOxH90fUPhBzfr0E7MM6wTnKOIkYGns/vHkeS+ulpZX6l8oI= Received: by 10.70.8.2 with SMTP id 2mr914840wxh; Fri, 30 Sep 2005 07:49:20 -0700 (PDT) Received: by 10.70.40.15 with HTTP; Fri, 30 Sep 2005 07:49:20 -0700 (PDT) Message-ID: <8c5248a80509300749s120f6badv679ea2efdfe35ada@mail.gmail.com> Date: Fri, 30 Sep 2005 10:49:20 -0400 From: "Harel M. Williams" Reply-To: "Harel M. Williams" Sender: harelw@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: TA: Jelani - Comments for Reading MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_2382_27898129.1128091760761" Status: RO X-UID: 2623 Content-Length: 726 X-Keywords: NonJunk NotJunk ------=_Part_2382_27898129.1128091760761 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline As reflected on the tutorial multiple choice problems, I don't understand the notation for surjective and total relation, altough I understand the definition of the terms. ~H ------=_Part_2382_27898129.1128091760761 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline As reflected on the tutorial multiple choice problems, I don't understand the notation for surjective and total relation, altough I understand the definition of the terms.

~H
------=_Part_2382_27898129.1128091760761-- From mrivas03@MIT.EDU Fri Sep 30 10:52:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEqKw0027293 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:52:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEqJ1O004157 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:52:19 -0400 (EDT) Received: from el-ternero.mit.edu (TDCIP95.MIT.EDU [18.237.0.95]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEqEh2010132 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:52:15 -0400 (EDT) Message-Id: <6.2.3.4.2.20050930104220.02eaecb0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 30 Sep 2005 10:46:45 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: [Sayan] Reading Course Notes Week 4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.978 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2624 Content-Length: 401 X-Keywords: NonJunk NotJunk The concept of Bijection in Isomorphic and non-isomorphic graphs was not too clear from the course notes. Such as finding the number of bijections between two isomorphic graphs. It would be particulary helpful if that is gone over in class. I found topological sorting of the section notes interesting, especially the claim that all partial orders have topological sorts. Thanks, Manuel Rivas From rshearer@MIT.EDU Fri Sep 30 10:53:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UErVw0027408 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:53:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UErT1O005376 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:53:29 -0400 (EDT) Received: from [18.240.6.247] (MCCORMICK-FIVE-O-TWO.MIT.EDU [18.240.6.247]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UErM0p010641 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:53:22 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: [Jelani] Week 4 Comment Date: Fri, 30 Sep 2005 10:53:09 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2625 Content-Length: 317 X-Keywords: NonJunk NotJunk I had a lot of trouble with the tutor problem relating to Isomorphism, especially the one about how to prove that two graphs are not isomorphic. I feel like it's actually not very hard but I am just assuming it's more complicated than it actually is. Hopefully class today will make things more clear. Rachel From nancyk@MIT.EDU Fri Sep 30 10:54:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEsxw0027474 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 10:54:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEsw1O007088; Fri, 30 Sep 2005 10:54:58 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEspj3011286; Fri, 30 Sep 2005 10:54:52 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8UEsprV001156; Fri, 30 Sep 2005 10:54:51 -0400 Received: from WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU (WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU [18.139.6.237]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 10:54:51 -0400 Message-ID: <20050930105451.9n56e01s07c4ckc4@webmail.mit.edu> Date: Fri, 30 Sep 2005 10:54:51 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 4 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2626 Content-Length: 386 X-Keywords: NonJunk NotJunk Hi, On page 7 of the Week 4 notes, directly under equation (7) is some discussion about the identity function. I do not understand why it follows that the partial order binary relation of I_A is the same relation as the proper subset relation. I wished there was more explanation between those two statements, as I could not make the logical jump from one to the other. Thanks, Nancy From letrec@MIT.EDU Fri Sep 30 10:56:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEuTw0027574 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:56:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEuS1O008795 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:56:28 -0400 (EDT) Received: from [18.237.0.82] (TDCIP82.MIT.EDU [18.237.0.82]) (authenticated bits=0) (User authenticated as letrec@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEuKr9011979 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:56:21 -0400 (EDT) Message-ID: <433D520F.7010207@mit.edu> Date: Fri, 30 Sep 2005 10:56:15 -0400 From: Alton Torregano User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: 6042-student: Hanson Week 4 comments X-Enigmail-Version: 0.90.0.0 X-Enigmail-Supports: pgp-inline, pgp-mime Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.924 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2627 Content-Length: 238 X-Keywords: NonJunk NotJunk Could I have the section on bijective mappings, and their counting made more explicit. Also, in relation to the tutor problems, could the meaning of: RA=B AR=B, etc...be explained more fully, as the syntax appeared rather foreign. ~ From dshin@MIT.EDU Fri Sep 30 11:02:16 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UF2Gw0027980 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:02:16 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF2EAW000848; Fri, 30 Sep 2005 11:02:14 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF2B0v003288; Fri, 30 Sep 2005 11:02:12 -0400 (EDT) Message-ID: <433D536A.1070806@mit.edu> Date: Fri, 30 Sep 2005 11:02:02 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: mike anderson CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] weekly comment References: <5B7D6F99-9F9E-4306-B7CA-326B336C68BD@mit.edu> In-Reply-To: <5B7D6F99-9F9E-4306-B7CA-326B336C68BD@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.625 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2628 Content-Length: 343 X-Keywords: NotJunk Try the very next sentence in the notes. g agrees with R iff for all distinct a,b, aRb iff g(a) is a proper subset of g(b) DS mike anderson wrote: > From Page 7 (section 3.1) > > I don't really understand this. Especially the "there is a g such > that R agrees w/ ...". What does agrees w/ mean in this context? > > > -mike anderson From dshin@MIT.EDU Fri Sep 30 11:03:15 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UF3Fw0028260 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:03:15 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF3E6K023996 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:03:14 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF3ErH019531; Fri, 30 Sep 2005 11:03:14 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF360v003342; Fri, 30 Sep 2005 11:03:06 -0400 (EDT) Message-ID: <433D53A1.2080004@mit.edu> Date: Fri, 30 Sep 2005 11:02:57 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-probs@theory.lcs.mit.edu Subject: Re: Reading Comments on LN4 [David]- Harrison Hall References: <884bfc7b6180f5205264705c4878ea84@mit.edu> In-Reply-To: <884bfc7b6180f5205264705c4878ea84@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2629 Content-Length: 668 X-Keywords: NotJunk Hopefully, you will get the practice you need in the next pset. DS Harrison King Hall wrote: > While I understand the axioms that define a partial order(e.g. > transitivity and antisymmetry) I could not explain to you in a way > coherent way what one is. While it is a type of regression I don't > own the material until I can explain it to someone else or even > myself. Right now I cant do that with partial orders. As such I am > not really able to understand all of the proofs in the lecture notes > that use partial orders or their properties to prove some property. > Some additional explanation of a partial order would be awesome. > -Harrison > From dshin@MIT.EDU Fri Sep 30 11:06:27 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UF6Rw0029551 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:06:27 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF6P6K027841 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:06:25 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF6PrH019686; Fri, 30 Sep 2005 11:06:25 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF6M3a008178; Fri, 30 Sep 2005 11:06:23 -0400 (EDT) Message-ID: <433D5465.5050706@mit.edu> Date: Fri, 30 Sep 2005 11:06:13 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] Email comments for reading References: <20050930094733.2gko3s8aj808kw8s@webmail.mit.edu> In-Reply-To: <20050930094733.2gko3s8aj808kw8s@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.782 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2630 Content-Length: 512 X-Keywords: NotJunk For any set set S, R(S) is the set {R(s) : s in S}. So, R{a} (I guess R({a}) would be better notation...) is just {R(a)}. DS Katherine A Romer wrote: >p. 9: >"Theorem: A binary relation is a partial order iff it is transitive and >antisymmetric.... To prove the theorem in the right to left direction, assume R >is transitive and antisymmetric. Define g(a) ::=R{a} union {a}" > >I was confused by the notation R{a} union {a}: does this mean: "the set >containing a and all the elements that map to a"? > > From dshin@MIT.EDU Fri Sep 30 11:08:52 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UF8qw0029709 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:08:52 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF8nAW001170; Fri, 30 Sep 2005 11:08:49 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF8f3a008311; Fri, 30 Sep 2005 11:08:42 -0400 (EDT) Message-ID: <433D54F0.2010809@mit.edu> Date: Fri, 30 Sep 2005 11:08:32 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: jacques CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 4 comments References: <20050930040736.fk9hd1m30p8oowwc@webmail.mit.edu> In-Reply-To: <20050930040736.fk9hd1m30p8oowwc@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2631 Content-Length: 375 X-Status: A X-Keywords: NotJunk asymmetric simply means "not symmetric". If you know what "symmetric" means, you understand asymmetry as well. DS jacques wrote: >In the lecture notes, page 8, there is a very good definition of the property of >antisymmetry. However, asymmetry is mentioned in the lecture slides but never >really defined in the lecture notes. That's something I'd like to see. > > From dshin@MIT.EDU Fri Sep 30 11:14:30 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UFETw0030019 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:14:29 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8UFESwb004236 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:14:28 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UFESrH020111; Fri, 30 Sep 2005 11:14:28 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UFEK0v004143; Fri, 30 Sep 2005 11:14:20 -0400 (EDT) Message-ID: <433D5643.8060607@mit.edu> Date: Fri, 30 Sep 2005 11:14:11 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yaser Khan CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 4 comments References: <200509301444.j8UEidEa007018@outgoing.mit.edu> In-Reply-To: <200509301444.j8UEidEa007018@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------020809080009010409040107" X-Spam-Score: -2.098 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2632 Content-Length: 8614 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------020809080009010409040107 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Great question! There CERTAINLY are short cuts that hold in certain cases. For example, one short cut is to just count the number of vertices and edges in each graph. If the numbers don't match up, I've come up with a short cut to show they are not isomorphic. Or, let's say I'm trying to show they are isomorphic, and each graph has exactly one node of degree 10. Well, if there is a bijection, those two nodes have to map to each other, so that narrows down my search for bijections. In fact, it's interesting to note that we've never been able to come up with a "difficult class" of graph-pairs, for which we're not able to solve graph-isomorphism by short-cuts! As for the height-age demonstration, imagine that all the people in the class are standing in order according to age. Then, given that fixed list of people, we define the following partial order: a Hi David, > > > > On the topic of isomorphism, I read that there is no generalizable > short-cut method to determine if two graphs are isomorphic, short of > exhaustively examining each possible bijection. During lecture, could > you please elaborate on whether there are some short cuts that hold in > certain cases, or what some of the other properties of isomorphic > bjiection are? > > > > Also, I still don't understand what the take-away message of the > height- age demonstration was... > > > > Thanks! > > > > _Yaser > --------------020809080009010409040107 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Great question!

There CERTAINLY are short cuts that hold in certain cases.  For example, one short cut is to just count the number of vertices and edges in each graph.  If the numbers don't match up, I've come up with a short cut to show they are not isomorphic. 

Or, let's say I'm trying to show they are isomorphic, and each graph has exactly one node of degree 10.  Well, if there is a bijection, those two nodes have to map to each other, so that narrows down my search for bijections.

In fact, it's interesting to note that we've never been able to come up with a "difficult class" of graph-pairs, for which we're not able to solve graph-isomorphism by short-cuts! 

As for the height-age demonstration, imagine that all the people in the class are standing in order according to age.  Then, given that fixed list of people, we define the following partial order:  a<b if a is shorter than b.  Dilworth's Lemma claims that we should be able to find either a long increasing subsequence or a long decreasing subsequence.  Those were the two sets of people you saw during the demonstration.

DS

Yaser Khan wrote:

Hi David,

 

On the topic of isomorphism, I read that there is no generalizable short-cut method to determine if two graphs are isomorphic, short of exhaustively examining each possible bijection. During lecture, could you please elaborate on whether there are some short cuts that hold in certain cases, or what some of the other properties of isomorphic bjiection are?

 

Also, I still don’t understand what the take-away message of the height- age demonstration was…

 

Thanks!

 

_Yaser

--------------020809080009010409040107-- From antonk@MIT.EDU Fri Sep 30 11:33:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UFXPw0031511 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:33:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UFXNrh017209 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:33:24 -0400 (EDT) Received: from quick-w20-copy2.mit.edu (QUICK-W20-COPY2.MIT.EDU [18.187.0.202]) (authenticated bits=56) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UFXJA4027896 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:33:20 -0400 (EDT) Received: (from antonk@localhost) by quick-w20-copy2.mit.edu (8.12.9) id j8UFXIZ0013576; Fri, 30 Sep 2005 11:33:18 -0400 (EDT) Subject: [David] 3 From: Anton Katz To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Fri, 30 Sep 2005 11:33:16 -0400 Message-Id: <1128094396.13555.3.camel@quick-w20-copy2.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2633 Content-Length: 338 X-Keywords: NonJunk NotJunk Hi, chapter 5.2 discusses isomorphism. I understand the first example with the square graph, but I want to hear more about the star graph. The same problem was on the tutorial questions and I had to try a lot of numbers before getting the answer. Thank you and sorry for being late, i simply forgot about the email, hope its ok. Anton From dshin@MIT.EDU Fri Sep 30 11:47:03 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UFl3w0001948 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:47:03 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UFl1AW002960; Fri, 30 Sep 2005 11:47:01 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UFkn0v006168; Fri, 30 Sep 2005 11:46:50 -0400 (EDT) Message-ID: <433D5DE0.1020305@mit.edu> Date: Fri, 30 Sep 2005 11:46:40 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] 3 References: <1128094396.13555.3.camel@quick-w20-copy2.mit.edu> In-Reply-To: <1128094396.13555.3.camel@quick-w20-copy2.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2634 Content-Length: 500 X-Keywords: NotJunk I can show you in more detail if you come by office hours today at 1 (24-319), or if you come to class a little early. DS Anton Katz wrote: >Hi, > >chapter 5.2 discusses isomorphism. >I understand the first example with the square graph, but I want to hear >more about the star graph. >The same problem was on the tutorial questions and I had to try a lot of >numbers before getting the answer. > >Thank you and sorry for being late, i simply forgot about the email, >hope its ok. > >Anton > > From alisonc@MIT.EDU Fri Sep 30 11:59:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UFxNw0005749 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 11:59:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UFxLrh016780 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 11:59:22 -0400 (EDT) Received: from [18.238.1.185] (ALISONC.MIT.EDU [18.238.1.185]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UFxGC5009082 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 11:59:16 -0400 (EDT) Message-ID: <433D60E2.70403@mit.edu> Date: Fri, 30 Sep 2005 11:59:30 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Lecture notes 4 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2635 Content-Length: 214 X-Keywords: NonJunk NotJunk I found the parallel task scheduling and topological sorting most interesting. It seems to have really fascinating direct applications to computer science, parallel processing and so forth, which is really neat. From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:07:37 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UG7bw0007066; Fri, 30 Sep 2005 12:07:37 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UG7anX014059; Fri, 30 Sep 2005 12:07:37 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UG7a1m014056; Fri, 30 Sep 2005 12:07:36 -0400 Date: Fri, 30 Sep 2005 12:07:36 -0400 (EDT) From: Hanson Zhou To: "Xiaoran (Sharon) Zhang" cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Week 4 comments In-Reply-To: <20050930003434.peberlpwka0osskk@webmail.mit.edu> Message-ID: References: <20050930003434.peberlpwka0osskk@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2636 Content-Length: 867 X-Keywords: NotJunk Go to office hours to get this cleared up. One helpful thing might be to review the definition of a bijection, and to reexamine how it applies in the case of graph isomorphism. -Hanson On Fri, 30 Sep 2005, Xiaoran (Sharon) Zhang wrote: > I was confused about isomorphism for undirected graphs. Page 17, section 5. I am > still unsure which characteristics to evaluate when determining whether two > graphs are isomorphic or not. The correspondence between the vertex of the > graphs are not clear to me. > > Sharon > > ************************************************** > Xiaoran (Sharon) Zhang > Class of 2008 > Department of Biology & > Department of Electrical Engineering and Computer Science > Massachusetts Institute of Technology > 320 Memorial Drive, > Cambridge, MA 02139 > E-mail: xiaoranz@mit.edu > ************************************************** > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:08:56 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UG8uw0007161; Fri, 30 Sep 2005 12:08:56 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UG8uU6014070; Fri, 30 Sep 2005 12:08:56 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UG8u55014067; Fri, 30 Sep 2005 12:08:56 -0400 Date: Fri, 30 Sep 2005 12:08:56 -0400 (EDT) From: Hanson Zhou To: Lohith Kini cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 4 Reading Comments In-Reply-To: <433CC791.7090107@mit.edu> Message-ID: References: <433CC791.7090107@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2637 Content-Length: 360 X-Keywords: NotJunk This is might be a bit tricky...go to office hours if you're still confused. -Hanson On Fri, 30 Sep 2005, Lohith Kini wrote: > Hi, > > I was totally confused about parallel task scheduling and theorem 4.11. > I didn't understand the corollaries either (because of this). I would > like it if you could go over that a little bit more. > > Thanks, > Lohith > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:12:24 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGCOw0007426; Fri, 30 Sep 2005 12:12:24 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGCOI6014093; Fri, 30 Sep 2005 12:12:24 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGCOH3014090; Fri, 30 Sep 2005 12:12:24 -0400 Date: Fri, 30 Sep 2005 12:12:24 -0400 (EDT) From: Hanson Zhou To: Elizabeth Reid cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Reading Comments In-Reply-To: <348E403C-2681-4A5D-BE9B-0F623B985801@MIT.EDU> Message-ID: References: <348E403C-2681-4A5D-BE9B-0F623B985801@MIT.EDU> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2638 Content-Length: 288 X-Keywords: NotJunk Why would you want to maximize it? -Hanson On Fri, 30 Sep 2005, Elizabeth Reid wrote: > If they could talk a bit about parallel task scheduling in lecture > that would be nice. I was wondering if there was any mathematical way > to maximize it, or if it's more of a guess and check? > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:15:10 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGF9w0007984; Fri, 30 Sep 2005 12:15:09 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGF9GZ014110; Fri, 30 Sep 2005 12:15:09 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGF9kT014107; Fri, 30 Sep 2005 12:15:09 -0400 Date: Fri, 30 Sep 2005 12:15:09 -0400 (EDT) From: Hanson Zhou To: Alton Torregano cc: 6042-probs@theory.csail.mit.edu Subject: Re: 6042-student: Hanson Week 4 comments In-Reply-To: <433D520F.7010207@mit.edu> Message-ID: References: <433D520F.7010207@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2639 Content-Length: 399 X-Keywords: NotJunk Read section 1.2 to understand that notation. Go to office hours if you are still confused. -Hanson On Fri, 30 Sep 2005, Alton Torregano wrote: > Could I have the section on bijective mappings, and their counting made > more explicit. Also, in relation to the tutor problems, could the > meaning of: > > RA=B > AR=B, etc...be explained more fully, as the syntax appeared rather foreign. > > ~ > From pjs@MIT.EDU Fri Sep 30 12:18:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGIUw0008880 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 12:18:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UGITrh006051 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 12:18:29 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UGILp0016426 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 12:18:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UGILZj032653; Fri, 30 Sep 2005 12:18:21 -0400 Received: from STRATTON-FOUR-TWENTY-EIGHT.MIT.EDU (STRATTON-FOUR-TWENTY-EIGHT.MIT.EDU [18.187.6.173]) (User authenticated as pjs@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 12:18:21 -0400 Message-ID: <20050930121821.ldch0z2q034s4goo@webmail.mit.edu> Date: Fri, 30 Sep 2005 12:18:21 -0400 From: Paul J Steiner To: 6042-probs@theory.lcs.MIT.EDU Subject: email comments [sayan] MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2640 Content-Length: 233 X-Keywords: NonJunk NotJunk I found the notion of topological sorting on page 13 the most surprising in the notes. It allows for the surprisingly elegant notions of minimum/maximum on a graph, such as 1/0 for divisibility on the positive integers. PJ Steiner From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:22:50 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGMow0010575; Fri, 30 Sep 2005 12:22:50 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGMo3v014152; Fri, 30 Sep 2005 12:22:50 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGMoZp014149; Fri, 30 Sep 2005 12:22:50 -0400 Date: Fri, 30 Sep 2005 12:22:50 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 4 comments In-Reply-To: <20050930105451.9n56e01s07c4ckc4@webmail.mit.edu> Message-ID: References: <20050930105451.9n56e01s07c4ckc4@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2641 Content-Length: 553 X-Keywords: NotJunk Go to office hours to get this clarified. It is essentially the definition of partial order. -Hanson On Fri, 30 Sep 2005, Nancy L Keuss wrote: > Hi, > > On page 7 of the Week 4 notes, directly under equation (7) is some discussion > about the identity function. I do not understand why it follows that the > partial order binary relation of I_A is the same relation as the proper subset > relation. I wished there was more explanation between those two statements, as > I could not make the logical jump from one to the other. > > Thanks, > Nancy > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:24:33 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGOXw0011446; Fri, 30 Sep 2005 12:24:33 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGOXjM014165; Fri, 30 Sep 2005 12:24:33 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGOXc2014162; Fri, 30 Sep 2005 12:24:33 -0400 Date: Fri, 30 Sep 2005 12:24:33 -0400 (EDT) From: Hanson Zhou To: Vera Carr cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Binary Relations Reading In-Reply-To: <433D5BE5.7010309@mit.edu> Message-ID: References: <433D5BE5.7010309@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2642 Content-Length: 421 X-Keywords: NotJunk Yes strict is irreflexive...I believe that's what it also says on page 7. -Hanson On Fri, 30 Sep 2005, Vera Carr wrote: > The section that I found most difficult to understand was the one on > Partial Orders (pg 7). I had difficulty recognizing them in the online > tutor. > > Also, when they are defining weak and strict partial orders for both > they say iff it is reflexive. For strict did they mean irreflexive? > From scholtz@MIT.EDU Fri Sep 30 13:35:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UHZlw0026948 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 13:35:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UHZkrh022648 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 13:35:46 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UHZdVu012450 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 13:35:39 -0400 (EDT) Message-Id: <200509301735.j8UHZdVu012450@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 4 Comments Date: Fri, 30 Sep 2005 13:35:38 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C5C3.D85FAA90" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXF5V7dinHKt8iBQBGAULZqwMcDow== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2643 Content-Length: 1908 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C5C3.D85FAA90 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I found section 3.4 on page 9 to be most difficult. Its not very easy to understand by just reading it, but I think doing more practice problems will help me understand it a lot easier. ------=_NextPart_000_0000_01C5C5C3.D85FAA90 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I found section 3.4 on page 9 to be most = difficult. Its not very easy to understand by just reading it, but I think doing = more practice problems will help me understand it a lot = easier.

------=_NextPart_000_0000_01C5C5C3.D85FAA90-- From minilek@theory.csail.mit.edu Sat Oct 1 11:58:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j91FwQw0011245 for <6042-probs@theory.lcs.mit.edu>; Sat, 1 Oct 2005 11:58:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j91FwNTF023261; Sat, 1 Oct 2005 11:58:23 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j91FwHRX004195 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sat, 1 Oct 2005 11:58:22 -0400 (EDT) Message-ID: <433EB23B.2030705@theory.csail.mit.edu> Date: Sat, 01 Oct 2005 11:58:51 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Rachel Shearer Subject: Re: [Jelani] Week 4 Comment References: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> In-Reply-To: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.564 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2644 Content-Length: 746 X-Keywords: NonJunk NotJunk Is it clearer now after Friday's lecture? One of the strategies, for instance, could be to count vertices of a certain degree (then apply what you proved in problem 1 in the class problems). This can only catch some non-isomorphic graphs and may produce false positives, but it can usually at least cut down on the n! different potential isomorphisms that need eliminating. -Jelani Rachel Shearer wrote: > I had a lot of trouble with the tutor problem relating to Isomorphism, > especially the one about how to prove that two graphs are not > isomorphic. I feel like it's actually not very hard but I am just > assuming it's more complicated than it actually is. Hopefully class > today will make things more clear. > > Rachel > From meyer@csail.mit.edu Sat Oct 1 11:58:40 2005 Return-Path: Received: from smtp105.mail.sc5.yahoo.com (smtp105.mail.sc5.yahoo.com [66.163.169.225]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j91Fwdw0011253 for <6042-probs@theory.lcs.mit.edu>; Sat, 1 Oct 2005 11:58:39 -0400 Received: (qmail 64281 invoked from network); 1 Oct 2005 15:58:38 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp105.mail.sc5.yahoo.com with SMTP; 1 Oct 2005 15:58:38 -0000 Message-ID: <433EB22E.5040000@csail.mit.edu> Date: Sat, 01 Oct 2005 11:58:38 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yasin Ozcan CC: 6042-probs@theory.lcs.mit.edu, Hanson M Zhou Subject: Re: [hanson]reading comments References: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> In-Reply-To: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 2645 Content-Length: 781 X-Keywords: The notation and answers to TP4.4 are given explicitly in Week 4 Notes, sections 1.2 & 1.3. Did you see that and not understand, or miss it? I have to agree that the Week 4 Notes were longer than I like (despite substantial cutting from earlier versions :-( ). But just saying "too long" is not the kind of comment we look for: we expect you to identify a specific topic that caught your attention. I'll look forward to a more detailed email comment next week. Regards, A. Yasin Ozcan wrote: > Hi, > > The only comments I have about this week's readings is that it was too > long. > > Also, I am a little bit confused by the notation in the True/False > questions in the Tutor problems, i.e. Question 4.4. Can we go over it > in the lecture > > thanks > > yasin From hmzhou@MIT.EDU Sat Oct 1 13:23:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j91HNDw0021719 for <6042-probs@theory.csail.MIT.EDU>; Sat, 1 Oct 2005 13:23:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j91HNBTF025653; Sat, 1 Oct 2005 13:23:12 -0400 (EDT) Received: from all-night-tool.mit.edu (ALL-NIGHT-TOOL.MIT.EDU [18.7.16.70]) (authenticated bits=56) (User authenticated as hmzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j91HN9Na013821 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sat, 1 Oct 2005 13:23:10 -0400 (EDT) Received: (from hmzhou@localhost) by all-night-tool.mit.edu (8.12.9) id j91HN9TR008305; Sat, 1 Oct 2005 13:23:09 -0400 (EDT) Date: Sat, 1 Oct 2005 13:23:09 -0400 (EDT) From: Hanson M Zhou To: ereid@MIT.EDU cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Reading Comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2646 Content-Length: 872 X-Keywords: Depends on number of processors. If you have as many as you want, you are only constrained by the longest chain, and it is also the best you can do. Otherwise, it gets more complicated but I'm sure there are algorithms out there for parallel task scheduling. -Hanson On Fri, 30 Sep 2005, Elizabeth Reid wrote: > Sorry, I meant *minimize* the time. Is there any method that will > always find the shortest time possible, or is it just kind of logic > mixed with guess and check? > > > On Sep 30, 2005, at 12:12 PM, Hanson Zhou wrote: > > > Why would you want to maximize it? > > > > -Hanson > > > > On Fri, 30 Sep 2005, Elizabeth Reid wrote: > > > > > >> If they could talk a bit about parallel task scheduling in lecture > >> that would be nice. I was wondering if there was any mathematical way > >> to maximize it, or if it's more of a guess and check? > >> > > > > From meyer@imap.theory.csail.mit.edu Sat Oct 01 11:49:25 2005 Message-ID: <433EB004.2070606@csail.mit.edu> Date: Sat, 01 Oct 2005 11:49:24 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Clinton Blackburn CC: Jelani Nelson Subject: Re: Jelani: Reading #4 References: <200509300509.j8U59WYL022496@outgoing.mit.edu> In-Reply-To: <200509300509.j8U59WYL022496@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 350 X-UID: 2647 X-Keywords: sounds like we should get together and talk about this. Are you free Wednesday after class? (I'd offer to meet sooner, but have religious holidays Mon eve through Wed afternoon). Clinton Blackburn wrote: > Im just totally confused by this subject. I dont know > > --- > > Clinton Blackburn > > DLP Have you seen it? > From meyer@imap.theory.csail.mit.edu Thu Sep 29 12:02:30 2005 Message-ID: <433C1019.6070903@csail.mit.edu> Date: Thu, 29 Sep 2005 12:02:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jing He CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] relations and graph theory reading comments References: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> In-Reply-To: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> Content-Type: text/plain; charset=GB2312 Content-Transfer-Encoding: 7bit Status: RO Content-Length: 693 X-UID: 2648 X-Keywords: 4.16 says there is a chain of size t, or an antichain of size n/t, for ANY t. So let t=sqrt(n) , and you get a chain of size sqrt(n) or an antichain of size n/sqrt(n), which also equals sqrt(n). So you 're sure to have a sqrt(n) size chain or antichain, which is what 4.17 says. Does that help? regards, A. Jing He wrote: >The most difficult part of this section for me to understand was what a total >order was, since it wasn't given the formal definitions that the partial order >came with in the reading. I also didn't understand the reasoning for the >"proof" of Corollary 4.17 on page 16. The last section on graphs (and >applications to computing) was the most interesting. > > From meyer@imap.theory.csail.mit.edu Sat Oct 01 22:08:53 2005 BCC: Ronitt Rubinfeld Message-ID: <433F4136.2080206@csail.mit.edu> Date: Sat, 01 Oct 2005 22:08:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Clinton Blackburn CC: Jelani Nelson Subject: Re: Jelani: Reading #4 References: <200510011821.j91ILIFq020002@outgoing.mit.edu> In-Reply-To: <200510011821.j91ILIFq020002@outgoing.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 764 X-UID: 2649 X-Keywords: OK, see you wed after class regards, A. Clinton Blackburn wrote: > Yes, sir. I am free. > > Thanks. > --- > Clinton Blackburn > DLP - Have you seen it? > http://www.dlp.com > > -----Original Message----- > From: Prof. Albert R. Meyer [mailto:meyer@csail.mit.edu] > Sent: Saturday, October 01, 2005 11:49 AM > To: Clinton Blackburn > Cc: Jelani Nelson > Subject: Re: Jelani: Reading #4 > > sounds like we should get together and talk about this. Are you free > Wednesday after class? (I'd offer to meet sooner, but have religious > holidays Mon eve through Wed afternoon). > > Clinton Blackburn wrote: > > >>I'm just totally confused by this subject. I don't know. >> >>--- >> >>Clinton Blackburn >> >>DLP - Have you seen it? >> From meyer@imap.theory.csail.mit.edu Sat Oct 01 11:58:39 2005 Message-ID: <433EB22E.5040000@csail.mit.edu> Date: Sat, 01 Oct 2005 11:58:38 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yasin Ozcan CC: 6042-probs@theory.lcs.mit.edu, Hanson M Zhou Subject: Re: [hanson]reading comments References: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> In-Reply-To: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 782 X-UID: 2650 X-Keywords: The notation and answers to TP4.4 are given explicitly in Week 4 Notes, sections 1.2 & 1.3. Did you see that and not understand, or miss it? I have to agree that the Week 4 Notes were longer than I like (despite substantial cutting from earlier versions :-( ). But just saying "too long" is not the kind of comment we look for: we expect you to identify a specific topic that caught your attention. I'll look forward to a more detailed email comment next week. Regards, A. Yasin Ozcan wrote: > Hi, > > The only comments I have about this week's readings is that it was too > long. > > Also, I am a little bit confused by the notation in the True/False > questions in the Tutor problems, i.e. Question 4.4. Can we go over it > in the lecture > > thanks > > yasin From meyer@imap.theory.csail.mit.edu Thu Sep 29 18:17:10 2005 Message-ID: <433C67E9.1050704@csail.mit.edu> Date: Thu, 29 Sep 2005 18:17:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chieu Nguyen CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Jelani] Week 4 comments References: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> In-Reply-To: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 254 X-UID: 2651 X-Keywords: Glad you think so, but we'd like more specific reactions to particular sections of the notes that caught your attention. I'll look forward to fuller comments next week. regards, A. Chieu Nguyen wrote: > (Pages 11-21) Graphs are cool. > > --Chieu From meyer@imap.theory.csail.mit.edu Thu Sep 29 18:14:32 2005 Message-ID: <433C674B.7010806@csail.mit.edu> Date: Thu, 29 Sep 2005 18:14:35 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 4 Comments References: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 789 X-UID: 2652 X-Keywords: If they had the same elements, they'd be the same set, not two sets :-) But I grant you it would have been clearer to say "two distinct sets," and I'm editing the notes to say that right now. Thanks for the comment. Regards, A. Praveen Pamidimukkala wrote: > In Section 3.2, the last 2 sentences read, "So < and < are total > orders on [the set of all Real Numbers]. On the other hand, the subset > relation is generally not total: any two finite sets of the same size > will be incomparable under [the contained-in-or-equal subset > relation]." Don't they mean just the [proper subset relation]? If > the two finite sets contain the exact same elements, then aren't they > comparable under [the contain-in-or-equal subset relation]? > > --Praveen Pamidimukkala > Team 7 From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:14:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3EWaJ024329 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:14:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3EUgk021105 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:14:30 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-TEN.MIT.EDU [18.244.5.10]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3EPe2011146 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:14:29 -0400 (EDT) Message-Id: <5.2.1.1.2.20050921210935.032f39d0@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 21 Sep 2005 21:14:20 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 1.08 X-Spam-Level: * (1.08) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8M3EWaJ024329 Status: RO X-Status: A Content-Length: 417 X-UID: 2653 X-Keywords: NonJunk NotJunk I found it interesting that the three proof methods: induction, strong induction, and the well ordering principle, can be used to prove the same theorems. I would like to understand why this is so. The equivalence between induction and strong induction is relatively simple, but I wouldnt know how to prove that the well ordering principle can be used in place of the other methods. Sergio Bacallado Group 1 From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:17:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3HXaJ024524 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:17:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3HVgk023397 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:17:32 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3HRg9012075 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:17:28 -0400 (EDT) Message-Id: <5.2.1.1.2.20050921231343.00bb83f8@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 21 Sep 2005 23:17:31 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.046 X-Spam-Level: * (1.046) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 187 X-UID: 2654 X-Keywords: NonJunk NotJunk I found the well ordered proof interesting, because we've been using it so much without even thinking about it. Even though it seems useless, it does have an important purpose. Jehan From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:33:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3X4aJ026901 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 21 Sep 2005 23:33:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3X2gk005276 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 21 Sep 2005 23:33:03 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3WuDO014644 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 21 Sep 2005 23:32:56 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8M3WuBI010631; Wed, 21 Sep 2005 23:32:56 -0400 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 21 Sep 2005 23:32:56 -0400 Message-ID: <20050921233256.1iyrb4m3kuq880sw@webmail.mit.edu> Date: Wed, 21 Sep 2005 23:32:56 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: required email comments, week 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 263 X-UID: 2655 X-Keywords: NonJunk NotJunk Week 3, Section 4: The Well Ordering Principle p.12: "Well Ordering proofs typically involve proof by contradiction, so using it is not always the best approach." Are there proofs that can be done only by contradiction and not induction? (Or not as easily.) From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:57:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3vbaJ028119 for <6042-probs@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 23:57:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3vagk025327 for <6042-probs@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 23:57:36 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3vXKt018713 for <6042-probs@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 23:57:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8M3vXxh006080; Wed, 21 Sep 2005 23:57:33 -0400 Received: from 18.218.1.221 ([18.218.1.221]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 21 Sep 2005 23:57:33 -0400 Message-ID: <20050921235733.499kbjv219gk04ws@webmail.mit.edu> Date: Wed, 21 Sep 2005 23:57:33 -0400 From: Neil M Dowgun To: 6042-probs@theory.csail.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.3 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 657 X-UID: 2656 X-Keywords: NonJunk NotJunk I completely understand everything in this week's readings, although I had to reread the false horse proof (section 2.5) before I really got how we were trying to prove it. It occured to me that if you take any set of elements for whom any pair of elements share a certain trait, then every element in that set must have that trait. Is this called by any certain name? It seems obvious, but so does the Well Ordering Principle. Also, I look forward to learning the difference between choosability and colorability. And I found "The country Inductia, whose unit of currency is the Strong" to be hilariously unimaginative. Keep up the good work. Neil Dowgun From meyer@imap.theory.csail.mit.edu Thu Sep 22 01:10:28 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M5ASaJ003872 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:10:28 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M5ARTW006405 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:10:27 -0400 (EDT) Received: from [18.239.7.198] (MACGREGOR-SEVEN-O-NINE.MIT.EDU [18.239.7.198]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M5AJ3a002326 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:10:19 -0400 (EDT) Message-ID: <43323CB8.7030406@mit.edu> Date: Thu, 22 Sep 2005 01:10:16 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 69 X-UID: 2657 X-Keywords: NotJunk I woul like the proof of product of primes explained again in class. From meyer@imap.theory.csail.mit.edu Thu Sep 22 02:17:49 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M6HnaJ013713 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:17:49 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M6Hn4k007672 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:17:49 -0400 (EDT) Received: from [18.241.6.223] (NEW-FOUR-SEVENTY-EIGHT.MIT.EDU [18.241.6.223]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M6Ha0v003901 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:17:41 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <0621A36B-A6F2-4FA7-93E8-E3D3FB7FBEFD@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: Week 3 Email Comments Date: Thu, 22 Sep 2005 02:25:57 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 223 X-UID: 2658 X-Keywords: NonJunk NotJunk I found the passage on the well ordering principle to be the most surprising because i dont see how such a simple principle can be used to provide proofs that are shorter than using proof by induction. Hamidou Soumare From meyer@imap.theory.csail.mit.edu Thu Sep 22 02:28:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M6ShaJ015053 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 02:28:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M6SfuR009330 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 02:28:41 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M6SZrQ004609 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 02:28:35 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8M6SYeO018484; Thu, 22 Sep 2005 02:28:34 -0400 Received: from MACGREGOR-TWO-FIFTEEN.MIT.EDU (MACGREGOR-TWO-FIFTEEN.MIT.EDU [18.239.5.215]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 02:28:34 -0400 Message-ID: <20050922022834.uxo9bn34oi4g40go@webmail.mit.edu> Date: Thu, 22 Sep 2005 02:28:34 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: Week 3 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 317 X-UID: 2659 X-Keywords: NonJunk Reading comments for this week on induction reading: The thing that I found hardest in this weeks reading were the concepts of strong induction, presented on pages 9 and 10, and the product of primes proof on page 10. I would appreciate going over the definition of strong induction and this proof in class. -Dave From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:28:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MESfaJ025546 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 10:28:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MESeWi026370; Thu, 22 Sep 2005 10:28:40 -0400 (EDT) Received: from [18.96.6.34] (SIMMONS-TWO-EIGHTY-NINE.MIT.EDU [18.96.6.34]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MESXpP011947 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Thu, 22 Sep 2005 10:28:33 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: Lecture Notes 3 Comments Date: Thu, 22 Sep 2005 10:30:29 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 363 X-UID: 2660 X-Keywords: NonJunk NotJunk After reading the notes I still can't differentiate readily between when or how to use induction and strong induction. I can tell you what the difference is, but the application of each is baffling to me. Further I can't figure out when there is an error in the logic of an induction proof, some added time on that would be very beneficial. -Harrison Hall From meyer@imap.theory.csail.mit.edu Wed Sep 21 17:37:07 2005 Message-ID: <4331D283.7020202@csail.mit.edu> Date: Wed, 21 Sep 2005 17:37:07 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama Subject: Re: Week 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 550 X-UID: 2661 X-Keywords: No, I don't think it would "make it more clear": it's better illustrated with examples. Let me know if you still disagree after Friday lecture. Regards, A Akari Kameyama wrote: > In other words, any theorem that can be proved with strong induction > could also be proved with ordinary > induction (using a slightly more complicated indcution hypothesis). > -page 9 > > An example of proving the same thing with ordinary then strong (or vice > versa) would help to make it more clear how one can be better than the > other. > --Apple-Ma From meyer@imap.theory.csail.mit.edu Wed Sep 21 02:11:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8L6BLaJ025131 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 02:11:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8L6BKNi023056 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 02:11:20 -0400 (EDT) Received: from [18.234.1.87] (DP-EIGHTY-SEVEN.MIT.EDU [18.234.1.87]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8L6BCNT026169 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 02:11:12 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-1-799499547 From: Akari Kameyama Subject: Week 3 Comments Date: Wed, 21 Sep 2005 02:11:11 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.695 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 2608 X-UID: 2662 X-Keywords: NonJunk NotJunk --Apple-Mail-1-799499547 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed In other words, any theorem that can be proved with strong induction could also be proved with ordinary induction (using a slightly more complicated indcution hypothesis). -page 9 An example of proving the same thing with ordinary then strong (or vice versa) would help to make it more clear how one can be better than the other. --Apple-Mail-1-799499547 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
=A0In other=A0words, any theorem that can be proved with = strong induction could also be proved with ordinary=A0
induction (using a = slightly more complicated indcution hypothesis).=A0 =A0-page 9

