From juang@MIT.EDU Mon Oct 3 18:46:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j93MkNw0004518 for <6042-probs@theory.csail.mit.edu>; Mon, 3 Oct 2005 18:46:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j93MkM4u019890 for <6042-probs@theory.csail.mit.edu>; Mon, 3 Oct 2005 18:46:22 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j93MkFuB012547 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 3 Oct 2005 18:46:15 -0400 (EDT) Message-ID: <4341B493.5010504@mit.edu> Date: Mon, 03 Oct 2005 18:45:39 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-IMAPbase: 1126748715 788 NonJunk $Label4 $Label1 $Label2 $Label3 $Label5 Junk $MDNSent $Forwarded NotJunk X-UID: 466 Content-Length: 167 X-Keywords: NonJunk I was most surprised by section 5.3 (Classifying Polyhedra) on page 14 of the reading. The proof that only 5 regular polyhedra exist was enlightening. Jason Juang. From rnjacobs@MIT.EDU Mon Oct 3 20:36:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j940aZw0026615 for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 20:36:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j940aZmx026365 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 20:36:35 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j940aXok003749 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 20:36:33 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j940aXUv024591; Mon, 3 Oct 2005 20:36:33 -0400 Message-Id: <200510040036.j940aXUv024591@white-meteo.mit.edu> To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading response Date: Mon, 03 Oct 2005 20:36:32 -0400 From: r n jacobs X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 467 Content-Length: 346 X-Keywords: NonJunk It took a while for me to find the definition of "matching" since it was only boldface in a paragraph. Are the tutor questions specifically supposed to reflect the reading? It seems strange that the material was half already covered and half required doing the reading. Otherwise, everything seems straightforward this week. - Robert Jacobs From benlu@MIT.EDU Mon Oct 3 22:16:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j942GOw0012374 for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 22:16:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j942GM0M029415 for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 22:16:22 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j942GG1B022907 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 3 Oct 2005 22:16:16 -0400 (EDT) Message-ID: <4341E5F1.5070306@mit.edu> Date: Mon, 03 Oct 2005 22:16:17 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 5 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 468 Content-Length: 347 X-Keywords: NonJunk Hey Jelani, I was delighted to see the formal definitions and comparisons between graphs, trees, cycles, and paths. I've only worked with them while implementing abstract data types, so it's kind of cool to see them as formal mathematical concepts. I guess I wish I had taken this class before 6.034 and 6.170. Maybe even before 6.001. ~Ben From pgroudas@MIT.EDU Mon Oct 3 23:14:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j943Epw0017536 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 23:14:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j943En1i004809 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 23:14:49 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j943ElFf004308 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 3 Oct 2005 23:14:47 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j943Ek2V009584; Mon, 3 Oct 2005 23:14:46 -0400 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 3 Oct 2005 23:14:46 -0400 Message-ID: <20051003231446.pvxuc6vde1us480c@webmail.mit.edu> Date: Mon, 3 Oct 2005 23:14:46 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments for wed, oct 5 TA Hanson MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.351 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 469 Content-Length: 291 X-Keywords: NonJunk I am interested in the trapping mentioned on page 6. Where it says that we shouldn't start with an n-edge graphe and then add an edge, but do the shrink-down, grow-back method. It mentions that we'll see an example in class, but I'd like something more formal in the notes. -Paul Groudas From ksindi@MIT.EDU Mon Oct 3 23:40:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j943exw0021023 for <6042-probs@theory.csail.MIT.EDU>; Mon, 3 Oct 2005 23:40:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j943ev1i020432 for <6042-probs@theory.csail.MIT.EDU>; Mon, 3 Oct 2005 23:40:58 -0400 (EDT) Received: from w20-575-2.mit.edu (W20-575-2.MIT.EDU [18.187.0.21]) (authenticated bits=56) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j943epWN009396 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Mon, 3 Oct 2005 23:40:51 -0400 (EDT) Received: (from ksindi@localhost) by w20-575-2.mit.edu (8.12.9) id j943epH5009341; Mon, 3 Oct 2005 23:40:51 -0400 Subject: [David] Week 5 Comments From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 03 Oct 2005 23:40:51 -0400 Message-Id: <1128397251.8343.18.camel@w20-575-2.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 470 Content-Length: 472 X-Keywords: NonJunk "In graph terms, our goal is to find a matching for the girls; that is, a subset of edges such that exactly one edge is incident to each girl and at most one edge is incident to each boy.For example, here is one possible matching for the girls:" I found this definition of matching to be confusing and the example not so helpful. I likes Eurler's formula and noticed the analogy of it with de Rham or chech cohomology. It woul'd be fun to see more than one proof of it. From sheldons@MIT.EDU Tue Oct 4 00:13:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j944Dnw0026042 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 00:13:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j944Dl1i008729 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 00:13:47 -0400 (EDT) Received: from [10.0.0.2] (209-6-159-26.c3-0.smr-ubr3.sbo-smr.ma.cable.rcn.com [209.6.159.26]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j944DeNE015205 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 00:13:41 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) References: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Sheldon Chan Subject: [David] Week 5 Comments Date: Tue, 4 Oct 2005 00:13:38 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.734) X-Spam-Score: 0.501 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 471 Content-Length: 264 X-Keywords: NonJunk I found Page 14, Section 5.3 Classifying Polyhedra to be most interesting. The concept of discovering polyhedra, based on specific restrictions and Euler's formula, is interesting, given it is shown to exist mathematically, prior to visualizing it. Sheldon From ereid@MIT.EDU Tue Oct 4 01:21:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j945Liw0002593 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 01:21:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j945Lg1i015424 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 01:21:43 -0400 (EDT) Received: from [18.242.6.96] (NEXT-THREE-FIFTY-ONE.MIT.EDU [18.242.6.96]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j945LU1t024443 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 01:21:40 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <5F196C36-0B0A-4F42-A808-0CCB751DBBA3@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] reading comment Date: Tue, 4 Oct 2005 01:21:26 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.081 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 472 Content-Length: 199 X-Keywords: NonJunk I think it might be helpful if we could go over classifying polyhedra in class. I didn't quite follow the math, although if I have time I'll look at it tomorrow when I'm less tired... Elizabeth From cwong08@MIT.EDU Tue Oct 4 01:30:50 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j945Uow0004056 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 01:30:50 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j945UnIk003460 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 01:30:49 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j945Ueol013568 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 01:30:46 -0400 (EDT) Message-ID: <43421374.9040704@mit.edu> Date: Tue, 04 Oct 2005 01:30:28 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [TA-name] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.944 