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 X-IMAPbase: 1126748715 889 NonJunk $Label4 $Label1 $Label2 $Label3 $Label5 Junk $MDNSent $Forwarded NotJunk Status: RO X-UID: 787 Content-Length: 259 X-Status: A X-Keywords: NonJunk 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: 788 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: 789 Content-Length: 61 X-Status: A X-Keywords: 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: 790 Content-Length: 210 X-Keywords: NonJunk 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: 791 Content-Length: 339 X-Keywords: NonJunk "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: 792 Content-Length: 186 X-Keywords: 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: 793 Content-Length: 374 X-Keywords: NonJunk 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: 794 Content-Length: 327 X-Keywords: 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: 795 Content-Length: 129 X-Status: A X-Keywords: 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: 796 Content-Length: 836 X-Status: A X-Keywords: NonJunk 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: 797 Content-Length: 162 X-Keywords: NonJunk 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: 798 Content-Length: 2233 X-Status: A X-Keywords: 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: 799 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: 800 Content-Length: 425 X-Keywords: NonJunk 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: 801 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: 802 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 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 > 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: 803 X-Keywords: NonJunk 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: 804 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: 805 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: 806 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: 807 Content-Length: 2671 X-Keywords: NonJunk 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: 808 Content-Length: 1395 X-Keywords: 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: 809 Content-Length: 98 X-Keywords: 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: 810 Content-Length: 1131 X-Keywords: 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: 811 Content-Length: 308 X-Keywords: NonJunk 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: 812 Content-Length: 126 X-Keywords: 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: 813 Content-Length: 231 X-Keywords: NonJunk 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: 814 Content-Length: 136 X-Keywords: NonJunk 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: 815 Content-Length: 250 X-Keywords: NonJunk 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: 816 Content-Length: 225 X-Keywords: NonJunk 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: 817 Content-Length: 283 X-Keywords: NonJunk 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: 818 Content-Length: 731 X-Keywords: 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: 819 Content-Length: 236 X-Keywords: 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: 820 Content-Length: 752 X-Keywords: NonJunk 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
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
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: 821 Content-Length: 228 X-Keywords: NonJunk 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: 822 Content-Length: 2347 X-Keywords: NonJunk 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: 824 Content-Length: 261 X-Keywords: NonJunk 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: 825 Content-Length: 174 X-Keywords: NonJunk 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: 826 Content-Length: 383 X-Keywords: NonJunk 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: 827 Content-Length: 359 X-Keywords: 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: 828 Content-Length: 219 X-Keywords: NonJunk 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: 829 Content-Length: 258 X-Keywords: NonJunk 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: 830 Content-Length: 3712 X-Status: A X-Keywords: 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: 831 Content-Length: 108 X-Keywords: NonJunk 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: 832 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: 833 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: 834 Content-Length: 494 X-Keywords: 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: 835 Content-Length: 404 X-Keywords: NonJunk 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: 836 Content-Length: 190 X-Keywords: NonJunk 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: 837 Content-Length: 193 X-Keywords: NonJunk 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: 838 Content-Length: 162 X-Keywords: NonJunk 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: 839 Content-Length: 330 X-Keywords: NonJunk 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: 840 Content-Length: 168 X-Keywords: NonJunk 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: 841 Content-Length: 717 X-Keywords: NonJunk 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: 842 Content-Length: 205 X-Keywords: 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: 843 Content-Length: 428 X-Status: A X-Keywords: 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: 844 Content-Length: 228 X-Keywords: 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: 845 Content-Length: 53 X-Keywords: 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: 846 Content-Length: 38 X-Keywords: NonJunk 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: 847 Content-Length: 3513 X-Keywords: NonJunk 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: 848 Content-Length: 294 X-Keywords: NonJunk 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: 849 Content-Length: 1365 X-Keywords: NonJunk 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: 850 Content-Length: 505 X-Keywords: NonJunk 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: 851 Content-Length: 449 X-Keywords: NonJunk 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 López 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: 852 Content-Length: 120 X-Keywords: NonJunk 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: 853 Content-Length: 509 X-Status: A X-Keywords: NonJunk 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: 854 Content-Length: 184 X-Keywords: NonJunk 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: 855 Content-Length: 556 X-Keywords: NonJunk 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: 856 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: 857 Content-Length: 255 X-Keywords: NonJunk 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: 858 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: 859 Content-Length: 375 X-Keywords: NonJunk 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: 860 Content-Length: 1911 X-Keywords: 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: 861 Content-Length: 101 X-Keywords: 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: 862 Content-Length: 497 X-Keywords: 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: 863 Content-Length: 180 X-Keywords: NonJunk 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: 864 Content-Length: 251 X-Keywords: NonJunk 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: 865 Content-Length: 118 X-Keywords: NonJunk 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: 866 Content-Length: 394 X-Keywords: NonJunk 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: 867 Content-Length: 658 X-Keywords: 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: 868 Content-Length: 1082 X-Keywords: NonJunk ------=_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: 869 Content-Length: 5441 X-Keywords: NonJunk 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: 870 Content-Length: 85 X-Keywords: 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: 871 Content-Length: 1075 X-Keywords: > 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: 872 Content-Length: 563 X-Keywords: 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: 873 Content-Length: 324 X-Keywords: 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: 874 Content-Length: 605 X-Keywords: 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: 875 Content-Length: 787 X-Keywords: 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: 876 Content-Length: 585 X-Keywords: 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: 877 Content-Length: 1680 X-Keywords: NonJunk 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: 878 Content-Length: 675 X-Keywords: NonJunk 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: 879 Content-Length: 258 X-Keywords: NonJunk 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: 880 Content-Length: 598 X-Keywords: 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: 881 Content-Length: 1079 X-Keywords: 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: 882 Content-Length: 3267 X-Keywords: 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: 883 Content-Length: 1537 X-Keywords: NonJunk ------=_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: 884 Content-Length: 262 X-Keywords: NonJunk 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: 885 Content-Length: 181 X-Status: A X-Keywords: NonJunk 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: 886 Content-Length: 238 X-Status: A X-Keywords: NonJunk 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: 887 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: 888 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: 889 Content-Length: 541 X-Keywords: 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 httpd@web02.rcahost.net Tue Nov 1 11:58:42 2005 Return-Path: Received: from web02.rcahost.net (66-55-78-18.yourhostingprovider.net [66.55.78.18] (may be forged)) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jA1Gwf5o007045 for <6042-probs@theory.lcs.mit.edu>; Tue, 1 Nov 2005 11:58:42 -0500 Received: from web02.rcahost.net (localhost [127.0.0.1]) by web02.rcahost.net (8.12.11/8.12.11) with ESMTP id jA1I02K7019988 for <6042-probs@theory.lcs.mit.edu>; Tue, 1 Nov 2005 12:00:02 -0600 Received: (from httpd@localhost) by web02.rcahost.net (8.12.11/8.12.11/Submit) id jA1I02vr019986; Tue, 1 Nov 2005 12:00:02 -0600 Date: Tue, 1 Nov 2005 12:00:02 -0600 Message-Id: <200511011800.jA1I02vr019986@web02.rcahost.net> To: 6042-probs@theory.lcs.mit.edu Subject: Notification of Limited Account Access X-PHP-Script: securepage.emcwebhosting.com/index1.php for 172.182.117.52 From: PayPal Account Review Department Reply-To: MIME-Version: 1.0 Content-Type: text/html Content-Transfer-Encoding: 8bit Untitled Document
As part of our security measures, we regularly screen activity in the PayPal system. We recently noticed the following issue on your account:

We recently received a report of unauthorized credit card use associated with this account. As a precaution, we have limited access to your PayPal account in order to protect against future unauthorized transactions.

Case ID Number: PP-075-481-830

For your protection, we have limited access to your account until additional security measures can be completed. We apologize for any inconvenience this may cause.

To review your account and some or all of the information that PayPal used to make its decision to limit your account access, please visit the Resolution Center by following the link below:
https://www.paypal.com/cgi-bin/webscr?cmd=login-run
If, after reviewing your account information, you seek further clarification regarding your account access, please contact PayPal by visiting the Help Center and clicking "Contact Us"


We thank you for your prompt attention to this matter. Please understand that this is a security measure intended to help protect you and your account. We apologize for any inconvenience.

Sincerely,
PayPal Account Review Department

PayPal Email ID PP3417

Accounts Management as outlined in our User Management , Paypal will
periodically send you information about site changes and enhancements

Visit our Privacy Policy and User Agreement if you have any questions :
http://www.paypal.com/cgi-bin/webscr?cmd=p/gen/ua/policy_privacy-outside

From httpd@web02.rcahost.net Tue Nov 1 11:58:43 2005 Return-Path: Received: from web02.rcahost.net (66-55-78-18.yourhostingprovider.net [66.55.78.18] (may be forged)) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id jA1Gwg5o007060 for <6042-probs@theory.csail.mit.edu>; Tue, 1 Nov 2005 11:58:43 -0500 Received: from web02.rcahost.net (localhost [127.0.0.1]) by web02.rcahost.net (8.12.11/8.12.11) with ESMTP id jA1I03FJ019995 for <6042-probs@theory.csail.mit.edu>; Tue, 1 Nov 2005 12:00:03 -0600 Received: (from httpd@localhost) by web02.rcahost.net (8.12.11/8.12.11/Submit) id jA1I03Ym019993; Tue, 1 Nov 2005 12:00:03 -0600 Date: Tue, 1 Nov 2005 12:00:03 -0600 Message-Id: <200511011800.jA1I03Ym019993@web02.rcahost.net> To: 6042-probs@theory.csail.mit.edu Subject: Notification of Limited Account Access X-PHP-Script: securepage.emcwebhosting.com/index1.php for 172.182.117.52 From: PayPal Account Review Department Reply-To: MIME-Version: 1.0 Content-Type: text/html Content-Transfer-Encoding: 8bit Untitled Document
As part of our security measures, we regularly screen activity in the PayPal system. We recently noticed the following issue on your account:

We recently received a report of unauthorized credit card use associated with this account. As a precaution, we have limited access to your PayPal account in order to protect against future unauthorized transactions.

Case ID Number: PP-075-481-830

For your protection, we have limited access to your account until additional security measures can be completed. We apologize for any inconvenience this may cause.

To review your account and some or all of the information that PayPal used to make its decision to limit your account access, please visit the Resolution Center by following the link below:
https://www.paypal.com/cgi-bin/webscr?cmd=login-run
If, after reviewing your account information, you seek further clarification regarding your account access, please contact PayPal by visiting the Help Center and clicking "Contact Us"


We thank you for your prompt attention to this matter. Please understand that this is a security measure intended to help protect you and your account. We apologize for any inconvenience.

Sincerely,
PayPal Account Review Department

PayPal Email ID PP3417

Accounts Management as outlined in our User Management , Paypal will
periodically send you information about site changes and enhancements

Visit our Privacy Policy and User Agreement if you have any questions :
http://www.paypal.com/cgi-bin/webscr?cmd=p/gen/ua/policy_privacy-outside