An example of = proving the same thing with ordinary then strong (or vice versa) would = help to make it more clear how one can be better than the = other.
= --Apple-Ma From meyer@imap.theory.csail.mit.edu Wed Sep 21 17:34:16 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8LLYGaJ006626 for <6042-staff@theory.csail.mit.edu>; Wed, 21 Sep 2005 17:34:16 -0400 Received: (qmail 30545 invoked from network); 21 Sep 2005 21:34:14 -0000 Received: from unknown (HELO ?128.30.49.31?) (armeyer10@128.30.49.31 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 21 Sep 2005 21:34:14 -0000 Message-ID: <4331D1D6.1080403@csail.mit.edu> Date: Wed, 21 Sep 2005 17:34:14 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: week 3 Notes, p.7 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 167 X-UID: 2663 X-Keywords: Any further comment now you've seen one? regards, A. Amanda Seybold wrote: > I would like to hear more about these other induction hypotheses > mentioned on page 7. From meyer@imap.theory.csail.mit.edu Wed Sep 21 21:35:20 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M1ZKaJ006925 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:35:20 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M1ZJmK001314 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:35:19 -0400 (EDT) Received: from [127.0.0.1] (NEXT-FOUR-EIGHTY-EIGHT.MIT.EDU [18.242.6.233]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M1ZB0w025235 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:35:16 -0400 (EDT) Message-ID: <43320A4C.6010508@mit.edu> Date: Wed, 21 Sep 2005 21:35:08 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 135 X-UID: 2664 X-Keywords: NonJunk NotJunk I'm confused on how the Well Ordering Principle (pg. 12) is actually useful. I get the idea, but what do I use it with? -chris wong From meyer@imap.theory.csail.mit.edu Wed Sep 21 19:47:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8LNlsaJ021544 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 19:47:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8LNlqGh009789 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 19:47:52 -0400 (EDT) Received: from [18.113.6.208] (ROGERS-FOUR-SIXTY-THREE.MIT.EDU [18.113.6.208]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8LNlkNO006187 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 19:47:47 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Sheldon Chan Subject: Comments on Reading - Week 3 Date: Wed, 21 Sep 2005 19:47:45 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 149 X-UID: 2665 X-Keywords: NonJunk NotJunk I find the concept of strong induction to still be confusing. From what I've gathered, it's simply an induction proof that has more assumptions. From meyer@imap.theory.csail.mit.edu Wed Sep 21 22:00:54 2005 Return-Path: Received: from smtp012.mail.yahoo.com (smtp012.mail.yahoo.com [216.136.173.32]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8M20raJ011895 for <6042-staff@theory.csail.mit.edu>; Wed, 21 Sep 2005 22:00:54 -0400 Received: (qmail 4121 invoked from network); 22 Sep 2005 02:00:53 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp012.mail.yahoo.com with SMTP; 22 Sep 2005 02:00:52 -0000 Message-ID: <43321058.4010507@csail.mit.edu> Date: Wed, 21 Sep 2005 22:00:56 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sheldon Chan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Comments on Reading - Week 3 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 233 X-UID: 2666 X-Keywords: yes that's right; so what's confusing you? regards, A. Sheldon Chan wrote: > I find the concept of strong induction to still be confusing. From > what I've gathered, it's simply an induction proof that has more > assumptions. From meyer@imap.theory.csail.mit.edu Wed Sep 21 21:59:56 2005 Return-Path: Received: from smtp016.mail.yahoo.com (smtp016.mail.yahoo.com [216.136.174.113]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8M1xtaJ011781 for <6042-staff@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:59:55 -0400 Received: (qmail 29449 invoked from network); 22 Sep 2005 01:59:55 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp016.mail.yahoo.com with SMTP; 22 Sep 2005 01:59:54 -0000 Message-ID: <4332101D.3040004@csail.mit.edu> Date: Wed, 21 Sep 2005 21:59:57 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Wong CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <43320A4C.6010508@mit.edu> In-Reply-To: <43320A4C.6010508@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 292 X-UID: 2667 X-Keywords: wait till lecture& class probs on fri for some more examples. ask me or your TA again after if that's not enough. regards, A. Chris Wong wrote: > I'm confused on how the Well Ordering Principle (pg. 12) is actually > useful. I get the idea, but what do I use it with? > > -chris wong > From meyer@imap.theory.csail.mit.edu Wed Sep 21 22:13:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M2DnaJ016804 for <6042-staff@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 22:13:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M2Dmht002686; Wed, 21 Sep 2005 22:13:48 -0400 (EDT) Received: from m56-129-17.mit.edu (M56-129-17.MIT.EDU [18.56.0.46]) (authenticated bits=56) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M2Dktj000246 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 21 Sep 2005 22:13:47 -0400 (EDT) Received: from localhost (sheldons@localhost) by m56-129-17.mit.edu (8.12.9) with ESMTP id j8M2Dkqp008288; Wed, 21 Sep 2005 22:13:46 -0400 Date: Wed, 21 Sep 2005 22:13:45 -0400 (EDT) From: Sheldon Chan To: "Prof. Albert R. Meyer" cc: <6042-staff@theory.csail.MIT.EDU> Subject: Re: Comments on Reading - Week 3 In-Reply-To: <43321058.4010507@csail.mit.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 780 X-UID: 2668 X-Keywords: NotJunk Perhaps its simply an expectation that the "strong" in the name implies that it should be more direct and supportive evidence in the proof. It makes sense making assumptions allows us to more easily prove something, but counterintuitive in producing a sound proof, since it could leave more room for faulty assumptions. I think it would be good if there were more examples or a clarification of the criteria for such assumptions made in a strong induction proof. Thanks, Sheldon On Wed, 21 Sep 2005, Prof. Albert R. Meyer wrote: > yes that's right; so what's confusing you? > regards, A. > > Sheldon Chan wrote: > > > I find the concept of strong induction to still be confusing. From > > what I've gathered, it's simply an induction proof that has more > > assumptions. > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 01:32:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M5WZaJ007009 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:32:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M5WXTW006762; Thu, 22 Sep 2005 01:32:33 -0400 (EDT) Received: from [192.168.1.101] (c-65-96-167-243.hsd1.ma.comcast.net [65.96.167.243]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M5WP0v002688; Thu, 22 Sep 2005 01:32:25 -0400 (EDT) Message-ID: <433241E3.2050008@csail.mit.edu> Date: Thu, 22 Sep 2005 01:32:19 -0400 From: sayan mitra User-Agent: Mozilla Thunderbird 1.0.6-1.1.fc4 (X11/20050720) X-Accept-Language: en-us, en MIME-Version: 1.0 CC: Sheldon Chan Subject: Re: Comments on Reading - Week 3 References: <43321058.4010507@csail.mit.edu> In-Reply-To: <43321058.4010507@csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.297 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 717 X-UID: 2669 X-Keywords: NonJunk NotJunk Compare the two induction rules, leaving aside the base cases: Regular induction: [P(n) => P(n+1) ] => \forall n P(n) Strong induction: [P(1),...,P(n) => P(n+1)] => \forall n P(n) Now the antecedent of the implication inside the [ ]'s is stronger for strong induction, making the [ ] implication itself weaker, right ? Therefore, if you look at the overall statement, the antecedent of the strong induction principle is weaker, and so, the strong induction as a rule (the outer implication) is stronger. Makes sense ? -Sayan > Sheldon Chan wrote: > >> I find the concept of strong induction to still be confusing. From >> what I've gathered, it's simply an induction proof that has more >> assumptions. > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:37:54 2005 Message-ID: <4332C1C2.8010709@csail.mit.edu> Date: Thu, 22 Sep 2005 10:37:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Arup Sarma CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> In-Reply-To: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 866 X-UID: 2670 X-Keywords: Yes, could use simple induction to PROVE that P(0),...,P(n) hold if you've proved P(0) and P(k) ==> P(k+1) for all k Section 3, page 10: > I found this difficult to understand, because I'm still not sure what the difference between simple and strong induction is. Doesn't simple induction imply that P(0), P(1), ..., P(n) are true as well? > > |Arup| > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:41:10 2005 Message-ID: <4332C289.2080605@csail.mit.edu> Date: Thu, 22 Sep 2005 10:41:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sheldon Chan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Comments on Reading - Week 3 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1046 X-UID: 2671 X-Keywords: not sure what kind of further "clarification" you're looking for, but you'll see several more examples Friday. Let me know if you're not satisfied after you've seen them. regards, A. Sheldon Chan wrote: >Perhaps its simply an expectation that the "strong" in the name implies >that it should be more direct and supportive evidence in the proof. >It makes sense making assumptions allows us to more easily prove >something, but counterintuitive in producing a sound proof, since it could >leave more room for faulty assumptions. I think it would be good if >there were more examples or a clarification of the criteria for such >assumptions made in a strong induction proof. > >Thanks, > >Sheldon > >On Wed, 21 Sep 2005, Prof. Albert R. Meyer wrote: > > > >>yes that's right; so what's confusing you? >>regards, A. >> >>Sheldon Chan wrote: >> >> >> >>>I find the concept of strong induction to still be confusing. From >>>what I've gathered, it's simply an induction proof that has more >>>assumptions. >>> >>> >> >> > > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:00:30 2005 Message-ID: <4332C711.2000700@csail.mit.edu> Date: Thu, 22 Sep 2005 11:00:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-staff@theory.csail.mit.edu Subject: Re: Email comments for reading References: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> In-Reply-To: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 642 X-UID: 2672 X-Keywords: we used to teach this, and students found it more confusing than helpful. Since it's not important, except foundationally, I now leave out the details. If you're really curious, I can sketch it for you after class some time. regards, A. Katherine A Romer wrote: >"...there's a routine way to transform any proof using the Well Ordering >Principle into a proof using Strong Induction, and vice-versa." (p.11) >I'd like to see an example in lecture of how one of the theorems we've proved >using induction could be proved using the Well Ordering Principle--I'm having a >hard time seeing how this might be done. > >Katherine Romer > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:38:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MEciaJ028852 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 10:38:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MEchWi007792 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 10:38:43 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MEcaBT016270 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 10:38:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8MEcao7001649; Thu, 22 Sep 2005 10:38:36 -0400 Received: from BURTON-FIFTY-THREE.MIT.EDU (BURTON-FIFTY-THREE.MIT.EDU [18.247.5.53]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 10:38:36 -0400 Message-ID: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> Date: Thu, 22 Sep 2005 10:38:36 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 360 X-UID: 2673 X-Keywords: NotJunk "...there's a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa." (p.11) I'd like to see an example in lecture of how one of the theorems we've proved using induction could be proved using the Well Ordering Principle--I'm having a hard time seeing how this might be done. Katherine Romer From meyer@imap.theory.csail.mit.edu Wed Sep 21 12:12:46 2005 21 12: 12:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8LGCkaJ007328 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 12:12:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8LGCjJU007908 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 12:12:45 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8LGCbKt026857 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 12:12:38 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 21 Sep 2005 12:11:37 -0400 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -1.486 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 87 X-UID: 2674 X-Keywords: NotJunk I would like to hear more about these other induction hypotheses mentioned on page 7. From meyer@imap.theory.csail.mit.edu Wed Sep 21 17:38:52 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <4331D2EB.2090000@csail.mit.edu> Date: Wed, 21 Sep 2005 17:38:51 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama Subject: Re: Week 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 560 X-UID: 2675 X-Keywords: No, I don't think it would "make it more clear": I think it's better illustrated with examples. Let me know if you still disagree after Friday lecture. Regards, A Akari Kameyama wrote: > In other words, any theorem that can be proved with strong induction > could also be proved with ordinary > induction (using a slightly more complicated indcution hypothesis). > -page 9 > > An example of proving the same thing with ordinary then strong (or vice > versa) would help to make it more clear how one can be better than the > other. > --Apple-Ma From meyer@imap.theory.csail.mit.edu Thu Sep 22 05:45:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M9jWaJ010784 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 05:45:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M9jV3p005920 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 05:45:31 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-SEVEN-FORTY-FOUR.MIT.EDU [18.238.5.233]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M9jRUC012432 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 05:45:28 -0400 (EDT) Message-ID: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Comments Week 3 Date: Thu, 22 Sep 2005 05:45:30 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 201 X-UID: 2676 X-Keywords: NonJunk NotJunk I'm a bit confused about the difference between simple induction and strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 qualifies as strong induction rather than simple. --Chieu From meyer@imap.theory.csail.mit.edu Thu Sep 22 02:54:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M6snaJ019565 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:54:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M6sluR026305 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:54:48 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-TWO-SIXTY-ONE.MIT.EDU [18.247.6.6]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M6sOOn006135 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:54:35 -0400 (EDT) Message-Id: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 22 Sep 2005 02:54:24 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 2.165 X-Spam-Level: ** (2.165) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 233 X-UID: 2677 X-Keywords: NonJunk NotJunk Section 3, page 10: I found this difficult to understand, because I'm still not sure what the difference between simple and strong induction is. Doesn't simple induction imply that P(0), P(1), ..., P(n) are true as well? |Arup| From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:16:04 2005 Message-ID: <4332CA84.4050107@csail.mit.edu> Date: Thu, 22 Sep 2005 11:15:16 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sergio Bacallado CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <5.2.1.1.2.20050921210935.032f39d0@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20050921210935.032f39d0@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 704 X-UID: 2678 X-Keywords: we used to teach this, and students found it more confusing than helpful. Since it's not important, except foundationally, I now leave out the details. If you're really curious, I can sketch it for you after class some time. regards, A. Sergio Bacallado wrote: > I found it interesting that the three proof methods: induction, strong > induction, and the well ordering principle, can be used to prove the > same theorems. I would like to understand why this is so. The > equivalence between induction and strong induction is relatively > simple, but I wouldnt know how to prove that the well ordering > principle can be used in place of the other methods. > > Sergio Bacallado > Group 1 > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 13:55:55 2005 Message-ID: <4332F032.9090004@csail.mit.edu> Date: Thu, 22 Sep 2005 13:56:02 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-staff@theory.csail.mit.edu Subject: Re: Lecture Notes 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 576 X-UID: 2679 X-Keywords: If you're still feeling shaky about using and/or debugging induction proofs, I suggest you take it up in office hours with your TA at the next opportunity. regards, A. Harrison King Hall wrote: > After reading the notes I still can't differentiate readily between when > or how to use induction and strong induction. I can tell you what the > difference is, but the application of each is baffling to me. Further I > can't figure out when there is an error in the logic of an induction > proof, some added time on that would be very beneficial. > -Harrison Hall > From meyer@imap.theory.csail.mit.edu Thu Sep 22 14:28:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MISnaJ003988 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:28:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MISmbH022504 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:28:48 -0400 (EDT) Received: from KAMIL.mit.edu (SENIOR-TWO-TWELVE.MIT.EDU [18.244.5.212]) (authenticated bits=0) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MISjf5018067 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:28:46 -0400 (EDT) Message-Id: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 22 Sep 2005 14:28:31 -0400 To: 6042-probs@theory.csail.mit.edu From: Kamil Sindi Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 800 X-UID: 2680 X-Keywords: NonJunk NotJunk "2.5 A Faulty Induction Proof False Theorem. All horses are the same color. Notice that no n is mentioned in this assertion, so we're going to have to reformulate it in a way that makes an n explicit. In particular, we'll (falsely) prove that False Theorem 2.4. In every set of n 1 horses, all are the same color. This a statement about all integers n 1 rather 0, so it's natural to use a slight variation on induction: prove P(1) in the base case and then prove that P(n) implies P(n + 1) for all n 1 in the inductive step. This is a perfectly valid variant of induction and is not the problem with the proof below." I would like to see more mathematical faulty inductions and see why certain assertions by induction are false; I didn't think the "horse theorems" were good examples. From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:33:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJXBaJ017447 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 15:33:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJX9bH022421 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 15:33:09 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MJX5wB013648 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 15:33:05 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8MJX5ic016466; Thu, 22 Sep 2005 15:33:05 -0400 Received: from DKE-ONE-TWENTY-ONE.MIT.EDU (DKE-ONE-TWENTY-ONE.MIT.EDU [18.233.1.121]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 15:33:05 -0400 Message-ID: <20050922153305.irm4vf76xf0os4sg@webmail.mit.edu> Date: Thu, 22 Sep 2005 15:33:05 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.031 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 342 X-UID: 2681 X-Keywords: NonJunk NotJunk I thought the reading was interesting, but there was one portion that was a little confusing. On page 5, the paragraph under equation (3) does the equivalent of a lot of fast talk and hand-waving, and while if you go through it it is most certainly correct, if one or two steps were actually written out it would have helped. -Paul Groudas From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:38:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJcHaJ019068 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:38:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJcGbH027575 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:38:16 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MJcCMB015618 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:38:12 -0400 (EDT) Message-ID: <000201c5bfad$2acafa40$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Thu, 22 Sep 2005 15:38:10 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 436 X-UID: 2682 X-Keywords: NonJunk NotJunk I have a question about the horse color problem. What if you start with the base case of n=2? Can you from there begin the same logic and show that all horses are again the same color? What would be the problem in this proof? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:03:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MK3naJ022727 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 16:03:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MK3lbH021838 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 16:03:48 -0400 (EDT) Received: from m12-182-13.mit.edu (M12-182-13.MIT.EDU [18.19.0.44]) (authenticated bits=56) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MK3f7v025558 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 16:03:42 -0400 (EDT) Received: (from bens@localhost) by m12-182-13.mit.edu (8.12.9) id j8MK3fBM015703; Thu, 22 Sep 2005 16:03:41 -0400 Subject: Comments From: Benjamin Schwartz To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 22 Sep 2005 16:03:41 -0400 Message-Id: <1127419421.15420.1.camel@m12-182-13.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 262 X-UID: 2683 X-Keywords: NotJunk On page 10, the comment that strong induction is not more powerful than weak induction is surprising. There is a conflict between logic, which says that every true statement is provable in infinitely many ways, and intuition, which demands One True Proof. -Ben From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:18:07 2005 Message-ID: <43331182.4000009@csail.mit.edu> Date: Thu, 22 Sep 2005 16:18:10 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kamil Sindi CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> In-Reply-To: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1020 X-UID: 2684 X-Keywords: you'll see quite a few more false proofs, including by induction,as the term progresses. But why was the "horses" example not good? Regards, A. Kamil Sindi wrote: > "2.5 A Faulty Induction Proof > > False Theorem. All horses are the same color. > Notice that no n is mentioned in this assertion, so we're going to > have to reformulate it in a way > that makes an n explicit. In particular, we'll (falsely) prove that > > False Theorem 2.4. In every set of n 1 horses, all are the same color. > This a statement about all integers n 1 rather 0, so it's natural to > use a slight variation on > induction: prove P(1) in the base case and then prove that P(n) > implies P(n + 1) for all n 1 in > the inductive step. This is a perfectly valid variant of induction and > is not the problem with the > proof below." > > I would like to see more mathematical faulty inductions and see why > certain assertions by induction are false; I didn't think the "horse > theorems" were good examples. > > > From - Thu From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:57:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJvPaJ021444 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:57:25 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJvNbc008734 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:57:24 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJvNOD021232; Thu, 22 Sep 2005 15:57:23 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8MJvF3a011824; Thu, 22 Sep 2005 15:57:16 -0400 (EDT) Message-ID: <43330C83.6020107@mit.edu> Date: Thu, 22 Sep 2005 15:56:51 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Andrew Shafer Subject: Re: Week 3 Comments References: <000201c5bfad$2acafa40$6d04f712@ajshafer> In-Reply-To: <000201c5bfad$2acafa40$6d04f712@ajshafer> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 791 X-UID: 2685 X-Keywords: NotJunk Great question! The answer is, there is nothing wrong with the proof! But think about what that proves: "IF any set of 2 horses is the same color, THEN all horses are the same color." Since the IF part is false, the statement as a whole is vacuously true, regardless of what comes in the THEN part. Andrew Shafer wrote: > I have a question about the horse color problem. > > What if you start with the base case of n=2? Can you from there begin > the same logic and show that all horses are again the same color? > What would be the problem in this proof? > > -Andrew > ---------------------------- > Illegitmitatum Non Carborundum Est > Andrew Shafer, MIT Blog > http://shaferandrew.blogspot.com > Si hoc legere scis numium eruditionis habes. > ---------------------------- > From meyer@imap.theory.csail.mit.edu Thu Sep 22 14:40:47 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MIegaJ007051 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:40:43 -0400 Received: (qmail 64372 invoked from network); 22 Sep 2005 17:55:53 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 22 Sep 2005 17:55:53 -0000 Message-ID: <4332F032.9090004@csail.mit.edu> Date: Thu, 22 Sep 2005 13:56:02 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-staff@theory.csail.mit.edu Subject: Re: Lecture Notes 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 575 X-UID: 2686 X-Keywords: If you're still feeling shaky about using and/or debugging induction proofs, I suggest you take it up in office hours with your TA at the next opportunity. regards, A. Harrison King Hall wrote: > After reading the notes I still can't differentiate readily between when > or how to use induction and strong induction. I can tell you what the > difference is, but the application of each is baffling to me. Further I > can't figure out when there is an error in the logic of an induction > proof, some added time on that would be very beneficial. > -Harrison Hall > From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:54:39 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJsdaJ021206 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:54:39 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJsbbc004073 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:54:38 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJrFOD021032; Thu, 22 Sep 2005 15:53:15 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8MJr70v009280; Thu, 22 Sep 2005 15:53:08 -0400 (EDT) Message-ID: <43330B8B.3070807@mit.edu> Date: Thu, 22 Sep 2005 15:52:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kamil Sindi Subject: Re: Week 3 Comments References: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> In-Reply-To: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 439 X-UID: 2687 X-Keywords: NotJunk Did you think they weren't good examples because the flaws were too obvious? If so, I think you're in great shape - that means you have a good grasp of induction. If it was something else, please let us know it was about the examples that bothered you. DS > I would like to see more mathematical faulty inductions and see why > certain assertions by induction are false; I didn't think the "horse > theorems" were good examples. From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:22:28 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MKMRaJ025372 for ; Thu, 22 Sep 2005 16:22:27 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EIXaN-0003G0-Me for meyer@csail.mit.edu; Thu, 22 Sep 2005 16:22:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MKMQbH009881 for ; Thu, 22 Sep 2005 16:22:26 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MKMNmt002614 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for ; Thu, 22 Sep 2005 16:22:24 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: In-Reply-To: <4331D1D6.1080403@csail.mit.edu> References: <4331D1D6.1080403@csail.mit.edu> Date: Thu, 22 Sep 2005 16:21:24 -0400 To: "Albert R. Meyer" From: Amanda Seybold Subject: Re: week 3 Notes, p.7 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.099 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO X-Status: A Content-Length: 292 X-UID: 2688 X-Keywords: It would be nice to see more? Perhaps ones with more than one variable, so one has to figure out what to assign n to. >Any further comment now you've seen one? >regards, A. > >Amanda Seybold wrote: >>I would like to hear more about these other induction hypotheses >>mentioned on page 7. From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:25:32 2005 Message-ID: <4333133D.2030402@csail.mit.edu> Date: Thu, 22 Sep 2005 16:25:33 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold CC: 6042-staff@theory.csail.mit.edu Subject: Re: week 3 Notes, p.7 References: <4331D1D6.1080403@csail.mit.edu> In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 464 X-UID: 2689 X-Keywords: you'll see more tmw, and lots during the term: induction is a pretty basic proof technique we use all the time. regards, A Amanda Seybold wrote: > It would be nice to see more? Perhaps ones with more than one > variable, so one has to figure out what to assign n to. > >> Any further comment now you've seen one? >> regards, A. >> >> Amanda Seybold wrote: >> >>> I would like to hear more about these other induction hypotheses >>> mentioned on page 7. >> From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:58:54 2005 Message-ID: <43331B11.9020408@csail.mit.edu> Date: Thu, 22 Sep 2005 16:58:57 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Paul Groudas CC: 6042-staff@theory.csail.mit.edu Subject: Re: reading comments References: <20050922153305.irm4vf76xf0os4sg@webmail.mit.edu> In-Reply-To: <20050922153305.irm4vf76xf0os4sg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 475 X-UID: 2690 X-Keywords: I agree, and I just revised a bit. See if new version helps. Thx for the feedback. regards, A. Paul Groudas wrote: >I thought the reading was interesting, but there was one portion that was a >little confusing. On page 5, the paragraph under equation (3) does the >equivalent of a lot of fast talk and hand-waving, and while if you go through >it it is most certainly correct, if one or two steps were actually written out >it would have helped. > >-Paul Groudas > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:41:11 2005 Return-Path: Received: from smtp013.mail.yahoo.com (smtp013.mail.yahoo.com [216.136.173.57]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MEfAaJ029152 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 10:41:10 -0400 Received: (qmail 65890 invoked from network); 22 Sep 2005 14:41:09 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp013.mail.yahoo.com with SMTP; 22 Sep 2005 14:41:09 -0000 Message-ID: <4332C289.2080605@csail.mit.edu> Date: Thu, 22 Sep 2005 10:41:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sheldon Chan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Comments on Reading - Week 3 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1045 X-UID: 2691 X-Keywords: not sure what kind of further "clarification" you're looking for, but you'll see several more examples Friday. Let me know if you're not satisfied after you've seen them. regards, A. Sheldon Chan wrote: >Perhaps its simply an expectation that the "strong" in the name implies >that it should be more direct and supportive evidence in the proof. >It makes sense making assumptions allows us to more easily prove >something, but counterintuitive in producing a sound proof, since it could >leave more room for faulty assumptions. I think it would be good if >there were more examples or a clarification of the criteria for such >assumptions made in a strong induction proof. > >Thanks, > >Sheldon > >On Wed, 21 Sep 2005, Prof. Albert R. Meyer wrote: > > > >>yes that's right; so what's confusing you? >>regards, A. >> >>Sheldon Chan wrote: >> >> >> >>>I find the concept of strong induction to still be confusing. From >>>what I've gathered, it's simply an induction proof that has more >>>assumptions. >>> >>> >> >> > > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:45:30 2005 Return-Path: Received: from smtp107.mail.sc5.yahoo.com (smtp107.mail.sc5.yahoo.com [66.163.169.227]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MEjSaJ029551 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 10:45:28 -0400 Received: (qmail 9436 invoked from network); 22 Sep 2005 14:29:22 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp107.mail.sc5.yahoo.com with SMTP; 22 Sep 2005 14:29:21 -0000 Message-ID: <4332BFC5.1000706@csail.mit.edu> Date: Thu, 22 Sep 2005 10:29:25 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chieu Nguyen CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments Week 3 References: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> In-Reply-To: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 545 X-UID: 2692 X-Keywords: The k and m in the proof are only known to be SMALLER than n+2 (in fact, they cannot equal n+2). Strong induction let's us assume both k & m are products of primes. Simple induction would only let us assume that n+2 was a product of primes, which isn't much use in attacking n+3. Make sense now? Regards, A. Chieu Nguyen wrote: > I'm a bit confused about the difference between simple induction and strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 qualifies as strong induction rather than simple. > > --Chieu From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:54:07 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MEs6aJ030723 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 10:54:07 -0400 Received: (qmail 95926 invoked from network); 22 Sep 2005 14:37:52 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 22 Sep 2005 14:37:51 -0000 Message-ID: <4332C1C2.8010709@csail.mit.edu> Date: Thu, 22 Sep 2005 10:37:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Arup Sarma CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> In-Reply-To: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 865 X-UID: 2693 X-Keywords: Yes, could use simple induction to PROVE that P(0),...,P(n) hold if you've proved P(0) and P(k) ==> P(k+1) for all k Section 3, page 10: > I found this difficult to understand, because I'm still not sure what the difference between simple and strong induction is. Doesn't simple induction imply that P(0), P(1), ..., P(n) are true as well? > > |Arup| > From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:00:31 2005 Return-Path: Received: from smtp017.mail.yahoo.com (smtp017.mail.yahoo.com [216.136.174.114]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MF0UaJ031242 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 11:00:30 -0400 Received: (qmail 33082 invoked from network); 22 Sep 2005 15:00:30 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp017.mail.yahoo.com with SMTP; 22 Sep 2005 15:00:29 -0000 Message-ID: <4332C711.2000700@csail.mit.edu> Date: Thu, 22 Sep 2005 11:00:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-staff@theory.csail.mit.edu Subject: Re: Email comments for reading References: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> In-Reply-To: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 641 X-UID: 2694 X-Keywords: we used to teach this, and students found it more confusing than helpful. Since it's not important, except foundationally, I now leave out the details. If you're really curious, I can sketch it for you after class some time. regards, A. Katherine A Romer wrote: >"...there's a routine way to transform any proof using the Well Ordering >Principle into a proof using Strong Induction, and vice-versa." (p.11) >I'd like to see an example in lecture of how one of the theorems we've proved >using induction could be proved using the Well Ordering Principle--I'm having a >hard time seeing how this might be done. > >Katherine Romer > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:01:14 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MF1EaJ031327 for ; Thu, 22 Sep 2005 11:01:14 -0400 Received: from theory.lcs.mit.edu ([128.30.51.92] helo=theory.csail.mit.edu) by rozz.csail.mit.edu with esmtp (Exim 4.50) id 1EISZW-0005RL-1J for meyer@csail.mit.edu; Thu, 22 Sep 2005 11:01:14 -0400 Received: from lyrebird.csail.mit.edu (lyrebird.csail.mit.edu [128.30.51.62]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MF1DaJ031324 for ; Thu, 22 Sep 2005 11:01:13 -0400 Received: from lyrebird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by lyrebird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8MF1DDW010858 for ; Thu, 22 Sep 2005 11:01:13 -0400 Received: (from ronitt@localhost) by lyrebird.csail.mit.edu (8.12.11/8.12.11/Submit) id j8MF1DGi010857 for meyer@csail.mit.edu; Thu, 22 Sep 2005 11:01:13 -0400 Date: Thu, 22 Sep 2005 11:01:13 -0400 From: Ronitt Rubinfeld Message-Id: <200509221501.j8MF1DGi010857@lyrebird.csail.mit.edu> To: meyer@csail.mit.edu Subject: Re: ps3 comments X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 4009 X-UID: 2695 X-Keywords: NotJunk Hi Albert, I have been working on ps3 in the week4 directory. It doesn't latex well, because I still have the problem including the pdf diagram. However, if you use latex (instead of pdflatex, which doesn't work at all), you can get something without the figure. Here are some comments on questions/modifications that came up: Relations course notes: in section 5 of the spring 04 notes, lemma 5.3 says that every partial order is reflexive and transitive, but I think it would be good to specify that this lemma only applies to weak partial orders. problem 1: I had them do both weak and strict partial orders in part (b). problem 2: for part (a), should we have them prove it is a maximum length chain? for part (b), changed wording, so that hopefully it is clear that they don't have to prove that the antichain is maximum. Do they have the math background to prove that it is at least as big as 2^n/n? problem 4 didn't change anything, since will cover DAG's in class. best, Ronitt From meyer@csail.mit.edu Wed Sep 21 09:33:54 2005 Date: Wed, 21 Sep 2005 09:33:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Ronitt Rubinfeld Subject: ps3 comments References: <200509211033.j8LAXi38023080@lyrebird.csail.mit.edu> In-Reply-To: <200509211033.j8LAXi38023080@lyrebird.csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=5.0 tests=AWL,BAYES_00 autolearn=ham version=2.63 Prob 1 is ok if reworded to say "which are p.o.'s? which are equiv rels? which are equiv rels on their domain?" because I'd prefer to focus on these kinds of relations we use and deemphasize pure relational axioms/properties in in isolation. For this reason, I wouldn't use Prob 2 (unless we can rework it into a false proof like the one that was in lecture). Prob 3(b) seems very tricky TO PROVE as Eric pointed out -- he said it requires Sperner's Lemma or similar cleverness to show you can' t beat the size n choose n/2 antichain of subsets of size n/2 (though I suspect there might be a more elementary induction argument lurking around there). But I like your idea to derive a 2^n/n lower bound, which is not a bad bound: using Stirling's formula I get 2^n/sqrt{2 pi n}. I like prob 4, though for 4(c) I'd give them the power of 2 chain and ask that they prove it's maximal. Then for 4(d), I'd say: "let c be the length of the power of 2 chain. By Dilworth Lemma, there must be an antichain of length 2^n/c. Describe one." But these probs are all theoretical, and it would be nice to have something reflecting possible applications, eg, scheduling. Take a look at F05 pset5, prob 5. It's writeup uses a lot of DAG/graph terminology, but shouldn't be hard to express purely in partial order language. If you go with this one, I'd suggest dropping Prob 3 above, which seems unmotivated. I'll work on finding an intro-to-graphs problem or two. If you agree to go with probs 1, 4, and F04 ps5, prob 5, that should do it for a reasonable ps4. regards, A. P.S. please delete latex-macros-ronitt from devel/week3 (baffled me for a while why my edits to course.dat were not working.) Much better to link to the single copy of the macros in pub/staff/ than make your own that can get out of sync. Ronitt Rubinfeld wrote: >on problem 3 of the proposed homework, >there is a partial order on subsets of {1..n}. >couldn't we get them to lower bound the >size of a max length antichain by dilworth's >thm by 2^n/(n+1) (since there is no chain of length > n+1) > >maybe problem 3 and problem 4 are too similar... >in that case, I would probably go with problem 4. > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 17:18:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MLImaJ004507 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 17:18:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MLIkag027351 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 17:18:46 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MLIhV7022281 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 17:18:44 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8E59DBA2-C7A8-4296-9EE0-5AD3A6AB043C@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: week 3 reading Date: Thu, 22 Sep 2005 17:17:46 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 310 X-UID: 2696 X-Keywords: NonJunk NotJunk I thought the reading on induction and strong induction were pretty clear, but I'm still not sure how the Well Ordering Principle is used to prove things. More examples would've been nice. It would've been cool also to see one thing proved three different ways, just to compare and contrast. ~Crystal From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:07:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MM7RaJ014538 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 18:07:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MM7Pag001883 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 18:07:25 -0400 (EDT) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MM7Njq005029 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 18:07:24 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Message-Id: Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rian Hunter Subject: Assigned Reading Date: Thu, 22 Sep 2005 18:07:23 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.475 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8MM7RaJ014538 Status: RO X-Status: A Content-Length: 2457 X-UID: 2697 X-Keywords: NonJunk NotJunk From page 12 of the reading: "4 The Well Ordering Principle Another proof method closely related to induction depends on the Well Ordering Principle. Every nonempty subset of natural numbers has a smallest element. The Well Ordering Principle looks nothing like the induction axiom, and it may seem obvious but useless. But as for obvious, note that this axiom would be false if the set of non-negative integers, N, were replaced by, say, the set, Z, of all integers, or the set, Q+, of positive rational numbers. Neither of these sets has a least element. So the Well Ordering Principle does capture something special about the natural numbers. As for useless, it turns out that there’s a routine way to transform any proof using the Well Or- dering Principle into a proof using Strong Induction, and vice-versa. (We won’t take the time to describe the transformations, though they are not hard.) So Well Ordering could have been used instead of induction in all the previous proofs. In fact, looking back, we implicitly relied on the Well Ordering Principle in the proof in Week 2 Notes that √2 is irrational. That proof assumed that any rational number, q, could be written as a fraction in lowest terms, that is, q = m/n where m and n are integers with no common factors. How do we know this is always possible? First, we can assume m ≥ 0 (otherwise, replace m/n by −m/(−n)), so the set of natural numbers, m, such that q = m/n for some integer, n, is not empty. Therefore, by Well Ordering, there must be a least natural number, m0 , such that q = m0 /n0 for some integer, n0 . Now if m0 and n0 had a common factor, p > 1, then (m0 /p)/(n0 /p) would be another way to express q as a quotient of integers. But since 0 ≤ (m0 /p) < m0 . this contradicts the minimality of m0 . We’ve using the Well-ordering Principle on the sly from early on! Mathematicians often use Well Ordering because it often leads to shorter proofs than induction. On the other hand, Well Ordering proofs typically involve proof by contradiction, so using it is not always the best approach. The choice of method is really a matter of style—but style does matter." This seemed to be the most difficult passage to grasp simply because it does not concretely explain what "The Well Ordering Principle" is, because of this I would personally like to have this discussed or elaborating upon in lecture. Thanks. Rian Hunter From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:22:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMMiaJ015609 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:22:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMMgag011659 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:22:43 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMMdBv008253 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:22:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8MMMdU0028571; Thu, 22 Sep 2005 18:22:39 -0400 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 18:22:39 -0400 Message-ID: <20050922182239.wlp6xdcmx8mc4ko4@webmail.mit.edu> Date: Thu, 22 Sep 2005 18:22:39 -0400 From: cbossard@MIT.EDU To: "6042-probs@theory.lcs.mit.edu" <6042-probs@theory.lcs.MIT.EDU> Subject: Week 3 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.563 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 419 X-UID: 2698 X-Keywords: NonJunk NotJunk passage: Section 4 The Well Ordering Principle pg 12 The principle seems obvious and easy enough but I would like to see more examples on how it could be used or even an example on how it could be used instead of induction Also I am having trouble understanding being vacuously true from page 11. It seems like a contradiction that when the hypothesis for something is false the whole thing is true. Cynthia Bossard From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:53:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMrraJ020654 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:53:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMrqag000617 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:53:52 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMroZQ014107 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:53:50 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8MMrn53001025; Thu, 22 Sep 2005 18:53:49 -0400 Received: from MDMURRAY.MIT.EDU (MDMURRAY.MIT.EDU [18.234.0.116]) (User authenticated as a_lopez@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 18:53:49 -0400 Message-ID: <20050922185349.n11mg3bfhr4gwsw8@webmail.mit.edu> Date: Thu, 22 Sep 2005 18:53:49 -0400 From: a_lopez@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: week 3 readings MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 1.031 X-Spam-Level: * (1.031) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 749 X-UID: 2699 X-Keywords: NonJunk NotJunk "Despite the name, strong induction is actually no more powerful than ordinary induction. In other words, any theorem that can be proved with strong induction could also be proved with ordinary induction (using a slightly more complicated indcution hypothesis)." I found this very surprising, I have seen induction and strong induction before and always thought that strong induction was more powerful, that is, that some problems could be solved using strong induction but not ordinary induction. I have never actually seen a problem be given two different solutions: one with strong induction and another with ordinary induction. I wonder how much stronger the hypothesis is in ordinary induction than it is in strong induction. Adriana Lopez From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:56:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMuiaJ020819 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:56:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMugag002198 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:56:42 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMue7V014659 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:56:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8MMueKp006855; Thu, 22 Sep 2005 18:56:40 -0400 Received: from EASTCAMPUS-NINE-FORTY-EIGHT.MIT.EDU (EASTCAMPUS-NINE-FORTY-EIGHT.MIT.EDU [18.238.6.181]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 18:56:40 -0400 Message-ID: <20050922185640.8zrcp3xy8hcs004w@webmail.mit.edu> Date: Thu, 22 Sep 2005 18:56:40 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 216 X-UID: 2700 X-Keywords: NotJunk I thought the section on simple induction was just fine. I am still having trouble understanding the well ordering principle, and might like to see a different example of strong induction in class. -Rob Crowell From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:57:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMvxaJ020892 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 18:57:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMvvag003027 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 18:57:57 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMvEUG014749 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 18:57:35 -0400 (EDT) Message-ID: <433336C9.7020102@mit.edu> Date: Thu, 22 Sep 2005 18:57:13 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.086 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 480 X-UID: 2701 X-Keywords: NonJunk NotJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 I found the strong induction examples (3.2 and 3.3 on pages 10 and 11) a bit confusing. The idea of strong induction was cleared up somewhat on the problem set, however. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDMzbJlO3j8HLL0+4RAsMfAKDB95MzFZ0PTvenM6U2S+dlGjktqwCg/a8m nwfl5hh1oHFh8b0QTtHFEtc= =Itmf -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Thu Sep 22 19:15:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MNFWaJ021962 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 19:15:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MNFVag012949 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 19:15:31 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MNFTjO017569 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 19:15:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8MNFTSO004043; Thu, 22 Sep 2005 19:15:29 -0400 Received: from SENIOR-TWO-O-SIX.MIT.EDU (SENIOR-TWO-O-SIX.MIT.EDU [18.244.5.206]) (User authenticated as berke@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 19:15:29 -0400 Message-ID: <20050922191529.rynea7i9aflwgcg4@webmail.mit.edu> Date: Thu, 22 Sep 2005 19:15:29 -0400 From: Allison Berke To: 6042-probs@theory.lcs.MIT.EDU Subject: reading questions MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.518 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 297 X-UID: 2702 X-Keywords: NonJunk NotJunk I found the section on the well-ordering principle (page 12) most interesting. I'm concurrently enrolled in 18.100b, and we've covered some proofs that use this principle, but never had it directly named. I'd be interested to see what other kinds of proofs can use this principle as a main tenet. From meyer@imap.theory.csail.mit.edu Thu Sep 22 19:33:44 2005 Message-ID: <43333F5B.4040601@csail.mit.edu> Date: Thu, 22 Sep 2005 19:33:47 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Rian Hunter CC: 6042-staff@theory.csail.mit.edu Subject: Re: Assigned Reading References: In-Reply-To: Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 584 X-UID: 2703 X-Keywords: " it does not concretely explain what "The Well Ordering Principle" is" I thought the box labeled "The Well Ordering Principle" was about as concrete as it gets. Can you explain what kind of further statement of the Principle you're looking for? Rian Hunter wrote: > From page 12 of the reading: .... > > This seemed to be the most difficult passage to grasp simply because > it does not concretely explain what "The Well Ordering Principle" is, > because of this I would personally like to have this discussed or > elaborating upon in lecture. Thanks. > > Rian Hunter From meyer@imap.theory.csail.mit.edu Thu Sep 22 20:05:34 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N05YaJ030875 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:34 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N05X4C009109 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:33 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N05W3E028720 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:32 -0400 (EDT) Received: from ebrum.mit.edu (NEW-SIX-SIXTY-THREE.MIT.EDU [18.241.7.152]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N05Q0w019037 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:27 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922200232.02503eb0@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 22 Sep 2005 20:05:25 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: -0.685 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 267 X-UID: 2704 X-Keywords: NonJunk $Label4 NotJunk Theorem 2.2 was an interesting point for me, I was aware that Fibonacci numbers had weird properties but this was even beyond my expectations. By the way, the theorem states that For all n greater than or equal to 2, F_0^2+F_1^2+...+F_n^2=F_n+F_(n+1) best yasin From meyer@imap.theory.csail.mit.edu Thu Sep 22 20:43:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N0h7aJ003599 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:43:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N0h5QQ000091 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:43:05 -0400 (EDT) Received: from [18.239.5.240] (MACGREGOR-TWO-FORTY.MIT.EDU [18.239.5.240]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N0h1DB000578 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:43:03 -0400 (EDT) Message-ID: <43334F94.6010402@mit.edu> Date: Thu, 22 Sep 2005 20:43:00 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 3 Reading Assignment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.909 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 341 X-UID: 2705 X-Keywords: NonJunk NotJunk Hi, After reading the week's lecture notes, I found the /well-ordering principle/ the most confusing topic to date. I didn't fully understand its connection with Induction and I was confused with the Tutor problem related to this principle. I would appreciate it if you could possibly explain this topic in more detail. Thanks, Lohith From meyer@imap.theory.csail.mit.edu Thu Sep 22 21:04:13 2005 Message-ID: <43335495.7060805@csail.mit.edu> Date: Thu, 22 Sep 2005 21:04:21 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-staff@theory.csail.mit.edu Subject: [Fwd: Re: Reading Comments Week 3] Content-Type: text/plain; charset=iso-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1190 X-UID: 2706 X-Keywords: -------- Original Message -------- Subject: Re: Reading Comments Week 3 Date: Thu, 22 Sep 2005 20:20:47 -0400 From: Chieu Nguyen To: Prof. Albert R. Meyer References: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> <4332BFC5.1000706@csail.mit.edu> Dear Prof. Meyer, That was very helpful. Thanks for clearing things up. --Chieu ----- Original Message ----- From: "Prof. Albert R. Meyer" To: "Chieu Nguyen" Cc: <6042-staff@theory.csail.mit.edu> Sent: Thursday, September 22, 2005 10:29 AM Subject: Re: Reading Comments Week 3 > The k and m in the proof are only known to be SMALLER than n+2 (in fact, > they cannot equal n+2). Strong induction let's us assume both k & m are > products of primes. Simple induction would only let us assume that n+2 > was > a product of primes, which isn't much use in attacking n+3. > > Make sense now? > > Regards, A. > > Chieu Nguyen wrote: > > > I'm a bit confused about the difference between simple induction and > strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 > qualifies as strong induction rather than simple. > > > > --Chieu > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 20:20:52 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N0KpaJ032742 for ; Thu, 22 Sep 2005 20:20:51 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EIbJ5-0006R6-KS for meyer@csail.mit.edu; Thu, 22 Sep 2005 20:20:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N0KoQQ018217 for ; Thu, 22 Sep 2005 20:20:50 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-SEVEN-FORTY-FOUR.MIT.EDU [18.238.5.233]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N0KjQ6027500 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for ; Thu, 22 Sep 2005 20:20:46 -0400 (EDT) Message-ID: <002401c5bfd4$a6432f40$e905ee12@addressisp.com> From: "Chieu Nguyen" To: "Prof. Albert R. Meyer" References: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> <4332BFC5.1000706@csail.mit.edu> Subject: Re: Reading Comments Week 3 Date: Thu, 22 Sep 2005 20:20:47 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=response X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.882 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 897 X-UID: 2707 X-Keywords: NonJunk $Forwarded NotJunk Dear Prof. Meyer, That was very helpful. Thanks for clearing things up. --Chieu ----- Original Message ----- From: "Prof. Albert R. Meyer" To: "Chieu Nguyen" Cc: <6042-staff@theory.csail.mit.edu> Sent: Thursday, September 22, 2005 10:29 AM Subject: Re: Reading Comments Week 3 > The k and m in the proof are only known to be SMALLER than n+2 (in fact, > they cannot equal n+2). Strong induction let's us assume both k & m are > products of primes. Simple induction would only let us assume that n+2 > was > a product of primes, which isn't much use in attacking n+3. > > Make sense now? > > Regards, A. > > Chieu Nguyen wrote: > > > I'm a bit confused about the difference between simple induction and > strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 > qualifies as strong induction rather than simple. > > > > --Chieu > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 21:25:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N1PraJ009865 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:25:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N1PqQQ023263 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:25:52 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N1PnAC006779 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:25:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8N1Pnas029585; Thu, 22 Sep 2005 21:25:49 -0400 Received: from BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (User authenticated as hectorb@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 21:25:49 -0400 Message-ID: <20050922212549.kxmlkccxu1kcs80g@webmail.mit.edu> Date: Thu, 22 Sep 2005 21:25:49 -0400 From: Hector Beltran To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 3 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 315 X-UID: 2708 X-Keywords: NonJunk NotJunk "...there?s a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa. So Well Ordering could have been used instead of induction in all the previous proofs." I would like to see some more examples of this, it's confusing to me. -Hector Beltran From meyer@imap.theory.csail.mit.edu Thu Sep 22 21:27:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N1RAaJ009907 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:27:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N1R9QQ023912 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:27:09 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N1R6up006951 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:27:06 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8N1R69C021223; Thu, 22 Sep 2005 21:27:06 -0400 Received: from NEXT-FIVE-O-THREE.MIT.EDU (NEXT-FIVE-O-THREE.MIT.EDU [18.242.6.248]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 21:27:06 -0400 Message-ID: <20050922212706.54lp2oenk9444o08@webmail.mit.edu> Date: Thu, 22 Sep 2005 21:27:06 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: E-mail Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.804 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 308 X-UID: 2709 X-Keywords: NonJunk NotJunk Reference: Section 3.3 Pg 11 I clearly understood normal induction, however I did have some difficulty with strong induction (especially this problem) and would like to discuss it in more detail in class. I did understand it after close reading but I was hoping we could go over it in class. -Rahul Shroff From meyer@imap.theory.csail.mit.edu Thu Sep 22 22:12:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N2CtaJ014084 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 22:12:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N2CrQQ020487 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 22:12:53 -0400 (EDT) Received: from [18.227.1.178] ([18.227.1.178]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N2CmjR013844 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 22:12:50 -0400 (EDT) Message-ID: <43336488.5090707@mit.edu> Disposition-Notification-To: Scot Frank Date: Thu, 22 Sep 2005 22:12:24 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.577 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 265 X-UID: 2710 X-Keywords: NonJunk $MDNSent NotJunk Hi, Even after reading about Strong Inductions twice, I did not feel the example was explained clearly in class. Perhaps it is my own fault but I was not able to make more sense of it even from the reading. I think I will come to office hours. Thanks, Scot From meyer@imap.theory.csail.mit.edu Thu Sep 22 22:17:10 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N2HAaJ014512 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:10 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N2H94C027571 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:10 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N2H93E001590 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:09 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N2H63a025969 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:06 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <9B3CB3F4-7DBB-4BE2-ACA3-E68CD73454BF@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-34-958248826 From: Michael Murray Subject: E-mail question Date: Thu, 22 Sep 2005 22:17:00 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.523 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 7896 X-UID: 2711 X-Keywords: NonJunk NotJunk --Apple-Mail-34-958248826 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed "First, we can assume m 0 (otherwise, replace m/n by =E2=88=92m/(=E2=88=92= n)), =20 so the set of natural numbers, m, such that q =3D m/n for some integer, n, is not empty. Therefore, by =20= Well Ordering, there must be a least natural number, m0, such that q =3D m0/n0 for some integer, =20= n0. Now if m0 and n0 had a common factor, p > 1, then (m0/p)/(n0/p) would be another way to =20 express q as a quotient of integers. But since 0 (m0/p) < m0. this contradicts the minimality =20 of m0." I did not fully understand this paragraph because i do not =20 understand how we reach a contradiction. This was the part in the =20 notes that I found most confusing. MIchael Murray --Apple-Mail-34-958248826 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
"First, we can assume m =C2=A0 0 (otherwise, replace m/n by =E2=88=92m/(=E2=88= =92n)), so the set of natural numbers,
m, such that q =3D = m/n for some integer, n, is not empty. Therefore, by Well Ordering, = there must
be a least natural number, = m0, such that q =3D = m0/n0 = for some integer, = n0. Now if = m0 = and = n0 = had
a common factor, p > 1, then = (m0/p)/(n0/p) would be = another way to express q as a quotient of
integers. But = since 0 =C2=A0 = (m0/p) < = m0. this contradicts = the minimality of m0."

I did not fully = understand this paragraph=C2=A0 because i do not understand how we reach = a contradiction.=C2=A0 This was the part in the notes that I found most = confusing.

MIchael = Murray



= --Apple-Mail-34-958248826-- From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:12:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3CXaJ018919 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 23:12:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3CVk8023598 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 23:12:31 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3CTs1023533 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 23:12:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N3CTsd028402; Thu, 22 Sep 2005 23:12:29 -0400 Received: from comm575-0401-dhcp142.bu.edu (comm575-0401-dhcp142.bu.edu [168.122.167.142]) (User authenticated as brevzin@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 23:12:29 -0400 Message-ID: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> Date: Thu, 22 Sep 2005 23:12:29 -0400 From: Boris E Revzin To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 449 X-UID: 2712 X-Keywords: NonJunk NotJunk Regarding the induction lecture, it seems excessive to me to have to write out the entire induction template. By showing a base case and P(n)->P(n+1), it is pretty clear to anyone reading the proof that it is by induction. I just want to know if it's enough to show P(0), P(n)->P(n+1), QED. I just think that's all that is necessary, oftentimes writing out the template will take more time than writing out the actual proof... Boris "Barry" Revzin From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:15:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3FwaJ019046 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:15:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3Fuk8025673 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:15:57 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3FrDc024086 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:15:53 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8N3FrgJ018437; Thu, 22 Sep 2005 23:15:53 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 21:15:53 -0600 Message-ID: <20050922211553.u61uh3a33cw0g8k4@webmail.mit.edu> Date: Thu, 22 Sep 2005 21:15:53 -0600 From: John Marrero To: 6042-probs@theory.csail.MIT.EDU Subject: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 459 X-UID: 2713 X-Keywords: NonJunk NotJunk Page 10, "Strong Induction" This is the only thing I had a problem with, since it seems so similar to simple induction. I found it somewhat difficult to see the difference between the two. If there's an extra example that can be seen in lecture, I'm sure it'd clarify things! Page 12, "The Well Ordering Principle" Interesting; I wonder if there's a principle that says there's a greatest element in a nonempty subset? Just made me think. - John Marrero From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:36:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3aQaJ022235 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:36:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3aOk8006811 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:36:25 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3aMN0027189 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:36:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8N3aMYA021006; Thu, 22 Sep 2005 23:36:22 -0400 Received: from EASTCAMPUS-NINE-FIFTY-THREE.MIT.EDU (EASTCAMPUS-NINE-FIFTY-THREE.MIT.EDU [18.238.6.186]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 23:36:22 -0400 Message-ID: <20050922233622.h94vhwbllhugw8oc@webmail.mit.edu> Date: Thu, 22 Sep 2005 23:36:22 -0400 From: Forrest O Green To: 6042-probs@theory.csail.MIT.EDU Subject: `Required reading comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 201 X-UID: 2714 X-Keywords: NonJunk NotJunk Page 12, well ordering principle (section 4) I am pretty sure that I underrstood everything, but a little bit more work on the well ordering principle could potentially be helpful. -Forrest Green From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:49:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3nkaJ025685 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:49:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3nik8014142 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:49:44 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-ONE.MIT.EDU [18.247.5.1]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3nb5j029156 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:49:40 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 22 Sep 2005 23:49:50 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.117 X-Spam-Level: * (1.117) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 578 X-UID: 2715 X-Keywords: NonJunk NotJunk I found the whole topic of induction to be such an elegant yet sound way to prove stuff. For example the courtyard problem amazed me since without knowing about induction I would still be brute forcing my way to tile the whole courtyard with L-shaped tiles. I'm only a bit confused on reasoning and claims which would make a proof using induction wrong. It could be made much clearer when induction would not be valid. Josh Monzon Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:53:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3rhaJ025922 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:53:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3rgk8016571 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:53:42 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-ONE-O-ONE.MIT.EDU [18.247.5.101]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3rTT1029809 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:53:38 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922235226.02a6c168@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 22 Sep 2005 23:53:31 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 150 X-UID: 2716 X-Keywords: NonJunk NotJunk I would like to see more practices in Induction in class, and more particular cases when induction fails (like the horse example in class) - Steve From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:57:05 2005 Message-ID: <43337D19.3030007@csail.mit.edu> Date: Thu, 22 Sep 2005 23:57:13 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Boris E Revzin CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> In-Reply-To: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 743 X-UID: 2717 X-Keywords: we'll relax the template discipline in a week or so, but for now and through ps2, please string along with us and use the template. A surprising number of students in the class need the template when they begin learning induction proofs. regards, A. Boris E Revzin wrote: > Regarding the induction lecture, it seems excessive to me to have to write out > the entire induction template. By showing a base case and P(n)->P(n+1), it is > pretty clear to anyone reading the proof that it is by induction. I just want > to know if it's enough to show P(0), P(n)->P(n+1), QED. I just think that's all > that is necessary, oftentimes writing out the template will take more time than > writing out the actual proof... > > Boris "Barry" Revzin From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:02:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N42vaJ028277 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:02:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N42tk8021784 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:02:55 -0400 (EDT) Received: from TNG.mit.edu (BURTON-SIX-THIRTY-SEVEN.MIT.EDU [18.247.7.126]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N42pYh001291 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:02:53 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922235948.01f56998@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 23 Sep 2005 00:02:58 -0400 To: 6042-probs@theory.csail.mit.edu From: Tony Ng Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 398 X-UID: 2718 X-Keywords: NonJunk NotJunk The "All Horses are the Same Color", starting on page 9, proof kind of bothered me. The flaw with the proof is that it is not true for n = 1. However, is it legitimate to try to prove some add horse function (which doesn't seem really related to mathematics and numbers)? I can imagine someone clever figuring out some proof that satisfy the n=1 case and prove all horses are the same color. From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:12:46 2005 Message-ID: <433380C7.9000400@csail.mit.edu> Date: Fri, 23 Sep 2005 00:12:55 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Tony Ng CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <5.2.1.1.2.20050922235948.01f56998@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050922235948.01f56998@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 711 X-UID: 2719 X-Keywords: I don't know what you mean when you say "try to prove some add horse function." Propositions get proved; functions don't get proved, they get applied. But you seem to be saying that a clever person could prove something that wasn't true. Do you really mean that? Regards, A. Tony Ng wrote: > The "All Horses are the Same Color", starting on page 9, proof kind of > bothered me. The flaw with the proof is that it is not true for n = 1. > However, is it legitimate to try to prove some add horse function (which > doesn't seem really related to mathematics and numbers)? I can imagine > someone clever figuring out some proof that satisfy the n=1 case and > prove all horses are the same color. > From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:06:23 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N46NaJ029504 for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:06:23 -0400 Received: (qmail 46094 invoked from network); 23 Sep 2005 04:06:22 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 04:06:22 -0000 Message-ID: <43337F48.10907@csail.mit.edu> Date: Fri, 23 Sep 2005 00:06:32 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-staff@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 918 X-UID: 2720 X-Keywords: Induction is always valid :-). What you have to watch out for are proofs that claim to be by induction but do it wrong: they skip some cases or make a mistake in proving the cases. We'll discuss additional mistakes in induction proofs as the term progresses. regards, A. Joshua Jen C. Monzon wrote: > I found the whole topic of induction to be such an elegant yet sound way > to prove stuff. For example the courtyard problem amazed me since > without knowing about induction I would still be brute forcing my way to > tile the whole courtyard with L-shaped tiles. I'm only a bit confused on > reasoning and claims which would make a proof using induction wrong. It > could be made much clearer when induction would not be valid. > > Josh Monzon > > > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:59:30 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N3xTaJ028047 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 23:59:30 -0400 Received: (qmail 33870 invoked from network); 23 Sep 2005 03:59:28 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 03:59:28 -0000 Message-ID: <43337DAA.7030202@csail.mit.edu> Date: Thu, 22 Sep 2005 23:59:38 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Michael Murray CC: 6042-staff@theory.csail.mit.edu Subject: Re: E-mail question References: <9B3CB3F4-7DBB-4BE2-ACA3-E68CD73454BF@mit.edu> In-Reply-To: <9B3CB3F4-7DBB-4BE2-ACA3-E68CD73454BF@mit.edu> Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 878 X-UID: 2721 X-Keywords: we let m0 be the smallest numerator in a fraction for and rational number, q, and then we found a smaller one -- that's the contradiction. Does that help? regards, A. Michael Murray wrote: > "First, we can assume m 0 (otherwise, replace m/n by −m/(−n)), so the > set of natural numbers, > m, such that q = m/n for some integer, n, is not empty. Therefore, by > Well Ordering, there must > be a least natural number, m0, such that q = m0/n0 for some integer, n0. > Now if m0 and n0 had > a common factor, p > 1, then (m0/p)/(n0/p) would be another way to > express q as a quotient of > integers. But since 0 (m0/p) < m0. this contradicts the minimality of m0." > > I did not fully understand this paragraph because i do not understand > how we reach a contradiction. This was the part in the notes that I > found most confusing. > > MIchael Murray > > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:57:06 2005 Return-Path: Received: from smtp106.mail.sc5.yahoo.com (smtp106.mail.sc5.yahoo.com [66.163.169.226]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N3v6aJ027853 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 23:57:06 -0400 Received: (qmail 69752 invoked from network); 23 Sep 2005 03:57:05 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp106.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 03:57:04 -0000 Message-ID: <43337D19.3030007@csail.mit.edu> Date: Thu, 22 Sep 2005 23:57:13 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Boris E Revzin CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> In-Reply-To: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 742 X-UID: 2722 X-Keywords: we'll relax the template discipline in a week or so, but for now and through ps2, please string along with us and use the template. A surprising number of students in the class need the template when they begin learning induction proofs. regards, A. Boris E Revzin wrote: > Regarding the induction lecture, it seems excessive to me to have to write out > the entire induction template. By showing a base case and P(n)->P(n+1), it is > pretty clear to anyone reading the proof that it is by induction. I just want > to know if it's enough to show P(0), P(n)->P(n+1), QED. I just think that's all > that is necessary, oftentimes writing out the template will take more time than > writing out the actual proof... > > Boris "Barry" Revzin From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:42:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4gqaJ002453 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 00:42:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4gpk8013572 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 00:42:51 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N4gntJ007021 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 00:42:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N4gnm1006146; Fri, 23 Sep 2005 00:42:49 -0400 Received: from MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU (MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU [18.239.7.231]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 00:42:49 -0400 Message-ID: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> Date: Fri, 23 Sep 2005 00:42:49 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: week 3 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 170 X-UID: 2723 X-Keywords: NonJunk NotJunk Proofs by induction are completely foriegn to me. Even after class, I still have no clue how to figure out where th flaws in the tutor quesiton #3 is. -Daniel Gutierrez From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:48:34 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4mYaJ002720 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:34 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4mXgh014819 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:33 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4mWTI004839 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:33 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N4mP0v028194 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:30 -0400 (EDT) Message-Id: <200509230448.j8N4mP0v028194@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 00:48:22 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00AB_01C5BFD8.82A1CC50" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcW/+gZWJmeNNMF0R8G6OUMmTW+Gnw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 1.957 X-Spam-Level: * (1.957) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2231 X-UID: 2724 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_00AB_01C5BFD8.82A1CC50 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Page 12's discussion of the Well Ordering Principle is quite fascinating. How do you prove the principle, in a general sense? Straightforward subsets (like even numbers) are pretty obviously true, but what about more complicated sets? Chris Yang ------=_NextPart_000_00AB_01C5BFD8.82A1CC50 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Page 12’s discussion of the Well Ordering = Principle is quite fascinating.  How do you prove the principle, in a general = sense?  Straightforward subsets (like even numbers) are pretty obviously true, = but what about more complicated sets?