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 473 Content-Length: 147 X-Keywords: NonJunk On page 10, when defining bipartite graphs, it is unclear what an "odd cycle" is. I also don't really understand the polyhedra part on page 14. From shauni@MIT.EDU Tue Oct 4 02:04:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9464vw0008088 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 02:04:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9464qAK005490 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 02:04:55 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j945v8Va027745 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 01:57:08 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j945v8m1020221; Tue, 4 Oct 2005 01:57:08 -0400 Received: from SIMMONS-FOUR-THIRTY-TWO.MIT.EDU (SIMMONS-FOUR-THIRTY-TWO.MIT.EDU [18.96.6.177]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 01:57:08 -0400 Message-ID: <20051004015708.9yq7mgg4d1k4c84k@webmail.mit.edu> Date: Tue, 4 Oct 2005 01:57:08 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Reading comments week 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 474 Content-Length: 188 X-Keywords: NonJunk Planar Graphs (pp. 11-12) I'd like to have the section on planar graphs explained more fully in lecture; when can a graph be planar and what are the special properties of a planar graph? From mike_a@MIT.EDU Tue Oct 4 02:38:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j946cow0014670 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 02:38:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j946cmAF021124 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 02:38:48 -0400 (EDT) Received: from [18.238.6.106] (EASTCAMPUS-EIGHT-SEVENTY-THREE.MIT.EDU [18.238.6.106]) (authenticated bits=0) (User authenticated as mike_a@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j946ckf8000647 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 02:38:47 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: mike anderson Subject: [David] Weekly reading comments Date: Tue, 4 Oct 2005 02:38:45 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.23 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 475 Content-Length: 244 X-Status: A X-Keywords: NonJunk I am a little confused by theorem 4.2 (page 10), regarding bipartite graphs and odd cycles. Does odd cycles mean there are an odd number of paths? this doesn't seem to agree w/ the diagram, and there is no proof for this theorem. -mike From hectorb@MIT.EDU Tue Oct 4 02:50:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j946ocw0016106 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 02:50:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j946oaAF026289; Tue, 4 Oct 2005 02:50:36 -0400 (EDT) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j946oT6W001371 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 4 Oct 2005 02:50:29 -0400 (EDT) Message-ID: <4340D452.4060103@mit.edu> Date: Mon, 03 Oct 2005 02:48:50 -0400 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.01 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 476 Content-Length: 534 X-Keywords: NonJunk "A 1994 University of Chicago study entitled The Social Organization of Sexuality found that on average men have 74% more opposite-gender partners than women, confirming a view that men are more promiscuous. But whoever reported this finding was lying or confused!" -pg 1 I don't think the reporters were lying. The data was probably gathered by surveys, and the ones lying were the women. The men might have exaggerated a little, but women never admit to having multiple sexual partners. This is why the study was incorrect. From rehughes@MIT.EDU Tue Oct 4 03:55:16 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j947tGw0023254 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:16 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j947tEEY001747 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:15 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j947tE9j024191 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:14 -0400 (EDT) Received: from [18.244.6.55] (SENIOR-THREE-TEN.MIT.EDU [18.244.6.55]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j947t2ok018101 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 03:55:02 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <156e4dedbd1233a85dd168b76bee12c6@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.csail.mit.edu From: Richard Hughes Subject: [Julani] Week 5 Comments Date: Tue, 4 Oct 2005 03:55:02 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 477 Content-Length: 300 X-Keywords: NonJunk I find the leap of logic in section 5.3 (the classifying polyhedra) and the history of the Pythagoras cult quite interesting. I'd love to hear more about how the ancients went through this basic series of important logical deductions and what we can (and can't) learn from them. Richard Hughes From lye@MIT.EDU Tue Oct 4 07:40:36 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94BeZw0023112 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 07:40:36 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94BeYYf027392 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 07:40:34 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94BeSuk015851 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 07:40:28 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j94BeSgk028766; Tue, 4 Oct 2005 07:40:28 -0400 Received: from RANDOM-FOUR-FORTY.MIT.EDU (RANDOM-FOUR-FORTY.MIT.EDU [18.243.6.185]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 07:40:28 -0400 Message-ID: <20051004074028.ryue99jas8w04gkg@webmail.mit.edu> Date: Tue, 4 Oct 2005 07:40:28 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 478 Content-Length: 287 X-Status: A X-Keywords: NonJunk Section 2.1, pg. 5: "Simple cycles will be cycles that don't cross themselves." How are non-simple cycles defined? If a non-simple cycle were taken as a subgraph, would there be a graph it is always isomorphic to? (e.g. for the simple graph, it is always isomorphic to C_n) Lunduo Ye From hkhall@MIT.EDU Tue Oct 4 08:22:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94CMew0030918 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 08:22:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94CMcHS020600; Tue, 4 Oct 2005 08:22:38 -0400 (EDT) Received: from [18.96.6.59] (SIMMONS-THREE-FOURTEEN.MIT.EDU [18.96.6.59]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94CMVkI022224 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Tue, 4 Oct 2005 08:22:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: LN5 Reading Comments [David] Date: Tue, 4 Oct 2005 08:24:31 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 479 Content-Length: 502 X-Keywords: NonJunk This is actually the first section that I feel really comfortable with, probably because there can exist a concrete visualization in the form of a graph. The only part of the reading that started to give me trouble was the section on translating planar graphs into polyhedra. While Euler's Formula is straightforward I was having difficulty proving that there were only 5 that satisfied the criterion (right?). Anyway, I found this section to be cake otherwise, at least I hope so. -Harrison From mukkala@MIT.EDU Tue Oct 4 09:28:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94DSWw0009047 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:28:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94DSVHS012493 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:28:31 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94DSOGw011902 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:28:25 -0400 (EDT) Message-Id: <6.2.1.2.2.20051004091356.01d77828@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Tue, 04 Oct 2005 09:28:35 -0400 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 480 Content-Length: 677 X-Status: A X-Keywords: NonJunk In section 5.2, it says: "Every edge on a cycle in the graph borders two faces and is traversed by each of the two cycles bounding these two faces, and not by any other face boundary. If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one 'going' and the other 'coming'), and not by any other face boundary." I thought every edge touched two faces. I understand that in this case you were probably referring to edges that touched the outside face. However, I still don't understand how the edge is traversed twice by the cycled in both directions. Why can't it just be traversed once? Thanks, Praveen Pamidimukkala From ajshafer@MIT.EDU Tue Oct 4 09:48:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Dmiw0015499 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:48:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94DmgHS001860 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:48:42 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94Dmdbr020763 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 09:48:40 -0400 (EDT) Message-ID: <000c01c5c8ea$61468080$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Tue, 4 Oct 2005 09:49:01 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.135 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 481 Content-Length: 361 X-Keywords: NonJunk I was confused by the classifying polyhedra section. Does this depend on some properties of projection? Can this be applied to higher dimensions? Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From mitras@drake.csail.mit.edu Tue Oct 4 14:33:11 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94IXAw0005405; Tue, 4 Oct 2005 14:33:11 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94IXAjQ017885; Tue, 4 Oct 2005 14:33:10 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94IXAJG017884; Tue, 4 Oct 2005 14:33:10 -0400 Date: Tue, 4 Oct 2005 14:33:10 -0400 From: Sayan Mitra Message-Id: <200510041833.j94IXAJG017884@drake.csail.mit.edu> To: 6042-probs@theory.csail.MIT.EDU, shauni@MIT.EDU Subject: Re: [Sayan] Reading comments week 5 Status: RO X-UID: 484 Content-Length: 1054 X-Keywords: NonJunk A graph is planar if there exists a drawing of the graph (on a flat sheet of paper) such that no two of its edges interset. There are many nice properties of planar graph including Euler's formula (5.1). We will probably see more properties in class exercises. -s. >From shauni@MIT.EDU Tue Oct 4 02:04:57 2005 Date: Tue, 4 Oct 2005 01:57:08 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Reading comments week 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 474 Content-Length: 188 X-Keywords: NonJunk Planar Graphs (pp. 11-12) I'd like to have the section on planar graphs explained more fully in lecture; when can a graph be planar and what are the special properties of a planar graph? From mitras@drake.csail.mit.edu Tue Oct 4 14:40:48 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Iemw0008285; Tue, 4 Oct 2005 14:40:48 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94Iem2L017935; Tue, 4 Oct 2005 14:40:48 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94IemDZ017934; Tue, 4 Oct 2005 14:40:48 -0400 Date: Tue, 4 Oct 2005 14:40:48 -0400 From: Sayan Mitra Message-Id: <200510041840.j94IemDZ017934@drake.csail.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU, lye@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Status: RO X-UID: 485 Content-Length: 1277 X-Keywords: NonJunk Definition of simple cycles: Consider a cycle C represented by a sequence of vertices v_1,...,v_n. If no two vertices in the sequence are equal other than v_1 = v_n, then C is simple. If there exists v_i, v_j, j \notin {1,n}, such that v_i = v_j, then C is a nonsimple cycle. I did not understand the last part of the question. Are you asking if a nonsimple cycles always contain simple cycles ? That is true. -s. >From lye@MIT.EDU Tue Oct 4 07:40:36 2005 Date: Tue, 4 Oct 2005 07:40:28 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 478 Content-Length: 287 X-Keywords: NonJunk Section 2.1, pg. 5: "Simple cycles will be cycles that don't cross themselves." How are non-simple cycles defined? If a non-simple cycle were taken as a subgraph, would there be a graph it is always isomorphic to? (e.g. for the simple graph, it is always isomorphic to C_n) Lunduo Ye From mitras@drake.csail.mit.edu Tue Oct 4 14:48:22 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94ImMw0009174; Tue, 4 Oct 2005 14:48:22 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94ImM6q017976; Tue, 4 Oct 2005 14:48:22 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94ImMbu017975; Tue, 4 Oct 2005 14:48:22 -0400 Date: Tue, 4 Oct 2005 14:48:22 -0400 From: Sayan Mitra Message-Id: <200510041848.j94ImMbu017975@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, mukkala@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Status: RO X-UID: 486 Content-Length: 1714 X-Keywords: NonJunk This is probably best explained with a quick figure; however, let me try. You are right, the special case is when the edges are on a outer face. Think of a tree hanging from a bounded face. Indeed, every edge on this tree is also traversed twice; once in each direction. If you draw the circle that traverses this outer face (in which the tree lies). You will notice that every branch of this tree is traversed twice by the circle, once going up and once coming down. -s. >From mukkala@MIT.EDU Tue Oct 4 09:28:32 2005 X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Tue, 04 Oct 2005 09:28:35 -0400 To: 6042-probs@theory.csail.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 480 Content-Length: 677 X-Keywords: In section 5.2, it says: "Every edge on a cycle in the graph borders two faces and is traversed by each of the two cycles bounding these two faces, and not by any other face boundary. If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one 'going' and the other 'coming'), and not by any other face boundary." I thought every edge touched two faces. I understand that in this case you were probably referring to edges that touched the outside face. However, I still don't understand how the edge is traversed twice by the cycled in both directions. Why can't it just be traversed once? Thanks, Praveen Pamidimukkala From dangut@MIT.EDU Tue Oct 4 15:52:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94JqLw0025324 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 15:52:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94JqJ85004831 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 15:52:19 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94JqCCC001537 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 15:52:12 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j94JqC0M009445; Tue, 4 Oct 2005 15:52:12 -0400 Received: from MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU (MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU [18.239.7.231]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 15:52:12 -0400 Message-ID: <20051004155212.soy36pqy00gss00k@webmail.mit.edu> Date: Tue, 4 Oct 2005 15:52:12 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: Wed Oct 5 Reading assignment reply, R13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.556 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 487 Content-Length: 111 X-Status: A X-Keywords: I'm okay with this material, a sidenote question though is what is the difference between a Lemma and a proof? From petek@MIT.EDU Tue Oct 4 16:10:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KAbw0027404 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:10:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KAa85023702 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:10:36 -0400 (EDT) Received: from [18.51.6.17] (HAYDEN-TWO-SEVENTY-TWO.MIT.EDU [18.51.6.17]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KAXqY009544 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:10:34 -0400 (EDT) Message-ID: <4342E1BF.4010003@mit.edu> Date: Tue, 04 Oct 2005 16:10:39 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Comments for This Week Content-Type: multipart/alternative; boundary="------------040509040406040309010603" X-Spam-Score: -2.059 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 488 Content-Length: 2075 X-Keywords: NonJunk This is a multi-part message in MIME format. --------------040509040406040309010603 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit I'm having difficulty with Euler's formula and getting it into my head. Should I memorize it? Or should I understand it's concepts? Also.... I was going to comment on Theorem 4.2 being unsupported, until I found a great way to think of it. A graph can only truly be bipartite if it's chromatic number is 2 (I hope I have the vocab right). If there's an odd cycle, some adjacent nodes would need to be colored the same... (A B A --- cycle of length 3 has AA (no good )) -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------040509040406040309010603 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit I'm having difficulty with Euler's formula and getting it into my head.  Should I memorize it?  Or should I understand it's concepts?


Also....

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

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------040509040406040309010603-- From alisonc@MIT.EDU Tue Oct 4 16:11:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KBew0027511 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:11:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KBd85024789 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:11:39 -0400 (EDT) Received: from [18.238.5.231] (EASTCAMPUS-SEVEN-FORTY-TWO.MIT.EDU [18.238.5.231]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KBatC010003 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:11:37 -0400 (EDT) Message-ID: <4342E1FD.2070109@mit.edu> Date: Tue, 04 Oct 2005 16:11:41 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Lecture notes 5 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 489 Content-Length: 401 X-Status: A X-Keywords: NonJunk The initial example is interesting to me because if the researcher's methods were sound, then what it *really* proves is that men are much more likely to exaggerate their number of partners (or women to under-estimate). The section on coloring graphs is also really interesting: such a simple way of representing things and yet still complex in the method of doing it and the information gained. From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:37:37 2005 Message-ID: <4342E817.1080700@csail.mit.edu> Date: Tue, 04 Oct 2005 16:37:43 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: sayan mitra Subject: Re: [Sayan] Week 5 Comments References: <6.2.1.2.2.20051004091356.01d77828@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051004091356.01d77828@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 965 Status: RO X-UID: 490 X-Keywords: Think of a face that has a single edge "hanging" off its boundary. A cycle around the boundary has to go back & forth on the edge to get back to where it starts. The animation on the web illustrates this. Regards, A. Praveen Pamidimukkala wrote: > In section 5.2, it says: > > "Every edge on a cycle in the graph borders two faces and is > traversed by each of the two cycles bounding these two faces, and not by > any other face boundary. If an edge touches only one face, then it is > traversed twice by the cycle that borders the face (one 'going' and the > other 'coming'), and not by any other face boundary." > > I thought every edge touched two faces. I understand that in this case > you were probably referring to edges that touched the outside face. > However, I still don't understand how the edge is traversed twice by the > cycled in both directions. Why can't it just be traversed once? > > Thanks, > Praveen Pamidimukkala > From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:46:27 2005 Message-ID: <4342EA27.7070602@csail.mit.edu> Date: Tue, 04 Oct 2005 16:46:31 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: mike anderson CC: David Shin Subject: Re: [David] Weekly reading comments References: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> In-Reply-To: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 443 Status: RO X-UID: 492 X-Keywords: "odd" means "odd length" (number of edge traversals) in this context; for simple cycles, this is the same as the number of distinct vertices on it. regards, A. mike anderson wrote: > I am a little confused by theorem 4.2 (page 10), regarding bipartite > graphs and odd cycles. Does odd cycles mean there are an odd number of > paths? this doesn't seem to agree w/ the diagram, and there is no proof > for this theorem. > > > -mike From meyer@imap.theory.csail.mit.edu Tue Oct 04 14:25:31 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94IPUw0003896; Tue, 4 Oct 2005 14:25:30 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j94IPU5K017838; Tue, 4 Oct 2005 14:25:30 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j94IPUgL017837; Tue, 4 Oct 2005 14:25:30 -0400 Date: Tue, 4 Oct 2005 14:25:30 -0400 From: Sayan Mitra Message-Id: <200510041825.j94IPUgL017837@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, cwong08@MIT.EDU Subject: Re: [TA-name] Week 5 Comments Cc: 6042-staff@theory.csail.mit.edu Status: RO Content-Length: 993 X-UID: 493 X-Keywords: NonJunk An odd cycle is a cycle with odd number of edges. In the polyhedra part, we are deriving some constraints on the number of vertices, edges, and faces of regular polyhedra, using Euler's formula. Which part exactly did you not understand ? -sayan >From cwong08@MIT.EDU Tue Oct 4 01:30:50 2005 Date: Tue, 04 Oct 2005 01:30:28 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [TA-name] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.944 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 473 Content-Length: 147 X-Keywords: On page 10, when defining bipartite graphs, it is unclear what an "odd cycle" is. I also don't really understand the polyhedra part on page 14. From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:43:11 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j94Kh9w0001547 for <6042-staff@theory.csail.mit.edu>; Tue, 4 Oct 2005 16:43:10 -0400 Received: (qmail 99189 invoked from network); 4 Oct 2005 20:43:09 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 4 Oct 2005 20:43:08 -0000 Message-ID: <4342E962.6030903@csail.mit.edu> Date: Tue, 04 Oct 2005 16:43:14 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Linda Ye CC: 6042-staff@theory.csail.mit.edu Subject: Re: [Sayan] Week 5 Comments References: <20051004074028.ryue99jas8w04gkg@webmail.mit.edu> In-Reply-To: <20051004074028.ryue99jas8w04gkg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 634 X-UID: 494 X-Keywords: I'm not sure that there is a most useful general def of "cycle", so I'm implicitly letting it be any path that begins and ends where it starts. You can't define it as isomorphic to a subgraph: how would you distinguish the simple cycle C_n from the nonsimple cycle that went around C_n twice? regards, A. Linda Ye wrote: > Section 2.1, pg. 5: > > "Simple cycles will be cycles that don't cross themselves." > > How are non-simple cycles defined? If a non-simple cycle were taken as a > subgraph, would there be a graph it is always isomorphic to? (e.g. for the > simple graph, it is always isomorphic to C_n) > > Lunduo Ye From kevin08@MIT.EDU Tue Oct 4 16:51:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KpHw0002803 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:51:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Kow85003807 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:50:58 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-ONE-SIXTY-THREE.MIT.EDU [18.242.5.163]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KorJ7026731 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:50:55 -0400 (EDT) Message-Id: <6.2.3.4.2.20051004164841.01e52848@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Tue, 04 Oct 2005 16:50:51 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.001 X-Spam-Level: * (1.001) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 495 Content-Length: 236 X-Keywords: NonJunk Hi, It seems to me that bipartite graphs are really important, but they're not discussed too heavily in the notes. Could you add an example or two of how they're applied to solve "common" computer science questions? Thanks, Kevin From avalys@MIT.EDU Tue Oct 4 16:53:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94KrBw0002932 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:53:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KqJ85005031 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:52:19 -0400 (EDT) Received: from [18.96.5.183] (SIMMONS-ONE-EIGHTY-THREE.MIT.EDU [18.96.5.183]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94KqDEd027320 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 16:52:13 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <248E756A-4D81-4C26-92D3-C276999202D0@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Required reading comments... Date: Tue, 4 Oct 2005 16:52:15 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 496 Content-Length: 195 X-Keywords: NonJunk I don't really have anything significant to say, the reading was fairly straightforward. The sections on Euler's Formula and polyhedrons were a bit difficult to follow, though. Alex Valys From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:53:42 2005 Message-ID: <4342EBDC.1060808@csail.mit.edu> Date: Tue, 