 

Chris Yang

------=_NextPart_000_00AB_01C5BFD8.82A1CC50-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:49:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4nHaJ002740 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 00:49:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4nGk8017911 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 00:49:16 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THIRTY.MIT.EDU [18.239.5.30]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N4n3Wn007703 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 00:49:08 -0400 (EDT) Message-Id: <200509230449.j8N4n3Wn007703@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: reading comments week 3 Date: Fri, 23 Sep 2005 00:49:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5BFD8.9A7CD2C0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcW/+h3wb19sjKSST4G8C60eQyp81Q== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2545 X-UID: 2725 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5BFD8.9A7CD2C0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit For me the most confusing part of the reading was the Well Ordering Principle. Particularly, how it was presented as a parallel method to induction, whereas said method merely depends on it. Additionally the tutor problems seemed to reinforce its presentation as a regular proof method rather than a tool used by one. Peter Bilodeau ------=_NextPart_000_0000_01C5BFD8.9A7CD2C0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

 

For me the most confusing part of the reading was the = Well Ordering Principle.  Particularly, how it was presented as a = parallel method to induction, whereas said method merely depends on it.  Additionally = the tutor problems seemed to reinforce its presentation as a regular proof method = rather than a tool used by one.

 

Peter Bilodeau

------=_NextPart_000_0000_01C5BFD8.9A7CD2C0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:56:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4uKaJ003207 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:56:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4uJk8022635 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:56:19 -0400 (EDT) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N4uHdS008479 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:56:17 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) In-Reply-To: <43333F5B.4040601@csail.mit.edu> References: <43333F5B.4040601@csail.mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <9BF45317-CBDD-41CA-B1B0-314DC9819D1E@mit.edu> Content-Transfer-Encoding: 7bit From: Rian Hunter Subject: Re: Assigned Reading Date: Fri, 23 Sep 2005 00:56:15 -0400 To: 6042-probs@theory.csail.mit.edu X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1337 X-UID: 2726 X-Keywords: NonJunk $Forwarded NotJunk Actually I guess the reading tricked me. I was used to the proof methods being explained how to be used step by step, while the Well Ordering Principle is just a simple statement. I thought the statement in the box "Every nonempty subset..." was a statement that could be proved using the Well Ordering Principle; I wasn't expecting that it was the Well Ordering Principle itself. Actually it's unclear to me whether or not the Well Ordering Principle is a proof method itself or just a principle used in other proof methods (proof by contradiction for example), but after re-reading and looking at the context it seems that it isn't. -rian On Sep 22, 2005, at 7:33 PM, Prof. Albert R. Meyer wrote: > " it does not concretely explain what "The Well Ordering Principle" > is" I thought the box labeled "The Well Ordering Principle" was > about as concrete as it gets. Can you explain what kind of further > statement of the Principle you're looking for? > > Rian Hunter wrote: > > >> From page 12 of the reading: .... >> >> This seemed to be the most difficult passage to grasp simply >> because it does not concretely explain what "The Well Ordering >> Principle" is, because of this I would personally like to have >> this discussed or elaborating upon in lecture. Thanks. >> >> Rian Hunter >> > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:03:31 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N53VaJ004283 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:03:31 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N53Vbd017229 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:03:31 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N53S3a001033 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:03:28 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j8N53Sqd027218; Fri, 23 Sep 2005 01:03:28 -0400 Message-Id: <200509230503.j8N53Sqd027218@white-meteo.mit.edu> To: 6042-probs@theory.lcs.mit.edu Subject: reading questions due sep 23 Date: Fri, 23 Sep 2005 01:03:28 -0400 From: r n jacobs X-Spam-Score: 1.729 X-Spam-Level: * (1.729) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 225 X-UID: 2727 X-Keywords: NonJunk NotJunk I found myself confused by the explanation of Strong Induction. Although maybe the fibbonaci proof helped that. (The definition made sense, but it took far to long for me to figure out the answer to the fibbonaci misproof) From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:12:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5CZaJ004634 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:12:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5CYk8001347 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:12:34 -0400 (EDT) Received: from AMMAR (NEW-FOUR-SIXTY-SEVEN.MIT.EDU [18.241.6.212]) (authenticated bits=0) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5CUZL010074 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:12:31 -0400 (EDT) Message-ID: <002b01c5bffd$66bd72d0$d406f112@AMMAR> From: "Adnaan" To: <6042-probs@theory.lcs.mit.edu> Subject: email comments for 24/9 Date: Fri, 23 Sep 2005 01:12:30 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0028_01C5BFDB.DEC6F110" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.595 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1176 X-UID: 2728 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0028_01C5BFDB.DEC6F110 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Hi,=20 I found the reading hard especially the "Well order reading" part. I = didn't understand how it is related to induction Adnaan Jiwaji ------=_NextPart_000_0028_01C5BFDB.DEC6F110 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Hi,
 
I found the reading hard especially the = "Well order=20 reading" part. I didn't understand how it is related to = induction
 
Adnaan = Jiwaji
------=_NextPart_000_0028_01C5BFDB.DEC6F110-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:27:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5R0aJ005427 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:27:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5Qxk8008147 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:26:59 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-THREE-FORTY-SEVEN.MIT.EDU [18.241.6.92]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5Qrrn011667 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:26:55 -0400 (EDT) Message-Id: <6.2.1.2.2.20050923012004.039a2e20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 23 Sep 2005 01:26:52 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 328 X-UID: 2729 X-Keywords: NonJunk NotJunk Passage: 2.4 Courtyard Tiling Page: 6 I found this passage most surprising because it used induction to prove a geometric predicate. I knew that induction could be used to prove mathematical predicates but I did not think it was possible to use induction to prove a geometric predicate. Sincerely, Timothy M. Mwangi From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:28:15 2005 Message-ID: <43339279.90703@csail.mit.edu> Date: Fri, 23 Sep 2005 01:28:25 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez CC: 6042-staff@theory.csail.mit.edu Subject: Re: week 3 comments References: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> In-Reply-To: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 274 X-UID: 2730 X-Keywords: I encourage you to go to TA office hours and go over this. regards,A. Daniel A Gutierrez wrote: > Proofs by induction are completely foriegn to me. Even after class, I still > have no clue how to figure out where th flaws in the tutor quesiton #3 is. > -Daniel Gutierrez From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:26:15 2005 Message-ID: <43339201.3070700@csail.mit.edu> Date: Fri, 23 Sep 2005 01:26:25 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-staff@theory.csail.mit.edu Subject: [Fwd: Re: Assigned Reading] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1588 X-UID: 2731 X-Keywords: -------- Original Message -------- Subject: Re: Assigned Reading Date: Fri, 23 Sep 2005 00:56:15 -0400 From: Rian Hunter To: 6042-probs@theory.csail.mit.edu References: <43333F5B.4040601@csail.mit.edu> Actually I guess the reading tricked me. I was used to the proof methods being explained how to be used step by step, while the Well Ordering Principle is just a simple statement. I thought the statement in the box "Every nonempty subset..." was a statement that could be proved using the Well Ordering Principle; I wasn't expecting that it was the Well Ordering Principle itself. Actually it's unclear to me whether or not the Well Ordering Principle is a proof method itself or just a principle used in other proof methods (proof by contradiction for example), but after re-reading and looking at the context it seems that it isn't. -rian On Sep 22, 2005, at 7:33 PM, Prof. Albert R. Meyer wrote: > " it does not concretely explain what "The Well Ordering Principle" > is" I thought the box labeled "The Well Ordering Principle" was > about as concrete as it gets. Can you explain what kind of further > statement of the Principle you're looking for? > > Rian Hunter wrote: > > >> From page 12 of the reading: .... >> >> This seemed to be the most difficult passage to grasp simply >> because it does not concretely explain what "The Well Ordering >> Principle" is, because of this I would personally like to have >> this discussed or elaborating upon in lecture. Thanks. >> >> Rian Hunter >> > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:06:22 2005 Message-ID: <43337F48.10907@csail.mit.edu> Date: Fri, 23 Sep 2005 00:06:32 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-staff@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 919 X-UID: 2732 X-Keywords: Induction is always valid :-). What you have to watch out for are proofs that claim to be by induction but do it wrong: they skip some cases or make a mistake in proving the cases. We'll discuss additional mistakes in induction proofs as the term progresses. regards, A. Joshua Jen C. Monzon wrote: > I found the whole topic of induction to be such an elegant yet sound way > to prove stuff. For example the courtyard problem amazed me since > without knowing about induction I would still be brute forcing my way to > tile the whole courtyard with L-shaped tiles. I'm only a bit confused on > reasoning and claims which would make a proof using induction wrong. It > could be made much clearer when induction would not be valid. > > Josh Monzon > > > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:28:18 2005 Return-Path: Received: from smtp101.mail.sc5.yahoo.com (smtp101.mail.sc5.yahoo.com [216.136.174.139]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N5SGaJ005485 for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:28:16 -0400 Received: (qmail 59927 invoked from network); 23 Sep 2005 05:28:15 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp101.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 05:28:15 -0000 Message-ID: <43339279.90703@csail.mit.edu> Date: Fri, 23 Sep 2005 01:28:25 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez CC: 6042-staff@theory.csail.mit.edu Subject: Re: week 3 comments References: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> In-Reply-To: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 273 X-UID: 2733 X-Keywords: I encourage you to go to TA office hours and go over this. regards,A. Daniel A Gutierrez wrote: > Proofs by induction are completely foriegn to me. Even after class, I still > have no clue how to figure out where th flaws in the tutor quesiton #3 is. > -Daniel Gutierrez From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:36:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5a8aJ005889 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:36:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5a6k8012554 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:36:07 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5a4Pt013770 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:36:04 -0400 (EDT) Message-ID: <43339446.1070908@mit.edu> Date: Fri, 23 Sep 2005 01:36:06 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: (no subject) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.098 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 264 X-UID: 2734 X-Keywords: NonJunk NotJunk From this week's notes, I found the proof that we went over last class re: the tiling (Section 2.4, page 8) to be the most surprising, since it was a very elegant and easy to prove answer to what I thought initially was going to be a hard to prove conjecture. From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:36:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5aqaJ005906 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:36:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5apk8012968 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:36:51 -0400 (EDT) Received: from ccbibmx30 (NEW-EIGHTY-TWO.MIT.EDU [18.241.5.82]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5ajP8013932 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:36:46 -0400 (EDT) Message-Id: <200509230536.j8N5ajP8013932@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 01:36:34 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5BFDF.3C0C2BD0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 thread-index: AcXAAMIwN13vJ5m8Spus8/pOCtkcnw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.419 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 3341 X-UID: 2735 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5BFDF.3C0C2BD0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Well Ordering is well.not understandable. If we study it tomorrow (later today) in class, perhaps I'll get it. Now the passage just looks like a bunch of words that I do not comprehend. Could you provide an example, or two? --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0000_01C5BFDF.3C0C2BD0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Well Ordering is well…not understandable. If we = study it tomorrow (later today) in class, perhaps I’ll get it. Now the = passage just looks like a bunch of words that I do not comprehend. Could you = provide an example, or two?

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

------=_NextPart_000_0000_01C5BFDF.3C0C2BD0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:46:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5kraJ007386 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:46:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5kqk8017589 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:46:52 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5knKh016117 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:46:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8N5knrx018267; Fri, 23 Sep 2005 01:46:49 -0400 Received: from NEW-THREE-SIXTY-SEVEN.MIT.EDU (NEW-THREE-SIXTY-SEVEN.MIT.EDU [18.241.6.112]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 01:46:49 -0400 Message-ID: <20050923014649.s2van2mf2o000ckw@webmail.mit.edu> Date: Fri, 23 Sep 2005 01:46:49 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: comment MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_3wt9ryflgz40" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.329 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 767 X-UID: 2736 X-Keywords: NonJunk NotJunk This message is in MIME format. --=_3wt9ryflgz40 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit Sorry there isn't as much this time, its just rather more familiar material. --=_3wt9ryflgz40 Content-Type: text/plain; charset=ISO-8859-1; name="week3comment.txt" Content-Disposition: attachment; filename="week3comment.txt" Content-Transfer-Encoding: 7bit Induction was covered in about as much detail as a highschool can cover it. I'm familiar with it, although its been a while. The in class problems were much easier for me than those of previous lectures. Strong induction is new, but its not hard to grasp. So, I think I understand pretty well whats going on with induction. --=_3wt9ryflgz40-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:53:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5rTaJ007760 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:53:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5rBk8020639 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:53:11 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5r8dY017163 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:53:08 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N5r86m012850; Fri, 23 Sep 2005 01:53:08 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 01:53:08 -0400 Message-ID: <20050923015308.8xqusatei7vowwow@webmail.mit.edu> Date: Fri, 23 Sep 2005 01:53:08 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP2 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 129 X-UID: 2737 X-Keywords: NonJunk NotJunk I would like to see an application of the well ordering principle in lecture (possibly in a proof or solution to a problem) Lana From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:00:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N60HaJ008158 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:00:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N60DaW022733 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:00:16 -0400 (EDT) Received: from [18.238.5.210] (EASTCAMPUS-SEVEN-TWENTY-ONE.MIT.EDU [18.238.5.210]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5tn2I017355 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:55:49 -0400 (EDT) Message-ID: <433398EC.90508@mit.edu> Date: Fri, 23 Sep 2005 01:55:56 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.537 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 234 X-UID: 2738 X-Keywords: NonJunk NotJunk I found the discussion of the Well-Ordering principle (last page, p. 12) most surprising. Although the statement itself makes perfect sense, it's really interesting how many other things depend on this seemingly-obvious principle. From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:27:02 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6R2aJ010644 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:27:02 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6R1Cg019151 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:27:01 -0400 (EDT) Received: from [127.0.0.1] (EASTCAMPUS-EIGHT-NINETY-EIGHT.MIT.EDU [18.238.6.131]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N6Qt3c003956 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:26:58 -0400 (EDT) Message-ID: <4333A02A.8090800@mit.edu> Date: Fri, 23 Sep 2005 02:26:50 -0400 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Lecture notes 3 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 176 X-UID: 2739 X-Keywords: NonJunk NotJunk On page 12 of the notes, you discuss the well ordering principle (section 4). Please discuss this further in class, as well as applications of this principle. ~Isaac Charny From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:31:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6VQaJ010928 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:26 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6VPad010737 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:25 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6VOdU007124 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:25 -0400 (EDT) Received: from MACGREGOR-TWO-EIGHTEEN.MIT.EDU (MACGREGOR-TWO-EIGHTEEN.MIT.EDU [18.239.5.218]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N6VL0v001607 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:21 -0400 (EDT) Subject: Reading Comments #3 From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 23 Sep 2005 02:31:16 -0400 Message-Id: <1127457077.6912.5.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 402 X-UID: 2740 X-Keywords: NonJunk NotJunk One thing I found that was particularly surprising about the reading was the discussion of the Well Ordering Principle on page 12. While (aside from the described caveats) it does seem obvious, I found it particularly interesting that the Well Ordering Principle could be used to replace strong induction. Though it's only out of curiosity, it might be good to see an example or two of this. -- Matt From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:32:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6WLaJ010963 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 02:32:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6WKaQ006639 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 02:32:20 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N6WCDb019621 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 02:32:13 -0400 (EDT) Message-Id: <200509230632.j8N6WCDb019621@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 02:32:09 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5BFE6.FFAC9190" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXACEy/bhMaT0tDTvuaHAYgTNLwvg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.833 X-Spam-Level: * (1.833) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2018 X-UID: 2741 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5BFE6.FFAC9190 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Could you please go over 2.5 A Faulty Induction Proof (pages 8-9 of ln3). I'm not sure I understand it. ------=_NextPart_000_0000_01C5BFE6.FFAC9190 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Could you please go over 2.5 A Faulty = Induction Proof (pages 8-9 of ln3). I’m not sure I understand = it.

------=_NextPart_000_0000_01C5BFE6.FFAC9190-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:38:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6c3aJ011162 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 02:38:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6c1aQ009102 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 02:38:02 -0400 (EDT) Received: from grumpy-fuzzball.mit.edu (GRUMPY-FUZZBALL.MIT.EDU [18.7.16.79]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N6bxXl019980 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 02:37:59 -0400 (EDT) Received: (from medrano@localhost) by grumpy-fuzzball.mit.edu (8.12.9) id j8N6bxAY023978; Fri, 23 Sep 2005 02:37:59 -0400 (EDT) Date: Fri, 23 Sep 2005 02:37:59 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: Induction Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 197 X-UID: 2742 X-Keywords: NonJunk NotJunk Induction is more formulaic than deductive proofs and thus much easier to grasp. There is not much else I can say about it. The reading was straightforward and easy to understand. Jesus Medrano From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:57:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6vpaJ013208 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:57:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6voaQ017326 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:57:50 -0400 (EDT) Received: from MACGREGOR-FOUR-EIGHTY.MIT.EDU (MACGREGOR-FOUR-EIGHTY.MIT.EDU [18.239.6.225]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N6vfxu021004 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:57:48 -0400 (EDT) Subject: tp3 reading From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 23 Sep 2005 02:57:36 -0400 Message-Id: <1127458656.18591.1.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.227 X-Spam-Level: *** (3.227) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 147 X-UID: 2743 X-Keywords: NonJunk NotJunk While the well-ordering principle (p. 12) is fairly straight forward, I am wondering if we can see some more examples of what it can be used for. From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:06:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N76VaJ013853 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:06:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N76UaQ021064 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:06:30 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-ONE-O-THREE.MIT.EDU [18.242.5.103]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N76QnE021478 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:06:28 -0400 (EDT) Message-Id: <6.2.3.4.2.20050923030000.01c13d90@po10@mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 23 Sep 2005 03:06:24 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: Comments for Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.544 X-Spam-Level: * (1.544) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 383 X-UID: 2744 X-Keywords: NonJunk NotJunk Hi Prof Meyer and Rubinfeld, I found sections 3.2 and 3.3 on pages 10-12 to be confusing. I didn't grasp the usage of strong induction in these proofs, though I had some vague idea of how they came into play (the P(n+2) for the primes and argument: Case (n+1 > 35): etc ... were very confusing). If you could go into detail about them, then that would be great. Thanks, Kevin From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:29:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N7TQaJ015468 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 03:29:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N7TPaQ000593 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 03:29:25 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N7TMBD022472 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 03:29:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N7TM4m017698; Fri, 23 Sep 2005 03:29:22 -0400 Received: from SIMMONS-SEVEN-TWENTY-NINE.MIT.EDU (SIMMONS-SEVEN-TWENTY-NINE.MIT.EDU [18.96.7.218]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 03:29:22 -0400 Message-ID: <20050923032922.urcb7e9e0zk0wcoc@webmail.mit.edu> Date: Fri, 23 Sep 2005 03:29:22 -0400 From: shauni@MIT.EDU To: 6042-probs@theory.csail.MIT.EDU Subject: week 3 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 291 X-UID: 2745 X-Keywords: NonJunk NotJunk I would like to have the principle of strong induction, specifically the Product of Primes example (pp. 10-11) explained more fully. I am a bit confused as to why you can assume more with the strong induction principle, and also when to use strong induction as opposed to regular induction. From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:51:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N7paaJ019295 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:51:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N7pZaQ009148 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:51:35 -0400 (EDT) Received: from [18.240.5.41] (MCCORMICK-FORTY-ONE.MIT.EDU [18.240.5.41]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N7pXat023357 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:51:33 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Transfer-Encoding: 7bit Message-Id: <66636106d07b6a5641402dcc17e625a2@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: Comment on Induction Reading Date: Fri, 23 Sep 2005 03:51:31 -0400 X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 236 X-UID: 2746 X-Keywords: NonJunk NotJunk I thought the sections on Strong Induction were confusing and the example given didn't make sense to me. I would appreciate more time spent on this in class, especially as it was a large part of the tutor problems. Rachel Shearer From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:58:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N7wAaJ019523 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:58:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N7w9aQ011735 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:58:09 -0400 (EDT) Received: from [18.242.5.26] (NEXT-TWENTY-SIX.MIT.EDU [18.242.5.26]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N7vwva023615 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:58:06 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: reading comment Date: Fri, 23 Sep 2005 03:57:56 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.074 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 105 X-UID: 2747 X-Keywords: NonJunk NotJunk I'm still a little confused on the Well Ordering Principle on page 12 of the reading and how it works. From meyer@imap.theory.csail.mit.edu Fri Sep 23 04:11:33 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N8BWaJ020744 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 04:11:33 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N8BWCg021007 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 04:11:32 -0400 (EDT) Received: from MACGREGOR-THREE-SIXTY-TWO.MIT.EDU (MACGREGOR-THREE-SIXTY-TWO.MIT.EDU [18.239.6.107]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N8BT3a006750 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 04:11:29 -0400 (EDT) Subject: Comments for Course Notes, Week 3 (Induction) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 23 Sep 2005 04:11:28 -0400 Message-Id: <1127463088.5662.12.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 0.978 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 265 X-UID: 2748 X-Keywords: NonJunk NotJunk I found section 4, The Well Ordering Principle, (located on page 12) very surprising. I was surprised that Strong Induction can be transformed into the Well Ordering Principle (and vice versa). I would be interested in seeing how this is done. Moira Racich From meyer@imap.theory.csail.mit.edu Fri Sep 23 04:21:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N8LWaJ021609 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 04:21:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N8LUaQ021053 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 04:21:30 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N8LNgA024466 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 04:21:23 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8N8LNUR028297; Fri, 23 Sep 2005 04:21:23 -0400 Received: from EASTCAMPUS-NINE-THIRTY-THREE.MIT.EDU (EASTCAMPUS-NINE-THIRTY-THREE.MIT.EDU [18.238.6.166]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 04:21:23 -0400 Message-ID: <20050923042123.0f7ianeqbykg4cog@webmail.mit.edu> Date: Fri, 23 Sep 2005 04:21:23 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: Email comments, Week 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 437 X-UID: 2749 X-Keywords: NonJunk NotJunk On page 12, the notes say: "...it turns out that there's a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa. (We won't take the time to describe the transformations...)" This surprised me at first, and it is definitely intruiging to me. When I have the time, if it isn't covered in class, I think I'll look it up or try it myself. From meyer@imap.theory.csail.mit.edu Fri Sep 23 05:17:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N9HMaJ026642 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 05:17:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N9HKR8012198 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 05:17:21 -0400 (EDT) Received: from el-ternero.mit.edu (SIMMONS-THREE-TWELVE.MIT.EDU [18.96.6.57]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N9HGCS026528 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 05:17:18 -0400 (EDT) Message-Id: <6.2.3.4.2.20050923051508.02eeaea0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 23 Sep 2005 05:17:14 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: Reading #3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 331 X-UID: 2750 X-Keywords: NonJunk NotJunk The difference between strong induction and simple induction is not too clear. If possible, it would be great to have a brief explanation of the proof of Lemma 3.1 Every integer greater than 1 is a product of primes and why if this is implied why can't we prove that there are infinite number of primes? Thanks, Manuel Rivas From meyer@imap.theory.csail.mit.edu Fri Sep 23 07:51:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NBpgaJ009624 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 07:51:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NBpUR8015621 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 07:51:30 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NBpRMm006476 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 07:51:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8NBpRZ7021318; Fri, 23 Sep 2005 07:51:27 -0400 Received: from PSK-THIRTEEN.MIT.EDU (PSK-THIRTEEN.MIT.EDU [18.217.1.13]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 07:51:27 -0400 Message-ID: <20050923075127.hm55454ciipwg0cg@webmail.mit.edu> Date: Fri, 23 Sep 2005 07:51:27 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: email comments 9-23 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.358 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 388 X-UID: 2751 X-Keywords: NonJunk NotJunk I felt this week's reading was fairly straightforward, so I don't have much to really complain about. It might be nice to see the same proposition get proven by both strong induction and regular induction, since on p11 (product of primes) it says that if you can do one you can do the other ... just to compare, I guess, see what situation is best to use which one on, etc. -Ryan Young From meyer@imap.theory.csail.mit.edu Fri Sep 23 08:03:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NC3baJ010186 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 08:03:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NC0Liv019135; Fri, 23 Sep 2005 08:03:35 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NBvWdV007257; Fri, 23 Sep 2005 07:57:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8NBvWbR021862; Fri, 23 Sep 2005 07:57:32 -0400 Received: from WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU (WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU [18.139.6.237]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 07:57:32 -0400 Message-ID: <20050923075732.0ri6h7bw77wookws@webmail.mit.edu> Date: Fri, 23 Sep 2005 07:57:32 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: Reading assignment -- Week 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 562 X-UID: 2752 X-Keywords: NonJunk NotJunk Hi, The proof of Theorem 2.2 on pages 5-6 (in the "Fibonacci Identity" section) surprised me. Before the proof you do some work until you reach a useful assertion (equation 5), and then you begin the proof in the opposite direction, using your new assertion as a link between P(n) and P(n+1). For the beginner like me, it was good to see this backwards "scratch work," because a lot of times proofs look so clean and clever but are not logical flows that would have occurred to me in the direction in which they're presented in texts, papers, etc. Nancy Keuss From meyer@imap.theory.csail.mit.edu Fri Sep 23 08:39:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NCdVaJ019321 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 08:39:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NCdTgZ012318 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 08:39:29 -0400 (EDT) Received: from [18.237.0.82] (TDCIP82.MIT.EDU [18.237.0.82]) (authenticated bits=0) (User authenticated as letrec@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NCdRje014777 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 08:39:27 -0400 (EDT) Message-ID: <4333F77E.4070805@mit.edu> Date: Fri, 23 Sep 2005 08:39:26 -0400 From: Alton Torregano User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: pp 11 of 12: Strong Induction X-Enigmail-Version: 0.90.0.0 X-Enigmail-Supports: pgp-inline, pgp-mime Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 0.203 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 453 X-UID: 2753 X-Keywords: NonJunk NotJunk The section on Inductia proved a little difficult as I'm generally unexposed to proof by induction, by cases. Though the proof makes sense, I feel that maybe strong induction hides alot of the machinery and therefore makes it harder to process. Also, I was confused as to whether or not the cases need be made explicit (formally) before use in the inductive step. If possible, I would like for this to be explained more fully in the next lecture. From meyer@imap.theory.csail.mit.edu Fri Sep 23 08:44:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NCiSaJ019989 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 08:44:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NCiOgZ015356; Fri, 23 Sep 2005 08:44:24 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NCiINK015818; Fri, 23 Sep 2005 08:44:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8NCiHX3032458; Fri, 23 Sep 2005 08:44:17 -0400 Received: from PIKA-ONE-O-NINE.MIT.EDU (PIKA-ONE-O-NINE.MIT.EDU [18.214.1.109]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 08:44:17 -0400 Message-ID: <20050923084417.6p3anl8wc5ws8k08@webmail.mit.edu> Date: Fri, 23 Sep 2005 08:44:17 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: 18.062J Reading Assignment 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.226 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 275 X-UID: 2754 X-Keywords: NotJunk 18.062J Reading Assignment 3 On pg. 10 of the reading, the proof that every integer greater than 1 is a product of primes confused me. This reminded me of the false proof that every horse has the same color. I don't see how P(0) is true implies P(1) is true. ~Thu Duong From meyer@imap.theory.csail.mit.edu Fri Sep 23 09:44:34 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NDiYaJ027805 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:44:34 -0400 Received: by xproxy.gmail.com with SMTP id t12so118039wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 06:44:28 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=A+dKziQ0h8wHcu8lFOrgyT4iqypclIDvt35vMd1vjbSB4zE6xlbMFB4SRSEW5AMGE9knBTscOli1R0SfauoXKLMEGIqRaKGhOzPxOMgP+ZM5plypATXHoxDUUBW0mxMDmTWzTQv21i8mPvpm/LI12OxEST//fVALw0W9Wronu60= Received: by 10.70.8.10 with SMTP id 10mr878399wxh; Fri, 23 Sep 2005 06:44:27 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Fri, 23 Sep 2005 06:44:27 -0700 (PDT) Message-ID: <4f2613e50509230644463a87ed@mail.gmail.com> Date: Fri, 23 Sep 2005 09:44:27 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8NDiYaJ027805 Status: RO Content-Length: 823 X-UID: 2755 X-Keywords: NonJunk NotJunk I don't really undertand the horses of the same color theorem. I understand that the proof fails to show that all horses are of the same color since it operates on two different sets, and fails to show that the first set implies the other. Howver, my problem lies with the assumption that the first n horses are the same color and then that the last n horses are of the same color on pages 7 and 8 of the course notes. It seems analagous to assuming that the charachteristic is true for an n size set and then assuming it it true for the n+1 element and through these assumption, proving that the assumed charachteristic was true for all elements to begin with when, in fact, it was just assumed to be true. In summation, it seems as though an assumption is being proven with another assumption, which bothers me. -zozer From meyer@imap.theory.csail.mit.edu Fri Sep 23 09:45:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NDjPaJ028581 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 09:45:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NDjOgZ003264 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 09:45:24 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SEVENTY-SEVEN.MIT.EDU [18.96.5.77]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NDjKJI006239 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 09:45:20 -0400 (EDT) Message-Id: <200509231345.j8NDjKJI006239@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 09:45:11 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.502 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 109 X-UID: 2756 X-Keywords: NonJunk NotJunk In section 4, I understand the Well Ordering Principle, but what is its significance to induction? - David From meyer@imap.theory.csail.mit.edu Fri Sep 23 09:49:45 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NDnjaJ029643 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:49:45 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NDnhaD020199 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:49:43 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NDmrtp016795 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:48:53 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NDmp0v016185 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:48:51 -0400 (EDT) Message-ID: <43342422.4040309@mit.edu> Date: Fri, 23 Sep 2005 09:49:54 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Induction Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.775 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 284 X-UID: 2757 X-Keywords: NonJunk NotJunk Strong Induction pg 9 Having learned a bit about induction before I initially thought Strong Induction would be more difficult to do because you're working towards a more specific goal. However, seeing its applications I was surprised at how much simpler and more powerful it is. From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:32:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEWJaJ006241 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:32:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEWIgZ016696 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:32:18 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEWEi0025629 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:32:15 -0400 (EDT) Message-Id: <200509231432.j8NEWEi0025629@outgoing.mit.edu> From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: slightly unclear Date: Fri, 23 Sep 2005 10:32:10 -0400 Organization: Massachusetts Institute of Technology MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0028_01C5C02A.0E8BF5A0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXAS5UjcrZJ0cLeSk+dn4YSWAQa8g== X-Spam-Score: 0.2 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 4054 X-UID: 2758 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0028_01C5C02A.0E8BF5A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi, Two topics are still slightly unclear for me: Well-ordering and how is it possible to transfer way of proof to another way. Is it done by math or is there a template to be used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using induction? Second topic is the false proof, I understand that the problem with the horses proof was that we did not take into account different horses at P(1) both in the tutorial problems and in the proof in the reading I don't fully understand where exactly the proof becomes false. Thank you in advance, Anton. ------=_NextPart_000_0028_01C5C02A.0E8BF5A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi,

 

Two topics are still slightly unclear for = me:

Well-ordering and how is it possible to transfer way = of proof to another way. Is it done by math or is there a template to be = used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using = induction?

 

Second topic is the false proof, I understand that = the problem with the horses proof was that we did not take into account = different horses at P(1) both in the tutorial problems and in the proof in the reading I = don’t fully understand where exactly the proof becomes = false.

 

Thank you in advance,

 

Anton.

 

------=_NextPart_000_0028_01C5C02A.0E8BF5A0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:45:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEjlaJ007782 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 10:45:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEjjgZ029968 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 10:45:45 -0400 (EDT) Received: from [18.113.6.244] (ROGERS-FOUR-NINETY-NINE.MIT.EDU [18.113.6.244]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEjgoo002091 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 10:45:42 -0400 (EDT) Message-ID: <43341516.9040905@mit.edu> Date: Fri, 23 Sep 2005 10:45:42 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments Content-Type: multipart/alternative; boundary="------------030804090609060605060201" X-Spam-Score: -2.059 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1672 X-UID: 2759 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. --------------030804090609060605060201 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Section 3.1 I'd like to see this discussed in lecture, as I don't think we've seen it yet and it'd be good to illuminate the difference between itself and regular induction. Incidentally, I also think that these readings and tutor problems should be announced in lecture as well :) -Pete -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------030804090609060605060201 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Section 3.1
   I'd like to see this discussed in lecture, as I don't think we've seen it yet and it'd be good to illuminate the difference between itself and regular induction.  Incidentally, I also think that these readings and tutor problems should be announced in lecture as well :)

-Pete
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------030804090609060605060201-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:46:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEksaJ007860 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:46:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEkqgZ001129 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:46:53 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEkkhZ002563 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:46:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8NEkki9031645; Fri, 23 Sep 2005 10:46:46 -0400 Received: from BURTON-FOUR-THIRTY-EIGHT.MIT.EDU (BURTON-FOUR-THIRTY-EIGHT.MIT.EDU [18.247.6.183]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 10:46:46 -0400 Message-ID: <20050923104646.p180njih9bn28so4@webmail.mit.edu> Date: Fri, 23 Sep 2005 10:46:46 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading 3: Induction MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.236 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 353 X-UID: 2760 X-Keywords: NonJunk NotJunk I was wondering if it would be possible to go over Strong Induction more thoroughly in class. While I understand that in principle, it is a more general method of induction, I do not exactly understand how it can be proved and, well, exactly how one goes about using strong induction. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:48:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEmsaJ008459 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:48:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEmqgZ002947 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:48:52 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEmnip003458 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:48:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8NEmnRi031973; Fri, 23 Sep 2005 10:48:49 -0400 Received: from BURTON-FOUR-THIRTY-EIGHT.MIT.EDU (BURTON-FOUR-THIRTY-EIGHT.MIT.EDU [18.247.6.183]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 10:48:49 -0400 Message-ID: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> Date: Fri, 23 Sep 2005 10:48:49 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading 3: Induction MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.543 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 552 X-UID: 2761 X-Keywords: NonJunk NotJunk I forgot to include in my last e-mail: I refer to the passage that defines Strong Induction at the top of page 10. Specifically, the part that confuses me is: "The only change from the ordinary induction principle is that strong induction allows you to assume more stuff in the inductive step of your proof! ...In a strong induction argument, you may assume that P(0), P(1), ..., and P(n) are all true when you go to prove P(n + 1)." Why are you allowed to assume all these cases? - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:51:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEp6aJ009052 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:51:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEp4gZ005057 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:51:04 -0400 (EDT) Received: from Charizard (NEW-TWO-SIXTY-SIX.MIT.EDU [18.241.6.11]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEp1vk004328 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:51:01 -0400 (EDT) Message-Id: <200509231451.j8NEp1vk004328@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: 6.042 reading email Date: Fri, 23 Sep 2005 10:50:57 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0010_01C5C02C.AE570640" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXATjTD8vh5ACiSQ5C6Q+arJVUQcA== X-Spam-Score: 2.247 X-Spam-Level: ** (2.247) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 5265 X-UID: 2762 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0010_01C5C02C.AE570640 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Dear .042, It would be great if you give some more examples on the Well Ordered Problems theorem- in particular, how to convert it from Strong and back. On paper it seems reasonable, but it'll sink in better with more examples. Thanks! _Yaser ------=_NextPart_000_0010_01C5C02C.AE570640 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Dear = .042,

 

It would be great if you give = some more examples on the Well Ordered Problems theorem- in particular, how to = convert it from Strong and back. On paper it seems reasonable, but it’ll sink = in better with more examples.

 

Thanks!

 

_Yaser

------=_NextPart_000_0010_01C5C02C.AE570640-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:01:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF1CaJ012412 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:01:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF0cgZ013969 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:00:38 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF0XE5008616 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:00:33 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8NF0XT0001624; Fri, 23 Sep 2005 11:00:33 -0400 Received: from KS-ONE-TWENTY-EIGHT.MIT.EDU (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 11:00:33 -0400 Message-ID: <20050923110033.vv23cwplepw00wgc@webmail.mit.edu> Date: Fri, 23 Sep 2005 11:00:33 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: email comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 185 X-UID: 2763 X-Keywords: NonJunk NotJunk The part I found most confusing was the difference between simple and strong induction. I understand both, but I am confused about how to distinguish between the two like on the pset. From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:01:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF1haJ012432 for <6.042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 11:01:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF10gZ014315 for <6.042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 11:01:01 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF0v8w008815 for <6.042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 11:00:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8NF0v0s015240; Fri, 23 Sep 2005 11:00:57 -0400 Received: from RLE-13-099.MIT.EDU (RLE-13-099.MIT.EDU [18.62.13.99]) (User authenticated as kushan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 11:00:57 -0400 Message-ID: <20050923110057.yqvtqs8i88o0ks4o@webmail.mit.edu> Date: Fri, 23 Sep 2005 11:00:57 -0400 From: Kushan K Surana To: 6.042-probs@theory.csail.MIT.EDU Subject: Reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.559 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 87 X-UID: 2764 X-Keywords: NonJunk NotJunk I am unclear about the well ordering principle. How is it relevant for proofs? Kushan From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:04:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF4BaJ012567 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 11:04:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF2ugZ015795 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 11:02:56 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF2qCu009651 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 11:02:54 -0400 (EDT) Message-Id: <6.2.1.2.2.20050923110050.01e50f50@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 23 Sep 2005 11:02:56 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 2.267 X-Spam-Level: ** (2.267) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 271 X-UID: 2765 X-Keywords: NonJunk NotJunk In section 3.2 of the reading, I don't understand the proof entirely. The step that troubles me is the one where we just jump from "k-2 is a natural number less than n+1" to "(k-2) +2 is a product of primes by induction hypothesis." I could use more explanation. From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:04:14 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF4EaJ012572 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:04:14 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EIp5y-0002ue-BU for 6042-probs@theory.lcs.MIT.EDU; Fri, 23 Sep 2005 11:04:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF3sgZ016553 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:03:54 -0400 (EDT) Received: from m56-129-2.mit.edu (M56-129-2.MIT.EDU [18.56.0.31]) (authenticated bits=56) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF3kvg010038 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:03:47 -0400 (EDT) Received: (from juang@localhost) by m56-129-2.mit.edu (8.12.9) id j8NF3k54024750; Fri, 23 Sep 2005 11:03:46 -0400 (EDT) Subject: Reading comments From: Jason K Juang To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Date: Fri, 23 Sep 2005 11:03:45 -0400 Message-Id: <1127487825.24672.1.camel@m56-129-2.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit Status: RO Content-Length: 264 X-UID: 2766 X-Keywords: NonJunk NotJunk On page 10: "The only change from the ordinary induction principle is that strong induction allows you to assume more stuff in the inductive step of your proof!" So then why do we bother making a distinction between simple induction and strong induction? Jason. From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:10:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NFAOaJ013397 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:10:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NFALOS021319 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:10:23 -0400 (EDT) Received: from m56-129-19.mit.edu (M56-129-19.MIT.EDU [18.56.0.48]) (authenticated bits=56) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF8aRR012117 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:08:37 -0400 (EDT) Received: (from jstritar@localhost) by m56-129-19.mit.edu (8.12.9) id j8NF8aMv028897; Fri, 23 Sep 2005 11:08:36 -0400 Subject: Comments From: Jonathan E Stritar To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Fri, 23 Sep 2005 11:08:36 -0400 Message-Id: <1127488116.28781.1.camel@m56-129-19.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 278 X-UID: 2767 X-Keywords: NonJunk NotJunk Referring to 2.4 Courtyard Tiling (page 6). I think this is really cool. Its interesting how with induction you can prove that a certain solution exists even though it does not show you exactly how to get there (in this case how you would lay the L shaped tiles). Jon Stritar From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:11:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NFBSaJ013871 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:11:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NFALPg021319 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:11:27 -0400 (EDT) Received: from Drache (NEW-ONE-EIGHTY-NINE.MIT.EDU [18.241.5.189]) (authenticated bits=0) (User authenticated as natalia3@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF7VCx011688 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:07:32 -0400 (EDT) Message-Id: <200509231507.j8NF7VCx011688@outgoing.mit.edu> From: "Natalia Chernenko" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 11:07:25 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0003_01C5C02E.FBCA87B0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXAUFwbpPeJFVRLSbyy6PxLm4BY/A== X-Spam-Score: 1.155 X-Spam-Level: * (1.155) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2593 X-UID: 2768 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0003_01C5C02E.FBCA87B0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit The most interesting part of the reading was the tiling problem (section 2.4). I really like visual techniques and it was fun to see a mathematical proof of a graphic layout. I am still a bit confused as to the difference between strong induction and simple induction, though. Natalia Chernenko ------=_NextPart_000_0003_01C5C02E.FBCA87B0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

The most interesting part of the reading was the = tiling problem (section 2.4). I really like visual techniques and it was fun to = see a mathematical proof of a graphic layout.

 

I am still a bit confused as to the difference = between strong induction and simple induction, though. =

 

Natalia Chernenko

------=_NextPart_000_0003_01C5C02E.FBCA87B0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 16:53:43 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.199]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NKrhaJ006138 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:53:43 -0400 Received: by xproxy.gmail.com with SMTP id t5so179888wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:53:38 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=L0mhzDsLYHNz935TsABX8nVV6UhuWfqN3iMlcjYRiXeib3AJyj/LGp5kPxspv8FoSwbniuKTWQuQpLN6D0qljXAkiBU5rm45Dv0Zl2K17zE1a0fX7h2RvhbJ7QI2hT8xzRQfyiEb9k6XeC3hzqXow7ztU8mFevT01nrLcy0JTsQ= Received: by 10.70.31.20 with SMTP id e20mr1211195wxe; Fri, 23 Sep 2005 13:53:38 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Fri, 23 Sep 2005 13:53:38 -0700 (PDT) Message-ID: <3c33fe380509231353f100e4c@mail.gmail.com> Date: Fri, 23 Sep 2005 16:53:38 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: Reading Comments - 9/23 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_11355_12228901.1127508818431" Status: RO Content-Length: 2466 X-UID: 2769 X-Keywords: NonJunk NotJunk ------=_Part_11355_12228901.1127508818431 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, My comments on this week's reading below. I'm sorry for the tardiness, I didn't check the course website to see when the reading comments were due. = I learned my lesson and will check the website more often in the future. Thanks for your understanding. The part I found most difficult in the reading was in the last section, using the Well Ordering Principle to transform a proof into Proof by Strong Induction. Taken from the reading: "As for useless, it turns out that there's a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa. (We won't take the time to describe the transformations, though they are not hard.)" (page 12). I don't quite understand how this transformation can take place, as when doing Proof by Induction, the base case we choose is usually the least element (like 0). I think I'd be cool to see some examples of applying the Principle in Proofs by Strong Induction. Thanks, Shreyes Seshasai Group 7 ------=_Part_11355_12228901.1127508818431 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

My comments on this week's reading below.  I'm sorry for the tardiness, I didn't check the course website to see when the reading comments were due.  I learned my lesson and will check the website more often in the future.  Thanks for your understanding.

The part I found most difficult in the reading was in the last section, using the Well Ordering Principle to transform a proof into Proof by Strong Induction.  Taken from the reading: "As for useless, it turns out that there's a routine way to transform any proof using the Well Ordering
Principle into a proof using Strong Induction, and vice-versa. (We won't ta= ke the time to
describe the transformations, though they are not hard.)" (page 12). I don't quite understand how this transformation can take place, as when doing Proof by Induction, the base case we choose is usually the least element (like 0).  I think I'd be cool to see some examples of applying the Principle in Proofs by Strong Induction.

Thanks,
Shreyes Seshasai
Group 7


------=_Part_11355_12228901.1127508818431-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 16:55:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NKt7aJ006267 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:55:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NKsqew020139 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:54:52 -0400 (EDT) Received: from aston (BOOKX.MIT.EDU [18.241.0.191]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NKsoPC023095 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:54:50 -0400 (EDT) Message-Id: <200509232054.j8NKsoPC023095@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: Comments on Reading Date: Fri, 23 Sep 2005 16:55:44 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0043_01C5C05F.A3DAF8B0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXAgSpy3FbpZA9xSPi0TpwbBqKYaQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2227 X-UID: 2770 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0043_01C5C05F.A3DAF8B0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I found the comment on page 11 about the ability to prove anything with ordinary induction that you can with strong induction extremely interesting. I'm not so clear, though, on how that's possible. - Aston ------=_NextPart_000_0043_01C5C05F.A3DAF8B0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I found the comment on page 11 about the ability to = prove anything with ordinary induction that you can with strong induction = extremely interesting. I’m not so clear, though, on how that’s = possible.