04 Oct 2005 16:53:48 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alison Cichowlas CC: Hanson M Zhou Subject: Re: [Hanson] Lecture notes 5 comments References: <4342E1FD.2070109@mit.edu> In-Reply-To: <4342E1FD.2070109@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 604 Status: RO X-UID: 497 X-Keywords: Alternatively, it could be that the researchers used a sample that had 74% more women, though how they would select such a skewed sample is puzzling. regards, A. Alison Cichowlas wrote: > The initial example is interesting to me because if the researcher's > methods were sound, then what it *really* proves is that men are much > more likely to exaggerate their number of partners (or women to > under-estimate). The section on coloring graphs is also really > interesting: such a simple way of representing things and yet still > complex in the method of doing it and the information gained. From hmzhou@blackbird.csail.mit.edu Tue Oct 4 17:01:20 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94L1Kw0004550; Tue, 4 Oct 2005 17:01:20 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j94L1JUA019203; Tue, 4 Oct 2005 17:01:20 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j94L1JwK019200; Tue, 4 Oct 2005 17:01:19 -0400 Date: Tue, 4 Oct 2005 17:01:19 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu, 6042-staff@theory.csail.mit.edu Subject: Re: [Hanson] Week 5 Comments In-Reply-To: <6.2.3.4.2.20051004164841.01e52848@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051004164841.01e52848@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 498 Content-Length: 443 X-Keywords: NonJunk Bipartite matchings are widely applicable. They actually use matching algorithms to assign med students to hospitals or something like that. -Hanson On Tue, 4 Oct 2005, Kevin Wang wrote: > Hi, > > It seems to me that bipartite graphs are really important, but > they're not discussed too heavily in the notes. Could you add an > example or two of how they're applied to solve "common" computer > science questions? > > Thanks, > Kevin > > From meyer@imap.theory.csail.mit.edu Tue Oct 04 17:01:51 2005 Return-Path: Received: from smtp014.mail.yahoo.com (smtp014.mail.yahoo.com [216.136.173.58]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j94L1mw0004654 for <6042-staff@theory.csail.mit.edu>; Tue, 4 Oct 2005 17:01:49 -0400 Received: (qmail 43242 invoked from network); 4 Oct 2005 21:01:48 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp014.mail.yahoo.com with SMTP; 4 Oct 2005 21:01:47 -0000 Message-ID: <4342EDC2.8010304@csail.mit.edu> Date: Tue, 04 Oct 2005 17:01:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: David Shin CC: 6042-staff@theory.csail.mit.edu Subject: Re: [David] Weekly reading comments References: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> <4342EA27.7070602@csail.mit.edu> <4342EAF5.7030806@mit.edu> In-Reply-To: <4342EAF5.7030806@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1064 X-UID: 499 X-Keywords: good question. i thought to cc 6042-staff selectively on answers that I thought might be of general interest; i think I've only done this one or twice so far this week. i have also been cc'ing the specific TA named by the student when I reply (as well as filing my reply in probs), tho i wasn't sure if this was a good thing to do. Shall I stop cc'ing you and just leave my replies in the probs file? regards, A. David Shin wrote: > Professor Meyer, > > Is there a reason you are CC'ing 6042-staff or the TA's instead of > 6042-probs? > > Prof. Albert R. Meyer wrote: > >> "odd" means "odd length" (number of edge traversals) in this context; >> for simple cycles, this is the same as the number of distinct vertices >> on it. >> regards, A. >> >> mike anderson wrote: >> >>> I am a little confused by theorem 4.2 (page 10), regarding bipartite >>> graphs and odd cycles. Does odd cycles mean there are an odd number >>> of paths? this doesn't seem to agree w/ the diagram, and there is no >>> proof for this theorem. >>> >>> >>> -mike >> >> From meyer@imap.theory.csail.mit.edu Tue Oct 04 16:50:40 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Kodw0002736 for ; Tue, 4 Oct 2005 16:50:39 -0400 Received: from south-station-annex.mit.edu ([18.72.1.2]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EMtkF-0008Nm-K7 for meyer@csail.mit.edu; Tue, 04 Oct 2005 16:50:39 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j94KocsS029501 for ; Tue, 4 Oct 2005 16:50:38 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94KocSF024732 for ; Tue, 4 Oct 2005 16:50:38 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j94KoGok001100 for ; Tue, 4 Oct 2005 16:50:17 -0400 (EDT) Message-ID: <4342EAF5.7030806@mit.edu> Date: Tue, 04 Oct 2005 16:49:57 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Prof. Albert R. Meyer" Subject: Re: [David] Weekly reading comments References: <9A75E368-866E-41A7-8F61-A079FB446309@mit.edu> <4342EA27.7070602@csail.mit.edu> In-Reply-To: <4342EA27.7070602@csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO X-Status: A Content-Length: 591 X-UID: 500 X-Keywords: Professor Meyer, Is there a reason you are CC'ing 6042-staff or the TA's instead of 6042-probs? Prof. Albert R. Meyer wrote: > "odd" means "odd length" (number of edge traversals) in this context; > for simple cycles, this is the same as the number of distinct vertices > on it. > regards, A. > > mike anderson wrote: > >> I am a little confused by theorem 4.2 (page 10), regarding bipartite >> graphs and odd cycles. Does odd cycles mean there are an odd number >> of paths? this doesn't seem to agree w/ the diagram, and there is no >> proof for this theorem. >> >> >> -mike > From arup@MIT.EDU Tue Oct 4 17:24:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94LOnw0009829 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:24:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94LOm2e003074 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:24:48 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-TWO-TWENTY-SEVEN.MIT.EDU [18.247.5.227]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94LOiNa009331 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:24:45 -0400 (EDT) Message-Id: <6.2.3.4.2.20051004172123.047bf878@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Tue, 04 Oct 2005 17:24:41 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Week 5 Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.045 X-Spam-Level: * (1.045) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 501 Content-Length: 299 X-Keywords: Section 2.5, page 6 (and used later as well): I think I don't completely understand the "shrink-down, grow-back" style of induction. It seems like it assumes P(n+1) before proving it. Also, what are examples of the logical errors that can occur if this style of argument isn't used? |Arup| From yaser@MIT.EDU Tue Oct 4 17:54:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94LsQw0018482 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:54:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94LsP2e027001 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:54:25 -0400 (EDT) Received: from Charizard (NEW-SIX-EIGHTY-THREE.MIT.EDU [18.241.7.172]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94LsMXg019580 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 17:54:22 -0400 (EDT) Message-Id: <200510042154.j94LsMXg019580@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: week 5 reading question Date: Tue, 4 Oct 2005 17:54:20 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0078_01C5C90C.A5C85F90" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXJLixKBLADFgtcQGuzIpTlXRtBpg== X-Spam-Score: 0.7 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 502 Content-Length: 5964 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0078_01C5C90C.A5C85F90 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi David, When it comes to k-coloring, is there a direct relation between k-coloring of vertices (as outlined in the reading) and the more general map-coloring problem? I.e., on the tutor problem there was a graph that had a chromatic value of 3, but would have a chromatic value of 2 if we were attempting to fill in edges or areas rather than vertices. Are areas between edges EVER considered legitimate or useful in such graphs? Thanks! Take care, _Yaser ------=_NextPart_000_0078_01C5C90C.A5C85F90 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

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

 

Thanks! =


Take care,

 

_Yaser

------=_NextPart_000_0078_01C5C90C.A5C85F90-- From adnaan@MIT.EDU Tue Oct 4 17:55:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Ltkw0018539 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 17:55:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Ltj2e027978 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 17:55:45 -0400 (EDT) Received: from m12-182-12.mit.edu (M12-182-12.MIT.EDU [18.19.0.43]) (authenticated bits=56) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94LtgAr019988 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 17:55:43 -0400 (EDT) Received: (from adnaan@localhost) by m12-182-12.mit.edu (8.12.9) id j94LtgGY003447; Tue, 4 Oct 2005 17:55:42 -0400 Subject: Week 5 comments-Hanson From: Adnaan N Jiwaji To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 04 Oct 2005 17:55:42 -0400 Message-Id: <1128462942.3180.4.camel@m12-182-12.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 503 Content-Length: 47 X-Keywords: The concept of a bipartite graph is confusing From natalia3@MIT.EDU Tue Oct 4 18:11:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94MBxw0020951 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 18:11:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94MBw2e009596 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 18:11:58 -0400 (EDT) Received: from hayden-1.mit.edu (HAYDEN-1.MIT.EDU [18.51.1.31]) (authenticated bits=56) (User authenticated as natalia3@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94MBmrG024178 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 18:11:49 -0400 (EDT) Received: (from natalia3@localhost) by hayden-1.mit.edu (8.12.9) id j94MBmMA020998; Tue, 4 Oct 2005 18:11:48 -0400 Subject: [Jelani] Week 5 Comments From: Natalia N Chernenko To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 04 Oct 2005 18:11:48 -0400 Message-Id: <1128463908.20774.11.camel@hayden-1.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 504 Content-Length: 219 X-Keywords: NonJunk It would be nice to discuss planar graphs (Section 5, pp. 11-13) more in lecture, especially Euler's formula (Section 5.1). I found the derivation of the formula a little confusing (induction proof). Natalia Chernenko From dnreshef@MIT.EDU Tue Oct 4 18:59:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Mxmw0027491 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 18:59:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Mxl2e013327 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 18:59:47 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94MxF76005997 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 18:59:15 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j94MxFgC022215; Tue, 4 Oct 2005 18:59:15 -0400 Received: from M66-064-09.MIT.EDU (M66-064-09.MIT.EDU [18.63.4.61]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 18:59:15 -0400 Message-ID: <20051004185915.x4fwuy4o38ws0ckg@webmail.mit.edu> Date: Tue, 4 Oct 2005 18:59:15 -0400 From: David N Reshef To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments for 10/5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 505 Content-Length: 260 X-Keywords: NonJunk Hey, This weeks reading was actually more interesting than usual, and I am really interested in learning more about graph theory. Hall's marriage thm on page 15 was a little confusing. Also, I wouldn't mind discussing connectedness further in class. -Dave From aston@MIT.EDU Tue Oct 4 19:20:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94NK8w0030978 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:20:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94NK72e026668 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:20:07 -0400 (EDT) Received: from aston (BOOKX.MIT.EDU [18.241.0.191]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NK4i1009988 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:20:05 -0400 (EDT) Message-Id: <200510042320.j94NK4i1009988@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: Email comments Date: Tue, 4 Oct 2005 19:20:11 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_007B_01C5C918.A4515980" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 thread-index: AcXJOirdtoR8vEvLRjiOxlM0LBMM8g== X-Spam-Score: 1.722 X-Spam-Level: * (1.722) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 506 Content-Length: 2251 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_007B_01C5C918.A4515980 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I found pages 14 and 15, the explanation of the limitations on regular 3d shapes, to be theoretically very cool and interesting, but I got lost when the equations started appearing, even after a couple read throughs. - Aston ------=_NextPart_000_007B_01C5C918.A4515980 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

 