 

         =    - Aston

------=_NextPart_000_0043_01C5C05F.A3DAF8B0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:25:07 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHP7aJ002589 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:07 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6id003556 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6HR026006; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHP20v027960; Fri, 23 Sep 2005 13:25:03 -0400 (EDT) Message-ID: <43343A52.6060506@mit.edu> Date: Fri, 23 Sep 2005 13:24:34 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason K Juang Subject: Re: Reading comments References: <1127487825.24672.1.camel@m56-129-2.mit.edu> In-Reply-To: <1127487825.24672.1.camel@m56-129-2.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 748 X-UID: 2771 X-Keywords: NotJunk My simple response would be that when you present a proof, you shouldn't state extraneous assumptions. There's nothing wrong with it logically, but it's not good style. For example, "a=b, therefore, a^2 = b^2." is good, but "a=b, and b=c, therefore, a^2 = b^2." is not really ideal. Distinguishing between strong and simple induction allows us to follow this "minimal-assumption" convention. Let me know if you're still confused. DS Jason K Juang wrote: >On page 10: "The only change from the ordinary induction principle is >that strong induction allows you to assume more stuff in the inductive >step of your proof!" > >So then why do we bother making a distinction between simple induction >and strong induction? > >Jason. > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:19:55 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHJtaJ001122 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHJsid025136 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHFlHR025650; Fri, 23 Sep 2005 13:15:47 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHFi3a000127; Fri, 23 Sep 2005 13:15:44 -0400 (EDT) Message-ID: <43343824.4060900@mit.edu> Date: Fri, 23 Sep 2005 13:15:16 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz Subject: Re: slightly unclear References: <200509231432.j8NEWEi0025629@outgoing.mit.edu> In-Reply-To: <200509231432.j8NEWEi0025629@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------000209080207090105030401" X-Spam-Score: -1.406 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 4899 X-UID: 2772 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------000209080207090105030401 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you. The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2. So, what we've proved is: P(1) P(3) -> P(4) P(4) -> P(5) P(5) -> P(6) P(6) -> P(7) ... Do you see that this does not imply P(n) for all n? DS Anton Katz wrote: > Hi, > > > > Two topics are still slightly unclear for me: > > Well-ordering and how is it possible to transfer way of proof to > another way. Is it done by math or is there a template to be used in > the transaction? Does this mean that everything that can be proved > using well-ordering can be proved using induction? > > > > Second topic is the false proof, I understand that the problem with > the horses proof was that we did not take into account different > horses at P(1) both in the tutorial problems and in the proof in the > reading I don't fully understand where exactly the proof becomes false. > > > > Thank you in advance, > > > > Anton. > > > --------------000209080207090105030401 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you.

The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2.  So, what we've proved is:

P(1)
P(3) -> P(4)
P(4) -> P(5)
P(5) -> P(6)
P(6) -> P(7)
...

Do you see that this does not imply P(n) for all n?

DS

Anton Katz wrote:

Hi,

 

Two topics are still slightly unclear for me:

Well-ordering and how is it possible to transfer way of proof to another way. Is it done by math or is there a template to be used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using induction?

 

Second topic is the false proof, I understand that the problem with the horses proof was that we did not take into account different horses at P(1) both in the tutorial problems and in the proof in the reading I don’t fully understand where exactly the proof becomes false.

 

Thank you in advance,

 

Anton.

 


--------------000209080207090105030401-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 12:44:05 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NGi5aJ027825; Fri, 23 Sep 2005 12:44:05 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j8NGi5rN027008; Fri, 23 Sep 2005 12:44:05 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j8NGi57E027007; Fri, 23 Sep 2005 12:44:05 -0400 Date: Fri, 23 Sep 2005 12:44:05 -0400 From: Sayan Mitra Message-Id: <200509231644.j8NGi57E027007@drake.csail.mit.edu> To: nedzel@MIT.EDU Subject: Re: Week 3 Comments Status: RO Content-Length: 898 X-UID: 2773 X-Keywords: NotJunk If a set W is well ordered then you can induct on its elements to prove the property in question. This may sound a little vague now but it will become clearer as we do more examples in class and in problem sets. Drop by in the office hours if you still have questions. -Sayan >From nedzel@MIT.EDU Fri Sep 23 09:45:25 2005 From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 09:45:11 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.502 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 In section 4, I understand the Well Ordering Principle, but what is its significance to induction? - David From meyer@imap.theory.csail.mit.edu Fri Sep 23 12:25:48 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NGPmaJ026211; Fri, 23 Sep 2005 12:25:48 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j8NGPmoJ026871; Fri, 23 Sep 2005 12:25:48 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j8NGPmVi026870; Fri, 23 Sep 2005 12:25:48 -0400 Date: Fri, 23 Sep 2005 12:25:48 -0400 From: Sayan Mitra Message-Id: <200509231625.j8NGPmVi026870@drake.csail.mit.edu> To: lana@mit.edu Subject: Re: TP2 Status: RO Content-Length: 83 X-UID: 2774 X-Keywords: NotJunk you'll see one very nice application of the well ordering principle today. -sayan From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:46:06 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.197]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NFk6aJ017921 for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:46:06 -0400 Received: by xproxy.gmail.com with SMTP id h28so824259wxd for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 08:46:01 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=XAk9zsQMl4HG2+TKdZixpVth6wDUlhQ3HM+T5Of7XSepP9tUsjaEY3ei9aOtK/BgMESnkU7lKY39WaG45Ro/B1CZpLf55zqa48cjdkQJMVkP/ilpFvzyIjTbD6sgy+UEaivqhSjf8w6R6llK/AOi8klG83vVARo14a9DQEc9CAs= Received: by 10.70.21.12 with SMTP id 12mr1128551wxu; Fri, 23 Sep 2005 08:46:01 -0700 (PDT) Received: by 10.70.52.3 with HTTP; Fri, 23 Sep 2005 08:46:01 -0700 (PDT) Message-ID: Date: Fri, 23 Sep 2005 11:46:01 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: r n jacobs Subject: Re: reading questions due sep 23 In-Reply-To: <200509231541.j8NFftZf007838@white-meteo.mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline References: <4333E709.5010506@theory.csail.mit.edu> <200509231541.j8NFftZf007838@white-meteo.mit.edu> Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8NFk6aJ017921 Status: RO Content-Length: 976 X-UID: 2775 X-Keywords: NotJunk Ok. If you find anything unclear later, feel free to ask. -Jelani On 9/23/05, r n jacobs wrote: > > The one on the tutor questions. I identified what was wrong fairly > fast (It wasn't proven for n=1), but it took me /forever/ to figure > out which line technically had the flaw. > > (I also still find the "every number is a product of primes" proof > (3.2, pg10) confusing. But I /think/ I understand strong induction > now.) > > - robert > > > Which fibonacci misproof? If you are talking about the fibonacci > > identity proven in the lecture notes, the identity is correct, and it is > > correctly proven using plain old induction (not strong induction). > > > > -Jelani > > > > r n jacobs wrote: > > > > >I found myself confused by the explanation of Strong > > >Induction. Although maybe the fibbonaci proof helped that. (The > > >definition made sense, but it took far to long for me to figure out > > >the answer to the fibbonaci misproof) > From meyer@imap.theory.csail.mit.edu Fri Sep 23 18:34:50 2005 Message-ID: <43348308.5040000@csail.mit.edu> Date: Fri, 23 Sep 2005 18:34:48 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Karena Tyan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading 3: Induction References: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> In-Reply-To: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1351 X-UID: 2776 X-Keywords: I hope today's class clarified things some, but here's one way to understand why all the extra P's can be assumed: think of the falling dominoes explanation of induction: if you knock over the 1st domino, and knocking over the nth domino will cause the next (n+1st) one to get knocked over, then you can conclude by simple induction that they all will get knocked over. But when the cascade of falling dominoes gets to the n+1st, all the previous ones will have been knocked over already, so it's perfectly ok to use that fact in reasoning about how the n+1st one falls. If you're still feeling shaky about this, I'll second Hanson's suggestion that you take it up with a TA is office hours. I'd also be happy to meet you after classs or make some other appt with you to talk about this. regards, A. Karena Tyan wrote: >I forgot to include in my last e-mail: I refer to the passage that defines >Strong Induction at the top of page 10. Specifically, the part that confuses >me is: > >"The only change from the ordinary induction principle is that strong induction >allows you to assume more stuff in the inductive step of your proof! ...In a >strong induction argument, you may assume that P(0), P(1), ..., and P(n) are >all true when you go to prove P(n + 1)." > >Why are you allowed to assume all these cases? > >- Karena > > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 18:23:07 2005 BCC: Ronitt Rubinfeld Message-ID: <43348049.9070206@csail.mit.edu> Date: Fri, 23 Sep 2005 18:23:05 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hanson Zhou Subject: Re: Reading 3: Induction References: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1028 X-UID: 2777 X-Keywords: I think your answer to this student is insulting: she has done the reading and is asking about it -- and your advice is to do the reading (again?). I'll respond to her further. regards, A. Hanson Zhou wrote: >You are essentially asking what strong induction is and you should read >about it. If it remains confusing, go to office hours to get this cleared >up. > >-Hanson > >On Fri, 23 Sep 2005, Karena Tyan wrote: > > > >>I forgot to include in my last e-mail: I refer to the passage that defines >>Strong Induction at the top of page 10. Specifically, the part that confuses >>me is: >> >>"The only change from the ordinary induction principle is that strong induction >>allows you to assume more stuff in the inductive step of your proof! ...In a >>strong induction argument, you may assume that P(0), P(1), ..., and P(n) are >>all true when you go to prove P(n + 1)." >> >>Why are you allowed to assume all these cases? >> >>- Karena >> >>-- >>410 Memorial Drive >>Cambridge, MA 02139 >>(585)957-5923 >> >> >> From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:25:09 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHP7aJ002589 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:07 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6id003556 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6HR026006; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHP20v027960; Fri, 23 Sep 2005 13:25:03 -0400 (EDT) Message-ID: <43343A52.6060506@mit.edu> Date: Fri, 23 Sep 2005 13:24:34 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason K Juang Subject: Re: Reading comments References: <1127487825.24672.1.camel@m56-129-2.mit.edu> In-Reply-To: <1127487825.24672.1.camel@m56-129-2.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 747 X-UID: 2778 X-Keywords: NotJunk My simple response would be that when you present a proof, you shouldn't state extraneous assumptions. There's nothing wrong with it logically, but it's not good style. For example, "a=b, therefore, a^2 = b^2." is good, but "a=b, and b=c, therefore, a^2 = b^2." is not really ideal. Distinguishing between strong and simple induction allows us to follow this "minimal-assumption" convention. Let me know if you're still confused. DS Jason K Juang wrote: >On page 10: "The only change from the ordinary induction principle is >that strong induction allows you to assume more stuff in the inductive >step of your proof!" > >So then why do we bother making a distinction between simple induction >and strong induction? > >Jason. > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:19:57 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHJtaJ001122 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHJsid025136 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHFlHR025650; Fri, 23 Sep 2005 13:15:47 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHFi3a000127; Fri, 23 Sep 2005 13:15:44 -0400 (EDT) Message-ID: <43343824.4060900@mit.edu> Date: Fri, 23 Sep 2005 13:15:16 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz Subject: Re: slightly unclear References: <200509231432.j8NEWEi0025629@outgoing.mit.edu> In-Reply-To: <200509231432.j8NEWEi0025629@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------000209080207090105030401" X-Spam-Score: -1.406 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,HTML_50_60, HTML_MESSAGE autolearn=no version=2.63 Status: RO Content-Length: 4899 X-UID: 2779 X-Keywords: NotJunk This is a multi-part message in MIME format. --------------000209080207090105030401 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you. The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2. So, what we've proved is: P(1) P(3) -> P(4) P(4) -> P(5) P(5) -> P(6) P(6) -> P(7) ... Do you see that this does not imply P(n) for all n? DS Anton Katz wrote: > Hi, > > > > Two topics are still slightly unclear for me: > > Well-ordering and how is it possible to transfer way of proof to > another way. Is it done by math or is there a template to be used in > the transaction? Does this mean that everything that can be proved > using well-ordering can be proved using induction? > > > > Second topic is the false proof, I understand that the problem with > the horses proof was that we did not take into account different > horses at P(1) both in the tutorial problems and in the proof in the > reading I don't fully understand where exactly the proof becomes false. > > > > Thank you in advance, > > > > Anton. > > > --------------000209080207090105030401 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you.

The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2.  So, what we've proved is:

P(1)
P(3) -> P(4)
P(4) -> P(5)
P(5) -> P(6)
P(6) -> P(7)
...

Do you see that this does not imply P(n) for all n?

DS

Anton Katz wrote:

Hi,

 

Two topics are still slightly unclear for me:

Well-ordering and how is it possible to transfer way of proof to another way. Is it done by math or is there a template to be used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using induction?

 

Second topic is the false proof, I understand that the problem with the horses proof was that we did not take into account different horses at P(1) both in the tutorial problems and in the proof in the reading I don’t fully understand where exactly the proof becomes false.

 

Thank you in advance,

 

Anton.

 


--------------000209080207090105030401-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 18:54:04 2005 Message-ID: <4334878A.6000409@csail.mit.edu> Date: Fri, 23 Sep 2005 18:54:02 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Angelique E Moscicki CC: 6042-staff@theory.csail.mit.edu Subject: Re: comment References: <20050923014649.s2van2mf2o000ckw@webmail.mit.edu> In-Reply-To: <20050923014649.s2van2mf2o000ckw@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 486 X-UID: 2780 X-Keywords: It was a dense 10 or so pages of reading, and even if you knew and understood it all, I can't believe you found it all equally well :-) explained, with enough examples, etc. Also, do you see how Well Ordering proofs can be converted to Induction proofs, and vice-versa?-- that was tricky. I'll look forward to more responsive comments from you next week. regards, A. Angelique E Moscicki wrote: >Sorry there isn't as much this time, its just rather more familiar material. > From meyer@imap.theory.csail.mit.edu Sat Sep 24 02:41:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8O6fiaJ014234 for <6042-probs@theory.lcs.mit.edu>; Sat, 24 Sep 2005 02:41:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8O6fgI1009021 for <6042-probs@theory.lcs.mit.edu>; Sat, 24 Sep 2005 02:41:42 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8O6fcl9029104 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 24 Sep 2005 02:41:39 -0400 (EDT) Message-ID: <433565AD.3040501@mit.edu> Date: Sat, 24 Sep 2005 06:41:49 -0800 From: Benjamin Lu User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [6.042] comments on reading 3 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.449 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 452 X-UID: 2781 X-Keywords: NonJunk NotJunk Hello, My apologies for the late comment, I completely forgot to send this earlier. "The Well Ordering Principle looks nothing like the induction axiom, and it may seem obvious but useless" -pg 12 I agreed with this sentiment, probably because using Well Ordering in proofs seems so backwards and counterintuitive. I think it would be cool to do a bunch of Well Ordering proofs to try and develop a little more intuition for them. Cheers, ~Ben From meyer@imap.theory.csail.mit.edu Sat Sep 24 17:00:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8OL0paJ016114 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 24 Sep 2005 17:00:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8OL0QNT009414 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 24 Sep 2005 17:00:50 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8OKxJcW006806 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 24 Sep 2005 16:59:19 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8OKxJKu003616; Sat, 24 Sep 2005 16:59:19 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sat, 24 Sep 2005 16:59:19 -0400 Message-ID: <20050924165919.924oi2ic8e50ckgc@webmail.mit.edu> Date: Sat, 24 Sep 2005 16:59:19 -0400 From: Rebecca Idell To: 6042-probs@theory.lcs.MIT.EDU Subject: 042 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.721 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 383 X-UID: 2782 X-Keywords: NonJunk $Forwarded NotJunk On page 3-4 i nthe reading is the template for the induction proof. When we do proofs in clas, I don't completely understand which parts of the proof correspond to which parts of the template. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From meyer@imap.theory.csail.mit.edu Mon Sep 26 08:23:25 2005 Message-ID: <4337E83E.8020403@csail.mit.edu> Date: Mon, 26 Sep 2005 08:23:26 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [6.042] comments on reading 3 References: <433565AD.3040501@mit.edu> In-Reply-To: <433565AD.3040501@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 550 X-UID: 2783 X-Keywords: Was Friday lecture cool enough? :-) Regards, A. Benjamin Lu wrote: > Hello, > > My apologies for the late comment, I completely forgot to send this > earlier. > > "The Well Ordering Principle looks nothing like the induction axiom, > and it may seem obvious but useless" -pg 12 > > I agreed with this sentiment, probably because using Well Ordering in > proofs seems so backwards and counterintuitive. I think it would be > cool to do a bunch of Well Ordering proofs to try and develop a little > more intuition for them. > > Cheers, > ~Ben From berke@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From berke@MIT.EDU Wed Sep 7 22:06:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8826CaJ026104 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Sep 2005 22:06:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8826AaH023833 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Sep 2005 22:06:11 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j88267sD020661 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Sep 2005 22:06:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j882678B016678; Wed, 7 Sep 2005 22:06:07 -0400 Received: from SENIOR-TWO-O-SIX.MIT.EDU (SENIOR-TWO-O-SIX.MIT.EDU [18.244.5.206]) (User authenticated as berke@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Sep 2005 22:06:07 -0400 Message-ID: <20050907220607.r9aa25eyjz4g4gg4@webmail.mit.edu> Date: Wed, 7 Sep 2005 22:06:07 -0400 From: Allison Berke To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Questions 1 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2784 Content-Length: 973 X-Keywords: NonJunk NotJunk Hi, I found the shortcomings of set theory (pgs. 12, 13, and Russell appendix) to be the most interesting. I'd like to know what the "simple ways" in which sets can be built are, and how they were decided upon. I'd also like to know more about the Banach-Tarski paradox: I understand what it implies, and how it can say that it's true (every piece must be infinitely "jagged", to an extent that goes beyond possible atomic subdivisions) but it would seem that infinite jaggedness would then be entirely uninteresting. How did the paradox become so famous? That the Axiom of Choice has to be assumed probably has something to do with it--especially for detractors of the Axiom--but it seems that even assuming the possibility of infinite subdivisions lends an absurdity to any conclusions that stem from it. If we can assume infinite subdivision of mathematical matter, can we propose infinite subdivision of mathematical time? And what would the point be? -Allison Berke From sheldons@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From sheldons@MIT.EDU Sat Sep 10 15:24:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8AJOXaJ029416 for <6042-probs@theory.csail.mit.edu>; Sat, 10 Sep 2005 15:24:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8AJOVDo000651 for <6042-probs@theory.csail.mit.edu>; Sat, 10 Sep 2005 15:24:32 -0400 (EDT) Received: from [10.0.0.2] (209-6-159-26.c3-0.smr-ubr3.sbo-smr.ma.cable.rcn.com [209.6.159.26]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8AJOOmO007145 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sat, 10 Sep 2005 15:24:25 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <1EC3B9B2-1A79-4E4E-9A99-D0AD840E87BC@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Sheldon Chan Subject: Required Reading Comments 1 Date: Sat, 10 Sep 2005 15:24:23 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2785 Content-Length: 326 X-Keywords: NonJunk NotJunk 1.4 Proof by Cases (page 3) I found that the example of proof by cases to be most difficult to comprehend. I'm not exactly sure why Case 1.2 falls logically under Case 1, since, it basically implies that at least 2 people did meet x and not 3. Why aren't the two cases 1. Two have met x, and 2. Three have not met x? From ksindi@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From ksindi@MIT.EDU Sat Sep 10 17:39:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ALdWaJ011814 for <6042-probs@theory.lcs.mit.edu>; Sat, 10 Sep 2005 17:39:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ALdUhu026648 for <6042-probs@theory.lcs.mit.edu>; Sat, 10 Sep 2005 17:39:31 -0400 (EDT) Received: from KAMIL.mit.edu (SENIOR-SIX-O-NINE.MIT.EDU [18.244.7.98]) (authenticated bits=0) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ALdLJg020131 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 10 Sep 2005 17:39:24 -0400 (EDT) Message-Id: <6.2.3.4.2.20050910170801.03b61eb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Sat, 10 Sep 2005 17:39:07 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kamil Sindi Subject: Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2786 Content-Length: 473 X-Keywords: NonJunk NotJunk " 2. Predicates (p. 4) A predicate is a proposition whose truth depends on the value of one or more variables." I found this passage surprising for its over-emphasis on mathematical terminology. However, I appreciated the 'simplicity by convention', which differentiates mathematics from many other disciplines. The example with P(n)= "n is a perfect square" made the concept of the predicate a little confusing, but reading on cleared all the misconceptions I had. From aeon@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From aeon@MIT.EDU Sun Sep 11 22:37:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8C2bAaJ000510 for <6042-probs@theory.csail.MIT.EDU>; Sun, 11 Sep 2005 22:37:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8C2b92D016621 for <6042-probs@theory.csail.MIT.EDU>; Sun, 11 Sep 2005 22:37:09 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8C2b7pU029283 for <6042-probs@theory.csail.MIT.EDU>; Sun, 11 Sep 2005 22:37:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8C2b7lk010499; Sun, 11 Sep 2005 22:37:07 -0400 Received: from NEW-FOUR-NINETY-THREE.MIT.EDU (NEW-FOUR-NINETY-THREE.MIT.EDU [18.241.6.238]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 11 Sep 2005 20:37:07 -0600 Message-ID: <20050911203707.c93buqpcs7400osk@webmail.mit.edu> Date: Sun, 11 Sep 2005 20:37:07 -0600 From: John Marrero To: 6042-probs@theory.csail.MIT.EDU Subject: LN2 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2787 Content-Length: 632 X-Keywords: NonJunk NotJunk Page 3, section "1.4 Proof by Cases" This was quite interesting, particularly the example given of the strangers and clubs. It seems like a useful method, but could it be used as easily for complicated expressions? If there's an example that addresses that issue, I'd very much like to see it. Page 10, section "3.3 Sequences" I understood what the idea behind the product operation was, but wasn't totally grasping it through the examples provided. Perhaps a step-by-step demonstration of it would help (on an example simple enough to actually be able to do this without tedium)? -- John Marrero MIT Class of '08 (305) 812.7042 From xiaoranz@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From xiaoranz@MIT.EDU Mon Sep 12 18:50:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8CMoSaJ012134 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 12 Sep 2005 18:50:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8CMoRVr019180 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 12 Sep 2005 18:50:27 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8CMoKu9025543 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 12 Sep 2005 18:50:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8CMoKFW031481; Mon, 12 Sep 2005 18:50:20 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 12 Sep 2005 18:50:19 -0400 Message-ID: <20050912185019.7e396ewb940kkkkg@webmail.mit.edu> Date: Mon, 12 Sep 2005 18:50:19 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Comment for Reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2788 Content-Length: 740 X-Keywords: NonJunk NotJunk Passage: 2.6 Validity, Page 7 of week 2 Reading. I thought the prove of validity of qualifier is difficult, because it isn't quite mathematically obvious to me and I would like to have it discussed more in depth in the lecture. Maybe an explanation about how to approach these problems would be best. I had some difficulties with the qualifier validity question on the tutor problems also. Xiaoran (Sharon) Zhang ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From lye@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From lye@MIT.EDU Mon Sep 12 20:58:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D0wNaJ029248 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 20:58:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D0wMX6012301 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 20:58:22 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D0wKMF019463 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 20:58:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8D0wKfl023507; Mon, 12 Sep 2005 20:58:20 -0400 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 12 Sep 2005 20:58:20 -0400 Message-ID: <20050912205820.tom5ow5cbrgo8400@webmail.mit.edu> Date: Mon, 12 Sep 2005 20:58:20 -0400 From: Linda Ye To: 6042-probs@theory.csail.MIT.EDU Subject: required email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2789 Content-Length: 511 X-Keywords: NonJunk NotJunk >From Course Notes, Week 2: Predicates & Sets Section 1.4, pg. 3: "We first argue that at least one of these two cases must hold. We?ll prove this by contradiction. Namely, suppose neither case holds. This means that at most 2 people in the group met x and at most 2 did not meet x. This leaves at least 1 of the remaining 5 people unaccounted for." You could explain the same idea with the pigeonhole principle: out of 5 people, at least 3 of them 1) all met x, or 2) all did not meet x. Linda (Lunduo) Ye From adnaan@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From adnaan@MIT.EDU Mon Sep 12 21:09:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D19MaJ032222 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:09:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D19KX6019582 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:09:20 -0400 (EDT) Received: from AdnaanJiwaji (NEW-THIRTY-NINE.MIT.EDU [18.241.5.39]) (authenticated bits=0) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D19IZ0021655 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:09:18 -0400 (EDT) Message-ID: <003401c5b7ff$b4da0e20$2705f112@AdnaanJiwaji> From: "Adnaan Jiwaji" To: <6042-probs@theory.csail.mit.edu> Subject: Week 2 Comments Date: Mon, 12 Sep 2005 20:08:52 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0031_01C5B7D5.CBB17950" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.71 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2790 Content-Length: 1424 X-Keywords: NonJunk NotJunk Paragraph 2.6 Validity page 7 The difference between a propositional formula and a predicate formula is not clear from the paragraph and would be good to have it clarified in class.  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0031_01C5B7D5.CBB17950 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Paragraph 2.6 Validity page 7 The difference between a propositional formula and a predicate formula = is not clear from the paragraph and would be good to have it clarified = in class. ------=_NextPart_000_0031_01C5B7D5.CBB17950 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Paragraph 2.6 Validity page = 7
 
The difference between a propositional = formula and=20 a predicate formula is not clear from the paragraph and would be good to = have it=20 clarified in class.
 
------=_NextPart_000_0031_01C5B7D5.CBB17950-- From cwong08@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From cwong08@MIT.EDU Mon Sep 12 21:42:24 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D1gOaJ005280 for <6042-probs@theory.lcs.mit.edu>; Mon, 12 Sep 2005 21:42:24 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D1gNAG007493 for <6042-probs@theory.lcs.mit.edu>; Mon, 12 Sep 2005 21:42:24 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8D1g93b025775 for <6042-probs@theory.lcs.mit.edu>; Mon, 12 Sep 2005 21:42:19 -0400 (EDT) Message-ID: <43262E65.7010401@mit.edu> Date: Mon, 12 Sep 2005 21:41:57 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: week 2 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2791 Content-Length: 137 X-Keywords: NonJunk NotJunk The stuff about breaking math by disproving ZFC axioms (page 12) was very intersting, and I would like to hear it elaborated in class. From mdmurray@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From mdmurray@MIT.EDU Mon Sep 12 21:50:42 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D1ogaJ006044 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:50:42 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D1ofAG007713 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:50:41 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8D1oc0v004705 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:50:38 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <27302626-1266-40D7-85F1-599792A4DDD1@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-3-92665577 From: Michael Murray Subject: Week 2 Comments Date: Mon, 12 Sep 2005 21:50:37 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.988 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2792 Content-Length: 9899 X-Keywords: NonJunk NotJunk 3.5 Functions A function assigns an element of one set, called the domain, to elements of another set, called the codomain. The notation f : A ! B indicates that f is a function with domain, A, and codomain, B. The familiar notation f(a) = b indicates that f assigns to a the element b. A function f : A ! B is: total if every element of A is assigned to some element of B; otherwise, f is called a partial function. surjective if every element of B is mapped to at least once, that is, 8b 2 B9a 2 A. f(a) = b. injective if every element of B is mapped to at most once. bijective if f is total, surjective, and injective. In particular, each element of B is mapped to exactly once. The names surjective and injective are hopelessly unmemorable and nondescriptive. Some authors us the term onto for surjective and one-to-one for injective, which are shorter but arguably no more memorable. Here are a couple examples: page 11 I found this section to be quite difficult to understand at first, it took me a while to get what was going on with f(A) and B. Now that I understand it though (thanks to another student in the class who helped me), I think it is a nice topic.  -=- MIME -=-  --Apple-Mail-3-92665577 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed 3.5 Functions A function assigns an element of one set, called the domain, to =20 elements of another set, called the codomain. The notation f : A ! B indicates that f is a function with domain, A, and codomain, B. The =20 familiar notation =93f(a) =3D b=94 indicates that f assigns to a the element b. A function f : A ! B is: =95 total if every element of A is assigned to some element of B; =20 otherwise, f is called a partial function. =95 surjective if every element of B is mapped to at least once, that =20= is, 8b 2 B9a 2 A. f(a) =3D b. =95 injective if every element of B is mapped to at most once. =95 bijective if f is total, surjective, and injective. In particular, =20= each element of B is mapped to exactly once. The names =93surjective=94 and =93injective=94 are hopelessly = unmemorable and =20 nondescriptive. Some authors us the term onto for surjective and one-to-one for injective, =20= which are shorter but arguably no more memorable. Here are a couple examples: page 11 I found this section to be quite difficult to understand at first, it =20= took me a while to get what was going on with f(A) and B. Now that I =20 understand it though (thanks to another student in the class who =20 helped me), I think it is a nice topic.= --Apple-Mail-3-92665577 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=WINDOWS-1252
3.5 = Functions
A = function assigns an element of one set, called the domain, to elements of another set, called = the
codomain. = The notation
f : A ! B
indicates that f = is a function with domain, A, and codomain, B. The familiar notation = =93f(a) =3D b=94
indicates that f assigns to a the element = b.
A = function f : A ! B is:
=95 total if every element of A is assigned to some = element of B; otherwise, f is called a partial
function.
=95 surjective if every element of B is mapped to at least = once, that is, 8b 2 B9a 2 A. f(a) =3D b.
=95 = injective = if every element = of B is mapped to at most once.
=95 = bijective = if f is total, = surjective, and injective. In particular, each element of B is mapped = to
exactly once.
The names =93surjective=94 and =93injective=94 = are hopelessly unmemorable and nondescriptive. = Some
authors us the term onto for surjective and one-to-one for injective, = which are shorter but arguably
no more memorable. = Here are a couple examples:

page = 11

I found this section to be quite difficult to = understand at first, it took me a while to get what was going on with = f(A) and B. Now that I understand it though (thanks to another student = in the class who helped me), I think it is a nice = topic.
= --Apple-Mail-3-92665577-- From ridell@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: iso-8859-1-unix Mail-from: From ridell@MIT.EDU Mon Sep 12 22:51:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D2pGaJ013616 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 22:51:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D2pFX6005702 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 22:51:15 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D2pCXC014129 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 22:51:12 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8D2pChT031088; Mon, 12 Sep 2005 22:51:12 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 12 Sep 2005 22:51:12 -0400 Message-ID: <20050912225112.gdmcwte32pkwkws8@webmail.mit.edu> Date: Mon, 12 Sep 2005 22:51:12 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: Rebecca Idell comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.226 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2793 Content-Length: 499 X-Keywords: NonJunk NotJunk 6.042 Comments On page 6, section 2.4 Order of Quantifiers, there is an example "every American has a dream." It is then translated into quantifier notation, but I do not understand the notation entirely, with lower and uppercase letters, and then creating some H(a,d), so I hope it will be further discussed in lecture. -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From shauni@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: iso-8859-1-unix Mail-from: From shauni@MIT.EDU Tue Sep 13 00:08:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D48eaJ020413 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:08:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D48cFB002382 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:08:38 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D48W8R001609 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:08:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8D48WCv017071; Tue, 13 Sep 2005 00:08:32 -0400 Received: from SIMMONS-FORTY-NINE.MIT.EDU (SIMMONS-FORTY-NINE.MIT.EDU [18.96.5.49]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 00:08:32 -0400 Message-ID: <20050913000832.ffo5y8fch7484cc4@webmail.mit.edu> Date: Tue, 13 Sep 2005 00:08:32 -0400 From: shauni@MIT.EDU To: 6042-probs@theory.csail.MIT.EDU Subject: Week 2 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2794 Content-Length: 173 X-Keywords: NonJunk NotJunk I would like to have the functions section (p.11) discussed more fully in lecture. I especially would like to go over the terms total, surjective, injective, and bijective. From dangut@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: iso-8859-1-unix Mail-from: From dangut@MIT.EDU Tue Sep 13 00:24:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D4O5aJ022133 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:24:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D4O4FB013139 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:24:04 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D4Nvt6004383 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:23:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8D4Nvct011692; Tue, 13 Sep 2005 00:23:57 -0400 Received: from MACGREGOR-THREE-EIGHTY-TWO.MIT.EDU (MACGREGOR-THREE-EIGHTY-TWO.MIT.EDU [18.239.6.127]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 00:23:57 -0400 Message-ID: <20050913002357.6dd1o18l5668sso4@webmail.mit.edu> Date: Tue, 13 Sep 2005 00:23:57 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: Week 2 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2795 Content-Length: 361 X-Keywords: NonJunk NotJunk I think the most difficult part for me is the part about sets (page 9) since I haven't seen too many before. A question I have about hte copmlement of a set, they say the complement of all real positive numbers is all negative numbers plus zero, what keeps the complement from having imaginary and fraction values? Since these are not part of the set anyways. From hkhall@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: undecided-unix Mail-from: From hkhall@MIT.EDU Tue Sep 13 00:42:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8D4gDaJ025558 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 00:42:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8D4gBFB025529; Tue, 13 Sep 2005 00:42:11 -0400 (EDT) Received: from [18.96.6.222] (SIMMONS-FOUR-SEVENTY-SEVEN.MIT.EDU [18.96.6.222]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D4g9hg007229 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Tue, 13 Sep 2005 00:42:10 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <80b4f15dbaa219a418cb1720b383f136@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: Comments on LN2 Date: Tue, 13 Sep 2005 00:44:01 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2796 Content-Length: 421 X-Keywords: NonJunk NotJunk I found section 2.1 of quantifying a predicate beginning on page 5 the most difficult concept in the reading. In my opinion this is due largely to the ambiguity inherent to the english language in its conversion to the language of mathematics and vice versa. While I don't want an english to math dictionary, more guidance in the conversion process would go a long way in understanding the method. -Harrison Hall From rnjacobs@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: undecided-unix Mail-from: From rnjacobs@MIT.EDU Tue Sep 13 13:21:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DHL7aJ030067 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 13:21:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DHL6uU027574 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 13:21:06 -0400 (EDT) Received: from m4-167-2.mit.edu (M4-167-2.MIT.EDU [18.53.0.86]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DHL4K9018842 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 13:21:04 -0400 (EDT) Received: (from rnjacobs@localhost) by m4-167-2.mit.edu (8.12.9) id j8DHL4Nt005486; Tue, 13 Sep 2005 13:21:04 -0400 (EDT) Message-Id: <200509131721.j8DHL4Nt005486@m4-167-2.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU Subject: Comments for Reading due 14 September Date: Tue, 13 Sep 2005 13:21:04 -0400 From: r n jacobs X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2797 Content-Length: 387 X-Keywords: NonJunk NotJunk "2.5 Negating Quantifiers ... (not all)x. P(x) is equivalent to (exists)x. (not) P(x)." (page 7) I find myself wondering if this is related to DeMorgan's law. Also, "1.4 Proof by Cases" (page 3) I found the phrasing of the "in a group of 6, at least 3 have met or 3 have not met" bewildering, although now that I've reread it I understand. I may have been tired. - Robert Jacobs From bilodeau@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: undecided-unix Mail-from: From bilodeau@MIT.EDU Tue Sep 13 15:28:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DJSNaJ030895 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 15:28:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DJSMt4022947 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 15:28:22 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THREE-SEVENTY-SEVEN.MIT.EDU [18.239.6.122]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DJSI4M018839 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 15:28:18 -0400 (EDT) Message-Id: <200509131928.j8DJSI4M018839@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: Date: Tue, 13 Sep 2005 15:28:11 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5B877.C0DE2490" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcW4mUdEpMXj9GHERfCQVlzjeOLRMw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 3.757 X-Spam-Level: *** (3.757) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2798 Content-Length: 14404 X-Keywords: NonJunk NotJunk “• The union of sets X and Y (denoted X [ Y ) contains all elements appearing in X or Y or both. Thus, X [ Y = {1, 2, 3, 4}. • The intersection of X and Y (denoted X \ Y ) consists of all elements that appear in both X and Y . So X \ Y = {2, 3}. • The difference of X and Y (denoted X −Y ) consists of all elements that are in X, but not in Y . Therefore, X − Y = {1} and Y − X = {4}.” The main thing that I find difficult about the reading is the large number of symbols. The quote above is an example of the introduction of more such symbols. Having not had set theory or logical analysis before, most of this is difficult and confusing to me; I think I’ll make myself a table with all of the operators and symbols for easy reference.  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5B877.C0DE2490 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable =20 =E2=80=9C=E2=80=A2 The union of sets X and Y (denoted X [ Y ) contains = all elements appearing in X or Y or both. Thus, X [ Y =3D {1, 2, 3, 4}. =E2=80=A2 The intersection of X and Y (denoted X \ Y ) consists of all = elements that appear in both X and Y . So X \ Y =3D {2, 3}. =E2=80=A2 The difference of X and Y (denoted X =E2=88=92Y ) consists of = all elements that are in X, but not in Y . Therefore, X =E2=88=92 Y =3D {1} and Y =E2=88=92 X =3D {4}.=E2=80=9D =20 The main thing that I find difficult about the reading is the large = number of symbols. The quote above is an example of the introduction of = more such symbols. Having not had set theory or logical analysis = before, most of this is difficult and confusing to me; I think = I=E2=80=99ll make myself a table with all of the operators and symbols = for easy reference. ------=_NextPart_000_0000_01C5B877.C0DE2490 Content-Type: text/html; charset="utf-8" Content-Transfer-Encoding: quoted-printable ------=_NextPart_000_0000_01C5B877.C0DE2490-- From iyzhang@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From iyzhang@MIT.EDU Tue Sep 13 16:04:37 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DK4baJ006668 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:37 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8DK4aau014081 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:36 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DK4a4M028799 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:36 -0400 (EDT) Received: from [18.239.6.65] (MACGREGOR-THREE-TWENTY.MIT.EDU [18.239.6.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8DK4S0v022239 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:28 -0400 (EDT) Message-ID: <432730C7.2030807@mit.edu> Date: Tue, 13 Sep 2005 16:04:23 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 2 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2799 Content-Length: 209 X-Keywords: NonJunk NotJunk I would like to discuss or see the Banach-Tarski theorem in the next class. Thanks, Irene p.s. in the function f : A -> B, A is A the codomain and B the domain? I think there may be an error in the reading From jstritar@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From jstritar@MIT.EDU Tue Sep 13 16:21:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DKLoaJ009396 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 16:21:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DKLnt4021662 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 16:21:49 -0400 (EDT) Received: from [18.245.6.203] (BAKER-FOUR-FIFTY-EIGHT.MIT.EDU [18.245.6.203]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DKLcgm013329 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 16:21:42 -0400 (EDT) Message-ID: <432734D0.8020005@mit.edu> Date: Tue, 13 Sep 2005 16:21:36 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.092 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2800 Content-Length: 523 X-Keywords: NonJunk NotJunk Lemma (Mapping Rule) on page 12. I thought the part pertaining to functions, and specifically to the sizes of the domain and codomains, was the most difficult part to understand. I guess I don't usually think of functions with respect to sets, and combined with a whole lot of new terminology, it takes a little while to get used to it. I guess comparing the sizes of the domains is like applying the function to every member of the starting set at the same time and comparing the results to the input. Jon Stritar From kevin08@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From kevin08@MIT.EDU Tue Sep 13 17:08:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DL8FaJ018070 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:08:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DL8Ebf009964 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:08:14 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-FOUR-THIRTY-NINE.MIT.EDU [18.242.6.184]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DL8C23003951 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:08:12 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913170501.021becf8@hesiod> X-Sender: kevin08@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 17:08:09 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: 6.042 Reading Feedback Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.052 X-Spam-Level: * (1.052) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2801 Content-Length: 555 X-Keywords: NonJunk NotJunk I thought that the theorem in Section 1.4 (Pages 3-4) regarding the clubs/groups of 3 people was interesting, and evoked pictures of the pigeonhole principle. If you could talk about this proof more thoroughly (with diagrams!!) in class, that would be very useful. Also, if you could introduce the pigeonhole principle and talk about whether it relates to this problem or not, that would be great. Thanks, Kevin Wang Massachusetts Institute of Technology - Class of 2008 Electrical Engineering and Computer Science http://web.mit.edu/kevin08/www From ereid@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From ereid@MIT.EDU Tue Sep 13 17:15:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DLFRaJ019591 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:15:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DLFQbf017327 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:15:26 -0400 (EDT) Received: from [18.242.5.26] (NEXT-TWENTY-SIX.MIT.EDU [18.242.5.26]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DLFITY006925 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:15:19 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8AE1BD52-4336-4C28-8297-410A72A8F212@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: reading response Date: Tue, 13 Sep 2005 17:15:11 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -0.596 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2802 Content-Length: 199 X-Keywords: NonJunk NotJunk I'm finding it difficult to understand the product of sequences. I don't follow how the elements are chosen for the sequences, and what the rules are on which sequence to take the elements from. From dowgun@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From dowgun@MIT.EDU Tue Sep 13 17:49:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DLnNaJ031041 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 17:49:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DLnMbf018932 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 17:49:22 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DLnFAL019383 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 17:49:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8DLnFO8030598; Tue, 13 Sep 2005 17:49:15 -0400 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 17:49:15 -0400 Message-ID: <20050913174915.4z7un1c55tmsgosw@webmail.mit.edu> Date: Tue, 13 Sep 2005 17:49:15 -0400 From: Neil M Dowgun To: 6042-probs@theory.csail.MIT.EDU Subject: response for 9/14 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2803 Content-Length: 951 X-Keywords: NonJunk NotJunk I feel that we should take some time in lecture to cover "functions" as applied to sets, which was written about in section 3.5 (pages 11-12) of the reading. It occurs to me that for a finite domain (and codomain), the only way to completely describe a set function is to specifically lay out which element of the codomain the domain is mapped too. I supposed that someone might have described a "sequential" function so that the first element of the domain maps to the first element of the codomain, but sets have no specific order, so this is impossible. Also, describing a function without knowing the codomain it is going to be mapped to seems impossible to me, unless you create the codomain FROM the application of the function to the domain. Is this allowed? Also, I don't see the difference between [ (upside-down A)x(backwards E)y(P(x,y)) ] and [ (backwards E)y(upside-down A)x(P(x,y)) ]. Does that change in order EVER matter? Neil Dowgun From hectorb@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From hectorb@MIT.EDU Tue Sep 13 17:56:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DLuGaJ032132 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:56:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DLuEbf024816; Tue, 13 Sep 2005 17:56:14 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DLu7xk021725; Tue, 13 Sep 2005 17:56:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8DLu7nS016371; Tue, 13 Sep 2005 17:56:07 -0400 Received: from BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (User authenticated as hectorb@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 17:56:07 -0400 Message-ID: <20050913175607.6p7cy2su6qe8gg84@webmail.mit.edu> Date: Tue, 13 Sep 2005 17:56:07 -0400 From: Hector Beltran To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 2 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2804 Content-Length: 326 X-Keywords: NonJunk NotJunk "When all variables in a formula are understood to take values from the same nonempty set D, it's conventional to omit mention of D. For example....It's easy to arrange for all variables to range over one domain." -pg 7 I did not find this explanation of variables over one domain so easy to understand. -Hector Beltran From lkini@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From lkini@MIT.EDU Tue Sep 13 17:57:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DLv5aJ032183 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:57:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DLv4bf025414 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:57:04 -0400 (EDT) Received: from quickstation-macgregor.mit.edu (QUICKSTATION-MACGREGOR.MIT.EDU [18.239.0.167]) (authenticated bits=56) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DLuxFn022016 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:57:01 -0400 (EDT) Received: (from lkini@localhost) by quickstation-macgregor.mit.edu (8.12.9) id j8DLuwL9025952; Tue, 13 Sep 2005 17:56:58 -0400 (EDT) Subject: 09/14/05 Reading Assignment From: Lohith G Kini To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 13 Sep 2005 17:56:56 -0400 Message-Id: <1126648616.25942.3.camel@quickstation-macgregor.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2805 Content-Length: 413 X-Keywords: NonJunk NotJunk Hi, The passage that I found most difficult to comprehend and would like to have explained in detail during lecture is the proof by cases of the theorem regarding a collection of 6 people which includes a club of 3 people or a group of 3 strangers [page 3 of lecture notes 2]. One strategy that might help explain [me] this point is to show other examples of proofs that use proof by cases. Thanks, Lohith Kini From icharny@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From icharny@MIT.EDU Tue Sep 13 18:42:04 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DMg4aJ005806 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 18:42:04 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DMg3UR017103 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 18:42:03 -0400 (EDT) Received: from [127.0.0.1] (EASTCAMPUS-EIGHT-NINETY-EIGHT.MIT.EDU [18.238.6.131]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8DMft0x029377 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 18:41:59 -0400 (EDT) Message-ID: <432755B1.50201@mit.edu> Date: Tue, 13 Sep 2005 18:41:53 -0400 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: 6.042 lecture notes 2 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.486 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2806 Content-Length: 167 X-Keywords: NonJunk NotJunk Pages 11-12 of the lecture notes discuss funtions and mappings (section 3.5). Please discuss this topic a little more thoroughly at the next lecture. Isaac Charny From avalys@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From avalys@MIT.EDU Tue Sep 13 18:43:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DMhdaJ005913 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 18:43:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DMhbbf001356 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 18:43:37 -0400 (EDT) Received: from [18.96.7.30] (SIMMONS-FIVE-FOURTY-ONE.MIT.EDU [18.96.7.30]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DMhISA005124 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 18:43:28 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <257831C0-3C54-4111-ABE3-8D4D18EAE1A6@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Alex Valys Subject: Week 2 Comments Date: Tue, 13 Sep 2005 18:43:19 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2807 Content-Length: 381 X-Keywords: NonJunk NotJunk Unhappily, I don't understand the specifics of how functions work, and what it actually means when the notes say: "a function _assigns_ elements of one set..." Also, on the tutor problems, I don't understand how the definition of a function relating A and B has any affect on A and B themselves - specifically, what the difference is between |A| and |f(a)|. Alex Valys From adamreed@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From adamreed@MIT.EDU Tue Sep 13 19:38:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8DNcFaJ013041 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 19:38:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8DNcEbf008737 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 19:38:14 -0400 (EDT) Received: from [18.228.0.23] (BOLIVAR.MIT.EDU [18.228.0.23]) (authenticated bits=0) (User authenticated as adamreed@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DNcCHq016048 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 19:38:12 -0400 (EDT) Message-ID: <4327632A.9000805@mit.edu> Date: Tue, 13 Sep 2005 19:39:22 -0400 From: Adam Reed User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050729) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 2 Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2808 Content-Length: 331 X-Keywords: NonJunk NotJunk I found the following passage on the Order of Quantifiers the most difficult part of the reading to understand: "Swapping quantifiers in Goldbach's Conjecture creates a patently false statement." p. 6 The example "Every American has a dream" was easy to understand, but I couldn't figure out the Goldbach example. -Adam Reed From rshroff@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From rshroff@MIT.EDU Tue Sep 13 20:21:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E0LpaJ019690 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 20:21:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E0LoEd006339 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 20:21:50 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E0LhWU024332 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 20:21:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8E0LgbS008878; Tue, 13 Sep 2005 20:21:42 -0400 Received: from NEXT-FOUR-EIGHTY.MIT.EDU (NEXT-FOUR-EIGHTY.MIT.EDU [18.242.6.225]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 20:21:42 -0400 Message-ID: <20050913202142.77mf8mn1guiscc4w@webmail.mit.edu> Date: Tue, 13 Sep 2005 20:21:42 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: E-mail Comments for assigned reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2809 Content-Length: 324 X-Keywords: NonJunk NotJunk Reference: Section 1.4 Proof by cases, Pg 3 I never really understood the concept of 'Proof by Cases' before I read the article. I found the example extremely interesting and it greatly helped me to understand better. However I would like it if we are exposed to some more examples on proofs by this method. -Rahul Shroff From yangc@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From yangc@MIT.EDU Tue Sep 13 20:24:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E0OQaJ020218 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 20:24:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E0OPEd008114 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 20:24:25 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E0OMna024924 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 20:24:23 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E0OM3h024139; Tue, 13 Sep 2005 20:24:22 -0400 Received: from MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) (User authenticated as yangc@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 20:24:22 -0400 Message-ID: <20050913202422.idtv2em8k288g0w8@webmail.mit.edu> Date: Tue, 13 Sep 2005 20:24:22 -0400 From: Christopher M Yang To: 6042-probs@theory.csail.MIT.EDU Subject: E-Mail Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.907 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2810 Content-Length: 284 X-Keywords: NonJunk NotJunk Page 12, Part 4 (Does all this really work?) Are we comfortable living in a universe where volume is plastic and fluid? I've heard a lot about Godel's Incompleteness Theorem and the ZFC axioms - I'd like to hear more about that, even if it is in the most cursory sense. Chris Yang From juang@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: raw-text-unix Mail-from: From juang@MIT.EDU Tue Sep 13 21:04:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E141aJ023132 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 21:04:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E140Ed004813 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 21:04:00 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E13wmb002604 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 21:03:59 -0400 (EDT) Message-ID: <432776ED.2070402@mit.edu> Date: Tue, 13 Sep 2005 21:03:41 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Reading comments Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2811 Content-Length: 671 X-Keywords: NonJunk NotJunk On page 12 of the reading: "No one knows whether the ZFC axioms are logically consistent; there is some possibility that one person might prove a proposition P and another might prove the proposition P. Then Math would be broken. This sounds like a crazy situation, but it has happened before. At the beginning of the 20th century, the logician Gotlob Frege made an initial attempt to axiomatize set theory using a few very plausible axioms. Several mathematicians most famously Bertrand Russell discovered that Freges axioms actually were self-contradictory!" I would like to see a further discussion of Russell's paradox in the next lecture. Jason Juang. From medrano@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From medrano@MIT.EDU Tue Sep 13 21:15:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E1FBaJ024636 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:15:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E1FAEd012648 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:15:10 -0400 (EDT) Received: from newhouse-3.mit.edu (NEWHOUSE-3.MIT.EDU [18.241.2.250]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1F7Fk004798 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:15:08 -0400 (EDT) Received: (from medrano@localhost) by newhouse-3.mit.edu (8.12.9) id j8E1F717008017; Tue, 13 Sep 2005 21:15:07 -0400 Date: Tue, 13 Sep 2005 21:15:07 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: required comments for reading Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2812 Content-Length: 330 X-Keywords: NonJunk NotJunk This weeks reading on proof by contradiction is very straight forward; however, it would be nice to work with more deductive proofs. I would like to also take more time dealing with multiple and mixed quantifiers in Logic. The set theory was also straight forward. Please clearly explain Russell's paradox. Jesus Medrano From jehan@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From jehan@MIT.EDU Tue Sep 13 21:31:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E1VpaJ027701 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:31:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E1VoEd024879 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:31:50 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1VkNo007928 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:31:47 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913212552.00bb6808@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 21:31:48 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: Week 2 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2813 Content-Length: 211 X-Keywords: NonJunk NotJunk 3.5 Function on page 11 Could you explain them a bit better in class. They seemed to be just tagged on the reading without much of an explanation and I am still a bit confused as to their use. Thanks Jehan From dnreshef@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From dnreshef@MIT.EDU Tue Sep 13 21:43:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E1hLaJ029619 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:43:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E1hJEd002762 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:43:20 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1hDbA010216 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:43:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8E1hDp8020185; Tue, 13 Sep 2005 21:43:13 -0400 Received: from MACGREGOR-FOUR-FIFTY-SIX.MIT.EDU (MACGREGOR-FOUR-FIFTY-SIX.MIT.EDU [18.239.6.201]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 21:43:13 -0400 Message-ID: <20050913214313.fx3vugr6f9a8ow0k@webmail.mit.edu> Date: Tue, 13 Sep 2005 21:43:13 -0400 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: READING COMMENTS MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 1.206 X-Spam-Level: * (1.206) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2814 Content-Length: 287 X-Keywords: NonJunk NotJunk READING COMMENTS: One thing that I found a little confusing was the difference between universal and existential quantification (pgs 5 and 6). Another confusing point was the business about Lemma (mapping rules) on page 12. I would appreciat disscusing this topic in lecture. -dave From pjs@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From pjs@MIT.EDU Tue Sep 13 21:44:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E1iZaJ029668 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:44:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E1iXEd003761 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:44:34 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1iQnN010466 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:44:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8E1iQIg020425; Tue, 13 Sep 2005 21:44:26 -0400 Received: from W20-575-76.MIT.EDU (W20-575-76.MIT.EDU [18.187.0.95]) (User authenticated as pjs@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 21:44:26 -0400 Message-ID: <20050913214426.jx0fhy62too4s0o0@webmail.mit.edu> Date: Tue, 13 Sep 2005 21:44:26 -0400 From: Paul J Steiner To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Assignment 1 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2815 Content-Length: 411 X-Keywords: NonJunk NotJunk I found the wrap up section "Does All This Really Work?" on pages 12 and 13 the most interesting part of the reading. Having done some cursory reading on some of the interesting consequences of accepting the Axiom of Choice, I wasn't surprised by the Banach-Tarski theorem, but it still remains tremendously fascinating! It's also a bit of a chuckle that mathematics as is cannot prove itself. ;) PJ Steiner From nedzel@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From nedzel@MIT.EDU Tue Sep 13 21:45:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E1jEaJ029720 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:45:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E1jDEd004241 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:45:13 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SIX-NINETY-FOUR.MIT.EDU [18.96.7.183]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1j9MZ010607 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:45:10 -0400 (EDT) Message-Id: <200509140145.j8E1j9MZ010607@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: Week 2 Comments Date: Tue, 13 Sep 2005 21:44:59 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.02 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2816 Content-Length: 161 X-Keywords: NonJunk NotJunk In section 2.4, I don't understand the Goldbach's conjecture example. It seems to me that the both orderings of the quantifiers have the same meaning. - David From lana@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From lana@MIT.EDU Tue Sep 13 21:49:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E1nIaJ029899 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:49:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E1nHEd007036 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:49:17 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1nAbW011387 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:49:10 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E1nAmt003121; Tue, 13 Sep 2005 21:49:10 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 21:49:10 -0400 Message-ID: <20050913214910.urnbjqms7jco40wg@webmail.mit.edu> Date: Tue, 13 Sep 2005 21:49:10 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP2 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 3.14 X-Spam-Level: *** (3.14) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2817 Content-Length: 383 X-Keywords: NonJunk NotJunk The most interesting section for me was the paragraph on Page 14 explaining Russel's paradox. I was very surprising to see that a seemingly simple and intuitive notion of a set cannot be assumed in any situation. I would love to see some known attempts of establishing rules in order to determine when the use of the concepts of sets is appropriate in lecture. Svetlana Godlenberg From moscicki@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From moscicki@MIT.EDU Tue Sep 13 22:30:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E2UaaJ007807 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 22:30:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E2UZEd006633 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 22:30:35 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E2UVFx019733 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 22:30:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E2UVD9032190; Tue, 13 Sep 2005 22:30:31 -0400 Received: from NEW-TWENTY-SIX.MIT.EDU (NEW-TWENTY-SIX.MIT.EDU [18.241.5.26]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 22:30:31 -0400 Message-ID: <20050913223031.8yjyyn9kjokgg448@webmail.mit.edu> Date: Tue, 13 Sep 2005 22:30:31 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 2 Reading Comments MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_4ytbw99wiav4" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.235 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2818 Content-Length: 4044 X-Keywords: NonJunk NotJunk Hope I did it right... Angelique Moscicki Week 2 Reading Comments *need to remember* to get contrapositive from positive, need to both negate and switch the order! Proof by cases procedure: 1. make some rules that have to be true concering the truth values of the statements in order for them to be consistent. 2. for a given value of a variable, try true or false to see if the statements are in fact consistent. The number of cases will be 2^n where n is the number of INDEPENDENT variables. Thus step 1 can make life easier. Predicates: A predicate is just like a Scheme predicate. Quantifiers: These are not so terribly new to me. Don't really have too many comments, only that they are always trickier when you actually use them than when they are explained in notes. Notes seem simple, useage somewhat harder. noting that the order is much more important that i had previously considered. Math data types: Used these before. Familiar with "element of" sign. Subsets(9): These are relatively new! Need to pay attention to these. Union: inclusive Intersection: exclusive Difference: (in the first) ^~ (in the second) Complement: domain - the set Power Set: familiar with it. All the subsets + the set Product of sets: set of the sequences where 1 element from each set All these things are not hard in themselves, just difficult to remember and keep straight, and it seems a little practice using them is required even though it doesn't at first appear to be the case. Set builder: just like filter from Scheme Functions: from earlier math, domain=domain, range=codomain. Total -> uses the whole domain Surjective -> all codomain used at least once Injective -> all codomain used at most once (this is a graphical function) Bijective -> one to one function; both of the above and therefore total.  -=- MIME -=-  This message is in MIME format. --=_4ytbw99wiav4 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit Hope I did it right... --=_4ytbw99wiav4 Content-Type: text/plain; charset=ISO-8859-1; name="week2comment.txt" Content-Disposition: attachment; filename="week2comment.txt" Content-Transfer-Encoding: 7bit Angelique Moscicki Week 2 Reading Comments *need to remember* to get contrapositive from positive, need to both negate and switch the order! Proof by cases procedure: 1. make some rules that have to be true concering the truth values of the statements in order for them to be consistent. 2. for a given value of a variable, try true or false to see if the statements are in fact consistent. The number of cases will be 2^n where n is the number of INDEPENDENT variables. Thus step 1 can make life easier. Predicates: A predicate is just like a Scheme predicate. Quantifiers: These are not so terribly new to me. Don't really have too many comments, only that they are always trickier when you actually use them than when they are explained in notes. Notes seem simple, useage somewhat harder. noting that the order is much more important that i had previously considered. Math data types: Used these before. Familiar with "element of" sign. Subsets(9): These are relatively new! Need to pay attention to these. Union: inclusive Intersection: exclusive Difference: (in the first) ^~ (in the second) Complement: domain - the set Power Set: familiar with it. All the subsets + the set Product of sets: set of the sequences where 1 element from each set All these things are not hard in themselves, just difficult to remember and keep straight, and it seems a little practice using them is required even though it doesn't at first appear to be the case. Set builder: just like filter from Scheme Functions: from earlier math, domain=domain, range=codomain. Total -> uses the whole domain Surjective -> all codomain used at least once Injective -> all codomain used at most once (this is a graphical function) Bijective -> one to one function; both of the above and therefore total. --=_4ytbw99wiav4-- From sergiob@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From sergiob@MIT.EDU Tue Sep 13 22:31:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E2VIaJ007869 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 22:31:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E2VGEd007151 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 22:31:17 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-FIVE-THIRTY-THREE.MIT.EDU [18.244.7.22]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E2V50T019881 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 22:31:09 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913202016.034af478@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 20:30:49 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: Week 2 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 1.08 X-Spam-Level: * (1.08) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8E2VIaJ007869 Status: RO X-UID: 2819 Content-Length: 523 X-Keywords: NonJunk NotJunk I found the most shocking part of the reading assignment to be the comment on the Banach-Tarski Theorem on page 13. Its amazing that the same set of axioms that induces familiar arithmetic propositions like 3+3=6 also induces a theorem that states that a sphere can be split into 6 parts which can be rearranged to form 2 spheres of the same volume of the original. It is mind-boggling that math, as we know it, is based on a system of axioms with such obscure consequences. Sergio Bacallado MIT ID# 955.518.187 From ozcan@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From ozcan@MIT.EDU Tue Sep 13 22:56:40 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E2ueaJ012903 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:40 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8E2udQ2024508 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:39 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E2uciB009959 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:38 -0400 (EDT) Received: from ebrum.mit.edu (NEW-ONE-SIXTY-NINE.MIT.EDU [18.241.5.169]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E2uV3b002374 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913225322.01fca0c8@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 22:56:26 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2820 Content-Length: 190 X-Keywords: NonJunk NotJunk Hi, Can we go over the proof of the theorem on page 3: Every collection of 6 people includes a club of 3 people or a group of 3 strangers. It was kind of confusing. Thanks Yasin Ozcan From tonyng@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From tonyng@MIT.EDU Tue Sep 13 22:56:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E2usaJ013181 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E2ufEd025305 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:41 -0400 (EDT) Received: from TNG.mit.edu (BURTON-SIX-THIRTY-SEVEN.MIT.EDU [18.247.7.126]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E2uYf8025268 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:35 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913225217.01f02390@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 22:56:48 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: Reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2821 Content-Length: 506 X-Keywords: NonJunk NotJunk I found Proof by Cases (Page 3, Section 1.4) particularly interesting and surprising. I never tried proving something by starting with a statement that I am not sure of, then analyzing what happens if it is true and what happens if it is false. Intuitively, I always thought that if I have a statement that I am not sure of, I cannot get anything from it, but apparently something can be concluded if the statement leads to the same thing when the statement is true and when the statement is false. From aston@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From aston@MIT.EDU Tue Sep 13 23:04:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E34qaJ014932 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:04:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E34iuK029697 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:04:51 -0400 (EDT) Received: from aston (BOOKX.MIT.EDU [18.241.0.191]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E32trL026519 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:02:56 -0400 (EDT) Message-Id: <200509140302.j8E32trL026519@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Response Date: Tue, 13 Sep 2005 23:02:54 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW42MziUpWksdORSru3lDnRuw1dNQ== X-Spam-Score: 2.224 X-Spam-Level: ** (2.224) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2822 Content-Length: 348 X-Keywords: NonJunk NotJunk The swap of the quantifiers in Goldbach's Conjecture (page 6) slowed me down for a second in my reading as I tried to clarify what, exactly, was weird about it (why it was false), until I figured out that, as stated, we would need magical numbers p and q that sum to every even number. I'm not sure this was well clarified in the notes. - Aston From hzhou@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From hzhou@MIT.EDU Tue Sep 13 23:06:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E36QaJ015001 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:06:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E36Ou8000475 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:06:25 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-ONE-O-ONE.MIT.EDU [18.247.5.101]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E368gH027089 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:06:09 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913225747.02a5c170@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 23:06:11 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: Interesting Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2823 Content-Length: 77 X-Keywords: NonJunk NotJunk I would like to see more discussion about the set notations. - Steven Zhou From jjmonzon@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From jjmonzon@MIT.EDU Tue Sep 13 23:22:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E3MiaJ018030 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:22:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E3Mhu8012407 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:22:43 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-ONE.MIT.EDU [18.247.5.1]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E3MZAk000172 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:22:36 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913220929.0272e7d8@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 23:22:43 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2824 Content-Length: 737 X-Keywords: NonJunk NotJunk I found it very confusing to understand section 2.4 and 2.5 (page 7-8) regarding validity of propositions. I find it hard to translate English propositions to math symbols and letters and vice versa. I also find negating quantifiers to be confusing if we put it in front of the predicate or in front of the "for all", "there exist" symbols. I think that a few more examples presented in class would help rather using elegant statements using math symbols. I know the symbols make a statement more elegant concise but people who are not really used to it will find it even more confusing. Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From kromer@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From kromer@MIT.EDU Tue Sep 13 23:23:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E3NbaJ018281 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E3NZu8012994 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:36 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E3NOPa000352 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:25 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E3NOSc016989; Tue, 13 Sep 2005 23:23:24 -0400 Received: from BURTON-FIVE-SIXTY-SIX.MIT.EDU (BURTON-FIVE-SIXTY-SIX.MIT.EDU [18.247.7.55]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 23:23:24 -0400 Message-ID: <20050913232324.p5zmdou9s14oc8kc@webmail.mit.edu> Date: Tue, 13 Sep 2005 23:23:24 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2825 Content-Length: 396 X-Keywords: NonJunk NotJunk I found it surprising that "we must reject the axiom that every mathematically well-defined collection of objects is a set" (Course Notes, week 2, p. 14). The ZFC axioms are so general that I expected them to apply to every well-defined collection of objects. It seems strange that people used the concept of sets for years without being able to define precisely what a set it. -Katherine Romer From shreyes@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From shreyes@MIT.EDU Tue Sep 13 23:23:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E3NdaJ018286 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E3Ncu8013031 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:38 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E3NaGN000385 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E3Na3o007265; Tue, 13 Sep 2005 23:23:36 -0400 Received: from NEXT-FOUR-FORTY-ONE.MIT.EDU (NEXT-FOUR-FORTY-ONE.MIT.EDU [18.242.6.186]) (User authenticated as shreyes@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 23:23:36 -0400 Message-ID: <20050913232336.oftclmyihby8ocw8@webmail.mit.edu> Date: Tue, 13 Sep 2005 23:23:36 -0400 From: Shreyes Seshasai To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments Week 2 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2826 Content-Length: 864 X-Keywords: NonJunk $Label4 NotJunk Hi, This week's reading covered several topics which I was unfamiliar with, including sequences and surjective/injective functions. The part that I found most difficult (and would benefit from by seeing it in lecture) is the mapping rule. It is still a little unclear to me the relation between A and B if we know the type of the function. I was also confused about the following passage (page 12). "Everything about a function is captured by three sets: it domain, its codomain, and the set {(a, b) | f(a) = b} which is called the graph of f. Notice that the graph of a function does not determine by itself whether a function has any of the mapping properties above." This was a little confusing because looking at a graph seems to be the easiest way to quickly determine if the function is injective and/or surjective. See you in class, Shreyes Seshasai From crowell@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From crowell@MIT.EDU Tue Sep 13 23:42:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E3gHaJ019644 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:42:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E3gGu8025155 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:42:16 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E3g8k6004020 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:42:08 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E3g8uj009879; Tue, 13 Sep 2005 23:42:08 -0400 Received: from EASTCAMPUS-SEVEN-SIXTY.MIT.EDU (EASTCAMPUS-SEVEN-SIXTY.MIT.EDU [18.238.5.249]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 23:42:08 -0400 Message-ID: <20050913234208.h24nzpn7eqf4wg00@webmail.mit.edu> Date: Tue, 13 Sep 2005 23:42:08 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2827 Content-Length: 312 X-Keywords: NonJunk NotJunk I was most confused by section 7.1.2 on page 12 on "Implies." I have seen the other logical operations before (OR, AND, etc) but was unfamiliar with IMPLIES. At first the fact that a false P (in the relation P=>Q) made a true implication was troubling as well, as it seemed counter-intuitive. -Rob Crowell From brevzin@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From brevzin@MIT.EDU Wed Sep 14 00:00:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E403aJ023592 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 00:00:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E402u8007148 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 00:00:02 -0400 (EDT) Received: from TP306.mit.edu (PLP-ONE-TWENTY-FOUR.MIT.EDU [18.218.1.124]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E3xvDN007092 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 00:00:00 -0400 (EDT) Message-Id: <6.1.2.0.1.20050913234422.01a7abb0@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Tue, 13 Sep 2005 23:59:56 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: Reading Assignment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2828 Content-Length: 669 X-Keywords: NonJunk $Label4 NotJunk So it's pretty interesting that switching the orders of the definitions (such as the example with Goldbach's conjecture) provides such a VAST difference in meaning... what with "for each even integer n, there exist p,q primes such that p + q = n" and "there exist p,q primes such that for each even integer n, p + q = n". Sort of goes against common intuition that definitions are just definitions, sort of how one would expect all multiplication to be commutative. But you would be wrong. Also, in analysis, the set of natural numbers does refer to {1,2,3,...}. But there IS a footnote which states that in some systems they start at 0. (Boris) Barry Revzin From bakster@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From bakster@MIT.EDU Wed Sep 14 00:23:37 2005 Return-Path: Received: from bexxxley.mit.edu (BEXXXLEY.MIT.EDU [18.246.1.238]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E4NaaJ027310 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:23:36 -0400 Received: (from bakster@localhost) by bexxxley.mit.edu (8.12.9) id j8E4NaQv000708; Wed, 14 Sep 2005 00:23:36 -0400 Date: Wed, 14 Sep 2005 00:23:36 -0400 (EDT) From: Alexander G Bakst X-X-Sender: bakster@bexxxley To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Reading Comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 2829 Content-Length: 188 X-Keywords: NonJunk NotJunk I found Russel's Paradox very confusing - perhaps we could go over that in lecture? I would also like to go over section 3.4, set builder notation in class, if possible. Alexander Bakst From hsoumare@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From hsoumare@MIT.EDU Wed Sep 14 00:30:29 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E4UTaJ027981 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:30:29 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E4USpm025007 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:30:29 -0400 (EDT) Received: from [18.241.5.187] (NEW-ONE-EIGHTY-SEVEN.MIT.EDU [18.241.5.187]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E4UL0v013001 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:30:22 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v733) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: Week 2 Required Reading Comments Date: Wed, 14 Sep 2005 00:38:33 -0400 X-Mailer: Apple Mail (2.733) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2830 Content-Length: 260 X-Keywords: NonJunk NotJunk I would like the proof of the theorem about strangers and clubs on Page 3 to be explained in class because I did not understand it at all. I dont see how the cases discussed account for everything and how they prove the theorem. Thanks. Hamidou Soumare From letrec@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From letrec@MIT.EDU Wed Sep 14 00:40:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E4ecaJ030496 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:40:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E4ebu8002746 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:40:37 -0400 (EDT) Received: from [18.237.0.82] (TDCIP82.MIT.EDU [18.237.0.82]) (authenticated bits=0) (User authenticated as letrec@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E4eSpg013322 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:40:29 -0400 (EDT) Message-ID: <4327B6D3.8040906@mit.edu> Date: Wed, 14 Sep 2005 01:36:19 -0400 From: Alton Torregano User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Page 8, paragraph 2 X-Enigmail-Version: 0.90.0.0 X-Enigmail-Supports: pgp-inline, pgp-mime Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2831 Content-Length: 627 X-Keywords: NonJunk NotJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 It is clear that the implication doesn't work both ways for the generic case, However -- the interpretation of the converse of the statement initially felt a bit confusing as it used the domain of integers. At first the conclusion felt a bit trivial, but proves essential to constructing predicates across a single domain. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.2 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD4DBQFDJ7bR60r5GXIeeLsRAl0PAJ9EnuzaZaOUazy5lV9l3UH8pRIcTwCYqL7X be6ICpBORkWzzIw6+h3PZg== =bCLV -----END PGP SIGNATURE----- From sil_03@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From sil_03@MIT.EDU Wed Sep 14 00:58:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E4wAaJ032515 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:58:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E4w9u8013335 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:58:09 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E4w3Wv015546 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:58:03 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8E4w3K4012386; Wed, 14 Sep 2005 00:58:03 -0400 Received: from NEW-TWO-SEVENTY-ONE.MIT.EDU (NEW-TWO-SEVENTY-ONE.MIT.EDU [18.241.6.16]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 00:58:03 -0400 Message-ID: <20050914005803.w6euvys4rqfk8s08@webmail.mit.edu> Date: Wed, 14 Sep 2005 00:58:03 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: week 2 reading comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2832 Content-Length: 204 X-Keywords: NonJunk NotJunk Section 3.5(Functions) page 12: I did not understand the meaning of the graph of f which is {(a,b)|f(a)=b}. I'm not even sure how you would read it. A few examples in the next lecture might clarify it. From mracich@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mracich@MIT.EDU Wed Sep 14 01:08:57 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E58vaJ001981 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:57 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8E58uFj025698 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:56 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E58tiB012618 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:55 -0400 (EDT) Received: from MACGREGOR-FOUR-EIGHTY-FIVE.MIT.EDU (MACGREGOR-FOUR-EIGHTY-FIVE.MIT.EDU [18.239.6.230]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E58q0v014273 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:52 -0400 (EDT) Subject: Comments for Course Notes, Week 2 (Predicates & Sets) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 14 Sep 2005 01:08:49 -0400 Message-Id: <1126674529.8290.23.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.372 X-Spam-Level: * (1.372) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2833 Content-Length: 362 X-Keywords: NonJunk NotJunk I found all of section 3.5, Functions, (starting on page 11) to be very confusing. I found it difficult to grasp the concepts because I have never seen them before, and they come across as being very non-intuitive (and it doesn't help that the terms all sound alike). I would very much appreciate it if we went over this section in lecture. Moira Racich From pgroudas@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From pgroudas@MIT.EDU Wed Sep 14 01:13:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E5DmaJ002331 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 01:13:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E5Dku8022586 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 01:13:47 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E5DdIN017530 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 01:13:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E5Ddqk029686; Wed, 14 Sep 2005 01:13:39 -0400 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 01:13:39 -0400 Message-ID: <20050914011339.vdq8zinsl40k08gk@webmail.mit.edu> Date: Wed, 14 Sep 2005 01:13:39 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.351 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2834 Content-Length: 231 X-Keywords: NonJunk NotJunk I would like the examples of validity on page 8 to be explained more fully. I just feel that there should be an easy way to think about it that will click in my mind and the reading wasn't doing it for me. Thanks. -Paul Groudas From jeffhoff@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From jeffhoff@MIT.EDU Wed Sep 14 01:19:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E5JfaJ003533 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:19:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E5Jdu8025871 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:19:40 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E5JVNn018195 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:19:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20050914010639.028e6d70@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 14 Sep 2005 01:19:30 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Jeffrey D. Hoff" Subject: Reading Comments 1 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2835 Content-Length: 544 X-Keywords: NonJunk NotJunk Would like to have discussed more fully it in the next lecture. Page: 8 Under Mathematical Data Types Definition of Natural Numbers: In class and in the notes it has been said that the natural numbers include zero. However, a quick Google search [ http://www.google.com/search?hl=en&lr=&oi=defmore&q=define:Natural+Numbers ] shows that 9 out of 10 definitions are { 1,2,3... } (the one says could be either). Why is there multiple definitions and also why do we use the one that seems to be the odd definition? Thanks, Jeffrey D. Hoff From mpapi@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mpapi@MIT.EDU Wed Sep 14 01:27:00 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E5R0aJ005406 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:27:00 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8E5QxFj021691 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:27:00 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E5QxiB012985 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:26:59 -0400 (EDT) Received: from MACGREGOR-ONE-THIRTY.MIT.EDU (MACGREGOR-ONE-THIRTY.MIT.EDU [18.239.5.130]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E5Qt3a007030 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:26:55 -0400 (EDT) Subject: Reading comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: multipart/signed; micalg=pgp-sha1; protocol="application/pgp-signature"; boundary="=-fW9rLwW2Ge/MIKv1vKtw" Date: Wed, 14 Sep 2005 01:26:54 -0400 Message-Id: <1126675615.27426.6.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2836 Content-Length: 1456 X-Keywords: NonJunk NotJunk Reading comments: On page 12, just before section 4, where the "graph" of a function is discussed: "Everything about a function is captured by..." and also "Notice that the graph of a function does not determine by itself...". I understand this concept, for the most part. However, I would like to have this discussed more fully in the next lecture, particularly with regard to why it's significant, and maybe to better understand it through some examples. -- Matt This is a digitally signed message part  -=- MIME -=-  --=-fW9rLwW2Ge/MIKv1vKtw Content-Type: text/plain Content-Transfer-Encoding: quoted-printable Reading comments: On page 12, just before section 4, where the "graph" of a function is discussed: "Everything about a function is captured by..." and also "Notice that the graph of a function does not determine by itself...". I understand this concept, for the most part. However, I would like to have this discussed more fully in the next lecture, particularly with regard to why it's significant, and maybe to better understand it through some examples. -- Matt --=-fW9rLwW2Ge/MIKv1vKtw Content-Type: application/pgp-signature; name=signature.asc Content-Description: This is a digitally signed message part -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.2.5 (GNU/Linux) iD8DBQBDJ7SeiNjR4eJ+VcIRAhK6AKCnEyYtKSvz7pc2I8yeRkP8fM7bBACghWQB ipdl4FDqap4Nq+HjKW5ocrk= =WJ2y -----END PGP SIGNATURE----- --=-fW9rLwW2Ge/MIKv1vKtw-- From mwangi@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mwangi@MIT.EDU Wed Sep 14 01:57:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E5vMaJ013555 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 01:57:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E5v1u8017795 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 01:57:01 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-FOUR-SEVENTY-FIVE.MIT.EDU [18.241.6.220]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E5uwsX022355 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 01:56:58 -0400 (EDT) Message-Id: <6.2.1.2.2.20050914014400.03abde20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 14 Sep 2005 01:56:54 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: Week 2 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2837 Content-Length: 381 X-Keywords: NonJunk NotJunk Passage: 3.3 sequences Page: 10 I found this passage most difficult because of the idea of multiplying two sets. Even after reading the passage it remained unclear what it means to multiply two sets. The example did little to help since it multiplied the set of natural numbers by itself and it was not possible to tell which numbers came from which set in the answer. From alisonc@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From alisonc@MIT.EDU Wed Sep 14 02:04:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E64haJ014145 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 02:04:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E64e97021517 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 02:04:41 -0400 (EDT) Received: from [18.238.6.66] (EASTCAMPUS-EIGHT-THIRTY-THREE.MIT.EDU [18.238.6.66]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E611KV022729 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 02:01:02 -0400 (EDT) Message-ID: <4327BCA2.3020009@mit.edu> Date: Wed, 14 Sep 2005 02:01:06 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Reading comments -- Course Notes, Week 2 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2838 Content-Length: 383 X-Keywords: NonJunk NotJunk I found the section on quantifiers (pages 6&7) most difficult -- attempting to logically dissect pieces of English as naturally spoken (as opposed to carefully phrased verbose math-english) is really tough, so the examples took quite a bit of thought to really get them to sink in. (They were entertaining though -- I'd like to try skiing over magma sometime.) Alison Cichowlas From mrivas03@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mrivas03@MIT.EDU Wed Sep 14 02:36:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E6aYaJ017947 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 02:36:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E6aX95007730; Wed, 14 Sep 2005 02:36:33 -0400 (EDT) Received: from el-ternero.mit.edu (MRIVAS03.MIT.EDU [18.237.0.105]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E6aPxT025094 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 14 Sep 2005 02:36:27 -0400 (EDT) Message-Id: <6.2.3.4.2.20050914013211.02d8ec78@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 14 Sep 2005 02:36:05 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2839 Content-Length: 476 X-Keywords: NonJunk NotJunk In the reading page number 3 - Reading Number 2 {Predicates and Sets} - Passage Proof by Cases the split of cases into subcases became surprisingly difficult to follow. Explanation of Theorem and Proof would be helpful. The difference of the two assertions one which is valid and the other not valid on page 8 of Reading number 2 was not quite clear. The second proof showing why the second was not valid was clear but the first proof was not. Thanks, Manuel Rivas From ryan786@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From ryan786@MIT.EDU Wed Sep 14 03:02:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E72faJ022222 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:02:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E72e95020231 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:02:40 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E72bol026364 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:02:37 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E72bIx004010; Wed, 14 Sep 2005 03:02:37 -0400 Received: from PSK-TWENTY-EIGHT.MIT.EDU (PSK-TWENTY-EIGHT.MIT.EDU [18.217.1.28]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 03:02:37 -0400 Message-ID: <20050914030237.iqf7kklo2n8kwwwk@webmail.mit.edu> Date: Wed, 14 Sep 2005 03:02:37 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: required comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2840 Content-Length: 378 X-Keywords: NonJunk NotJunk I found the segment about functions (3.5 FUNCTIONS, p.11/14) most difficult. I had never seen surjective, injective, or bijective before, so I had to read it over a few times. I still don't fully grasp the definitions provided, although the visual representation helped. When doing the tutorial problems for this section, I had to constantly re-look at the pdf. -Ryan Young From scholtz@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From scholtz@MIT.EDU Wed Sep 14 03:18:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E7IHaJ024739 for <6042-probs@theory.csail.MIT.EDU>; Wed, 14 Sep 2005 03:18:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E7IF95027753 for <6042-probs@theory.csail.MIT.EDU>; Wed, 14 Sep 2005 03:18:15 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7ICb8027043 for <6042-probs@theory.csail.MIT.EDU>; Wed, 14 Sep 2005 03:18:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8E7ICC9021498; Wed, 14 Sep 2005 03:18:12 -0400 Received: from c-24-34-20-46.hsd1.ma.comcast.net (c-24-34-20-46.hsd1.ma.comcast.net [24.34.20.46]) (User authenticated as scholtz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 03:18:12 -0400 Message-ID: <20050914031812.6b9vnnk4s68g4440@webmail.mit.edu> Date: Wed, 14 Sep 2005 03:18:12 -0400 From: Edward A Scholtz To: 6042-probs@theory.csail.MIT.EDU Subject: Reading Comments - LN2 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 1.801 X-Spam-Level: * (1.801) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2841 Content-Length: 258 X-Keywords: NonJunk NotJunk I haven't had much experience with sets and it would help if you spent some time going over the notation. In section 3.2 on page 9, I understood the subset symbol but I was confused by the subset symbol without the bar underneath it (the two are not equal?) From rshearer@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From rshearer@MIT.EDU Wed Sep 14 03:21:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E7L6aJ024943 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:21:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E7L595029104 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:21:05 -0400 (EDT) Received: from [18.240.7.234] (MCCORMICK-SEVEN-FORTY-FIVE.MIT.EDU [18.240.7.234]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7KxaJ027171 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:21:03 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Transfer-Encoding: 7bit Message-Id: <15b0f8a81bd1e0c484a4ad7b01c2b04f@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: Comments Date: Wed, 14 Sep 2005 03:20:58 -0400 X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2842 Content-Length: 592 X-Keywords: NonJunk NotJunk "The product operation is one link between sets and sequences. A product of sets, S1 * S2 * S3 * ... SN, is a new set consisting of all sequences where the first component is drawn from S1, the second from S2, and so forth" I found the above passage, found on page 10 of the notes, to be confusing and I would like it to be discussed in the lecture. I think that if I saw a full example of the process one goes through to get the product of two sets I would be able to understand it. The way it is worded in the notes doesn't give me a good idea of the steps I should go through. From crowell@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From crowell@MIT.EDU Wed Sep 14 03:26:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E7QFaJ026457 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:26:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E7QE95001602 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:26:14 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7Q715027399 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:26:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8E7Q7Bt021796; Wed, 14 Sep 2005 03:26:07 -0400 Received: from EASTCAMPUS-SEVEN-SIXTY.MIT.EDU (EASTCAMPUS-SEVEN-SIXTY.MIT.EDU [18.238.5.249]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 03:26:07 -0400 Message-ID: <20050914032607.j658zx21e64g8wos@webmail.mit.edu> Date: Wed, 14 Sep 2005 03:26:07 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Re: Reading comments References: <20050913234208.h24nzpn7eqf4wg00@webmail.mit.edu> <20050914032442.lp7423xe2x6sc8o0@webmail.mit.edu> In-Reply-To: <20050914032442.lp7423xe2x6sc8o0@webmail.mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.041 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2843 Content-Length: 684 X-Keywords: NonJunk NotJunk It seems I misunderstood and sent comments for the first week reading. Here's my comment for the second week. I was most surprised by the closing remarks of the notes (page 13-14). It is amazing that so much depends on a mathematics which has no axiomatic foundation. -Rob Crowell Quoting Robert Crowell : > >> I was most confused by section 7.1.2 on page 12 on "Implies." I >> have seen the >> other logical operations before (OR, AND, etc) but was unfamiliar >> with IMPLIES. >> At first the fact that a false P (in the relation P=>Q) made a true >> implication was troubling as well, as it seemed counter-intuitive. >> >> -Rob Crowell >> >> >> >> > > > From kjhollen@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From kjhollen@MIT.EDU Wed Sep 14 03:30:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E7UfaJ026857 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:30:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E7Ud95003738 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:30:40 -0400 (EDT) Received: from MACGREGOR-ONE-O-NINE.MIT.EDU (MACGREGOR-ONE-O-NINE.MIT.EDU [18.239.5.109]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7UZ2m027611 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:30:38 -0400 (EDT) Subject: reading for September 14 From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 14 Sep 2005 03:30:35 -0400 Message-Id: <1126683035.24797.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 0.978 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2844 Content-Length: 462 X-Keywords: NonJunk NotJunk Hello, I found the passage about Sequences (on page 10 of ln2) somewhat confusing. I understand what a sequence is, but the multiplication of sets and sequences is something that is difficult to understand and that I hope will be discussed more in lecture. Also, the terms for functions in section 3.5 (on page 11) were explained well but they are brand new to me and I would really appreciate learning them in more depth in lecture. Kate Hollenbach kjhollen From petek@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From petek@MIT.EDU Wed Sep 14 05:02:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E92eaJ006335 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:02:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E92dDp016496 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:02:39 -0400 (EDT) Received: from [18.194.1.144] (SN-ONE-FORTY-FOUR.MIT.EDU [18.194.1.144]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E92bdw000865 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:02:38 -0400 (EDT) Message-ID: <4327E734.9050908@mit.edu> Date: Wed, 14 Sep 2005 05:02:44 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Comments Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -1.886 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2845 Content-Length: 885 X-Keywords: NonJunk NotJunk Latter half of section 2.6, on page 8:

The fact that mathematical symbols are not yet second-nature for me means that shuffling them around makes it difficult for me to follow what's being done.  Perhaps I need to take a second look later on, but it's hard for me to see why the last statement is not true save for the example given.


 
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Pete Kruskall
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From kkdb@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From kkdb@MIT.EDU Wed Sep 14 08:02:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EC2waJ032227 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:02:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EC2vko022772 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:02:57 -0400 (EDT) Received: from KKDB (ASHDOWN-FIVE-NINETY-FOUR.MIT.EDU [18.250.7.83]) (authenticated bits=0) (User authenticated as kkdb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EC2tux013786 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:02:55 -0400 (EDT) Message-ID: <001601c5b924$3dfb5590$5307fa12@KKDB> From: "Kaustuv DeBiswas" To: <6042-probs@theory.lcs.mit.edu> Subject: *Required* Comments for Reading Date: Wed, 14 Sep 2005 08:02:55 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0013_01C5B902.B661C0E0" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.431 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2846 Content-Length: 2469 X-Keywords: NonJunk NotJunk Passage 1.4 - Proof by Cases, Pg 3 I found the theorem surprising and would like the proof to be discussed in the next lecture. It was surprising because in a collection of 6 people, there might be a situation where nobody has met each other. The cases in the example need clarification. -------------------------------------------------------------------------------------------------------------------------------- Kaustuv DeBiswas | MIT - SMArchS - Computation | kkdb@mit.edu | +1 (617) 230 6471  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0013_01C5B902.B661C0E0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Passage 1.4 - Proof by Cases, Pg 3 I found the theorem surprising and would like the proof to be discussed = in the next lecture. It was surprising because in a collection of 6 = people, there might be a situation where nobody has met each other. The = cases in the example need clarification. -------------------------------------------------------------------------= ------------------------------------------------------- Kaustuv DeBiswas | MIT - SMArchS - Computation | kkdb@mit.edu | +1 = (617) 230 6471 ------=_NextPart_000_0013_01C5B902.B661C0E0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Passage 1.4 - Proof by Cases, Pg=20 3
I found the theorem surprising and = would like the=20 proof to be discussed in the next lecture. It was surprising because in = a=20 collection of 6 people, there might be a situation where nobody has met = each=20 other. The cases in the example need clarification.
 
----------------------------------------------------------------= ----------------------------------------------------------------
Kaust= uv=20 DeBiswas | MIT - SMArchS - Computation |  kkdb@mit.edu  | +1 (617) 230=20 6471
 
------=_NextPart_000_0013_01C5B902.B661C0E0-- From kktyan@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From kktyan@MIT.EDU Wed Sep 14 08:21:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ECLTaJ001524 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:21:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ECLRko006429 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:21:28 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ECLQxK018162 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:21:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8ECLQAj019872; Wed, 14 Sep 2005 08:21:26 -0400 Received: from BURTON-TWO-FORTY-ONE.MIT.EDU (BURTON-TWO-FORTY-ONE.MIT.EDU [18.247.5.241]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 08:21:26 -0400 Message-ID: <20050914082126.by2bkffais1mw444@webmail.mit.edu> Date: Wed, 14 Sep 2005 08:21:26 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.032 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8ECLTaJ001524 Status: RO X-UID: 2847 Content-Length: 1028 X-Keywords: NonJunk NotJunk Here are my comments on the first two readings for 6.042J: Good & bad proofs: In the reading for good and bad proofs, I found Theorem 5.1 (DeMorgan's Law for Sets) and its subsequent proof on page 8 most difficult to understand, primarily because I am unfamiliar with set notation and meaning. Overall, the entire reading and its concepts were not too difficult to understand; after looking up the meaning of the set notation online, I was able to understand the proof and theorem much more easily. Proofs by Contradiction and Cases/Predicate Logic: The most surprising part of this reading was the product operation on page 10: "The product operation is one link between sets and sequences. A product of sets, S1S2?Sn, is a new set consisting of all sequences where the first component is drawn from S1, the second from S2, and so forth." I had not known that one could take the product of two sets, and that the result was a set of sequences. - Karena Tyan -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From cbossard@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From cbossard@MIT.EDU Wed Sep 14 08:27:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ECRcaJ002520 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:27:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ECRbko010714 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:27:37 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ECRVMR019448 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:27:31 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8ECRVim004877; Wed, 14 Sep 2005 08:27:31 -0400 Received: from NEXT-SEVEN-THIRTEEN.MIT.EDU (NEXT-SEVEN-THIRTEEN.MIT.EDU [18.242.7.202]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 08:27:31 -0400 Message-ID: <20050914082731.93o0c5h9waec8ccw@webmail.mit.edu> Date: Wed, 14 Sep 2005 08:27:31 -0400 Disposition-Notification-To: cbossard@MIT.EDU X-Confirm-Reading-To: cbossard@mit.edu X-PMRQC: 1 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 2 Reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2848 Content-Length: 336 X-Keywords: NonJunk $MDNSent NotJunk passage: Section 3.5 Functions pgs. 11 - 12 I would like to see this dicussed more fully in class because I am having trouble with the injection, surjection, etc. concerning functions. It would be easier if I say some actual examples that had words or numbers assigned to a and b to help me make better sense of them. Cynthia Bossard From mmt@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From mmt@MIT.EDU Wed Sep 14 08:34:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ECYoaJ004011 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:34:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ECYnko015999 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:34:49 -0400 (EDT) Received: from all-night-tool.mit.edu (ALL-NIGHT-TOOL.MIT.EDU [18.7.16.70]) (authenticated bits=56) (User authenticated as mmt@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ECYlZ3021230 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:34:47 -0400 (EDT) Received: (from mmt@localhost) by all-night-tool.mit.edu (8.12.9) id j8ECYlrv013542; Wed, 14 Sep 2005 08:34:47 -0400 (EDT) Date: Wed, 14 Sep 2005 08:34:47 -0400 From: Mark M Tobenkin To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Comments Message-ID: <20050914123447.GB13098@all-night-tool.mit.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.5.6i X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2849 Content-Length: 512 X-Keywords: NonJunk NotJunk The most interesting aspect of the ln2.pdf reading for me was the Power Set/Sequence section of the reading. Particularly, there seems to be a relationship between the "power set" P(A), which implies AxA and the multiplication operator for sequences (NxN) which is implicit in the reading. Can we make more explicit what the actual members of a power set are, and the origins of the sequence/set multiplication is? -- As for a picture, if it isn't worth a thousand words, the hell with it. - Ad Reinhardt From ajshafer@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From ajshafer@MIT.EDU Wed Sep 14 08:45:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ECjhaJ006659 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:45:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ECjfko025575 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:45:41 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ECjXk2024285 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:45:34 -0400 (EDT) Message-ID: <000701c5b92a$30caa8c0$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.lcs.mit.edu> Subject: Set Functions: ln2.pdf Date: Wed, 14 Sep 2005 08:45:29 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2850 Content-Length: 393 X-Keywords: NonJunk NotJunk On page 11 you describe functions that map one set to another. I'm still confused by the four types: total, surjective, injective, and bijective. Can we get an example of each of these? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From lmccart@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: iso-8859-1-unix Mail-from: From lmccart@MIT.EDU Wed Sep 14 08:59:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ECxDaJ008551 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:59:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ECxBko007616 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:59:11 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ECxAer028412 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:59:10 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8ECxAO3012543; Wed, 14 Sep 2005 08:59:10 -0400 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 08:59:10 -0400 Message-ID: <20050914085910.oy8307ow1sgsooww@webmail.mit.edu> Date: Wed, 14 Sep 2005 08:59:10 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments 9/14 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2851 Content-Length: 344 X-Keywords: NonJunk NotJunk All of the material in this reading seemed pretty straight forward, but the passage I found most confusing was section 3.5: Functions. I understood the definitions for total, surjective, injective, and bijective, but I still found the tutor problem covering this information slightly confusing. I would therefore like this reviewed in class. From aka_kame@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: emacs-mule-unix Mail-from: From aka_kame@MIT.EDU Wed Sep 14 09:40:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EDeFaJ019548 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EDeEko019666 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:14 -0400 (EDT) Received: from [18.234.1.72] (DP-SEVENTY-TWO.MIT.EDU [18.234.1.72]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EDe7cs015660 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:08 -0400 (EDT) Message-ID: <43282837.1090503@mit.edu> Date: Wed, 14 Sep 2005 09:40:07 -0400 From: Akari Kameyama User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 2 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2852 Content-Length: 502 X-Keywords: NonJunk NotJunk A function f : A ! B is: total if every element of A is assigned to some element of B; otherwise, f is called a partial function. surjective if every element of B is mapped to at least once, that is, 8b 2 B9a 2 A. f(a) = b. injective if every element of B is mapped to at most once. bijective if f is total, surjective, and injective. In particular, each element of B is mapped to exactly once. This is confusing, it would be helpful if there were more specific examples given in lecture. From veracarr@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From veracarr@MIT.EDU Wed Sep 14 09:40:24 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EDeOaJ019582 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:24 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EDeORT006899 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:24 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8EDeL0v000765 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:22 -0400 (EDT) Message-ID: <432844A7.2080708@mit.edu> Date: Wed, 14 Sep 2005 09:41:27 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Comments #1 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2853 Content-Length: 403 X-Keywords: NonJunk NotJunk Functions (pg 11) I found this concept interesting in how they denote the relative size of the function's domain and codomain. However, the part I found difficult was when I had to apply these concepts, for example, in the tutor problems. I didn't understand the cases where surjection and injection ended up being equal to the other set. E.g. If f is a surjection/injection then |f(A)| ____ |B|. From mukkala@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From mukkala@MIT.EDU Wed Sep 14 10:13:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EEDsaJ025718 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:13:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EEDrko028258 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:13:53 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EEDjdA000919 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:13:47 -0400 (EDT) Message-Id: <6.2.1.2.2.20050914100717.01eb2288@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 14 Sep 2005 10:13:46 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: Reading Comments 9/14 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2854 Content-Length: 394 X-Keywords: NonJunk NotJunk In Section 2.4, entitled Order of Quantifiers, it explains how the order of different kinds of quantifiers changes the meaning of a proposition. I understand the example with the set of Americans and the set of Dreams. However, I don't understand how swapping quantifiers in Goldbach's Conjecture creates a "patently false statement"; it seems true to me. Thanks, Praveen Pamidimukkala From yaser@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: undecided-unix Mail-from: From yaser@MIT.EDU Wed Sep 14 10:16:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EEGBaJ026487 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:16:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EEG9ko001225 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:16:10 -0400 (EDT) Received: from Charizard (NEW-THREE-TWENTY-FIVE.MIT.EDU [18.241.6.70]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EEG4DU002194 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:16:06 -0400 (EDT) Message-Id: <200509141416.j8EEG4DU002194@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: .042 email on readings Date: Wed, 14 Sep 2005 10:16:02 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000C_01C5B915.511612A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW5NtTAkvN6oBFZQ7W8eW1sDHHcVg== X-Spam-Score: -0.229 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2855 Content-Length: 13771 X-Keywords: NonJunk $Forwarded NotJunk *Required* Comments for Reading: send email to 6042-probs@theory.lcs.mit.edu citing a passage in the reading - including its page number - and explaining, in at most three sentences, why you found it 1. most difficult, or 2. most surprising, or 3. would like to have discussed more fully it in the next lecture. Passage: "2.5 Negating Quantifiers" (page 7) I found this slightly difficult (#1 above) to follow near the expression of the logic equivalence (line marked "(1)") because the inversion, simple as it is, is not making intuitive sense to me. Could you please discuss this more fully in the next lecture? Thanks! (Please note, my athena id is actually "ykhan" in case you're using that for scorekeeping). Also- on the tutor, question TP 2.1.1 number 3: I am confident that the answer is R>P (getting an A implies you got an A on the final). However, I tried entering that and it didnd't accept. I then tried all sorts of combinations and permutations (and spent a long time doing it). In the end I got frustrated, wrote some random answer (in my case P>R as that was the last of the 20 some combos), and hit submit, only to find that the expected answer was "R->P". So could I get my point back as my original answer would have been correct had the expected answer not had the "-" in it? (In your notation note at the top, we were told NOT to put in the "-" in our "implies" answer). Thank you very much, and have wonderful time! _Yaser  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_000C_01C5B915.511612A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit *Required* Comments for Reading: send email to 6042-probs@theory.lcs.mit.edu citing a passage in the reading - including its page number - and explaining, in at most three sentences, why you found it 1. most difficult, or 2. most surprising, or 3. would like to have discussed more fully it in the next lecture. Passage: "2.5 Negating Quantifiers" (page 7) I found this slightly difficult (#1 above) to follow near the expression of the logic equivalence (line marked "(1)") because the inversion, simple as it is, is not making intuitive sense to me. Could you please discuss this more fully in the next lecture? Thanks! (Please note, my athena id is actually "ykhan" in case you're using that for scorekeeping). Also- on the tutor, question TP 2.1.1 number 3: I am confident that the answer is R>P (getting an A implies you got an A on the final). However, I tried entering that and it didnd't accept. I then tried all sorts of combinations and permutations (and spent a long time doing it). In the end I got frustrated, wrote some random answer (in my case P>R as that was the last of the 20 some combos), and hit submit, only to find that the expected answer was "R->P". So could I get my point back as my original answer would have been correct had the expected answer not had the "-" in it? (In your notation note at the top, we were told NOT to put in the "-" in our "implies" answer). Thank you very much, and have wonderful time! _Yaser ------=_NextPart_000_000C_01C5B915.511612A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

*Required* Comments for Reading: send email to =

6042-probs@theory.lcs.mit.edu

citing a passage in the reading = – including its page number – and explaining, in at most three = sentences, why you found it

  1. most difficult, or
  2. most surprising, or
  3. would like to have discussed more fully it in the next = lecture.

 

Passage: = “2.5 Negating Quantifiers” (page 7)

 

I found this = slightly difficult (#1 above) to follow near the expression of the logic = equivalence (line marked “(1)”) because the inversion, simple as it is, = is not making intuitive sense to me. Could you please discuss this more fully = in the next lecture?

 

Thanks!

 

(Please note, = my athena id is actually = “ykhan” in case you’re using that for scorekeeping)…

 

Also- on the = tutor, question TP 2.1.1 number 3:

I am = confident that the answer is R>P (getting an A implies you got an A on the final). = However, I tried entering that and it didnd’t = accept. I then tried all sorts of combinations and permutations (and spent a long = time doing it). In the end I got frustrated, wrote some random answer (in my = case P>R as that was the last of the 20 some combos), and hit submit, only = to find that the expected answer was “R->P”. =

 

So could I = get my point back as my original answer would have been correct had the = expected answer not had the “-“ in it? (In = your notation note at the top, we were told NOT to put in the “-“ in our “implies” = answer).

 

Thank you = very much, and have wonderful time!

 

_Yaser =

------=_NextPart_000_000C_01C5B915.511612A0-- From jacques@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: iso-8859-1-unix Mail-from: From jacques@MIT.EDU Wed Sep 14 10:23:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EENdaJ029455 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:23:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EENcko010437 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:23:38 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EENVII006014 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:23:31 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8EENVLR026872; Wed, 14 Sep 2005 10:23:31 -0400 Received: from EASTCAMPUS-NINE-THIRTY-THREE.MIT.EDU (EASTCAMPUS-NINE-THIRTY-THREE.MIT.EDU [18.238.6.166]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 10:23:31 -0400 Message-ID: <20050914102331.q6018mus1eck4884@webmail.mit.edu> Date: Wed, 14 Sep 2005 10:23:31 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: Comments for Reading: Week 2 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2856 Content-Length: 461 X-Keywords: NonJunk NotJunk On page 11 of Course Notes, Week 2: "surjective if every element of B is mapped to at least once, that is, ... injective if every element of B is mapped to at most once." I would like to see this covered in lecture, or more examples given, because I found the descriptions a little confusing (easy to mix up). As it says in the text, the names are "hopelessly unmemorable and nondescriptive," so anything to help me get a handle on which is which. From cvnguyen@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: undecided-unix Mail-from: From cvnguyen@MIT.EDU Wed Sep 14 10:33:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EEX8aJ032766 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:33:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EEX7ko022106 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:33:07 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-SEVEN-FORTY-FOUR.MIT.EDU [18.238.5.233]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EEX0Qv011056 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:33:01 -0400 (EDT) Message-ID: <001301c5b939$3777a6f0$e905ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Comments Week 2 Date: Wed, 14 Sep 2005 10:33:03 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.596 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2857 Content-Length: 364 X-Keywords: NonJunk NotJunk I found the discussion of Godel's incompleteness theorem on page 13 of the notes particularly worthy of discussion in class. From what I understand of it, the theorem demonstrates that it is impossible to know everything about a system from just the underlying rules of the system themselves. This has interesting philosophical consequences. --Chieu Nguyen From harelw@mit.edu Wed Sep 14 21:47:34 2005 X-Coding-System: undecided-unix Mail-from: From harelw@gmail.com Wed Sep 14 10:36:06 2005 Return-Path: Received: from zproxy.gmail.com (zproxy.gmail.com [64.233.162.192]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EEa6aJ000650 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:36:06 -0400 Received: by zproxy.gmail.com with SMTP id o1so133297nzf for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 07:36:01 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type; b=KAgeNdTclPl3aeL/GJz2ZPbcBKKyKuAwgnvGYx2DTjZPnKlcP6VX9bv7Zvdej8d9jgJlJupikIafN+EppdT8PJ6y6bbUwiV4Lx+35dQ/CA0TiWzDmw2ORCjIGBsdZxO5Ve+sH5SCqW+SaNd8HvVymZEqk5ZbU+rHjC1MxnEuoCw= Received: by 10.36.33.19 with SMTP id g19mr2103558nzg; Wed, 14 Sep 2005 07:35:53 -0700 (PDT) Received: by 10.36.109.9 with HTTP; Wed, 14 Sep 2005 07:35:52 -0700 (PDT) Message-ID: <8c5248a80509140735499af1ba@mail.gmail.com> Date: Wed, 14 Sep 2005 10:35:53 -0400 From: "Harel M. Williams" Reply-To: "Harel M. Williams" Sender: harelw@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Reading for 9/14 Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_20643_29364537.1126708553001" Status: RO X-UID: 2858 Content-Length: 863 X-Keywords: NonJunk NotJunk I found functions to be most difficult because there weren't many examples in the reading. Also, there are a lot of definitions to know. ~Harel Williams  -=- MIME -=-  ------=_Part_20643_29364537.1126708553001 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found functions to be most difficult because there weren't many examples= =20 in the reading. Also, there are a lot of definitions to know. ~Harel Williams ------=_Part_20643_29364537.1126708553001 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found functions to be most difficult because there weren't many examples in the reading. Also, there are a lot of definitions to know.

~Harel Williams
------=_Part_20643_29364537.1126708553001-- From hejing85@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: gb2312-unix Mail-from: From hejing85@MIT.EDU Wed Sep 14 10:43:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EEhKaJ006159 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:43:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EEhJko004390 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:43:19 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EEhGC5016020 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:43:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8EEhGFB024704; Wed, 14 Sep 2005 10:43:16 -0400 Received: from NEW-ONE-FIFTEEN.MIT.EDU (NEW-ONE-FIFTEEN.MIT.EDU [18.241.5.115]) (User authenticated as hejing85@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 10:43:16 -0400 Message-ID: <20050914104316.t28wssw0592s8c84@webmail.mit.edu> Date: Wed, 14 Sep 2005 10:43:16 -0400 From: Jing He To: 6042-probs@theory.lcs.MIT.EDU Subject: 6042 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=GB2312 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2859 Content-Length: 238 X-Keywords: NonJunk NotJunk I found that the passage on page 11 about the functions surjective, injective, and bijective the most difficult to understand. This could be because I had never heard of them before and it's hard to visualize how A maps to B. --Jing He From rehughes@MIT.EDU Wed Sep 14 21:47:35 2005 X-Coding-System: undecided-unix Mail-from: From rehughes@MIT.EDU Wed Sep 14 11:10:58 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EFAwaJ011829 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:58 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8EFAvRJ000461 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:57 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EFAv93028305 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:57 -0400 (EDT) Received: from [18.186.6.142] (TANG-ELEVEN-SIXTY-FIVE.MIT.EDU [18.186.6.142]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8EFAo0v006311 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:50 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) To: 6042-probs@theory.lcs.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-1-227077757 From: Richard Hughes Subject: Required Comments for Reading Date: Wed, 14 Sep 2005 11:10:49 -0400 X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2860 Content-Length: 782 X-Keywords: NonJunk NotJunk Page 11: Functions. This was easily the hardest simply because injunctive, surjective, and so on are such opaque terms. What are their root words? /no idea/ -=- MIME -=-  --Apple-Mail-1-227077757 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; format=flowed Page 11: Functions. This was easily the hardest simply because injunctive, surjective, and so on are such opaque terms. What are their root words? /no idea/ --Apple-Mail-1-227077757 Content-Transfer-Encoding: 7bit Content-Type: text/enriched; charset=US-ASCII Page 11: Functions. This was easily the hardest simply because injunctive, surjective, and so on are such opaque terms. What are their root words? /no idea/ --Apple-Mail-1-227077757-- From fgreen@MIT.EDU Wed Sep 14 21:47:35 2005 X-Coding-System: iso-8859-1-unix Mail-from: From fgreen@MIT.EDU Wed Sep 14 14:17:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EIHuaJ020108 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 14:17:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8EIHt0p026771 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 14:17:55 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EIHmrv017992 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 14:17:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8EIHmKQ015788; Wed, 14 Sep 2005 14:17:48 -0400 Received: from GREENONE.MIT.EDU (GREENONE.MIT.EDU [18.238.2.183]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 14:17:48 -0400 Message-ID: <20050914141748.0wct7vu18ug4c0ck@webmail.mit.edu> Date: Wed, 14 Sep 2005 14:17:48 -0400 From: Forrest O Green To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2861 Content-Length: 108 X-Keywords: NonJunk NotJunk I found page 11 section 3.5 to be somewhat confusing primarily on account of the terminology. -Forrest From manosdefierro@gmail.com Wed Sep 14 21:47:35 2005 X-Coding-System: undecided-unix Mail-from: From manosdefierro@gmail.com Wed Sep 14 14:46:01 2005 Return-Path: Received: from mintaka.lcs.mit.edu (mintaka.lcs.mit.edu [18.26.0.36]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EIk0aJ027513 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 14:46:00 -0400 Received: from wproxy.gmail.com (wproxy.gmail.com [64.233.184.197]) by mintaka.lcs.mit.edu (8.12.9/8.12.9) with ESMTP id j8EIk0SF046965 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 14:46:00 -0400 (EDT) (envelope-from manosdefierro@gmail.com) Received: by wproxy.gmail.com with SMTP id i5so12897wra for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:44:55 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=jE8tapmEaHjrVEtj3yqP/8agrYaZUOqii86iTAjCp+/JhMydUFutglKGjZi4irDxMb6iTiTp2xrivxCmMj3mLm/Q4lqB6yMlmxzTQr3SMjPICe41dafjSnCJkT+Ow4RBD5vTFS/DPE0XxIK39/uFxaC+udDxD0ZIyR/Rpie1TPw= Received: by 10.54.44.11 with SMTP id r11mr1133758wrr; Wed, 14 Sep 2005 03:09:01 -0700 (PDT) Received: by 10.54.151.13 with HTTP; Wed, 14 Sep 2005 03:09:01 -0700 (PDT) Message-ID: <2bac7c5d050914030950528d75@mail.gmail.com> Date: Wed, 14 Sep 2005 06:09:01 -0400 From: Mario Marrufo Reply-To: joeyrufo@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: comments for reading Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_17744_4553015.1126692541798" Status: RO X-UID: 2862 Content-Length: 31430 X-Keywords: NonJunk NotJunk I found the lemma on page 12 somewhat confusing as I am unfamiliar with the terms that were used ("surjective," etc.). (But basically, everything made sense.)  -=- MIME -=-  ------=_Part_17744_4553015.1126692541798 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found the lemma on page 12 somewhat confusing as I am unfamiliar with the= =20 terms that were used ("surjective," etc.). (But basically, everything made= =20 sense.) ------=_Part_17744_4553015.1126692541798 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found the lemma on page 12 somewhat confusing as I am unfamiliar with the terms that were used ("surjective," etc.). (But basically, everything made sense.)
------=_Part_17744_4553015.1126692541798-- X-Gmail-Received: 4f9137ba2b528ad8158b2bd0308727d64ab03014 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:45:18 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:45:18 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: nancyk@mit.edu Subject: Re: Reading assignment -- Weeks 1 & 2 Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44593_9862883.1127227518585" Delivered-To: minilek@gmail.com ------=_Part_44593_9862883.1127227518585 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Nancy, Regarding why false implies everything. A-->B in English basically means=20 "whenever A is true, B is true". But if A is actually the literal "False",= =20 then A is never true. So the statement that "whenever A is true, B is true"= =20 is what we call *vacuously true*. In other words, it is true that whenever = A=20 is true B is true, because A is never true. Let me know if this cleared=20 things up. -Jelani ------------------------------------------------------------------------ Hi, here are my Week 2 and Week 1 comments: Week 2 Reading -- "Predicates and Sets": Passage 3.5 "Functions -- Lemma (Mapping Rule)," pg 12: At first I struggled with the implication in the first bullet ( If f : A ->= =20 B is surjective, then |A| >=3D |B|. ). I thought the proposition needed to=20 mention surjective *and* total. Only after doing the online tutor problems did I realize that the implication is valid because a function can only have one y-value for each x-value (or only one resulting value for each argument to= =20 the function). --------------------------------------------------------------------------- Week 1 Reading -- "Proofs": Passage 7.1.2 "Implies," pg. 12: Even after careful reading, for some reason I still don't understand the=20 truth table results of "P implies Q" or "if P then Q." The summary statement at= =20 the end of 7.1.2 ("An implication is true when the if-part is false or the then-part is true") is a very helpful rule, but I found the basic logic of= =20 the table difficult. Why does false imply true and false imply false? -- Nancy Keuss ------=_Part_44593_9862883.1127227518585 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Nancy,

Regarding why false implies everything.  A-->B in English basically means "whenever A is true, B is true".  But if A i= s actually the literal "False", then A is never true.  So the statement that "whenever A is true, B is true" is what we call *vacuously true*.  In other words, it is true that whenever A is true B is true, because A is never true.  Let me know if this cleared things up.

-Jelani
------------------------------------------------------------------------ Hi, here are my Week 2 and Week 1 comments:


Week 2 Reading -- "Predicates and Sets":

Passage 3.5 "Functions -- Lemma (Mapping Rule)," pg 12:

At first I struggled with the implication in the first bullet ( If  f = : A -> B
is surjective, then |A| >=3D |B|. ). I thought the proposition needed to= mention
surjective *and* total. Only after doing the online tutor problems did I realize that the implication is valid because a function can only have one<= br> y-value for each x-value (or only one resulting value for each argument to = the
function).

---------------------------------------------------------------------------=


Week 1 Reading -- "Proofs":

Passage 7.1.2 "Implies," pg. 12:

Even after careful reading, for some reason I still don't understand the tr= uth
table results of "P implies Q" or "if P then Q." The su= mmary statement at the
end of 7.1.2 ("An implication is true when the if-part is false or the=
then-part is true") is a very helpful rule, but I found the basic logi= c of the
table difficult. Why does false imply true and false imply false?




-- Nancy Keuss X-Gmail-Received: 6801f1c6374cd99a2cb4f126dfc2ca53ec2247bc Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:37:02 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:37:02 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: benlu@mit.edu Subject: Re: [6.042] comment on reading 2 Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44512_31489427.1127227022695" Delivered-To: minilek@gmail.com ------=_Part_44512_31489427.1127227022695 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Ben, S =3D {x | P(x) } is the set of all elements x such that P(x) is true. If= =20 you've used Scheme before, you can think of it as "filter" in Scheme. So what the statement was saying was, a function is defined by its domain= =20 (where it maps from), codomain (where it maps to), and the set of pairs=20 (a,b) such that f(a) =3D b (how it maps the elements). -Jelani ---------------------------------------------------------------------------= ------------- "Everything about a function is captured by three sets: its domain, its=20 codomain, and the set {(a,b)|f(a)=3Db} which is called the graph of f." This was the most confusing thing in the reading for me because I have=20 never seen a formal definition of functions before and I am somewhat=20 unfamiliar with a lot of the notation. Specifically, what does the bar=20 between the sequence and the statement mean? How would that set be read=20 aloud? Cheers, ~Ben ------=_Part_44512_31489427.1127227022695 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Ben,

S =3D {x | P(x) } is the set of all elements x such that P(x) is true.  If you've used Scheme before, you can think of it as "filter" in Scheme.
So what the statement was saying was, a function is defined by its domain (where it maps from), codomain (where it maps to), and the set of pairs (a,b) such that f(a) =3D b (how it maps the elements).

-Jelani
---------------------------------------------------------------------------= -------------
"Everything about a function is captured by three sets: its domain, it= s
codomain, and the set {(a,b)|f(a)=3Db} which is called the graph of f."= ;

This was the most confusing thing in the reading for me because I have
never seen a formal definition of functions before and I am somewhat
unfamiliar with a lot of the notation. Specifically, what does the bar
between the sequence and the statement mean? How would that set be read aloud?

Cheers,
~Ben
X-Gmail-Received: 10f976bc5127ca2f355fd6774a0af853c6eb1314 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:30:49 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:30:49 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: clintonb@mit.edu Subject: Re: Reading comments Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44435_2315022.1127226649707" Delivered-To: minilek@gmail.com ------=_Part_44435_2315022.1127226649707 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Clinton, If A is equal to B, then A is a subset of B. A "proper subset" though is a= =20 subset that is not equal to the set itself, so A would not be a proper=20 subset of B. Let me know if there were any other confusing terms or=20 notation. -Jelani ---------------------------------------------------------------------------= ------------- Section 3.2, page 9. The subset notations are confusing. What exactly is the difference between = . Are there better examples? Never mind. I see now that the difference occurs in conditional statements. Although, if A is equal to B, does A not qualify as a subset of B?=20 Either way, I found this difficult as it took me a few read-throughs and this email to understand. ---- Clinton C. Blackburn MIT Class of 2008 X-Gmail-Received: 8e7b6752df12006814f31d1e89b0f3c2a5627d17 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:26:54 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:26:54 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: arup@mit.edu Subject: Re: Week 2 Comments Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44375_14408544.1127226414758" Delivered-To: minilek@gmail.com ------=_Part_44375_14408544.1127226414758 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Arup, In general, it suffices to prove something false by just giving a=20 counter-example. What did you have in mind when you said a more general way= ?=20 For example, even though it is not a validity it can still be true for some= =20 predicates P. Consider the predicate P(x,y) =3D True. -Jelani ---------------------------------------------------------------------------= -------- Section 2.6, page 8: I would like to have the last (false) assertion explained more=20 fully. While I understand the proof for why it would not be true, I=20 want to know if there is any more general way of understanding it=20 besides the example given, since at first glance it does not appear=20 to be incorrect. |Arup| ------=_Part_44375_14408544.1127226414758 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Arup,

In general, it suffices to prove something false by just giving a counter-example.  What did you have in mind when you said a more general way?  For example, even though it is not a validity it can still be true for some predicates P.  Consider the predicate P(x,y) =3D True.

-Jelani
---------------------------------------------------------------------------= --------
Section 2.6, page 8:

I would like to have the last (false) assertion explained more
fully.  While I understand the proof for why it would not be true, I <= br> want to know if there is any more general way of understanding it
besides the example given, since at first glance it does not appear
to be incorrect.

|Arup| X-Gmail-Received: 4e601b3a0fe19f7b8677f1feb577b592fd83af3e Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:21:59 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:21:59 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: sriaz@mit.edu Subject: Re: 6.042 Reading Comments Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44315_26985981.1127226119384" Delivered-To: minilek@gmail.com ------=_Part_44315_26985981.1127226119384 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sameer, Are you still confused by the definitions of surjective, injective, and=20 bijective? If so, let me know tomorrow in class and I'll try to better explain them. -Jelani ---------------------------------------------------------------------------= -------------- One aspect I would like to have explained in class further is (3.5=20 Functions, page 11) where surjectives, injectives, and bijectives are=20 explained. The terminology here is confusing, and it would be useful to=20 go over the Lemma and definitions once more. ------=_Part_44315_26985981.1127226119384 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sameer,

Are you still confused by the definitions of surjective, injective, and bij= ective?  If so, let me know tomorrow
in class and I'll try to better explain them.

-Jelani
---------------------------------------------------------------------------= --------------
One aspect I would like to have explained in class further is (3.5
Functions, page 11) where surjectives, injectives, and bijectives are
explained.  The terminology here is confusing, and it would be useful = to
go over the Lemma and definitions once more.
X-Gmail-Received: 7d3b52505cb1e28200f5b701439b3d04e7ca8049 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:19:02 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:19:02 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: scot@mit.edu Subject: Re: weekly problem comments Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44283_33339107.1127225942076" Delivered-To: minilek@gmail.com ------=_Part_44283_33339107.1127225942076 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Scot, The truth table for implies is basically saying True implies only True, and= =20 False implies anything. The contrapositive says if A implies B, then not B= =20 implies not A. Before looking at the truth tables, you can try to reason=20 about it. If A implies B, that means whenever A is true then B must be true= .=20 So if B isn't true, then A couldn't be true either (because if it were, it= =20 would imply B were true, which we said it isn't). Think about it for a=20 little, and let me know if you still have questions. -Jelani ---------------------------------------------------------------------------= ----- Hello, In Week 1 readings, on page 15, I found it more difficult to understand=20 the contrapositive. In lecture I thought it made sense, but even some=20 aspects of the 'implies' truth table also seem confusing. Scot Frank ------=_Part_44283_33339107.1127225942076 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Scot,

The truth table for implies is basically saying True implies only True, and False implies anything.  The contrapositive says if A implies B, then not B implies not A.  Before looking at the truth tables, you can try to reason about it.  If A implies B, that means whenever A is true then B must be true.  So if B isn't true, then A couldn't be true either (because if it were, it would imply B were true, which we said it isn't).  Think about it for a little, and let me know if you still have questions.

-Jelani
---------------------------------------------------------------------------= -----
Hello,

In Week 1 readings, on page 15, I found it more difficult to understand the contrapositive. In lecture I thought it made sense, but even some
aspects of the 'implies' truth table also seem confusing.

Scot Frank X-Gmail-Received: 6d7a8711d8a9fb9a496f04807a5d275c70884d27 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 07:11:39 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 10:11:39 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: natalia3@mit.edu Subject: Re: Required Reading Comments 1 Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44238_1308454.1127225499871" Delivered-To: minilek@gmail.com ------=_Part_44238_1308454.1127225499871 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Natalia, With long lists of statements, order matters and you should begin reading= =20 from the left That is to say: there exists a, there exists b, there exists c, for all d P(a,b,c,d) is the same as there exists a (there exists b (there exists c (for all d P(a,b,c,d)))) In this case parentheses didn't matter. But, parentheses in predicates=20 matter for cases such as the following: (P ^ (Q v R)) v S vs. P ^ (Q v R v S) The first says that either S is true, or P is true and one of Q and R is=20 true. The second says that P is true and one of Q,R,S is true. In particular, S=3DTrue, P=3DQ=3DR=3DFalse makes the first one evaluate to = true but=20 the second one evaluate to false. For 2.4.1 in the reading, the parentheses= =20 weren't incorrect, but they didn't add any extra information. Please let me know if you still have questions. -Jelani ---------------------------------------------------------------------------= ------------- The part I had the most trouble with was the notation. It became especially difficult in parts 2.2 through 2.4 (pp. 6-7). I am not sure how to interpret long lists of statements, such as in 2.4, and also how to read the meaning of parentheses when predicates are enclosed within them (2.4.1).=20 Natalia Chernenko ------=_Part_44238_1308454.1127225499871 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Natalia,

With long lists of statements, order matters and you should begin reading f= rom the left

That is to say:

there exists a, there exists b, there exists c, for all d P(a,b,c,d)

is the same as

there exists a (there exists b (there exists c (for all d P(a,b,c,d))))

In this case parentheses didn't matter.  But, parentheses in predicate= s matter for cases such as the following:

(P ^ (Q v R)) v S

vs.

P ^ (Q v R v S)

The first says that either S is true, or P is true and one of Q and R is tr= ue.
The second says that P is true and one of Q,R,S is true.

In particular, S=3DTrue, P=3DQ=3DR=3DFalse makes the first one evaluate to = true but the second one evaluate to false.  For 2.4.1 in the reading, the parentheses weren't incorrect, but they didn't add any extra information.

Please let me know if you still have questions.

-Jelani
---------------------------------------------------------------------------= -------------
The part I had the most trouble with was the notation. It became
especially difficult in parts 2.2 through 2.4 (pp. 6-7). I am not sure
how to interpret long lists of statements, such as in 2.4, and also how
to read the meaning of parentheses when predicates are enclosed within
them (2.4.1).

Natalia Chernenko 
X-Gmail-Received: 5283e43b5ac09356c15648a5775b77684063c451 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 06:53:52 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 09:53:52 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: fluff@mit.edu Subject: Re: week 2 reading Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44114_29818067.1127224432098" Delivered-To: minilek@gmail.com ------=_Part_44114_29818067.1127224432098 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Crystal, The case with 4 strangers and a pair who knows each other is actually=20 covered by our cases. We picked an x, and Case 2 says that at least 3 people have not met x. If= =20 there are 4 strangers and a pair who know each other, we will be in Case 2.= =20 The, we will get into Case 2.2 which says that at least one pair of people= =20 haven't met each other, and we will conclude that there is a group of at=20 least 3 strangers (which is true, since there are 4). Please let me know if this made things clearer. If you still have questions= ,=20 feel free to email me or ask me in class or office hours. -Jelani ---------------------------------------------------------------------------= - I was confused by the strangers/clubs proof by case analysis (p.=20 3-4), especially cases 1.2 and 2.2. How does one know there isn't a=20 set of, say, four strangers, and then a pair who knows each other? I=20 mean, I trust that it works, but I got lost in all the wording. I=20 think a visual depiction of the problem would have been effective.=20 (Also, just so you know, there is a small typo on p. 3; it says,=20 "Among... at least 3 have did not met x.") ~Crystal ------=_Part_44114_29818067.1127224432098 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Crystal,

The case with 4 strangers and a pair who knows each other is actually cover= ed by our cases.

We picked an x, and Case 2 says that at least 3 people have not met x.  If there are 4 strangers and a pair who know each other, we will be in Case 2.  The, we will get into Case 2.2 which says that at least one pair of people haven't met each other, and we will conclude that there is a group of at least 3 strangers (which is true, since there are 4).

Please let me know if this made things clearer.  If you still have questions, feel free to email me or ask me in class or office hours.

-Jelani
---------------------------------------------------------------------------= -
I was confused by the strangers/clubs proof by case analysis (p.  
3-4), especially cases 1.2 and 2.2. How does one know there isn't a  <= br> set of, say, four strangers, and then a pair who knows each other? I  =
mean, I trust that it works, but I got lost in all the wording. I   think a visual depiction of the problem would have been effective.   (Also, just so you know, there is a small typo on p. 3; it says,  
"Among... at least 3 have did not met x.")

~Crystal X-Gmail-Received: b1a97598e0cc28777be1763354a6e6c985299167 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 06:46:18 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 09:46:18 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: vixen@mit.edu Subject: Re: Cc: 6042-staff@theory.csail.mit.edu Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_44062_6233465.1127223978732" Delivered-To: minilek@gmail.com ------=_Part_44062_6233465.1127223978732 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline With set builder notation, we say things like S =3D {a in A | P (a)}. You c= an=20 read the | as a "such that", so what this is saying is that S consists of= =20 all elements a in A such that P(a) is true. Please let me know if you are still confused about the definitions of total= ,=20 surjective, injective, and bijective. If so, I'd be happy to go over them= =20 with you with examples. -Jelani ------------------------------------------------------------------------ I found set builder notation (pg 10) and functions (pg 11) most=20 confusing. I would not be able to describe a set using the words=20 total, surjective, injective, or bijective. ------=_Part_44062_6233465.1127223978732 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline With set builder notation, we say things like S =3D {a in A | P (a)}.  You can read the | as a "such that", so what this is = saying is that S consists of all elements a in A such that P(a) is true.

Please let me know if you are still confused about the definitions of total, surjective, injective, and bijective.  If so, I'd be happy to go over them with you with examples.

-Jelani

------------------------------------------------------------------------ I found set builder notation (pg 10) and functions (pg 11) most
confusing.  I would not be able to describe a set using the words
total, surjective, injective, or bijective. X-Gmail-Received: 333e6ea08698d973e0b8071ffc6963ae892c0158 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 06:35:33 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 09:35:33 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: a_lopez@mit.edu Subject: Re: week 1 and week 2 readings Cc: 6042-staff@theory.csail.mit.edu Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_43962_1209255.1127223333682" Delivered-To: minilek@gmail.com ------=_Part_43962_1209255.1127223333682 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Adriana, >> I was wondering, is there a way to say "P(n) is true for exactly one n" = ? Yes. You could say that there exists an n such that P(n), and for all m P(m= )=20 implies m =3D n. >> for exactly m n's where m is an integer greater than 1? This can be done also, and I'll let you think about it as practice. As a=20 hint, consider trying to modify the solution for "P(n) is true for exactly= =20 one n". -Jelani ---------------------------------------------------------------------- In the readings for week 2, on page 5: "Sometimes True: There exists an n such that P(n) is true. P(n) is true for some n. P(n) is true for at least one n." I was wondering, is there a way to say "P(n) is true for exactly one n" ?= =20 for at most one n? for exactly m n's where m is an integer greater than 1? I would like this to be discussed more fully in class. Adriana Lopez ------=_Part_43962_1209255.1127223333682 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Adriana,

>> I was wondering, is there a way to say "P(n) is true for exac= tly one n" ?
Yes.  You could say that there exists an n such that P(n), and for all= m P(m) implies m =3D n.

>> for exactly m n's where m is an integer greater than 1?
This can be done also, and I'll let you think about it as practice.  As a hint, consider trying to modify the solution for "P(n) is true for exactly one n".

-Jelani

----------------------------------------------------------------------
In the readings for week 2, on page 5:

"Sometimes True:
There exists an n such that P(n) is true.
P(n) is true for some n.
P(n) is true for at least one n."

I was wondering, is there a way to say "P(n) is true for exactly one n= " ? for at
most one n?  for exactly m n's where m is an integer greater than 1?
I would like this to be discussed more fully in class.

Adriana Lopez X-Gmail-Received: 470e0df7236341a7bff356b3bd22b43c65a00040 Received: by 10.70.52.3 with HTTP; Tue, 20 Sep 2005 06:26:14 -0700 (PDT) Message-ID: Date: Tue, 20 Sep 2005 09:26:14 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: bens@mit.edu Subject: Re: 3 sentences max Cc: 6042-staff@theory.csail.mit.edu Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_43914_15116132.1127222774227" Delivered-To: minilek@gmail.com ------=_Part_43914_15116132.1127222774227 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Notationally speaking, computability people have already come up with their= =20 own style for dealing with exactly this. They say a "language" is just a se= t=20 of strings over some alphabet. They then say an algorithm A "decides" a=20 language L if A always halts when given an input, and it returns true=20 ("accepts" the input) if the input was in L and it rejects otherwise. So=20 using this framework you could ask things like, "does there exist an=20 algorithm that decides the language consisting of all predicates?". If you're interested in this, you should read "Introduction to the Theory o= f=20 Computation" by Michael Sipser, or take 6.840, which he teaches. -Jelani ** ---------------------------------------------------------------------------= ------------ I think Predicate notation (pg. 4) is very cool, because it allows us to=20 write functions f: -> {T,F}. In particular, it would be cool=20 to think about predicates whose arguments are logical statements. That=20 might provide a good notation for reasoning about theorem-provers and=20 computability. -Ben ------=_Part_43914_15116132.1127222774227 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Notationally speaking, computability people have already come up with their own style for dealing with exactly this.  They say a "language" is just a set of strings over some alphabet.  The= y then say an algorithm A "decides" a language L if A always halts when = given an input, and it returns true ("accepts" the input) if the input = was in L and it rejects otherwise.  So using this framework you could ask things like, "does there exist an algorithm that decides the language consisting of all predicates?".

If you're interested in this, you should read "Intro= duction to the Theory of Computation" by Michael Sipser, or tak= e 6.840, which he teaches.

-Jelani

-------------------------------------------= --------------------------------------------
I think Predicate notation (pg. 4) is very cool, because it allows us to write functions f:<anything> -> {T,F}.  In particular, it wou= ld be cool
to think about predicates whose arguments are logical statements.  Tha= t
might provide a good notation for reasoning about theorem-provers and
computability.

-Ben From meyer@imap.theory.csail.mit.edu Thu Sep 15 15:13:56 2005 X-Account-Key: account2 X-UIDL: c6b8c09a35bd2a8ceb4a74d487ed2065 X-Apparently-To: armeyer10@yahoo.com via 66.218.78.124; Thu, 15 Sep 2005 12:07:49 -0700 X-Originating-IP: [128.30.51.92] Return-Path: Authentication-Results: mta336.mail.scd.yahoo.com from=MIT.EDU; domainkeys=neutral (no sig) Received: from 128.30.51.92 (EHLO theory.csail.mit.edu) (128.30.51.92) by mta336.mail.scd.yahoo.com with SMTP; Thu, 15 Sep 2005 12:07:49 -0700 Received: from harrier.csail.mit.edu (harrier.csail.mit.edu [128.30.51.159]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8FJ7maJ024425 for ; Thu, 15 Sep 2005 15:07:48 -0400 Received: from harrier.csail.mit.edu (localhost.localdomain [127.0.0.1]) by harrier.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j8FJ7m6j011431 for ; Thu, 15 Sep 2005 15:07:48 -0400 Received: (from meyer@localhost) by harrier.csail.mit.edu (8.13.1/8.13.1/Submit) id j8FJ7mBH011428; Thu, 15 Sep 2005 15:07:48 -0400 Resent-Message-Id: <200509151907.j8FJ7mBH011428@harrier.csail.mit.edu> Message-Id: <200509151907.j8FJ7mBH011428@harrier.csail.mit.edu> Resent-From: meyer@csail.mit.edu Resent-Date: 15 Sep 2005 15:07:48 -0400 Resent-To: armeyer10@yahoo.com Date: Thu, 15 Sep 2005 14:52:44 -0400 (EDT) From: Yuhsin Chen To: 6042-probs@theory.lcs.MIT.EDU Subject: mistake MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Status: RO X-Status: A Content-Length: 826 X-UID: 2863 X-Keywords: NotJunk I just realized that I didn't turn in my reading comment or the online tutor problem. I'm very sorry, I was mixed up on the dates. Now that I'm off on a bad start, would you suggest that I drop the class and take it again at some later date, or should I keep trying? Yuhsin (Joyce) Chen The comment I found most interesting in the reading over Sets and Functions was in section 2.4, page 6, regarding Order of Quantifiers. Had I submitted it in on time, I would've liked to have a chance to go over how order influences the logical flow in lecture. (not directly related to the reading, I also found prefixing "not" in front of a proposition surprising--"not all P are true" is equivalent to "there exists P that is false". Perhaps I am just very interested in how many ways a proposition can be transformed). From meyer@imap.theory.csail.mit.edu Thu Sep 15 15:24:31 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <4329CA79.3070405@csail.mit.edu> Date: Thu, 15 Sep 2005 15:24:41 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yuhsin Chen Subject: Re: mistake References: <200509151907.j8FJ7mBH011428@harrier.csail.mit.edu> In-Reply-To: <200509151907.j8FJ7mBH011428@harrier.csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1463 X-UID: 2864 X-Keywords: I won't penalize you for your late email comments. It's good that you asked. And for goodness sake, I hope you won't consider dropping again in the event of another such minor problem! (Do the Math: online tutor + email totals 5% of the final grade, of which the email is about half. There are about a dozen required emails, and being a day late would get you part credit anyway. So even if I did penalize you, it would be for an imperceptible 0.1% of your grade.) Anyway, it will be a pleasure to have you continue in the course. I hope you enjoy it. Regards, A. Yuhsin Chen wrote: > I just realized that I didn't turn in my reading comment or the online > tutor problem. I'm very sorry, I was mixed up on the dates. Now that I'm > off on a bad start, would you suggest that I drop the class and take it > again at some later date, or should I keep trying? > > Yuhsin (Joyce) Chen > > The comment I found most interesting in the reading over Sets and > Functions was in section 2.4, page 6, regarding Order of Quantifiers. > Had I submitted it in on time, I would've liked to have a chance to go > over how order influences the logical flow in lecture. (not directly > related to the reading, I also found prefixing "not" in front of a > proposition surprising--"not all P are true" is equivalent to "there > exists P that is false". Perhaps I am just very interested in how many > ways a proposition can be transformed). > > > > From meyer@imap.theory.csail.mit.edu Fri Sep 16 00:16:19 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <432A471D.1070107@csail.mit.edu> Date: Fri, 16 Sep 2005 00:16:29 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Zev Benjamin Subject: Re: Week 2 Comments References: <4327D7E0.3050401@mit.edu> In-Reply-To: <4327D7E0.3050401@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 822 Status: RO X-UID: 2865 X-Keywords: ZFC was introduced in Week 1 Notes, pp.4--5. Did you read them? Regards, A. Zev Benjamin wrote: > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > The most difficult section I found in the reading was 3.5 Functions > (page 11). The various terms and their definitions are rather > confusing. Section four mentions ZFC axioms without explaining what the > acronym expands to and without much pretext. It's odd that it jumps > into talking about proving mathematics without explaining what it is > talking about. > > > Zev > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.1 (GNU/Linux) > Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org > > iD8DBQFDJ9fglO3j8HLL0+4RAoiOAJ4xjTtvYLzW/F0y1C4uv1Xc8D6M8QCg91tt > 9iEB7LH7N65459bGCwf1ms8= > =0/lc > -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From zev@MIT.EDU Wed Sep 14 03:57:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E7vTaJ028229 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 03:57:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E7vS95016530 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 03:57:28 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7vKpI028640 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 03:57:25 -0400 (EDT) Message-ID: <4327D7E0.3050401@mit.edu> Date: Wed, 14 Sep 2005 03:57:20 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 2 Comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 666 X-UID: 2866 X-Keywords: NonJunk NotJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 The most difficult section I found in the reading was 3.5 Functions (page 11). The various terms and their definitions are rather confusing. Section four mentions ZFC axioms without explaining what the acronym expands to and without much pretext. It's odd that it jumps into talking about proving mathematics without explaining what it is talking about. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDJ9fglO3j8HLL0+4RAoiOAJ4xjTtvYLzW/F0y1C4uv1Xc8D6M8QCg91tt 9iEB7LH7N65459bGCwf1ms8= =0/lc -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Fri Sep 16 00:20:43 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <432A4824.40600@csail.mit.edu> Date: Fri, 16 Sep 2005 00:20:52 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz Subject: Re: would like to have discussed more fully it in the next lecture References: <200509141302.j8ED2CMA029410@outgoing.mit.edu> In-Reply-To: <200509141302.j8ED2CMA029410@outgoing.mit.edu> Content-Type: text/plain; charset=ISO-2022-JP Content-Transfer-Encoding: 7bit Content-Length: 492 Status: RO X-UID: 2867 X-Keywords: did wed lecture and problems help you sort this out? If not, go to TA's office hours and ask for more explanation. regards, A. Anton Katz wrote: > Page 6, > > 2.4 = order of Quantifiers. > > > > Seems pretty clear but had trouble with the last question in the tutor. > > > > When you write: > > //x////y// //Q//(//x//, //y//) //y////x// //Q//(//x//, //y//) > > > > How are the different components connected amongst each other? (and, or) > > > > > From meyer@imap.theory.csail.mit.edu Wed Sep 14 21:47:33 2005 X-Coding-System: iso-2022-jp-unix Mail-from: From antonk@MIT.EDU Wed Sep 14 09:02:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ED2KaJ009057 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:02:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8ED2Iko010544 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:02:19 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ED2CMA029410 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:02:12 -0400 (EDT) Message-Id: <200509141302.j8ED2CMA029410@outgoing.mit.edu> From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: would like to have discussed more fully it in the next lecture Date: Wed, 14 Sep 2005 09:02:04 -0400 Organization: Massachusetts Institute of Technology MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0090_01C5B90A.FA2AD610" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW5LIC6RAFOaPOOSYe/jJpvzqE/tg== X-Spam-Score: 1.293 X-Spam-Level: * (1.293) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 5382 X-UID: 2868 X-Keywords: NonJunk NotJunk Page 6, 2.4 = order of Quantifiers. Seems pretty clear but had trouble with the last question in the tutor. When you write: xy Q(x, y) yx Q(x, y) How are the different components connected amongst each other? (and, or)  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0090_01C5B90A.FA2AD610 Content-Type: text/plain; charset="iso-2022-jp" Content-Transfer-Encoding: 7bit Page 6, 2.4 = order of Quantifiers. Seems pretty clear but had trouble with the last question in the tutor. When you write: xy Q(x, y) yx Q(x, y) How are the different components connected amongst each other? (and, or) ------=_NextPart_000_0090_01C5B90A.FA2AD610 Content-Type: text/html; charset="iso-2022-jp" Content-Transfer-Encoding: quoted-printable

Page 6,

2.4 =3D order of = Quantifiers.

 

Seems pretty clear but had trouble with the last = question in the tutor.

 

When you write:

=1B$B"O=1B(Jx=1B$B"P=1B(Jy Q(x, = y) = =1B$B"*=1B(J =1B$B"P=1B(Jy=1B$B"O=1B(Jx Q(x, = y)

 

=

How are the different components connected amongst each other? (and, = or=1B$B!D=1B(J)

 

=

 

------=_NextPart_000_0090_01C5B90A.FA2AD610-- From meyer@imap.theory.csail.mit.edu Fri Sep 16 00:22:23 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <432A4888.3090706@csail.mit.edu> Date: Fri, 16 Sep 2005 00:22:32 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Zachary Adam Ozer Subject: Re: Email comment References: <4f2613e505091405417acd5051@mail.gmail.com> In-Reply-To: <4f2613e505091405417acd5051@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 657 Status: RO X-UID: 2869 X-Keywords: did wed lecture and problems help you sort this out? If not, go to TA's office hours and ask for more explanation. regards, A. Zachary Adam Ozer wrote: > My confusion arises from the order of quantifiers section (2.4) of > week 2 course notes. On page 6, the statement "Swapping quantifiers in > Goldbach's Conjecture creates a patently false statement." I suppose > that I don't understand why the statement which follows is indeed > patently false. Indeed, in English one could say either "I will stand > up every time I hear the bell" or "Every time I hear a bell I will > stand up." The statements are equivalent, why not so in math? > > -zozer > From meyer@imap.theory.csail.mit.edu Wed Sep 14 21:47:33 2005 X-Coding-System: iso-8859-1-unix Mail-from: From zacharyozer@gmail.com Wed Sep 14 08:41:21 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.200]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8ECfLaJ005412 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:41:21 -0400 Received: by xproxy.gmail.com with SMTP id h28so205050wxd for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:41:15 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=ZC06ZTDPwXsiXFAHxXHfLZC24r7fj5p2TpGY4hL0pSV5QqRI3GJn4Ss3PSotvjbE/hgJuyyFkUEyoQXAL1rzF9cKx4Qy6a/CLCxmzQ0yUeSMSu1ompCx05GXxEnElmjolB+n/QCZR/SYLGV3dlyOB3jsYQS1m/4vOa6uBUgJDA0= Received: by 10.70.43.12 with SMTP id q12mr188761wxq; Wed, 14 Sep 2005 05:41:15 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Wed, 14 Sep 2005 05:41:15 -0700 (PDT) Message-ID: <4f2613e505091405417acd5051@mail.gmail.com> Date: Wed, 14 Sep 2005 08:41:15 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email comment Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8ECfLaJ005412 Status: RO X-Status: A Content-Length: 481 X-UID: 2870 X-Keywords: NonJunk NotJunk My confusion arises from the order of quantifiers section (2.4) of week 2 course notes. On page 6, the statement "Swapping quantifiers in Goldbach's Conjecture creates a patently false statement." I suppose that I don't understand why the statement which follows is indeed patently false. Indeed, in English one could say either "I will stand up every time I hear the bell" or "Every time I hear a bell I will stand up." The statements are equivalent, why not so in math? -zozer From meyer@imap.theory.csail.mit.edu Fri Sep 16 00:26:31 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <432A4981.40208@csail.mit.edu> Date: Fri, 16 Sep 2005 00:26:41 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Knight W Fu Subject: Re: Required Reading Comment References: <20050914045543.6yddpubg7w0soo40@webmail.mit.edu> In-Reply-To: <20050914045543.6yddpubg7w0soo40@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 2170 Status: RO X-UID: 2871 X-Keywords: the "C" in ZFC stands for Zermelo-Frankel with "C"hoice. It's not really controversial -- virtually all mathematicians accept it an axiom; what was a surprise is that Choice does not follow from ZF; it really has to be added as an extra axiom. Regards, A. Knight W Fu wrote: > The ZFC Axioms > For the record, we list the axioms of Zermelo-Frankel Set Theory. > Essentially all of mathematics can be derived from these axioms together > with a few logical deduction rules. > Extensionality. Two sets are equal if they have the same members. In > formal logical notation, this would be stated as: > (8z. (z 2 x ! z 2 y)) ?! x = y. > Pairing. For any two sets x and y, there is a set, {x, y}, with x and y as > its only elements. > Union. The union of a collection, z, of sets is also a set. > 9u8x. (9y. x 2 y ^ y 2 z) ! x 2 u. > Infinity. There is an infinite set; specifically, a nonempty set, x, such > that for any set y 2 x, the set {y} is also a member of x > Subset. Given any set, x, and any proposition P(y), there is a set containing > precisely those elements y 2 x for which P(y) holds. > Power Set. All the subsets of a set form another set. > Replacement. The image of a set under a function is a set. > Foundation. For every non-empty set, x, there is a set y 2 x such that > x and y are disjoint. (In particular, this axiom prevents a set from > being a member of itself.) > Choice. We can choose one element from each set in a collection of > nonempty sets. More precisely, if f is a function on a set, and > the result of applying f to any element in the set is always > a nonempty set, then there is a ?choice? function g such that > g(y) 2 y for every y in the set. > We?re not going to be working with the ZFC axioms in this course. We > just thought you might like to see them. > > - The above passage was quoted from page 5 > > I had read a few books that mention the axioms of set theory, and I have > actually not seen the ZFC Axioms so completely listed. I have studied the > axioms of choice in a topology class, and I was told that it was very > controversial in the math world. I never thought that it was one of the ZFC > axioms. From meyer@imap.theory.csail.mit.edu Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From knightfu@MIT.EDU Wed Sep 14 04:59:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E8xmaJ006073 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 04:59:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8E8xjDr015094 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 04:59:46 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E8thku000659 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 04:55:44 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E8thcg030090; Wed, 14 Sep 2005 04:55:43 -0400 Received: from EASTCAMPUS-THREE-EIGHTY-FOUR.MIT.EDU (EASTCAMPUS-THREE-EIGHTY-FOUR.MIT.EDU [18.248.6.129]) (User authenticated as knightfu@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 04:55:43 -0400 Message-ID: <20050914045543.6yddpubg7w0soo40@webmail.mit.edu> Date: Wed, 14 Sep 2005 04:55:43 -0400 From: Knight W Fu To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 1819 X-UID: 2872 X-Keywords: NonJunk NotJunk The ZFC Axioms For the record, we list the axioms of Zermelo-Frankel Set Theory. Essentially all of mathematics can be derived from these axioms together with a few logical deduction rules. Extensionality. Two sets are equal if they have the same members. In formal logical notation, this would be stated as: (8z. (z 2 x ! z 2 y)) ?! x = y. Pairing. For any two sets x and y, there is a set, {x, y}, with x and y as its only elements. Union. The union of a collection, z, of sets is also a set. 9u8x. (9y. x 2 y ^ y 2 z) ! x 2 u. Infinity. There is an infinite set; specifically, a nonempty set, x, such that for any set y 2 x, the set {y} is also a member of x Subset. Given any set, x, and any proposition P(y), there is a set containing precisely those elements y 2 x for which P(y) holds. Power Set. All the subsets of a set form another set. Replacement. The image of a set under a function is a set. Foundation. For every non-empty set, x, there is a set y 2 x such that x and y are disjoint. (In particular, this axiom prevents a set from being a member of itself.) Choice. We can choose one element from each set in a collection of nonempty sets. More precisely, if f is a function on a set, and the result of applying f to any element in the set is always a nonempty set, then there is a ?choice? function g such that g(y) 2 y for every y in the set. We?re not going to be working with the ZFC axioms in this course. We just thought you might like to see them. - The above passage was quoted from page 5 I had read a few books that mention the axioms of set theory, and I have actually not seen the ZFC Axioms so completely listed. I have studied the axioms of choice in a topology class, and I was told that it was very controversial in the math world. I never thought that it was one of the ZFC axioms. From meyer@imap.theory.csail.mit.edu Fri Sep 16 00:38:35 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <432A4C53.7060001@csail.mit.edu> Date: Fri, 16 Sep 2005 00:38:43 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Irene Zhang Subject: Re: question about pset References: <432A47E1.5050909@mit.edu> In-Reply-To: <432A47E1.5050909@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 515 Status: RO X-UID: 2873 X-Keywords: the domain of f is A, and the codomain of f is the powerset of A. So what we know is that for each x in A, the value f(x) is a subset of A. (Technically, f(x) also might be undefined for certain x, since it's possible that f is not total. But it's ok to assume that f is total in this problem if that makes things clearer.) regards, A. Irene Zhang wrote: > In problem 4, could you explain the {x \in A | x \notin f(x)}? Would the > f(x) be the range of the function? or the codomain? > > Thanks, > Irene From meyer@imap.theory.csail.mit.edu Fri Sep 16 02:42:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8G6g2aJ012224 for <6042-meyer@theory.csail.mit.edu>; Fri, 16 Sep 2005 02:42:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8G6g1e5002020 for <6042-meyer@theory.csail.mit.edu>; Fri, 16 Sep 2005 02:42:01 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8G6fr3s023796 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-meyer@theory.csail.mit.edu>; Fri, 16 Sep 2005 02:41:58 -0400 (EDT) Message-ID: <432A6931.6060902@mit.edu> Date: Fri, 16 Sep 2005 02:41:53 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Subject: Re: Week 2 Comments References: <4327D7E0.3050401@mit.edu> <432A471D.1070107@csail.mit.edu> In-Reply-To: <432A471D.1070107@csail.mit.edu> X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1086 X-UID: 2874 X-Keywords: NotJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 Ah. I didn't see the lecture notes links on the calendar because I followed the week two lecture notes from the tutor and was confused when the week 1 tutor problem set linked to the week 2 lecture notes. Now that I found them, I'll review them. Thanks! Zev Prof. Albert R. Meyer wrote: > ZFC was introduced in Week 1 Notes, pp.4--5. Did you read them? > > Regards, A. > > Zev Benjamin wrote: > The most difficult section I found in the reading was 3.5 Functions > (page 11). The various terms and their definitions are rather > confusing. Section four mentions ZFC axioms without explaining what the > acronym expands to and without much pretext. It's odd that it jumps > into talking about proving mathematics without explaining what it is > talking about. > > > Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDKmkwlO3j8HLL0+4RAo63AKDJtKGT4SLpmhwUu1ah3BzmH/K4fwCgmjVh o24n6i/JN4G73khrrdavuZU= =r4VZ -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Fri Sep 16 01:30:48 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8G5UlaJ031715 for <6042-meyer@theory.csail.mit.edu>; Fri, 16 Sep 2005 01:30:47 -0400 Received: by xproxy.gmail.com with SMTP id i28so139155wxd for <6042-meyer@theory.csail.mit.edu>; Thu, 15 Sep 2005 22:30:42 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=ErcU0bar4YzPbNVZRSZD1Uyd5T81WGmuaMPKLJm/K1htimHg71FWzehlg7KQzWr2zif3rzpmqsxwttYYxx/GNc32h3W3sOWI2BjQ/dJ4FOZaf/ixMEkowp2p7px6LYuk/GhgkSEW1didtIEhjzYYuh5j+Y9BpeD7S0uIK/1jFtw= Received: by 10.70.26.3 with SMTP id 3mr753wxz; Thu, 15 Sep 2005 22:30:42 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Thu, 15 Sep 2005 22:30:42 -0700 (PDT) Message-ID: <4f2613e505091522304b587338@mail.gmail.com> Date: Fri, 16 Sep 2005 01:30:42 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Subject: Re: Email comment In-Reply-To: <432A4888.3090706@csail.mit.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline References: <4f2613e505091405417acd5051@mail.gmail.com> <432A4888.3090706@csail.mit.edu> Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8G5UlaJ031715 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 856 X-UID: 2875 X-Keywords: $Forwarded NotJunk Absolutely. I now understand why the statements are not equivalent. Thanks again for your help. -zozer On 9/16/05, Prof. Albert R. Meyer wrote: > did wed lecture and problems help you sort this out? If not, go to > TA's office hours and ask for more explanation. > > regards, A. > > Zachary Adam Ozer wrote: > > My confusion arises from the order of quantifiers section (2.4) of > > week 2 course notes. On page 6, the statement "Swapping quantifiers in > > Goldbach's Conjecture creates a patently false statement." I suppose > > that I don't understand why the statement which follows is indeed > > patently false. Indeed, in English one could say either "I will stand > > up every time I hear the bell" or "Every time I hear a bell I will > > stand up." The statements are equivalent, why not so in math? > > > > -zozer > > > From meyer@imap.theory.csail.mit.edu Fri Sep 16 12:04:51 2005 Message-ID: <432AED22.7070201@csail.mit.edu> Date: Fri, 16 Sep 2005 12:04:50 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-tas@theory.csail.mit.edu, Ronitt Rubinfeld Subject: [Fwd: Re: Email comment] Content-Type: multipart/mixed; boundary="------------040301010107080900080704" Status: RO Content-Length: 3235 X-UID: 2876 X-Keywords: This is a multi-part message in MIME format. --------------040301010107080900080704 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit The kind of email reply I hope for. regards, A. --------------040301010107080900080704 Content-Type: message/rfc822; name="Re: Email comment" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Re: Email comment" Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8G5UlaJ031715 for <6042-meyer@theory.csail.mit.edu>; Fri, 16 Sep 2005 01:30:47 -0400 Received: by xproxy.gmail.com with SMTP id i28so139155wxd for <6042-meyer@theory.csail.mit.edu>; Thu, 15 Sep 2005 22:30:42 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=ErcU0bar4YzPbNVZRSZD1Uyd5T81WGmuaMPKLJm/K1htimHg71FWzehlg7KQzWr2zif3rzpmqsxwttYYxx/GNc32h3W3sOWI2BjQ/dJ4FOZaf/ixMEkowp2p7px6LYuk/GhgkSEW1didtIEhjzYYuh5j+Y9BpeD7S0uIK/1jFtw= Received: by 10.70.26.3 with SMTP id 3mr753wxz; Thu, 15 Sep 2005 22:30:42 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Thu, 15 Sep 2005 22:30:42 -0700 (PDT) Message-ID: <4f2613e505091522304b587338@mail.gmail.com> Date: Fri, 16 Sep 2005 01:30:42 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-meyer <6042-meyer@theory.csail.mit.edu> Subject: Re: Email comment In-Reply-To: <432A4888.3090706@csail.mit.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline References: <4f2613e505091405417acd5051@mail.gmail.com> <432A4888.3090706@csail.mit.edu> Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8G5UlaJ031715 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 X-UID: 816 X-Keywords: Absolutely. I now understand why the statements are not equivalent. Thanks again for your help. -zozer On 9/16/05, Prof. Albert R. Meyer wrote: > did wed lecture and problems help you sort this out? If not, go to > TA's office hours and ask for more explanation. > > regards, A. > > Zachary Adam Ozer wrote: > > My confusion arises from the order of quantifiers section (2.4) of > > week 2 course notes. On page 6, the statement "Swapping quantifiers in > > Goldbach's Conjecture creates a patently false statement." I suppose > > that I don't understand why the statement which follows is indeed > > patently false. Indeed, in English one could say either "I will stand > > up every time I hear the bell" or "Every time I hear a bell I will > > stand up." The statements are equivalent, why not so in math? > > > > -zozer > > > --------------040301010107080900080704-- From meyer@imap.theory.csail.mit.edu Wed Nov 30 02:03:11 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jAU73A5o025795 for ; Wed, 30 Nov 2005 02:03:10 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EhLzi-0005F6-Mg for meyer@csail.mit.edu; Wed, 30 Nov 2005 02:03:10 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jAU739Ka023867 for ; Wed, 30 Nov 2005 02:03:09 -0500 (EST) Received: from [18.242.6.198] (NEXT-FOUR-FIFTY-THREE.MIT.EDU [18.242.6.198]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jAU732ZE003436 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for ; Wed, 30 Nov 2005 02:03:03 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) To: meyer@csail.mit.edu Message-Id: <19B745BC-2571-4E74-AC2B-B2DC24C28781@MIT.EDU> Content-Type: multipart/alternative; boundary=Apple-Mail-4-408157557 References: From: Elizabeth Reid Subject: Fwd: [Hanson] Reading comments Date: Wed, 30 Nov 2005 02:03:00 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.218 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.7 required=3.7 tests=BAYES_00,HTML_70_80, HTML_MESSAGE autolearn=no version=2.63 Status: RO Content-Length: 2658 X-UID: 2877 X-Keywords: NonJunk --Apple-Mail-4-408157557 Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit Begin forwarded message: > From: Elizabeth Reid > Date: November 22, 2005 4:15:53 AM EST > To: 6042-probs@theory.lcs.mit.edu > Subject: [Hanson] Reading comments > > > The a posteriori problems were a bit odd, but I think I get the > idea behind them now. > --Apple-Mail-4-408157557 Content-Type: text/html; charset=US-ASCII Content-Transfer-Encoding: quoted-printable