         =    - Aston

------=_NextPart_000_007B_01C5C918.A4515980-- From ridell@MIT.EDU Tue Oct 4 19:24:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94NO2w0001692 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 19:24:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94NO12e029068 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 19:24:01 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NNsCl010622 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 19:23:54 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j94NNsUs028364; Tue, 4 Oct 2005 19:23:54 -0400 Received: from STRATTON-THREE-SIXTY-SEVEN.MIT.EDU (STRATTON-THREE-SIXTY-SEVEN.MIT.EDU [18.187.6.112]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 19:23:54 -0400 Message-ID: <20051004192354.o5g222c08lo0o8c0@webmail.mit.edu> Date: Tue, 4 Oct 2005 19:23:54 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: week 5 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 507 Content-Length: 352 X-Keywords: NonJunk I am confused about the colorale graphs, and the theorem described on page 10. Could you explain more about how a graph with degree at most k is (k+1)-colorable. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From kktyan@MIT.EDU Tue Oct 4 19:31:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94NVXw0003556 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 19:31:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94NVV2e003901 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 19:31:32 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NVO8P012178 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 19:31:24 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j94NVOAd027956; Tue, 4 Oct 2005 19:31:24 -0400 Received: from BURTON-FOUR-THIRTY-FOUR.MIT.EDU (BURTON-FOUR-THIRTY-FOUR.MIT.EDU [18.247.6.179]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 19:31:24 -0400 Message-ID: <20051004193124.wo2hpl958jcwc84o@webmail.mit.edu> Date: Tue, 4 Oct 2005 19:31:24 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading 5 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 508 Content-Length: 1026 X-Keywords: I found the part about k-connected vertices most confusing. Namely, the passage on page 5: Definition 2.6. Two vertices in a graph are k-connected if they remain connected in any subgraph obtained by deleting k-1 edges. A graph is k-connected if every pair of its vertices are k-connected. How do you know what "k" is? What I mean to say is - to test for k-connectivity, you have to delete k-1 edges and see if the two vertices in question are still conected. I assume that this refers just to the edges that exist along some path between the two vertices, but I still don't quite understand how "k" is derived. Like, in Figure 1 on page 3, they say that B and E are 2-connected, which means you could delete 1 edge between the two vertices and still be connected, correct? But couldn't you delete the three edges (B-C) (C-D) and (D-E) and still be connected? I'm sorry, this was very long. I'm just confused over the definition of k-connectivity. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From bilodeau@MIT.EDU Tue Oct 4 19:50:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j94Notw0005696 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:50:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j94Noa2e015220 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:50:36 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THIRTY.MIT.EDU [18.239.5.30]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j94NoXBo015901 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 19:50:33 -0400 (EDT) Message-Id: <200510042350.j94NoXBo015901@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Comments [jelani] Date: Tue, 4 Oct 2005 19:50:26 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C91C.DE4183F0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJPmS638Nbv5hsQQaD++88v/IpRQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.419 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 509 Content-Length: 3141 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C91C.DE4183F0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Reading comments week 5: In this weeks reading, the section about using Euler's rule for planar graphs to examine regular polyhedra makes sense, but it seems to me that a figure with enough faces cannot be viewed through one face so that all edges are within it. I find this a little confusing, but I guess its just another logical leap based on geometrical intuition. Peter Bilodeau ------=_NextPart_000_0000_01C5C91C.DE4183F0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Reading comments week 5:

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

 

Peter Bilodeau

------=_NextPart_000_0000_01C5C91C.DE4183F0-- From hzhou@MIT.EDU Tue Oct 4 20:00:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9500Ew0007831 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:00:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9500Dkj020157 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:00:13 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-TWO-EIGHTY-THREE.MIT.EDU [18.247.6.28]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9500AWg017553 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:00:11 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004195656.02a9d758@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 20:00:14 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: email Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.055 X-Spam-Level: * (1.055) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 510 Content-Length: 455 X-Keywords: NonJunk Hi, I didn't really have any questions concerning the lecture notes, but the one example brought up in class was somewhat confusing. Prof Meyer said that the survey that stated 75% more guys had sexual interactions than girls is false. However, if you view guys/girls as vertices, the survey is simply saying that 75% more of the guys vertices has edges compared to girls vertices with edges? Thus the survey could have some validity?? - Steve From fluff@MIT.EDU Tue Oct 4 20:03:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95037w0008030 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:03:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95035kj021772 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:03:06 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95031Jo018009 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 20:03:01 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8BC5300F-BC1D-48A6-88A8-24CBAFC63466@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 5 reading Date: Tue, 4 Oct 2005 20:02:57 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 511 Content-Length: 791 X-Keywords: NonJunk Overall, I thought the reading was pretty understandable and for once had enough examples. I didn't get this thing in the proof of Euler's Formula (pg. 12). In the base case, it says that a graph with nothing but a single vertex has one face, the "outside" face. What the? How can something with no edges have faces? Also, I have a comment about something on page 6, in the second paragraph after Corollary 2.10. It talks about taking an n+1 graph, then generalizing about an n subgraph to generalize about the n+1 one in induction. This is something that we needed in the pset that was due on Monday (on the last problem). Suggestion: maybe in the future, this shrink-down/grow-back paradigm in induction with graphs can be introduced before the pset is due. ~Crystal From dowgun@MIT.EDU Tue Oct 4 20:10:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j950APw0008461 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:10:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j950AOkj026506 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:10:24 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j950AMlI019192 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:10:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j950AMYQ019786; Tue, 4 Oct 2005 20:10:22 -0400 Received: from PLP-FORTY-SIX.MIT.EDU (PLP-FORTY-SIX.MIT.EDU [18.218.1.46]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 20:10:22 -0400 Message-ID: <20051004201022.2x0to8rcggg8gsow@webmail.mit.edu> Date: Tue, 4 Oct 2005 20:10:22 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 512 Content-Length: 907 X-Keywords: NonJunk I was most interested in the section about planar embeddings (page 12) because I was sure that the notes were going to prove the idea behind many a child's logic puzzle - that all embedded graphs are 4-colorable. Surprisingly they did not! This left me to wonder if this theorem has ever actually been proven, although I know from experience that it is almost certainly true. In fact, the coloring of the faces of a planar embeddign never came up at all, so I thought maybe it would be better to replace the embedding with a graph of vertices where each vertex stands in for a face, and each edge crosses an edge that it is replacing. It is a feature of this graph that no edges will ever cross each other. Therefore, this is ANOTHER planar embedding, so it follows that if the faces of planar embeddings are 4-colorable, THEN THE VERTICES ARE TOO. Has anyone ever tried to prove THIS theorem? Neil Dowgun From crowell@MIT.EDU Tue Oct 4 20:44:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j950iZw0011422 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:44:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j950iYkj016309 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:44:34 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j950iW9G025125 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 20:44:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j950iW2U004130; Tue, 4 Oct 2005 20:44:32 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 20:44:32 -0400 Message-ID: <20051004204432.jnvky8j8wbxes44w@webmail.mit.edu> Date: Tue, 4 Oct 2005 20:44:32 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 5 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 513 Content-Length: 665 X-Keywords: I did not understand the proof of lemma 5.2 on page 13 (specifically the line in the proof that states 'If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one ?going? and the other ?coming?), and not by any other face boundary.' I am also not clear on part 1 of the proof of the Marriage theorem on page 16 (I don't understand how "In this case, we have some latitude: we pair an arbitrary girl with a boy she likes and send them both away" ensures that we don't violate the marriage condition (if, say, G1 likes B1 and G2 likes B1, B2, B3 and we first pair G2 with B1 and send them "on their way"). -Rob Crowell From vixen@MIT.EDU Tue Oct 4 20:53:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j950rIw0011967 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 20:53:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j950rHkj021328 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 20:53:17 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j950r904026587 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 20:53:10 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Tue, 4 Oct 2005 20:52:14 -0400 To: 6042-probs@theory.csail.mit.edu From: Amanda Seybold Subject: [Jelani] LN5 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 514 Content-Length: 141 X-Keywords: I was confused by the formal statement of Hall's Theorem on page 17. I would also like to see algorithms for matching in a bipartite graph. From yangc@MIT.EDU Tue Oct 4 21:18:58 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j951Ivw0013668 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:58 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j951IuNJ008422 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:56 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j951IuIY001811 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:56 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j951InMv011661 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 21:18:53 -0400 (EDT) Message-Id: <200510050118.j951InMv011661@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 5 Comments Date: Tue, 4 Oct 2005 21:18:45 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000A_01C5C929.346CB9F0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJSrsoqClu6QFGQyKK3y8HwBpGOQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 0.872 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 515 Content-Length: 2322 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_000A_01C5C929.346CB9F0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit On page 17, it says that there are very efficient algorithms for doing the Hall's Theorem matching. What are some examples of those? Also, what sorts of problems can be reduced to finding a matching? Thanks, Chris Yang ------=_NextPart_000_000A_01C5C929.346CB9F0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