Begin = forwarded message:

From: = Elizabeth Reid <ereid@MIT.EDU>
Date: = November 22, 2005 4:15:53 AM EST
Subject: [Hanson] Reading = comments
The a posteriori problems were a bit odd, but I = think I get the idea behind them now.


= --Apple-Mail-4-408157557-- From juang@MIT.EDU Sat Dec 3 23:07:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB447v6N000595 for <6042-probs@theory.lcs.mit.edu>; Sat, 3 Dec 2005 23:07:57 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB447tCH021544 for <6042-probs@theory.lcs.mit.edu>; Sat, 3 Dec 2005 23:07:55 -0500 (EST) Received: from [18.238.6.17] (EASTCAMPUS-SEVEN-EIGHTY-FOUR.MIT.EDU [18.238.6.17]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB447nWj021592 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 3 Dec 2005 23:07:49 -0500 (EST) Message-ID: <43926B8E.9080106@mit.edu> Date: Sat, 03 Dec 2005 23:07:42 -0500 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] Week 14 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-IMAPbase: 1126748715 1231 NonJunk $Label4 $Label1 $Label2 $Label3 $Label5 Junk $MDNSent $Forwarded NotJunk Status: RO X-UID: 1137 Content-Length: 331 X-Keywords: NonJunk NotJunk I don't have much comment on the reading, so I'll instead mention that Problem B4 on the Putnam today was, paraphrased: Let f(m,n) be the number of n-tuples of integers (x_1, x_2, ..., x_n) which satisfy |x_1| + |x_2| + ... + |x_n| <= m. Show that f(m,n) = f(n,m). ... Sounds like a good final exam question to me ;) Jason. From kevin08@MIT.EDU Sun Dec 4 05:29:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB4ATc6N011798 for <6042-probs@theory.lcs.mit.edu>; Sun, 4 Dec 2005 05:29:38 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB4ATbwU018983 for <6042-probs@theory.lcs.mit.edu>; Sun, 4 Dec 2005 05:29:37 -0500 (EST) Received: from kevlar.mit.edu (NEXT-FIVE-EIGHTY-SIX.MIT.EDU [18.242.7.75]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB4ATXsV012249 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 4 Dec 2005 05:29:35 -0500 (EST) Message-Id: <6.2.3.4.2.20051204052628.0296b7b8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Sun, 04 Dec 2005 05:29:34 -0500 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Week's reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1138 Content-Length: 236 X-Keywords: NonJunk NotJunk Hi, I had a general question about this week's readings--it seems that Chebyshev's Inequality (S7, Pg. 12) is always tighter than Markov's Inequality (S6, pg. 10). Is there a simple proof of this, or is it even true? Thanks, Kevin From ajshafer@MIT.EDU Sun Dec 4 11:50:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB4Go76N002555 for <6042-probs@theory.csail.mit.edu>; Sun, 4 Dec 2005 11:50:07 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB4Go6OG017200 for <6042-probs@theory.csail.mit.edu>; Sun, 4 Dec 2005 11:50:06 -0500 (EST) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB4Gnxfj007857 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sun, 4 Dec 2005 11:50:00 -0500 (EST) Message-ID: <000201c5f8f2$c4620760$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 14 Comments Date: Sun, 4 Dec 2005 11:50:00 -0500 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1139 Content-Length: 299 X-Keywords: NonJunk NotJunk Will we be required to know the expectation and variance for some standard random variables? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From ksindi@MIT.EDU Sun Dec 4 21:07:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB527g6N027943 for <6042-probs@theory.csail.MIT.EDU>; Sun, 4 Dec 2005 21:07:42 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB527fss018085 for <6042-probs@theory.csail.MIT.EDU>; Sun, 4 Dec 2005 21:07:41 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB527R8a011532 for <6042-probs@theory.csail.MIT.EDU>; Sun, 4 Dec 2005 21:07:27 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jB527RsX015403; Sun, 4 Dec 2005 21:07:27 -0500 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 4 Dec 2005 21:07:27 -0500 Message-ID: <20051204210727.dhabmn2m4bw4wksk@webmail.mit.edu> Date: Sun, 4 Dec 2005 21:07:27 -0500 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [DAVID] WEEK 14 COMMENTS MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.602 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1140 Content-Length: 296 X-Keywords: NotJunk I found everything well explained but still have trouble grasping the qualitative nature of the standard deviation; when to say, for example, "wow that's a high standard deviation!". I'd also like to hear in class, in a bit more detail, about when one can't use the Chebychev inequality. -Kamil From bens@MIT.EDU Mon Dec 5 01:42:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB56gN6N009230 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 01:42:23 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB56gMV9015195 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 01:42:22 -0500 (EST) Received: from [18.221.0.117] ([18.221.0.117]) (authenticated bits=0) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB56gKkr029488 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 01:42:20 -0500 (EST) Message-ID: <4393E13D.2060705@mit.edu> Date: Mon, 05 Dec 2005 01:42:05 -0500 From: "Benjamin M. Schwartz" User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051014) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1141 Content-Length: 108 X-Keywords: NotJunk The Chebyshev bound is awesome. Are there distributions for which Markov, chebyshev, etc. are always tight? From kjhollen@MIT.EDU Mon Dec 5 03:01:13 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB581D6N023388 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 03:01:13 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB581CK2025291 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 03:01:12 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB581CI2016221 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 03:01:12 -0500 (EST) Received: from MACGREGOR-FOUR-SIXTY-EIGHT.MIT.EDU (MACGREGOR-FOUR-SIXTY-EIGHT.MIT.EDU [18.239.6.213]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB5814Mu021999 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 03:01:04 -0500 (EST) Subject: [Hanson] confused about calculating variance From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Mon, 05 Dec 2005 03:01:02 -0500 Message-Id: <1133769662.8170.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1142 Content-Length: 372 X-Status: A X-Keywords: NonJunk NotJunk while the gambling game example for calculating variance in the notes was a good illustrating of the concept (p. 13), I found it difficult to apply the concept to the online tutor problem about the variance of the number of heads that appear in 100 coin flips. it might be helpful to explain how to do variance calculations with an example that has more than 2 outcomes. From meyer@imap.theory.csail.mit.edu Mon Dec 05 09:33:26 2005 Message-ID: <43944FBB.4050506@csail.mit.edu> Date: Mon, 05 Dec 2005 09:33:31 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kate Hollenbach Subject: Re: [Hanson] confused about calculating variance References: <1133769662.8170.2.camel@localhost.localdomain> In-Reply-To: <1133769662.8170.2.camel@localhost.localdomain> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 451 X-UID: 1143 X-Keywords: did you read ln14, sec 1.4? regards, A. Kate Hollenbach wrote: >while the gambling game example for calculating variance in the notes >was a good illustrating of the concept (p. 13), I found it difficult to >apply the concept to the online tutor problem about the variance of the >number of heads that appear in 100 coin flips. it might be helpful to >explain how to do variance calculations with an example that has more >than 2 outcomes. > > > From xiaoranz@MIT.EDU Mon Dec 5 10:48:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB5Fmb6N029091 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 5 Dec 2005 10:48:37 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB5Fmasu022640 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 5 Dec 2005 10:48:36 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB5FmUQB025497 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 5 Dec 2005 10:48:30 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jB5FmT7h006467; Mon, 5 Dec 2005 10:48:29 -0500 Received: from M56-129-20.MIT.EDU (M56-129-20.MIT.EDU [18.56.0.49]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 5 Dec 2005 10:48:29 -0500 Message-ID: <20051205104829.h5tkt7zd8m2so0wc@webmail.mit.edu> Date: Mon, 5 Dec 2005 10:48:29 -0500 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 14 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1144 Content-Length: 589 X-Keywords: NonJunk NotJunk I thought section 2.1 on expected number of meals to get each Racin' Rocket is an interesting problem and application. Also, The Markov and Chebyshev's Theorems provides good way to upper bound certain probabilities and helpful in checking result. Xiaoran Zhang ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From meyer@imap.theory.csail.mit.edu Mon Dec 05 10:32:54 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB5FWq6N021682 for ; Mon, 5 Dec 2005 10:32:53 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EjIKi-0003g2-Pz for meyer@csail.MIT.EDU; Mon, 05 Dec 2005 10:32:52 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB5FWpsu003680 for ; Mon, 5 Dec 2005 10:32:51 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB5FWm04018197 for ; Mon, 5 Dec 2005 10:32:48 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jB5FWmbA017919; Mon, 5 Dec 2005 10:32:48 -0500 Received: from dhcp-23-135.media.mit.edu (dhcp-23-135.media.mit.edu [18.85.23.135]) (User authenticated as kjhollen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 5 Dec 2005 10:32:48 -0500 Message-ID: <20051205103248.opf3mjvmlvcw8k0g@webmail.mit.edu> Date: Mon, 5 Dec 2005 10:32:48 -0500 From: Kate Hollenbach To: "Prof. Albert R. Meyer" Subject: Re: [Hanson] confused about calculating variance References: <1133769662.8170.2.camel@localhost.localdomain> <43944FBB.4050506@csail.mit.edu> In-Reply-To: <43944FBB.4050506@csail.mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO X-Status: A Content-Length: 716 X-UID: 1145 X-Keywords: NonJunk NotJunk yes, and it did help me figure out how to find the expectation of the number of heads. but I am still having trouble with translating it into the variance formulas found on p. 13 and 16. Quoting "Prof. Albert R. Meyer" : > did you read ln14, sec 1.4? > regards, A. > > Kate Hollenbach wrote: > >> while the gambling game example for calculating variance in the notes >> was a good illustrating of the concept (p. 13), I found it difficult to >> apply the concept to the online tutor problem about the variance of the >> number of heads that appear in 100 coin flips. it might be helpful to >> explain how to do variance calculations with an example that has more >> than 2 outcomes. >> >> > From hmzhou@blackbird.csail.mit.edu Mon Dec 5 11:15:57 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB5GFv6N026964; Mon, 5 Dec 2005 11:15:57 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jB5GFvoK030117; Mon, 5 Dec 2005 11:15:57 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jB5GFvnJ030114; Mon, 5 Dec 2005 11:15:57 -0500 Date: Mon, 5 Dec 2005 11:15:56 -0500 (EST) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week's reading comments In-Reply-To: <6.2.3.4.2.20051204052628.0296b7b8@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051204052628.0296b7b8@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1146 Content-Length: 602 X-Keywords: NotJunk Well...Chebyshev is nothing more than applying Markov's inequality in a certain way as you will see from the proof of it, so I don't think you can say it is stronger. However, typically, if the variance is known to you and small, then you can often do better using that than the usual form of Markov. -Hanson On Sun, 4 Dec 2005, Kevin Wang wrote: > Hi, > > I had a general question about this week's readings--it seems that > Chebyshev's Inequality (S7, Pg. 12) is always tighter than Markov's > Inequality (S6, pg. 10). Is there a simple proof of this, or is it even true? > > Thanks, > Kevin > > From meyer@imap.theory.csail.mit.edu Mon Dec 05 18:34:20 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB5NYJ6N001543 for <6042-staff@theory.csail.mit.edu>; Mon, 5 Dec 2005 18:34:19 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB5NYIw2005948 for <6042-staff@theory.csail.mit.edu>; Mon, 5 Dec 2005 18:34:18 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB5NYH9Q027861; Mon, 5 Dec 2005 18:34:17 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB5NYEMu018948; Mon, 5 Dec 2005 18:34:14 -0500 (EST) Message-ID: <4394CE74.1070101@mit.edu> Date: Mon, 05 Dec 2005 18:34:12 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Richard Hughes CC: 6042-staff@theory.csail.mit.edu Subject: Re: I can't figure out how to calculate variance References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,RCVD_IN_SORBS autolearn=no version=2.63 Status: RO Content-Length: 702 X-UID: 1147 X-Keywords: NotJunk Richard, You also know that if A and B are independent, then Var(A+B) = Var(A) + Var(B). Or, if A1, A2, ..., A99, A100 are mutually independent, then Var(A1+A2+...+A99+A100) = Var(A1)+Var(A2)+...+Var(A99)+Var(A100). DS Richard Hughes wrote: > Or, I can figure out how, but I can't actually DO it. > > I'm trying to do Tutor Problem 14.3 and it's giving me more trouble > than a tutor problem should. I honest to god cannot figure out how to > calculate the variance on a number of heads. I know the formula is > Var(R) = E[R^2] - E^2[R]. I know that E^2[R] = 2500. I absolutely > cannot figure out how to calculate R^2 or the expected value of it. > HELP ME PLEASE. :-( > > Richard > From meyer@imap.theory.csail.mit.edu Mon Dec 05 18:41:40 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB5Nfc6N003971 for <6042-staff@theory.csail.mit.edu>; Mon, 5 Dec 2005 18:41:38 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB5NfaTT000346; Mon, 5 Dec 2005 18:41:36 -0500 (EST) Received: from [18.24.1.16] (24-1-16.dynamic.csail.mit.edu [18.24.1.16]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB5NfXok006079; Mon, 5 Dec 2005 18:41:34 -0500 (EST) In-Reply-To: <4394CE74.1070101@mit.edu> References: <4394CE74.1070101@mit.edu> Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <546ea75c652249fb2cfeb63e00b0c955@mit.edu> Content-Transfer-Encoding: 7bit Cc: 6042-staff@theory.csail.mit.edu From: Richard Hughes Subject: Re: I can't figure out how to calculate variance Date: Mon, 5 Dec 2005 18:41:32 -0500 To: David Shin X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 828 X-UID: 1148 X-Keywords: NotJunk Thanks. That did it for me, I solved it. Richard On Dec 5, 2005, at 6:34 PM, David Shin wrote: > Richard, > > You also know that if A and B are independent, then Var(A+B) = Var(A) > + Var(B). > Or, if A1, A2, ..., A99, A100 are mutually independent, then > Var(A1+A2+...+A99+A100) = Var(A1)+Var(A2)+...+Var(A99)+Var(A100). > > DS > > Richard Hughes wrote: > >> Or, I can figure out how, but I can't actually DO it. >> >> I'm trying to do Tutor Problem 14.3 and it's giving me more trouble >> than a tutor problem should. I honest to god cannot figure out how >> to calculate the variance on a number of heads. I know the formula >> is Var(R) = E[R^2] - E^2[R]. I know that E^2[R] = 2500. I >> absolutely cannot figure out how to calculate R^2 or the expected >> value of it. HELP ME PLEASE. :-( >> >> Richard >> From meyer@imap.theory.csail.mit.edu Mon Dec 05 18:41:38 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB5Nfc6N003971 for <6042-staff@theory.csail.mit.edu>; Mon, 5 Dec 2005 18:41:38 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB5NfaTT000346; Mon, 5 Dec 2005 18:41:36 -0500 (EST) Received: from [18.24.1.16] (24-1-16.dynamic.csail.mit.edu [18.24.1.16]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB5NfXok006079; Mon, 5 Dec 2005 18:41:34 -0500 (EST) In-Reply-To: <4394CE74.1070101@mit.edu> References: <4394CE74.1070101@mit.edu> Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <546ea75c652249fb2cfeb63e00b0c955@mit.edu> Content-Transfer-Encoding: 7bit Cc: 6042-staff@theory.csail.mit.edu From: Richard Hughes Subject: Re: I can't figure out how to calculate variance Date: Mon, 5 Dec 2005 18:41:32 -0500 To: David Shin X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 829 X-UID: 1149 X-Keywords: NonJunk NotJunk Thanks. That did it for me, I solved it. Richard On Dec 5, 2005, at 6:34 PM, David Shin wrote: > Richard, > > You also know that if A and B are independent, then Var(A+B) = Var(A) > + Var(B). > Or, if A1, A2, ..., A99, A100 are mutually independent, then > Var(A1+A2+...+A99+A100) = Var(A1)+Var(A2)+...+Var(A99)+Var(A100). > > DS > > Richard Hughes wrote: > >> Or, I can figure out how, but I can't actually DO it. >> >> I'm trying to do Tutor Problem 14.3 and it's giving me more trouble >> than a tutor problem should. I honest to god cannot figure out how >> to calculate the variance on a number of heads. I know the formula >> is Var(R) = E[R^2] - E^2[R]. I know that E^2[R] = 2500. I >> absolutely cannot figure out how to calculate R^2 or the expected >> value of it. HELP ME PLEASE. :-( >> >> Richard >> From cwong08@MIT.EDU Mon Dec 5 20:29:29 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB61TT6N009942 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 20:29:29 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB61TSN1007031 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 20:29:28 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB61Rg9L001857 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 20:27:42 -0500 (EST) Received: from [127.0.0.1] (CWONG08.MIT.EDU [18.216.0.136]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB61RYMv026331 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 20:27:39 -0500 (EST) Message-ID: <4394E8E5.8090409@mit.edu> Date: Mon, 05 Dec 2005 20:27:01 -0500 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 14 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1150 Content-Length: 100 X-Keywords: NotJunk I would like clarification on when to use standard deviation and when to use variance.(pgs 14-16) From icharny@MIT.EDU Mon Dec 5 21:26:20 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB62QK6N000561 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 21:26:20 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB62QJN1023025 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 21:26:19 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB62QI9L003938 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 21:26:19 -0500 (EST) Received: from [127.0.0.1] (EASTCAMPUS-NINE-EIGHTY-THREE.MIT.EDU [18.238.6.216]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB62QCMw029663 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 21:26:16 -0500 (EST) Message-ID: <4394F6C5.7040907@mit.edu> Date: Mon, 05 Dec 2005 21:26:13 -0500 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 14 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1151 Content-Length: 130 X-Keywords: NonJunk NotJunk I don't really understand variance, what it is, and how to quickly calculate it. Please cover this in the next lecture. ~Isaac From vixen@MIT.EDU Mon Dec 5 23:40:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB64eO6N016167 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 23:40:24 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB64eMNc005741 for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 23:40:22 -0500 (EST) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB64eFC3004626 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 5 Dec 2005 23:40:16 -0500 (EST) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Mon, 5 Dec 2005 23:38:10 -0500 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Subject: LN14 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1152 Content-Length: 60 X-Keywords: NotJunk I was confused as to what "x" was in many of the equations. From hkhall@MIT.EDU Mon Dec 5 23:48:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB64mm6N016611 for <6042-probs@theory.csail.mit.edu>; Mon, 5 Dec 2005 23:48:48 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB64mkNc012499; Mon, 5 Dec 2005 23:48:46 -0500 (EST) Received: from [18.96.6.44] (SIMMONS-TWO-NINETY-NINE.MIT.EDU [18.96.6.44]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB64mcI6006440 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Mon, 5 Dec 2005 23:48:39 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <24b9be0d51a5c9b1398b83895ce0c687@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: [David] Week 14 Comments Date: Mon, 5 Dec 2005 23:48:37 -0500 To: 6042-probs@theory.csail.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1153 Content-Length: 751 X-Keywords: NotJunk David- Even after our talk today I still had trouble with variance and using it to apply Chebyshev's theorem. For that matter is is taking me a long time to figure out Markov's as well. I think that I will get it by the time we go over it in class. I think that my problem lies in the fact that I don't really get E[R-c] where R is a random variable and c is some constant. I think this can be explained by linearity of expectation and E[c]=c, but I am not sure. Well, that is all I have to say, but I need to tell you that there is a high probability that I will not be in class on Friday since I have an away swim meet. I will get back to you on the exact time that I will be leaving and whether it means I will miss class. -Harrison From medrano@MIT.EDU Tue Dec 6 03:16:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB68GD6N021623 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 03:16:13 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB68GBf0019816 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 03:16:11 -0500 (EST) Received: from SENIOR-THREE-TWENTY-ONE.MIT.EDU (SENIOR-THREE-TWENTY-ONE.MIT.EDU [18.244.6.66]) (authenticated bits=0) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB68Fv71001143 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 03:16:00 -0500 (EST) Subject: [sayan] final weekly question. finally From: Jesus Medrano To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Tue, 06 Dec 2005 03:15:58 +0000 Message-Id: <1133838958.10992.4.camel@localhost> Mime-Version: 1.0 X-Mailer: Evolution 2.2.3 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.184 X-Spam-Level: *** (3.184) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1154 Content-Length: 462 X-Status: A X-Keywords: NotJunk Hi Sayan, My question is: is the Markov Theorem actually used in practice? It does not seem to be a great bound. The last problem in the tutor problem to be true. Also I kept submitting the same answer it kept telling me it was wrong and eventually it said it was right. I had to kept with a friend to make sure I was doing it right and he submitted the same answer I did and it said it was wrong. So my question again is Why Markov? Why?! Jesus Medrano From lye@MIT.EDU Tue Dec 6 09:54:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB6Es56N018000 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 09:54:05 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB6Es3r1017665 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 09:54:04 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB6ErvdH023351 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 09:53:57 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jB6Erv4u015269; Tue, 6 Dec 2005 09:53:57 -0500 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 6 Dec 2005 09:53:57 -0500 Message-ID: <20051206095357.oo1wznsrikvwcgok@webmail.mit.edu> Date: Tue, 6 Dec 2005 09:53:57 -0500 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 14 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1155 Content-Length: 117 X-Status: A X-Keywords: NotJunk Is there a general way of calculating E[X^2] (like a nice compact formula)? What about for continuous distributions? From cbossard@MIT.EDU Tue Dec 6 12:00:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB6H0o6N016856 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 12:00:50 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB6H0m7G023929 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 12:00:48 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB6H0fvK023857 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 12:00:42 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jB6H0fkb022078; Tue, 6 Dec 2005 12:00:41 -0500 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 6 Dec 2005 12:00:41 -0500 Message-ID: <20051206120041.763rwy9139c4gkwg@webmail.mit.edu> Date: Tue, 6 Dec 2005 12:00:41 -0500 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 14 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1156 Content-Length: 256 X-Status: A X-Keywords: NonJunk NotJunk I am a little confused about Chebyshev's inequality in terms of standard deviation. Also on the TP I dont understand why to get 6,000 more than the other candidate you don't just need half + 6,000 but it keeps telling me that 3,006,000 is wrong. Cynthia From rnjacobs@MIT.EDU Tue Dec 6 13:54:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB6IsT6N019081 for <6042-probs@theory.csail.MIT.EDU>; Tue, 6 Dec 2005 13:54:29 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB6Is37G009720 for <6042-probs@theory.csail.MIT.EDU>; Tue, 6 Dec 2005 13:54:03 -0500 (EST) Received: from m2-225-10.mit.edu (M2-225-10.MIT.EDU [18.21.0.92]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB6Iruk6014081 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Tue, 6 Dec 2005 13:53:57 -0500 (EST) Received: (from rnjacobs@localhost) by m2-225-10.mit.edu (8.12.9) id jB6Is04d025614; Tue, 6 Dec 2005 13:54:00 -0500 Message-Id: <200512061854.jB6Is04d025614@m2-225-10.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Reading response Date: Tue, 06 Dec 2005 13:54:00 -0500 From: r n jacobs X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1157 Content-Length: 53 X-Keywords: NonJunk NotJunk Nope, nothing new in the reading. - Robert Jacobs From meyer@imap.theory.csail.mit.edu Tue Dec 06 15:28:38 2005 Message-ID: <4395F47E.3050109@csail.mit.edu> Date: Tue, 06 Dec 2005 15:28:46 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jesus Medrano Subject: Re: [sayan] final weekly question. finally References: <1133838958.10992.4.camel@localhost> In-Reply-To: <1133838958.10992.4.camel@localhost> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 882 X-UID: 1158 X-Keywords: Tutor behaved properly for me and answers are correct. Which part of the last problem did you find the tutor to be responding inconsistently on? Anyway, Markov rarely gives a good bound all by itself, but can be powerful when combined with other tricks. For example, by applying Markov to (R-m)^2, where m = E[R], we derive the Chebyshev bound about R. regards, A. Jesus Medrano wrote: > Hi Sayan, > > My question is: is the Markov Theorem actually used in practice? It does > not seem to be a great bound. The last problem in the tutor problem to > be true. Also I kept submitting the same answer it kept telling me it > was wrong and eventually it said it was right. I had to kept with a > friend to make sure I was doing it right and he submitted the same > answer I did and it said it was wrong. So my question again is > Why Markov? Why?! > > Jesus Medrano > From meyer@imap.theory.csail.mit.edu Tue Dec 06 15:31:01 2005 Message-ID: <4395F50D.9030304@csail.mit.edu> Date: Tue, 06 Dec 2005 15:31:09 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: cbossard@MIT.EDU Subject: Re: [David] Week 14 reading comments References: <20051206120041.763rwy9139c4gkwg@webmail.mit.edu> In-Reply-To: <20051206120041.763rwy9139c4gkwg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 466 X-UID: 1159 X-Keywords: The total # of votes is fixed, so if one candidate got 6000 over the mean, the other would have to get 6000 BELOW the mean, so the diff would be 12,000. regards, A. cbossard@MIT.EDU wrote: > I am a little confused about Chebyshev's inequality in terms of > standard deviation. > > Also on the TP I dont understand why to get 6,000 more than the other > candidate you don't just need half + 6,000 but it keeps telling me > that 3,006,000 is wrong. > > Cynthia From meyer@imap.theory.csail.mit.edu Tue Dec 06 15:37:36 2005 Message-ID: <4395F698.8010402@csail.mit.edu> Date: Tue, 06 Dec 2005 15:37:44 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Linda Ye Subject: Re: [Sayan] Week 14 Comments References: <20051206095357.oo1wznsrikvwcgok@webmail.mit.edu> In-Reply-To: <20051206095357.oo1wznsrikvwcgok@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 560 X-UID: 1160 X-Keywords: We don't do continuous distributions in discrete Math :-), though replacing our sums by integrals (of suitable kind) often allows discrete results to carry over to the continuous case. And no, there's no closed form formula for E[X^2] in terms of say, E[X]. But for the kind of simple X's we consider, it's usually possible to calculate E[X^2] from first principles using the def and linearity of E[]. regards, A Linda Ye wrote: > Is there a general way of calculating E[X^2] (like a nice compact formula)? What > about for continuous distributions? From meyer@imap.theory.csail.mit.edu Tue Dec 06 16:09:46 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB6L9j6N000382 for ; Tue, 6 Dec 2005 16:09:45 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1Ejk4H-0000Iz-FM for meyer@csail.mit.edu; Tue, 06 Dec 2005 16:09:45 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB6L9i9a004763 for ; Tue, 6 Dec 2005 16:09:44 -0500 (EST) Received: from SENIOR-THREE-TWENTY-ONE.MIT.EDU (SENIOR-THREE-TWENTY-ONE.MIT.EDU [18.244.6.66]) (authenticated bits=0) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB6L9YGU016808 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for ; Tue, 6 Dec 2005 16:09:37 -0500 (EST) Subject: Re: [sayan] final weekly question. finally From: Jesus Medrano To: "Prof. Albert R. Meyer" In-Reply-To: <4395F47E.3050109@csail.mit.edu> References: <1133838958.10992.4.camel@localhost> <4395F47E.3050109@csail.mit.edu> Content-Type: text/plain Date: Tue, 06 Dec 2005 16:09:35 +0000 Message-Id: <1133885376.27912.1.camel@localhost> Mime-Version: 1.0 X-Mailer: Evolution 2.2.3 X-Spam-Score: 1.695 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.5 required=3.7 tests=AWL,BAYES_00, DATE_IN_PAST_03_06 autolearn=no version=2.63 Status: RO Content-Length: 1180 X-UID: 1161 X-Keywords: NotJunk Part 3 of the last problem. I put 1000/1001 and 1000 / 1001 and many combinations with different spacing and I received the x mark on all of them. Then out of nowhere it worked. Jesus On Tue, 2005-12-06 at 15:28 -0500, Prof. Albert R. Meyer wrote: > Tutor behaved properly for me and answers are correct. Which part of > the last problem did you find the tutor to be responding inconsistently on? > > Anyway, Markov rarely gives a good bound all by itself, but can be > powerful when combined with other tricks. For example, by applying > Markov to (R-m)^2, where m = E[R], we derive the Chebyshev bound about R. > > regards, A. > > > Jesus Medrano wrote: > > Hi Sayan, > > > > My question is: is the Markov Theorem actually used in practice? It does > > not seem to be a great bound. The last problem in the tutor problem to > > be true. Also I kept submitting the same answer it kept telling me it > > was wrong and eventually it said it was right. I had to kept with a > > friend to make sure I was doing it right and he submitted the same > > answer I did and it said it was wrong. So my question again is > > Why Markov? Why?! > > > > Jesus Medrano > > From meyer@imap.theory.csail.mit.edu Tue Dec 06 16:59:11 2005 Message-ID: <439609AF.7010602@csail.mit.edu> Date: Tue, 06 Dec 2005 16:59:11 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jesus Medrano CC: 6042-online-tutor@theory.csail.mit.edu Subject: Re: [sayan] final weekly question. finally References: <1133838958.10992.4.camel@localhost> <4395F47E.3050109@csail.mit.edu> <1133885376.27912.1.camel@localhost> In-Reply-To: <1133885376.27912.1.camel@localhost> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 1321 Status: RO X-UID: 1162 X-Keywords: Ok, I'm cc'ing the tutor-master to watch for similar probs. thx for the info. regards, A. Jesus Medrano wrote: >Part 3 of the last problem. I put 1000/1001 and 1000 / 1001 and many >combinations with different spacing and I received the x mark on all of >them. Then out of nowhere it worked. > >Jesus > >On Tue, 2005-12-06 at 15:28 -0500, Prof. Albert R. Meyer wrote: > > >>Tutor behaved properly for me and answers are correct. Which part of >>the last problem did you find the tutor to be responding inconsistently on? >> >>Anyway, Markov rarely gives a good bound all by itself, but can be >>powerful when combined with other tricks. For example, by applying >>Markov to (R-m)^2, where m = E[R], we derive the Chebyshev bound about R. >> >>regards, A. >> >> >>Jesus Medrano wrote: >> >> >>>Hi Sayan, >>> >>>My question is: is the Markov Theorem actually used in practice? It does >>>not seem to be a great bound. The last problem in the tutor problem to >>>be true. Also I kept submitting the same answer it kept telling me it >>>was wrong and eventually it said it was right. I had to kept with a >>>friend to make sure I was doing it right and he submitted the same >>>answer I did and it said it was wrong. So my question again is >>>Why Markov? Why?! >>> >>>Jesus Medrano >>> >>> >>> From hectorb@MIT.EDU Tue Dec 6 18:25:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB6NP16N008100 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 18:25:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB6NP0xb029651 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 18:25:00 -0500 (EST) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB6NOvVa006661 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 18:24:58 -0500 (EST) Message-ID: <43961DC4.3060601@mit.edu> Date: Tue, 06 Dec 2005 18:24:52 -0500 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 14 Reading Comment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1163 Content-Length: 374 X-Keywords: NonJunk NotJunk "High variance is often associated with high risk. For example, in ten rounds of Game A, we expect to make $10, but could conceivably lose $10 instead. On the other hand, in ten rounds of game B, we also expect to make $10, but could actually lose more than $20,000!" -p.14 I found this example interesting. I enjoy reading the examples that involve gambling. -Hector From dowgun@MIT.EDU Tue Dec 6 18:42:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB6NgF6N013462 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 18:42:15 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB6NgExb015618 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 18:42:14 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB6NgCoB010923 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 6 Dec 2005 18:42:12 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jB6NgCwV016235; Tue, 6 Dec 2005 18:42:12 -0500 Received: from PLP-FORTY-SIX.MIT.EDU (PLP-FORTY-SIX.MIT.EDU [18.218.1.46]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 6 Dec 2005 18:42:12 -0500 Message-ID: <20051206184212.295phtwjiggowoos@webmail.mit.edu> Date: Tue, 6 Dec 2005 18:42:12 -0500 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] ln 14 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1164 Content-Length: 430 X-Status: A X-Keywords: NotJunk I feel like I must be misunderstanding variance, because it seems incredibly difficult to calculate for problems that don't have only two outcomes (say, the third problem on the tutor problems). I mean, if you flip a fair coin 100 times, then to get E(A - E(A))^2 you have to calculate the probability of each and every value of A coming up, right? And then add them all up? There has to be an easier way to do this. Thanks, Neil From meyer@imap.theory.csail.mit.edu Tue Dec 06 20:57:01 2005 Message-ID: <4396416E.6030703@csail.mit.edu> Date: Tue, 06 Dec 2005 20:57:02 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Neil M Dowgun Subject: Re: [Hanson] ln 14 References: <20051206184212.295phtwjiggowoos@webmail.mit.edu> In-Reply-To: <20051206184212.295phtwjiggowoos@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 603 Status: RO X-UID: 1165 X-Keywords: the variance is 100 times the variance of one flip (which is1/2(1-1/2) = 1/4) so the answer is 100(1/4)=25; see Notes 14, section 8.3. Neil M Dowgun wrote: >I feel like I must be misunderstanding variance, because it seems incredibly >difficult to calculate for problems that don't have only two outcomes (say, the >third problem on the tutor problems). I mean, if you flip a fair coin 100 times, >then to get E(A - E(A))^2 you have to calculate the probability of each and >every value of A coming up, right? And then add them all up? There has to be an >easier way to do this. >Thanks, Neil > > From petek@MIT.EDU Tue Dec 6 21:50:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB72o56N018847 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 21:50:05 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB72o4vM015882 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 21:50:04 -0500 (EST) Received: from [18.194.1.37] (SN-THIRTY-SEVEN.MIT.EDU [18.194.1.37]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB72nvrG018151 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 21:49:58 -0500 (EST) Message-ID: <43964DE4.2020005@mit.edu> Date: Tue, 06 Dec 2005 21:50:12 -0500 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Markov+ Chebsnsvlskjdfslky Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 4.562 X-Spam-Level: **** (4.562) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1166 Content-Length: 1030 X-Status: A X-Keywords: NonJunk NotJunk Despite the fact that my eyes / brain refuse to take the effort to actually pronounce the second man's name, I got through this week's notes in an ok manner.  I'm having trouble getting the last TP prob right.  Given standard dev = 2000, and you want the pr that gore gets 6,000 more votes, shouldn't the prob be bounded by (standard dev)^2 / (6,000)^2  =  1/9? 

Also, this week's notes is a good example of stuff that rather than make a good home in my head will probably make a much better one on the crib sheet we bring into the exam.

-P
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
508.843.5861 ::::Cell Phone::
617.536.9925 :::Sigma Nu:::::
From mracich@MIT.EDU Tue Dec 6 23:35:39 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB74Zd6N015096 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:35:39 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB74ZcSt011822 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:35:38 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB74ZcIj005445 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:35:38 -0500 (EST) Received: from MACGREGOR-TWO-FIFTY-NINE.MIT.EDU (MACGREGOR-TWO-FIFTY-NINE.MIT.EDU [18.239.6.4]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB74ZZok007795 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:35:35 -0500 (EST) Subject: [Sayan] Comments for Course Notes, Week 14 (Missed Expectations?) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Tue, 06 Dec 2005 23:35:34 -0500 Message-Id: <1133930134.8131.5.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1167 Content-Length: 274 X-Keywords: NotJunk I found all of section 7, "Chebyshev's Theorem", (starting on page 12) incredibly confusing. I also had a very hard time with the tutor problem questions that related to this material. I would appreciate it if this section was reviewed in depth. Thanks, Moira Racich From rian@MIT.EDU Tue Dec 6 23:40:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB74eo6N015450 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:40:50 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB74enYe009708 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:40:49 -0500 (EST) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB74ek0b010708 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:40:47 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rian Hunter Subject: [jelani] - week 14 comments Date: Tue, 6 Dec 2005 23:40:45 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1168 Content-Length: 585 X-Keywords: NotJunk This week's reading is very long, lots of concepts to learn and/or equations to memorize. I'd like to request that when the class is tested on this material, it isn't very very rigorous since to adequately absorb this material and apply it, it will take longer than a week or at least I think so. Variance and Chebyshev's Theorem seem clear and sensible to me but in class I would like to see more applications of these tools to gain a better grasp of the topics. Also way more examples of ways to calculate variance in different circumstances or problems. Rian Hunter From ridell@MIT.EDU Tue Dec 6 23:42:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB74gV6N015589 for <6042-probs@theory.csail.MIT.EDU>; Tue, 6 Dec 2005 23:42:31 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB74gUYe010921 for <6042-probs@theory.csail.MIT.EDU>; Tue, 6 Dec 2005 23:42:30 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB74gSMq011018 for <6042-probs@theory.csail.MIT.EDU>; Tue, 6 Dec 2005 23:42:28 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jB74gSii030633; Tue, 6 Dec 2005 23:42:28 -0500 Received: from NEXT-NINE.MIT.EDU (NEXT-NINE.MIT.EDU [18.242.5.9]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 6 Dec 2005 23:42:28 -0500 Message-ID: <20051206234228.nqov9g9yta1w0kg4@webmail.mit.edu> Date: Tue, 6 Dec 2005 23:42:28 -0500 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: tp 14 reading comments [hanson] MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1169 Content-Length: 271 X-Keywords: NonJunk NotJunk Can you give more examples of problems with Markov's thm from p10 of the reading? -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From tonyng@MIT.EDU Tue Dec 6 23:57:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB74vU6N020683 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:57:30 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB74vTYe021563 for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:57:29 -0500 (EST) Received: from TNG.mit.edu (BURTON-NINETY-TWO.MIT.EDU [18.247.5.92]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB74vMwr013673 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 6 Dec 2005 23:57:23 -0500 (EST) Message-Id: <5.2.1.1.2.20051206234656.0214fea0@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 06 Dec 2005 23:57:01 -0500 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments Week 14 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1170 Content-Length: 214 X-Keywords: NotJunk I have a question on theorem 4.2 on pg8. Given two independent random variables A and B, E[A*B] = E[A]*E[B]. However, is it true backwards? Does E[A]*E[B] = E[A*B] imply that A and B are independent? - Tony Ng From ereid@MIT.EDU Wed Dec 7 00:07:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB757r6N025934 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:07:53 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB757qYe028747 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:07:52 -0500 (EST) Received: from [18.242.6.198] (NEXT-FOUR-FIFTY-THREE.MIT.EDU [18.242.6.198]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB757kDO015962 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:07:50 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Reading Comment Date: Wed, 7 Dec 2005 00:07:43 -0500 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1171 Content-Length: 210 X-Status: A X-Keywords: NonJunk NotJunk The definition used for Variance and Standard Deviation seem like overly complicated ones. Maybe it's just the shorthand used, but it's a lot less intuitive this way than other ways that I have learned it. From jehan@MIT.EDU Wed Dec 7 00:40:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB75e16N005758 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:40:01 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB75e0Ye021180 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:40:00 -0500 (EST) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB75diYL020682 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:39:44 -0500 (EST) Message-Id: <5.2.1.1.2.20051207003453.00bc3678@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 07 Dec 2005 00:37:48 -0500 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 14 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1172 Content-Length: 72 X-Keywords: NonJunk NotJunk No questions. I thought Markov's approximation was pretty cool. jehan From avalys@MIT.EDU Wed Dec 7 00:42:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB75gr6N006269 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:42:53 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB75gqYe023163 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:42:52 -0500 (EST) Received: from [18.96.6.214] (SIMMONS-FOUR-SIXTY-NINE.MIT.EDU [18.96.6.214]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB75gn63021101 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:42:50 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: <2C7AD70A-F06B-4655-8E53-8759C266EA5D@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Required Reading Comments Date: Wed, 7 Dec 2005 00:42:51 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1173 Content-Length: 174 X-Keywords: NonJunk NotJunk I am a little unclear about how to use the alternative expression of variance described in section 8.2 (page 8). How is one supposed to find the value of E[R^2]? Alex From yangc@MIT.EDU Wed Dec 7 00:43:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB75hi6N006325 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:43:44 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB75hhYe023814 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:43:43 -0500 (EST) Received: from christope8f0c6 (MACGREGOR-ONE-TWENTY-FOUR.MIT.EDU [18.239.5.124]) (authenticated bits=0) (User authenticated as yangc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB75hcLr021194 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:43:41 -0500 (EST) Message-Id: <200512070543.jB75hcLr021194@outgoing.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 14 Comments Date: Wed, 7 Dec 2005 00:43:37 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0043_01C5FAC7.44623590" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcX68StIqMReESUKT/OUXquD7Sjetg== X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1174 Content-Length: 2191 X-Keywords: NonJunk NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0043_01C5FAC7.44623590 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Oh my God, I don't understand Variance or the two theorems about the upper/lower bounds at all. Please tell me we are going over this in class. Thanks, Chris Yang ------=_NextPart_000_0043_01C5FAC7.44623590 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Oh my God, I don’t understand Variance or the = two theorems about the upper/lower bounds at all.  Please tell me we = are going over this in class.

 

Thanks,

Chris Yang

------=_NextPart_000_0043_01C5FAC7.44623590-- From hzhou@MIT.EDU Wed Dec 7 00:46:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB75k76N006614 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:46:07 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB75k6Ye025398 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:46:06 -0500 (EST) Received: from hzhou.mit.edu (BURTON-NINETY-FIVE.MIT.EDU [18.247.5.95]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB75k4vd021573 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 00:46:04 -0500 (EST) Message-Id: <5.2.1.1.2.20051207003858.02aa40a0@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 07 Dec 2005 00:46:21 -0500 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: [David] tp-14 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1175 Content-Length: 263 X-Keywords: NonJunk NotJunk Hi, Topic is clear. I have a question for problem set number 5. With the answer i got for (a), if i were to solve for K that minimized that expression, I get a quadratic equation. Is it that complicated, or did I do part (a) wrong. Thanks. - Steven Zhou From mukkala@MIT.EDU Wed Dec 7 01:00:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB76066N012155 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 01:00:06 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7605Ye004427 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 01:00:05 -0500 (EST) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-FIFTY-TWO.MIT.EDU [18.235.1.152]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB75xvFU023437 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 00:59:58 -0500 (EST) Message-Id: <6.2.1.2.2.20051207001238.01f2fe20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 07 Dec 2005 00:59:55 -0500 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 14 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jB76066N012155 Status: RO X-UID: 1176 Content-Length: 535 X-Keywords: NotJunk Hi, My question pertains to the TP as opposed to the reading. In TP.14.3, it reads: "Suppose you flip a fair coin 100 times. The coin flips are all mutually independent. 3. What is the variance of the number of heads? Let Xi be the random variable whose value is 1 if the ith coin flip is heads. Then Var[Xi] = 1/2 - (1/2)2 = 1/4. So Var[X1 + ... + X100] = 100(1/4) = 25. (Note we can use additivity of variance here since our flips are independent.) ] " I don't understand the calculation for Var[Xi]. Thanks, Praveen From dnreshef@MIT.EDU Wed Dec 7 01:05:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB765h6N012498 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:05:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB765fYe008369 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:05:42 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB765Y79024272 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:05:34 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jB765YGE025250; Wed, 7 Dec 2005 01:05:34 -0500 Received: from MACGREGOR-EIGHTY-SIX.MIT.EDU (MACGREGOR-EIGHTY-SIX.MIT.EDU [18.239.5.86]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 01:05:34 -0500 Message-ID: <20051207010534.pihvlgyaiyckwgks@webmail.mit.edu> Date: Wed, 7 Dec 2005 01:05:34 -0500 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments 14 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1177 Content-Length: 287 X-Keywords: This week's reading was particularly technical and cumbersome. I had the most trouble understanding condeitional expectation and variance. Could we please go over variance real quickly in class, I know it's not a really hard concept, I just don't really understand it quite yet. -Dave From lmccart@MIT.EDU Wed Dec 7 01:11:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB76Bk6N013625 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:11:46 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB76BjYe012308 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:11:45 -0500 (EST) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB76Bh6K024949 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:11:43 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id jB76Bh5V009935; Wed, 7 Dec 2005 01:11:43 -0500 Received: from 18.153.1.172 ([18.153.1.172]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 01:11:43 -0500 Message-ID: <20051207011143.kx8gkdpnkrcwsccw@webmail.mit.edu> Date: Wed, 7 Dec 2005 01:11:43 -0500 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1178 Content-Length: 95 X-Keywords: NotJunk The topic of standard deviation was a little bit confusing but everything else was very clear. From ctsims@MIT.EDU Wed Dec 7 01:23:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB76NL6N019398 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 01:23:21 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB76NJYe019796 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 01:23:20 -0500 (EST) Received: from HowardRoark.mit.edu (EASTCAMPUS-EIGHT-TWELVE.MIT.EDU [18.238.6.45]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB76NGgC026240 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 01:23:17 -0500 (EST) Message-Id: <5.2.1.1.2.20051207011740.0303ae80@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 07 Dec 2005 01:21:06 -0500 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: [6.042] Reading Comment Lecture Notes 14 - Clayton Sims Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1179 Content-Length: 295 X-Keywords: NonJunk NotJunk I found that the definition of Variance using a the expectation of (random variable minus it's expected value) squared) defined on page 13, and used in the next 3 or 4 pages, to be hard to apply to some real world cases. It would be nice to have it more fully explained in class. -Clayton From jjmonzon@MIT.EDU Wed Dec 7 01:24:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB76OX6N019643 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 01:24:33 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB76OWYe020538 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 01:24:32 -0500 (EST) Received: from JOSHDESKTOP.mit.edu (BURTON-ONE-FORTY-SIX.MIT.EDU [18.247.5.146]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB76OQpx026362 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 01:24:29 -0500 (EST) Message-Id: <6.2.3.4.2.20051207012427.01d8a630@po9.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 07 Dec 2005 01:24:29 -0500 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: [David] - Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1180 Content-Length: 336 X-Status: A X-Keywords: NonJunk NotJunk I liked the section about inequalities (Markov and Chebyshev) - they are very great at giving bounds. I am just wondering if Central Limit Theorem is going to be introduced in this class. Josh Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From pgroudas@MIT.EDU Wed Dec 7 01:37:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB76bj6N028640 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:37:45 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB76biYe028619 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:37:44 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB76bgMG027648 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 01:37:42 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jB76bg2d027834; Wed, 7 Dec 2005 01:37:42 -0500 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 01:37:42 -0500 Message-ID: <20051207013742.lbmgldl6544kw4w0@webmail.mit.edu> Date: Wed, 7 Dec 2005 01:37:42 -0500 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: [HANSON] Week 14 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1182 Content-Length: 250 X-Keywords: NotJunk I thought the reading was okay, but shouldn't there have been a section on binomial standard deviation? Was this just covered already or something? I needed to look it up on the web so i could get the variance for one of the tutor problems. -Paul From shreyes19@gmail.com Wed Dec 7 01:43:15 2005 Return-Path: Received: from nproxy.gmail.com (nproxy.gmail.com [64.233.182.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB76hF6N030315 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 01:43:15 -0500 Received: by nproxy.gmail.com with SMTP id d4so87495nfe for <6042-probs@theory.lcs.mit.edu>; Tue, 06 Dec 2005 22:43:14 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=rINwFbOa1dAqS/8rKF3DDanQs+RotpMUQInBgRLEsw+QRpf+yZ5vEcp+tac9JPP8ENgxI6F2byJZ/huzSeYyfA73GcXYNQH+DzzIP0EtEvgU8dkdCYhxmnQG3PXlxcrqJWRaey3XC0xZl/2TTdoCEpjCU9jfNDeC4CGqe8BdaMU= Received: by 10.48.164.14 with SMTP id m14mr18498nfe; Tue, 06 Dec 2005 22:43:14 -0800 (PST) Received: by 10.48.127.11 with HTTP; Tue, 6 Dec 2005 22:43:14 -0800 (PST) Message-ID: <3c33fe380512062243t452e6f37ic72aa43637dbff44@mail.gmail.com> Date: Wed, 7 Dec 2005 01:43:14 -0500 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 14 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_13968_10334097.1133937794310" Status: RO X-UID: 1183 Content-Length: 2042 X-Keywords: NonJunk NotJunk ------=_Part_13968_10334097.1133937794310 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, Comments on the reading for this week: My comments deal with the two inequalities presented, Markov's (on page 10) and Chebyshev's (on page 12). The calculations required for both of these make sense to me and don't seem too difficult, but one thing I'd like to se= e discussed in class is the types of problems that each of these can help resolve, and suggestions on how to decide which inequality to choose when given a problem. I know Markov's inequality deals with the values greater than a specified number, but could Chebyshev's be used as well with careful selection of x in the inequality. (Hopefully the answer to this is better than the one given in 6.041 this morning, that Chebyshev's has an x^2 in th= e denominator, so when given a bound of that form, use that inequality). Thanks, Shreyes ------=_Part_13968_10334097.1133937794310 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

Comments on the reading for this week:

My comments deal with the two inequalities presented, Markov's (on page 10) and Chebyshev's (on page 12).  The calculations required for both of these make sense to me and don't seem too difficult, but one thing I'd like to see discussed in class is the types of problems that each of these can help resolve, and suggestions on how to decide which inequality to choose when given a problem.  I know Markov's inequality deals with the values greater than a specified number, but could Chebyshev's be used as well with careful selection of x in the inequality.  (Hopefully the answer to this is better than the one given in 6.041 this morning, that Chebyshev's has an x^2 in the denominator, so when given a bound of that form, use that inequality).


Thanks,
Shreyes
------=_Part_13968_10334097.1133937794310-- From jeffhoff@MIT.EDU Wed Dec 7 02:08:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB778Q6N007867 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 02:08:26 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB778PJh015920 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 02:08:25 -0500 (EST) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7766qN000145 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 02:06:47 -0500 (EST) Message-Id: <6.2.3.4.2.20051207020413.01e2a200@hesiod> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 07 Dec 2005 02:06:03 -0500 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 14 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1184 Content-Length: 165 X-Keywords: NonJunk NotJunk Page 13 Section 7.0 Variance! I don't get it. I can't calculate it. The binomial example didn't really help much. Can we get a better example with real numbers? From mpapi@MIT.EDU Wed Dec 7 03:04:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB784u6N021765 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:04:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB784tJh017910 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:04:55 -0500 (EST) Received: from mpapi.MIT.EDU (MPAPI.MIT.EDU [18.239.4.219]) (authenticated bits=0) (User authenticated as mpapi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB784qj7004187 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:04:53 -0500 (EST) Subject: [Jelani] week 14 comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 07 Dec 2005 03:08:59 -0500 Message-Id: <1133942939.5998.63.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1185 Content-Length: 226 X-Keywords: NonJunk NotJunk Section 7, on Chebyshev's theorem; I thought that the applications of Chebyshev's theorem (i.e. in the tutor problem) were pretty hard to grasp. I also don't really understand how it is derived from Markov's theorem. - Matt From mwangi@MIT.EDU Wed Dec 7 03:08:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB788Z6N022599 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:08:35 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB788WJh019825; Wed, 7 Dec 2005 03:08:32 -0500 (EST) Received: from IBM-1E80C73FC15.mit.edu (NEW-SIXTY.MIT.EDU [18.241.5.60]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB788Uge004373 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 7 Dec 2005 03:08:31 -0500 (EST) Message-Id: <6.2.1.2.2.20051207025356.03fd1eb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 07 Dec 2005 03:08:27 -0500 To: 6042-probs@theory.lcs.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 14 Comments Cc: mitras@MIT.EDU Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1186 Content-Length: 372 X-Keywords: NotJunk Passage: 7 Chebyshev's Theorem Page: 12 I found this passage most surprising because of the very low probability of a random variable deviating more than a few standard deviations from the mean. I found it especially surprising because Chebyshev's Theorem can be applied to any random variable, regardless of what it's PDF is. Sincerely, Timothy M. Mwangi From meyer@imap.theory.csail.mit.edu Wed Dec 07 03:22:16 2005 Message-ID: <43969BB9.2040402@csail.mit.edu> Date: Wed, 07 Dec 2005 03:22:17 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Pete Kruskall Subject: Re: [Jelani] Markov+ Chebsnsvlskjdfslky References: <43964DE4.2020005@mit.edu> In-Reply-To: <43964DE4.2020005@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 894 Status: RO X-UID: 1187 X-Keywords: if Gore got 6000 more votes, then he got 3000 ABOVE THE MEAN, and Bush got 3000 below, so the denominator should be (3000)^2. Regards, A. Pete Kruskall wrote: > Despite the fact that my eyes / brain refuse to take the effort to actually pronounce the second man's name, I got through this week's notes in an ok manner. I'm having trouble getting the last TP prob right. Given standard dev = 2000, and you want the pr that gore gets 6,000 more votes, shouldn't the prob be bounded by (standard dev)^2 / (6,000)^2 = 1/9? > > Also, this week's notes is a good example of stuff that rather than make a good home in my head will probably make a much better one on the crib sheet we bring into the exam. > > -P > >-- >Pete Kruskall >28 The Fenway >Boston, MA 02215 > >::::::::::::::::::::::::::::: >508.843.5861 ::::Cell Phone:: >617.536.9925 :::Sigma Nu::::: > From lana@MIT.EDU Wed Dec 7 03:41:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB78fE6N003973 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 03:41:14 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB78fCJh007498 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 03:41:13 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB78f6WB006051 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 03:41:06 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jB78f6mn006398; Wed, 7 Dec 2005 03:41:06 -0500 Received: from NEXT-FIVE-SIXTY-NINE.MIT.EDU (NEXT-FIVE-SIXTY-NINE.MIT.EDU [18.242.7.58]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 03:41:06 -0500 Message-ID: <20051207034106.2sfpeg8bro08co80@webmail.mit.edu> Date: Wed, 7 Dec 2005 03:41:06 -0500 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] TP14 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1188 Content-Length: 215 X-Keywords: NotJunk The part that I found to be the most intriguing was the symmetry in the alternative definition of variance, particularly because the link between the original definition and this version is far from intuitive. Lana From lkini@MIT.EDU Wed Dec 7 03:52:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB78qJ6N007253 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:52:19 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB78qIJh013853 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:52:18 -0500 (EST) Received: from [18.239.6.223] (MACGREGOR-FOUR-SEVENTY-EIGHT.MIT.EDU [18.239.6.223]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB78qA3s006610 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 03:52:12 -0500 (EST) Message-ID: <4396A2A8.4050800@mit.edu> Date: Wed, 07 Dec 2005 03:51:52 -0500 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 14 Lecture Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1189 Content-Length: 166 X-Keywords: NotJunk Hi Hanson, I found Chebyshev's Theorem most interesting cause it gives limits in the broadest of applications: from the birthday problem to the IQ example. Lohith From meyer@imap.theory.csail.mit.edu Wed Dec 07 04:12:18 2005 Message-ID: <4396A771.8040005@csail.mit.edu> Date: Wed, 07 Dec 2005 04:12:17 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Elizabeth Reid Subject: Re: [Hanson] Reading Comment References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 283 Status: RO X-UID: 1190 X-Keywords: What was your intuitive way? regards, A. Elizabeth Reid wrote: > The definition used for Variance and Standard Deviation seem like > overly complicated ones. Maybe it's just the shorthand used, but it's > a lot less intuitive this way than other ways that I have learned it. From meyer@imap.theory.csail.mit.edu Wed Dec 07 04:16:03 2005 Message-ID: <4396A854.8040201@csail.mit.edu> Date: Wed, 07 Dec 2005 04:16:04 -0500 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" Subject: Re: [David] - Reading Comments References: <6.2.3.4.2.20051207012427.01d8a630@po9.mit.edu> In-Reply-To: <6.2.3.4.2.20051207012427.01d8a630@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 557 Status: RO X-UID: 1191 X-Keywords: Central lim thm will be mentioned VERY briefly next week, along with an explanation of why it's not of central interest in engineering and computer science. regards, Joshua Jen C. Monzon wrote: > I liked the section about inequalities (Markov and Chebyshev) - they > are very great at giving bounds. I am just wondering if Central Limit > Theorem is going to be introduced in this class. > > Josh > > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From fluff@MIT.EDU Wed Dec 7 04:39:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB79dN6N022792 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 04:39:23 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB79dLJh008347 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 04:39:21 -0500 (EST) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB79dEkT008578 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 04:39:15 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v746.2) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 14 reading Date: Wed, 7 Dec 2005 04:37:08 -0500 X-Mailer: Apple Mail (2.746.2) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1192 Content-Length: 259 X-Keywords: NonJunk NotJunk I found all the examples about IQ really depressing. They just seem to highlight the fact that I don't understand something. I still don't intuitively get how you can just ignore dependence/independence for the linearity of expectation stuff. ~Crystal From benlu@MIT.EDU Wed Dec 7 08:09:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7D9B6N025772 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 08:09:11 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7D99uB004719 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 08:09:10 -0500 (EST) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7D97EG022723 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 08:09:07 -0500 (EST) Message-ID: <4396DEF3.4050003@mit.edu> Date: Wed, 07 Dec 2005 08:09:07 -0500 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051013) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 14 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.076 X-Spam-Level: *** (3.076) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1193 Content-Length: 354 X-Keywords: NotJunk Hey Jelani, It's cool to finally have a definition for variance and standard deviation! I've heard those terms thrown around so many times, and until now have had to rely on a purely intuitive understanding of what people meant. As for things that I don't understand, I mostly just need practice calculating those E(R^2) terms. Cheers, ~Ben Lu From kromer@MIT.EDU Wed Dec 7 10:01:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7F1f6N011374 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 10:01:41 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7F1euB025552 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 10:01:40 -0500 (EST) Received: from m12-182-19.mit.edu (M12-182-19.MIT.EDU [18.19.0.50]) (authenticated bits=56) (User authenticated as kromer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7F1Wt8027859 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 10:01:33 -0500 (EST) Received: (from kromer@localhost) by m12-182-19.mit.edu (8.12.9) id jB7F1WxY021379; Wed, 7 Dec 2005 10:01:32 -0500 (EST) Subject: [David] Email comments for Reading From: Katherine Romer To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 07 Dec 2005 10:01:31 -0500 Message-Id: <1133967691.21236.9.camel@m12-182-19.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1194 Content-Length: 742 X-Keywords: NonJunk NotJunk "the expected value of a sum of random variables is the sum of the expected values of the variables" (p. 1) When we consider two random variables at the same time, I understand how we get the expectation of their sum, but how do we expect the values of the variables to be distributed? For example, say we're rolling two six-sided dice. We expect them to sum to 7, and each die individualy has an expected value of 3.5. Is the expected maximum of the dice 3.5, because that's the expectation of each die? It seems likely that one die would have a higher value than the other on most rolls, so it seems that the expected maximum must be larger than 3.5 and the expected minimum must be less--how would you go about computing this? Katherine From arup@MIT.EDU Wed Dec 7 10:09:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7F976N016057 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:09:07 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7F95uB005291 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:09:05 -0500 (EST) Received: from [18.247.6.35] (BURTON-TWO-NINETY.MIT.EDU [18.247.6.35]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7F8qQA001138 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:08:58 -0500 (EST) Message-ID: <4396FAE7.8000005@mit.edu> Date: Wed, 07 Dec 2005 10:08:23 -0500 From: Arup Sarma User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Week 14 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1195 Content-Length: 239 X-Keywords: NonJunk NotJunk The coupon collector problem (section 2) is probably the most interesting section in this week's reading to me because I've seen most of the rest of it in my statistics class, I think. Most of it seemed pretty straightforward. |Arup| From ryoung@MIT.EDU Wed Dec 7 10:40:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7FeO6N031865 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 10:40:24 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7FeNuB017076 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 10:40:23 -0500 (EST) Received: from ryan.mit.edu (PSK-SIXTEEN.MIT.EDU [18.217.1.16]) (authenticated bits=0) (User authenticated as ryan786@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7FeF9l015586 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 10:40:16 -0500 (EST) Message-Id: <6.2.3.4.2.20051207102936.02f0feb0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 07 Dec 2005 10:40:17 -0500 To: 6042-probs@theory.lcs.mit.edu From: ryan Subject: [sayan] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1196 Content-Length: 177 X-Keywords: NonJunk NotJunk I had a really hard time with Chebyshev bounds, and couldn't even get them on the tutor problems for the longest time. I understand them now, but it took me a while. -Ryan From rshroff@MIT.EDU Wed Dec 7 10:43:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7FhO6N002751 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 10:43:24 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7FhMuB021219 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 10:43:23 -0500 (EST) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7FhGdr016955 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 10:43:16 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id jB7FhG6Z031822; Wed, 7 Dec 2005 10:43:16 -0500 Received: from QUICKSTATION-LOBBY-16-1.MIT.EDU (QUICKSTATION-LOBBY-16-1.MIT.EDU [18.55.1.64]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 10:43:16 -0500 Message-ID: <20051207104316.v98xqdxg8vyg40c4@webmail.mit.edu> Date: Wed, 7 Dec 2005 10:43:16 -0500 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Reading Assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1197 Content-Length: 119 X-Keywords: NonJunk NotJunk I found the properties of expectation and variance of trhe various distributions extremely interesting. -Rahul Shroff From zacharyozer@gmail.com Wed Dec 7 10:44:31 2005 Return-Path: Received: from zproxy.gmail.com (zproxy.gmail.com [64.233.162.204]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7FiV6N005151 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 10:44:31 -0500 Received: by zproxy.gmail.com with SMTP id o37so457467nzf for <6042-probs@theory.lcs.mit.edu>; Wed, 07 Dec 2005 07:44:30 -0800 (PST) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=YY/xrE0qiVsOEQlGP4bi59t1VY0dHGm34+jkJ5SM1CIuIWABYUX7e0nhZEG0BJ5jVb50Vm+CGbZZG2x+ORZ8TrzgIAw8M1FK5yS6+eANi4lrTf2/aS2sAQzUIL/SvSiRWHa9c+TtJW3MOP79yazmYKb+SyfthxivNWYoadVEtvE= Received: by 10.64.220.17 with SMTP id s17mr1841538qbg; Wed, 07 Dec 2005 07:44:30 -0800 (PST) Received: by 10.64.27.3 with HTTP; Wed, 7 Dec 2005 07:44:30 -0800 (PST) Message-ID: <4f2613e50512070744p626f014esa7e23e823b0e6752@mail.gmail.com> Date: Wed, 7 Dec 2005 10:44:30 -0500 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: =?WINDOWS-1252?Q?Markov=92s_Theorem_and_Chebyshev=92s_Theorem?= MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id jB7FiV6N005151 Status: RO X-UID: 1198 Content-Length: 267 X-Keywords: NotJunk I though that both of these sections of the notes were really well done. Having read them, I really felt like I understood the concepts and I'm eager to tackle the in class problems on them. But really, I just wanted to compliment y'all on an excellent job. -zozer From fgreen@MIT.EDU Wed Dec 7 10:53:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7FrM6N009810 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 10:53:22 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7FrKuB004558 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 10:53:20 -0500 (EST) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7FrDxQ021379 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 10:53:13 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id jB7FrDVV005993; Wed, 7 Dec 2005 10:53:13 -0500 Received: from FAIRCHILD-NINE-FORTY-SIX.MIT.EDU (FAIRCHILD-NINE-FORTY-SIX.MIT.EDU [18.108.5.179]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 10:53:13 -0500 Message-ID: <20051207105313.a4hytrdbkczk4wwo@webmail.mit.edu> Date: Wed, 7 Dec 2005 10:53:13 -0500 From: Forrest O Green To: 6042-probs@theory.csail.MIT.EDU Subject: julani, required reading comment 14, from Forrest Green MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1199 Content-Length: 105 X-Keywords: NotJunk I was really glad to finally figure out where variences and standerd deviations came from. -Forrest From yaser@MIT.EDU Wed Dec 7 11:03:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7G3k6N018943 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:03:46 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7G3VuB018869 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:03:31 -0500 (EST) Received: from Charizard (NEW-THREE-TWENTY-TWO.MIT.EDU [18.241.6.67]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7G3OnW026286 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:03:29 -0500 (EST) Message-Id: <200512071603.jB7G3OnW026286@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: tp14 reading comments Date: Wed, 7 Dec 2005 11:03:23 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0066_01C5FB1D.DAA49650" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcX7R7/7i1tAvm00TXO1OwL8l3VbgA== X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1200 Content-Length: 6275 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0066_01C5FB1D.DAA49650 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi David, On a small point, could we use Markov instead of Chebyshev's theorem in cases where the distribution is symmetric? (As I understand that it shows the 'distance' from the mean in either direction as opposed to Markov who simply shows the deviation 'above the mean'.) My actual comment is that the proof of variance of sums seems really intimidating, I hope that the lecture & lecture slides can clear up what exactly needs to be understood about its properties. Thanks! _Yaser ------=_NextPart_000_0066_01C5FB1D.DAA49650 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

On a small point, could we use = Markov instead of Chebyshev’s theorem in cases where = the distribution is symmetric? (As I understand that it shows the = ‘distance’ from the mean in either direction as opposed to Markov who simply shows = the deviation ‘above the mean’.)

 

My actual comment is that the = proof of variance of sums seems really intimidating, I = hope that the lecture & lecture slides can clear up what exactly needs to = be understood about its properties.

 

Thanks!

 

_Yaser

------=_NextPart_000_0066_01C5FB1D.DAA49650-- From mitras@theory.csail.mit.edu Wed Dec 7 11:06:22 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7G6M6N020704 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:06:22 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7G6LNj001095; Wed, 7 Dec 2005 11:06:21 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7G6CMu027472; Wed, 7 Dec 2005 11:06:13 -0500 (EST) Message-ID: <43970874.6070207@theory.csail.mit.edu> Date: Wed, 07 Dec 2005 11:06:12 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Andrew Shafer CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Sayan] Week 14 Comments References: <000201c5f8f2$c4620760$6d04f712@ajshafer> In-Reply-To: <000201c5f8f2$c4620760$6d04f712@ajshafer> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1201 Content-Length: 354 X-Keywords: NotJunk Yes. -sayan Andrew Shafer wrote: > Will we be required to know the expectation and variance for some > standard random variables? > > -Andrew > ---------------------------- > Illegitmitatum Non Carborundum Est > Andrew Shafer, MIT Blog > http://shaferandrew.blogspot.com > Si hoc legere scis numium eruditionis habes. > ---------------------------- > From meyer@imap.theory.csail.mit.edu Wed Dec 07 08:03:28 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7D3R6N025170 for ; Wed, 7 Dec 2005 08:03:27 -0500 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EjyxD-0003VX-GS for meyer@csail.mit.edu; Wed, 07 Dec 2005 08:03:27 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7D3QuB029979 for ; Wed, 7 Dec 2005 08:03:26 -0500 (EST) Received: from [18.194.1.37] (SN-THIRTY-SEVEN.MIT.EDU [18.194.1.37]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7D3NpD021775 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for ; Wed, 7 Dec 2005 08:03:24 -0500 (EST) Message-ID: <4396DDAC.1050202@mit.edu> Date: Wed, 07 Dec 2005 08:03:40 -0500 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Prof. Albert R. Meyer" Subject: Re: [Jelani] Markov+ Chebsnsvlskjdfslky References: <43964DE4.2020005@mit.edu> <43969BB9.2040402@csail.mit.edu> In-Reply-To: <43969BB9.2040402@csail.mit.edu> Content-Type: text/html; charset=ISO-8859-1 X-Spam-Score: 1.962 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.4 required=3.7 tests=BAYES_00,HTML_MESSAGE, HTML_TITLE_EMPTY,MIME_HTML_ONLY autolearn=no version=2.63 Status: RO Content-Length: 1718 X-UID: 1202 X-Keywords: NonJunk NotJunk Haha.   That would make more sense.  Thanks!

-P

Prof. Albert R. Meyer wrote:
if Gore got 6000 more votes, then he got 3000 ABOVE THE MEAN, and Bush got
3000 below, so the denominator should be (3000)^2.

Regards, A.

Pete Kruskall wrote:

> Despite the fact that my eyes / brain refuse to take the effort to actually pronounce the second man's name, I got through this week's notes in an ok manner.  I'm having trouble getting the last TP prob right.  Given standard dev = 2000, and you want the pr that gore gets 6,000 more votes, shouldn't the prob be bounded by (standard dev)^2 / (6,000)^2  =  1/9?  >
> Also, this week's notes is a good example of stuff that rather than make a good home in my head will probably make a much better one on the crib sheet we bring into the exam.
>
> -P
>
>--  >Pete Kruskall
>28 The Fenway
>Boston, MA 02215
>
>:::::::::::::::::::::::::::::
>508.843.5861 ::::Cell Phone::
>617.536.9925 :::Sigma Nu:::::
> 


-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
508.843.5861 ::::Cell Phone::
617.536.9925 :::Sigma Nu:::::
From hmzhou@blackbird.csail.mit.edu Wed Dec 7 11:12:09 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GC96N023451; Wed, 7 Dec 2005 11:12:09 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jB7GC8B1025648; Wed, 7 Dec 2005 11:12:08 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jB7GC8Kr025645; Wed, 7 Dec 2005 11:12:08 -0500 Date: Wed, 7 Dec 2005 11:12:07 -0500 (EST) From: Hanson Zhou To: Alex Valys cc: 6042-probs@theory.lcs.mit.edu Subject: Re: 6.042: [Hanson] Required Reading Comments In-Reply-To: <2C7AD70A-F06B-4655-8E53-8759C266EA5D@mit.edu> Message-ID: References: <2C7AD70A-F06B-4655-8E53-8759C266EA5D@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1203 Content-Length: 309 X-Keywords: NonJunk NotJunk It shows you an example on page 9...you simply use definition of expectation. -Hanson On Wed, 7 Dec 2005, Alex Valys wrote: > I am a little unclear about how to use the alternative expression of > variance described in section 8.2 (page 8). How is one supposed to > find the value of E[R^2]? > > Alex > > From alisonc@MIT.EDU Wed Dec 7 11:13:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GDZ6N023629 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 11:13:35 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GDXoV027679 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 11:13:33 -0500 (EST) Received: from sns23.mit.edu (SNS23.MIT.EDU [18.62.2.196]) (authenticated bits=56) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7GDV46000744 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 11:13:31 -0500 (EST) Received: (from alisonc@localhost) by sns23.mit.edu (8.12.9) id jB7GDVjm007524; Wed, 7 Dec 2005 11:13:31 -0500 Subject: [Hanson] comments on notes #14 From: Alison S Cichowlas To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 07 Dec 2005 11:13:30 -0500 Message-Id: <1133972010.7478.0.camel@sns23.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1204 Content-Length: 96 X-Keywords: NotJunk I found the section on variance sort of confusing -- more examples would have been really nice. From dshin@MIT.EDU Wed Dec 7 11:13:46 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GDj6N023636 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:13:45 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GDiNj001539; Wed, 7 Dec 2005 11:13:44 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GDaok012628; Wed, 7 Dec 2005 11:13:37 -0500 (EST) Message-ID: <43970A0F.8040800@mit.edu> Date: Wed, 07 Dec 2005 11:13:03 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine Romer CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Email comments for Reading References: <1133967691.21236.9.camel@m12-182-19.mit.edu> In-Reply-To: <1133967691.21236.9.camel@m12-182-19.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.263 X-Spam-Level: *** (3.263) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1205 Content-Length: 996 X-Keywords: NotJunk There is no clean formula for E[max{A,B}] given E[A] and E[B]. One of the in-class problems actually outlined a nice way to compute E[max{k independent die}] - it used the formula E[X] = \sum_i Pr[X>i] DS Katherine Romer wrote: >"the expected value of a sum of random variables is the sum of the >expected values of the variables" (p. 1) > >When we consider two random variables at the same time, I understand how >we get the expectation of their sum, but how do we expect the values of >the variables to be distributed? For example, say we're rolling two >six-sided dice. We expect them to sum to 7, and each die individualy has >an expected value of 3.5. Is the expected maximum of the dice 3.5, >because that's the expectation of each die? It seems likely that one die >would have a higher value than the other on most rolls, so it seems that >the expected maximum must be larger than 3.5 and the expected minimum >must be less--how would you go about computing this? > >Katherine > > From mitras@theory.csail.mit.edu Wed Dec 7 11:16:22 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GGM6N023996 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:16:22 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GGKNj001682; Wed, 7 Dec 2005 11:16:20 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GGGMu028185; Wed, 7 Dec 2005 11:16:16 -0500 (EST) Message-ID: <43970AD0.6010503@theory.csail.mit.edu> Date: Wed, 07 Dec 2005 11:16:16 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Isaac Charny CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Sayan] Week 14 Comments References: <4394F6C5.7040907@mit.edu> In-Reply-To: <4394F6C5.7040907@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1206 Content-Length: 296 X-Keywords: NotJunk Hope you'll attend today's lecture. An easy way to calculate variance is to use theorem 8.1, i.e., Var(X) = E(X^2) - (E(X))^2. -sayan Isaac Charny wrote: > I don't really understand variance, what it is, and how to quickly > calculate it. Please cover this in the next lecture. > > ~Isaac > From crowell@MIT.EDU Wed Dec 7 11:17:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GH86N024096 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:17:08 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GH7oV002420 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:17:07 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7GH59a002209 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:17:06 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jB7GH5wM018129; Wed, 7 Dec 2005 11:17:05 -0500 Received: from SNS25.MIT.EDU (SNS25.MIT.EDU [18.62.2.198]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 11:17:05 -0500 Message-ID: <20051207111705.6ltzed48owdck8w0@webmail.mit.edu> Date: Wed, 7 Dec 2005 11:17:05 -0500 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1207 Content-Length: 155 X-Keywords: NonJunk NotJunk I really found the examples (coupon collector etc) to be very interesting. These were talked about in lecture and are on 1.3 on page 3. -Rob Crowell From kktyan@MIT.EDU Wed Dec 7 11:18:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GIu6N024182 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:18:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GItoV004761 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:18:55 -0500 (EST) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7GIqRZ002989 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:18:52 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id jB7GIq0K019874; Wed, 7 Dec 2005 11:18:52 -0500 Received: from BURTON-THREE-SEVENTY.MIT.EDU (BURTON-THREE-SEVENTY.MIT.EDU [18.247.6.115]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 11:18:52 -0500 Message-ID: <20051207111852.p9kr5f3649jkckwk@webmail.mit.edu> Date: Wed, 7 Dec 2005 11:18:52 -0500 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments 14 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1208 Content-Length: 527 X-Keywords: NonJunk NotJunk I was a little confused over the definition of variance on page 13. Var[R]::= E[(R - E[R])^2] makes it seem as if it's the expected value of the squares of (R - E[R]), so I thought this meant calculating the sums of the squares and dividing by the probability of each happening. It seems to be more like the square of the expected (R - E[R]) and then the expected value of that...or something. In other words, I'm confused over the definition of variance. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From dshin@MIT.EDU Wed Dec 7 11:27:25 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GRO6N028629 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:27:25 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GRN7T014711 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:27:23 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GRNd9000122; Wed, 7 Dec 2005 11:27:23 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GRKMu029003; Wed, 7 Dec 2005 11:27:21 -0500 (EST) Message-ID: <43970D48.9030300@mit.edu> Date: Wed, 07 Dec 2005 11:26:48 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason Juang CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 14 Comments References: <43926B8E.9080106@mit.edu> In-Reply-To: <43926B8E.9080106@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.263 X-Spam-Level: *** (3.263) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1209 Content-Length: 487 X-Keywords: NotJunk Heh, Professor Meyer and I discussed this after class the other day. Were you able to solve it during the exam? DS Jason Juang wrote: > I don't have much comment on the reading, so I'll instead mention that > Problem B4 on the Putnam today was, paraphrased: > Let f(m,n) be the number of n-tuples of integers (x_1, x_2, ..., > x_n) which satisfy |x_1| + |x_2| + ... + |x_n| <= m. Show that f(m,n) > = f(n,m). > > ... Sounds like a good final exam question to me ;) > > Jason. From mitras@theory.csail.mit.edu Wed Dec 7 11:27:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GRQ6N028634 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:27:26 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GRP7T014734 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:27:25 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GLcd9029809; Wed, 7 Dec 2005 11:21:39 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GLVMu028657; Wed, 7 Dec 2005 11:21:31 -0500 (EST) Message-ID: <43970C0B.3030400@theory.csail.mit.edu> Date: Wed, 07 Dec 2005 11:21:31 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Svetlana Goldenberg CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Sayan] TP14 References: <20051207034106.2sfpeg8bro08co80@webmail.mit.edu> In-Reply-To: <20051207034106.2sfpeg8bro08co80@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1210 Content-Length: 412 X-Keywords: NotJunk By alternative definition I guess you mean theorem 8.1, right ? This also happens to be an easier definition to use when you are computing variance. -Sayan Svetlana Goldenberg wrote: >The part that I found to be the most intriguing was the symmetry in the >alternative definition of variance, particularly because the link between the >original definition and this version is far from intuitive. >Lana > > From dshin@MIT.EDU Wed Dec 7 11:27:27 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GRR6N028639 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:27:27 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GRP7V014735 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 11:27:26 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GQNd9000071; Wed, 7 Dec 2005 11:26:23 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GQLok013415; Wed, 7 Dec 2005 11:26:21 -0500 (EST) Message-ID: <43970D0C.9010302@mit.edu> Date: Wed, 07 Dec 2005 11:25:48 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Steven Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] tp-14 References: <5.2.1.1.2.20051207003858.02aa40a0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20051207003858.02aa40a0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1211 Content-Length: 460 X-Keywords: NotJunk In fact, it should be something even more complicated than a quadratic. As a hint, you will need to use the approximation ln(1+x)=x to get a closed form. DS Steven Zhou wrote: > Hi, > > Topic is clear. I have a question for problem set number 5. With the > answer i got for (a), if i were to solve for K that minimized that > expression, I get a quadratic equation. Is it that complicated, or > did I do part (a) wrong. Thanks. > > - Steven Zhou > From hmzhou@blackbird.csail.mit.edu Wed Dec 7 11:27:54 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GRs6N028720; Wed, 7 Dec 2005 11:27:54 -0500 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id jB7GRsLi025795; Wed, 7 Dec 2005 11:27:54 -0500 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id jB7GRsrl025792; Wed, 7 Dec 2005 11:27:54 -0500 Date: Wed, 7 Dec 2005 11:27:53 -0500 (EST) From: Hanson Zhou To: Karena Tyan cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Reading Comments 14 In-Reply-To: <20051207111852.p9kr5f3649jkckwk@webmail.mit.edu> Message-ID: References: <20051207111852.p9kr5f3649jkckwk@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 1212 Content-Length: 853 X-Keywords: NonJunk NotJunk Hmm...not sure what the question is here. Maybe it would help if you looked at it like this: Var[R] = E[(R-c)^2], for a constant c=E[R]. Now we are talking about essentially the expectation of the square of a random variable. Does that make sense? -Hanson On Wed, 7 Dec 2005, Karena Tyan wrote: > I was a little confused over the definition of variance on page 13. > > Var[R]::= E[(R - E[R])^2] makes it seem as if it's the expected value of > the squares of (R - E[R]), so I thought this meant calculating the sums > of the squares and dividing by the probability of each happening. It > seems to be more like the square of the expected (R - E[R]) and then the > expected value of that...or something. In other words, I'm confused > over the definition of variance. > > - Karena > > -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 > From dshin@MIT.EDU Wed Dec 7 11:29:09 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GT96N029063 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:29:09 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GT7Nj002403; Wed, 7 Dec 2005 11:29:07 -0500 (EST) Received: from [192.168.1.3] (c-65-96-190-64.hsd1.ma.comcast.net [65.96.190.64]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GSwMu029086; Wed, 7 Dec 2005 11:28:59 -0500 (EST) Message-ID: <43970DA9.1010004@mit.edu> Date: Wed, 07 Dec 2005 11:28:25 -0500 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 14 Comments References: <24b9be0d51a5c9b1398b83895ce0c687@mit.edu> In-Reply-To: <24b9be0d51a5c9b1398b83895ce0c687@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1213 Content-Length: 889 X-Keywords: NotJunk Harrison King Hall wrote: > David- > Even after our talk today I still had trouble with variance and using > it to apply Chebyshev's theorem. For that matter is is taking me a > long time to figure out Markov's as well. I think that I will get it > by the time we go over it in class. I think that my problem lies in > the fact that I don't really get E[R-c] where R is a random variable > and c is some constant. I think this can be explained by linearity of > expectation and E[c]=c, but I am not sure. Yes, thats exactly right - that's why E[R-c] = E[R] - E[c] = E[R] - c. > Well, that is all I have to say, but I need to tell you that there is > a high probability that I will not be in class on Friday since I have > an away swim meet. I will get back to you on the exact time that I > will be leaving and whether it means I will miss class. No prob. > -Harrison > From mitras@theory.csail.mit.edu Wed Dec 7 11:29:53 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GTr6N030561 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:29:53 -0500 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GTq7T018838 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:29:52 -0500 (EST) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GTpd9000254; Wed, 7 Dec 2005 11:29:52 -0500 (EST) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7GThok013631; Wed, 7 Dec 2005 11:29:44 -0500 (EST) Message-ID: <43970DF7.6050106@theory.csail.mit.edu> Date: Wed, 07 Dec 2005 11:29:43 -0500 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Sayan] Week 14 Comments References: <6.2.1.2.2.20051207001238.01f2fe20@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051207001238.01f2fe20@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1214 Content-Length: 819 X-Keywords: NotJunk Hi Praveen, Var[X_i] is calculated usinf theorem 8.1 which says that Var[X_i] = E[X_i^2] - (E[X_i])^2. E[X_i] = 1 x1/2 + 0 = 1/2 (E[X_i])^2 = 1/4 E[X_i^2] = 1^2 x 1/2 + 0 = 1/2 So, Var[X_i] = 1/2 - 1/4 =1/4. -sayan Praveen Pamidimukkala wrote: > Hi, > > My question pertains to the TP as opposed to the reading. In TP.14.3, > it reads: > > > "Suppose you flip a fair coin 100 times. The coin flips are all > mutually independent. > > 3. What is the variance of the number of heads? > > Let Xi be the random variable whose value is 1 if the ith coin flip is > heads. Then > Var[Xi] = 1/2 - (1/2)2 = 1/4. So Var[X1 + ... + X100] = 100(1/4) = 25. > (Note we can use additivity of variance here since our flips are > independent.) ] " > > I don't understand the calculation for Var[Xi]. > > Thanks, > Praveen > From sil_03@MIT.EDU Wed Dec 7 11:35:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GZH6N001383 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:35:17 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GXppL025487 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:35:16 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7G4Fg5026664 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 11:04:15 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jB7G4FU7016024; Wed, 7 Dec 2005 11:04:15 -0500 Received: from NEW-ONE-TWENTY-ONE.MIT.EDU (NEW-ONE-TWENTY-ONE.MIT.EDU [18.241.5.121]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 11:04:15 -0500 Message-ID: <20051207110415.ct97lpsjz3wcskoc@webmail.mit.edu> Date: Wed, 7 Dec 2005 11:04:15 -0500 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] week 14 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1215 Content-Length: 111 X-Keywords: NotJunk I'm confused about the Markov and Chebyshev upper bounds. Especially when you're given a lower bound. Silvia From aston@MIT.EDU Wed Dec 7 11:38:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7GcL6N001562 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:38:21 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7GcKoV001842 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:38:20 -0500 (EST) Received: from astonlaptop (NEW-THREE-TWENTY-SIX.MIT.EDU [18.241.6.71]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7GcB4Z011759 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 11:38:11 -0500 (EST) Message-Id: <200512071638.jB7GcB4Z011759@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 14 Comments Date: Wed, 7 Dec 2005 11:38:08 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5FB22.B3276350" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcX7TJpQoe53F9VkTxS7Xxt+0iciRg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1216 Content-Length: 2884 X-Keywords: NotJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5FB22.B3276350 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I had an oddly intuitive sense of Markov's theorem as far as the online questions went, though I realized once I saw the formula for it that I didn't really understand why what I was doing worked. The explanation is pretty nice (on pages 10 and 11). I was wondering, though, if there's some way to tell how well the Markov bound actually bounds examples. Is it due to the variance? - Aston ------=_NextPart_000_0000_01C5FB22.B3276350 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

I had an oddly intuitive sense of Markov’s = theorem as far as the online questions went, though I realized once I saw the = formula for it that I didn’t really understand why what I was doing worked. = The explanation is pretty nice (on pages 10 and 11). =

 

I was wondering, though, if there’s some way to = tell how well the Markov bound actually bounds examples. Is it due to the = variance?

 

         =    - Aston

------=_NextPart_000_0000_01C5FB22.B3276350-- From ozcan@MIT.EDU Wed Dec 7 12:09:04 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7H946N022813 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 12:09:04 -0500 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7H93Nj004624 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 12:09:03 -0500 (EST) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id jB7H8tol016374 for <6042-probs@theory.csail.mit.edu>; Wed, 7 Dec 2005 12:08:55 -0500 (EST) Message-Id: <5.2.1.1.2.20051207120612.02175200@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 07 Dec 2005 12:08:51 -0500 To: 6042-probs@theory.csail.mit.edu From: Yasin Ozcan Subject: [hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1217 Content-Length: 133 X-Keywords: Section 3, Conditional Expectation, Theorem 3.4 This is an amazing way of finding the expectation for pair-wise disjoint events. From jstritar@MIT.EDU Wed Dec 7 12:25:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7HP26N031269 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 12:25:02 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7HP0oV004642 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 12:25:00 -0500 (EST) Received: from JSTRITAR.MIT.EDU (JSTRITAR.MIT.EDU [18.245.4.29]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7HOq3T003023 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 12:24:53 -0500 (EST) Subject: [Sayan] Reading comments From: Jon Stritar To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 07 Dec 2005 12:25:19 -0500 Message-Id: <1133976319.1652.3.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1218 Content-Length: 339 X-Keywords: NonJunk Sorry for being a bit late... I actually did the reading/TP on Monday and forgot to send the email. 3. Conditional Expectation I like the law of total expectation. It seems like it usually comes in very handy. I feel like you might be able to make some cool AI using some sort of conditional expectations + bayesian probabilities. Jon From meyer@imap.theory.csail.mit.edu Wed Dec 07 10:56:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7Fuu6N014313 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:56:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7FutuB009231 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:56:55 -0500 (EST) Received: from [18.244.7.131] (SENIOR-SIX-FOURTY-TWO.MIT.EDU [18.244.7.131]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7FulJr023080 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:56:48 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <18f0f87df9571f2bca682ae228610a93@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6.042-online-tutor@theory.csail.mit.edu From: Rachel Shearer Subject: [Jelani] This Week's Comment Date: Wed, 7 Dec 2005 10:56:41 -0500 X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 96 X-UID: 1219 X-Keywords: NonJunk I don't really understand the standard deviation stuff. Lecture should help, I think. Rachel From nedzel@MIT.EDU Wed Dec 7 13:35:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7IZI6N001225 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Dec 2005 13:35:18 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7IZFoV005029; Wed, 7 Dec 2005 13:35:15 -0500 (EST) Received: from m38-370-5.mit.edu (M38-370-5.MIT.EDU [18.107.0.24]) (authenticated bits=56) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7IZ8cK000862 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 7 Dec 2005 13:35:08 -0500 (EST) Received: (from nedzel@localhost) by m38-370-5.mit.edu (8.12.9) id jB7IZ7An010815; Wed, 7 Dec 2005 13:35:07 -0500 Subject: [Sayan] Week 14 Comments From: David A Nedzel To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 07 Dec 2005 13:35:07 -0500 Message-Id: <1133980507.10767.1.camel@m38-370-5.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1220 Content-Length: 118 X-Keywords: NonJunk What is the significance of Markov and Chebyshev error bounds as they relate to statistical sampling theory? - David From dangut@MIT.EDU Wed Dec 7 13:44:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7Iih6N014728 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 13:44:43 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7IifoV017055 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 13:44:41 -0500 (EST) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7IiYhO004847 for <6042-probs@theory.csail.MIT.EDU>; Wed, 7 Dec 2005 13:44:34 -0500 (EST) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id jB7IiYX8009097; Wed, 7 Dec 2005 13:44:34 -0500 Received: from MAJI.MIT.EDU (MAJI.MIT.EDU [18.239.1.17]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 7 Dec 2005 13:44:34 -0500 Message-ID: <20051207134434.80w3lpulfndwk8os@webmail.mit.edu> Date: Wed, 7 Dec 2005 13:44:34 -0500 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: [Hansen] TP 14 Response MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1221 Content-Length: 134 X-Keywords: NonJunk I'm still a little confused, is variance related to independence? Does independence necessarily imply normal probability of variance? From pjs@MIT.EDU Wed Dec 7 15:37:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7KbO6N027614 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 15:37:24 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7KbMqY015947 for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 15:37:22 -0500 (EST) Received: from [18.24.1.245] (24-1-245.dynamic.csail.mit.edu [18.24.1.245]) (authenticated bits=0) (User authenticated as pjs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7KbAbr024969 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 7 Dec 2005 15:37:16 -0500 (EST) Message-ID: <439747F4.3080403@mit.edu> Date: Wed, 07 Dec 2005 15:37:08 -0500 From: PJ Steiner User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051130) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 1222 Content-Length: 90 X-Keywords: NonJunk I found the distinction between pairwise independence and independence interesting. pjs From meyer@imap.theory.csail.mit.edu Wed Dec 07 10:56:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jB7Fuu6N014313 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:56:56 -0500 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id jB7FutuB009231 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:56:55 -0500 (EST) Received: from [18.244.7.131] (SENIOR-SIX-FOURTY-TWO.MIT.EDU [18.244.7.131]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id jB7FulJr023080 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 7 Dec 2005 10:56:48 -0500 (EST) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <18f0f87df9571f2bca682ae228610a93@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6.042-online-tutor@theory.csail.mit.edu From: Rachel Shearer Subject: [Jelani] This Week's Comment Date: Wed, 7 Dec 2005 10:56:41 -0500 X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 97 X-UID: 1224 X-Keywords: NonJunk NotJunk I don't really understand the standard deviation stuff. Lecture should help, I think. Rachel

 

=E2=80=9C=E2=80=A2= The union of sets X and = Y (denoted X [ Y = ) contains all elements appearing in X or Y or

both. Thus, X [ Y = =3D {1, 2, 3, 4}.

=E2=80=A2 The intersection of X = and Y (denoted = X \ Y ) consists of all elements that appear in both X

and Y . So = X \ Y =3D {2, 3}.

=E2=80=A2 The difference of X = and Y (denoted = X =E2=88=92Y ) consists of all elements that are in X, but not in

Y . Therefore, X =E2=88=92 = Y = =3D {1} and Y =E2=88=92 = X = =3D {4}.=E2=80=9D

 

The main thing that I find difficult about the = reading is the large number of symbols.=C2=A0 The quote above is an example of the = introduction of more such symbols.=C2=A0 Having not had set theory or logical = analysis before, most of this is difficult and confusing to me; I think I=E2=80=99ll make = myself a table with all of the operators and symbols for easy = reference.