 

Thanks,

Chris Yang

------=_NextPart_000_000A_01C5C929.346CB9F0-- From hsoumare@MIT.EDU Tue Oct 4 22:04:07 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95247w0020051 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:07 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j95246NJ023812 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:06 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95245IY002799 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:06 -0400 (EDT) Received: from [18.241.6.193] (NEW-FOUR-FORTY-EIGHT.MIT.EDU [18.241.6.193]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95241Mu013525 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:04:01 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: [Sayan] Weekly Email Comment Date: Tue, 4 Oct 2005 22:12:35 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 516 Content-Length: 259 X-Status: A X-Keywords: NonJunk I found the passage on sex in America to be the most surprising because I never thought that you could prove mathematically, that the only reason men are more promiscuous than women is because of the differences in the population size. Hamidou Soumare From ozcan@MIT.EDU Tue Oct 4 22:05:22 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9525Mw0020194 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:05:22 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9525MOv014950 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:05:22 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9525Eol013757 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 22:05:14 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004220040.028accd0@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 22:05:02 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 517 Content-Length: 491 X-Keywords: Page 9, in Coloring Graphs: We can recast this scheduling problem as a question about coloring the vertices of a graph. Create a vertex for each course with a final exam. Put an edge between two vertices if some student is taking both courses. The above paragraph, and the section it comes from, explains how to model the scheduling problem with graphs. I was quite amazed (and surprised) how such a complex problem is modeled in a very simple and comprehensive way. best, Yasin From icharny@mit.edu Tue Oct 4 22:10:52 2005 Return-Path: Received: from auathena (EASTCAMPUS-SIX-NINETY-EIGHT.MIT.EDU [18.238.5.187]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j952Apw0020615 for <6042-probs@theory.csail.MIT.EDU>; Tue, 4 Oct 2005 22:10:51 -0400 Received: (from icharny@localhost) by auathena (8.12.9) id j952AnIj010271; Tue, 4 Oct 2005 22:10:49 -0400 Subject: [Sayan] Week 5 Comments From: Isaac E Charny To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 04 Oct 2005 22:10:48 -0400 Message-Id: <1128478249.24973.16.camel@auathena> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 Status: RO X-UID: 518 Content-Length: 123 X-Keywords: NonJunk I was very surprised that the faces, edges, and vertices were related by the simple equation v - e + f = 2. ~Isaac Charny From jjmonzon@MIT.EDU Tue Oct 4 22:15:41 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j952Ffw0020907 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:15:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j952Fdkj009018 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:15:39 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-TWO-TWELVE.MIT.EDU [18.247.5.212]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j952FKHI012253 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:15:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004221138.020ed008@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 22:15:24 -0400 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 519 Content-Length: 370 X-Keywords: I found the Hall's Marriage Theorem a bit confusing and I hope this would be discussed better in lecture. I just found it hard to read the notation the notes uses and a layman's explanation would be very much appreciated. Josh Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From brevzin@MIT.EDU Tue Oct 4 22:36:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j952aYw0025987 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:36:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j952aXkj021282 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:36:33 -0400 (EDT) Received: from BARRY.mit.edu ([18.218.1.202]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j952aPpO016637 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 22:36:26 -0400 (EDT) Message-Id: <6.1.2.0.1.20051004223238.019d6ce8@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Tue, 04 Oct 2005 22:36:33 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 520 Content-Length: 269 X-Keywords: I found the classifying polyhedra section very interesting, since I never knew that you could actually prove that there are only 5. I just took it for granted that they were there. Also, the proof was of Euler's formula was pretty hot. I enjoyed reading it. Barry From dshin@MIT.EDU Tue Oct 4 23:01:28 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9531Rw0028488 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:01:27 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9531Qjv017175 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:01:26 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j952w38U003989; Tue, 4 Oct 2005 22:58:03 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j952vrMu015392; Tue, 4 Oct 2005 22:57:53 -0400 (EDT) Message-ID: <43434111.8040108@mit.edu> Date: Tue, 04 Oct 2005 22:57:21 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yaser Khan CC: 6042-probs@theory.lcs.mit.edu Subject: Re: David: week 5 reading question References: <200510042154.j94LsMXg019580@outgoing.mit.edu> In-Reply-To: <200510042154.j94LsMXg019580@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------030405050302080903060109" X-Spam-Score: -2.448 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 521 Content-Length: 7118 X-Keywords: This is a multi-part message in MIME format. --------------030405050302080903060109 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Yes, you are touching upon some very deep concepts with this line of thought. Map coloring and "vertex coloring" are equivalent in some sense: given some map (say, a map of the USA), construct a graph G that has a vertex for each region in the map (US states), and an edge between u and v if the corresponding regions border each other. map <---> graph region <---> vertex border <---> edge Then, coloring the regions in the map is equivalent to coloring the vertices of the graph. DS Yaser Khan wrote: > Hi David, > > > > When it comes to k-coloring, is there a direct relation between > k-coloring of vertices (as outlined in the reading) and the more > general map-coloring problem? I.e., on the tutor problem there was a > graph that had a chromatic value of 3, but would have a chromatic > value of 2 if we were attempting to fill in edges or areas rather than > vertices. Are areas between edges EVER considered legitimate or useful > in such graphs? > > > > Thanks! > > > Take care, > > > > _Yaser > --------------030405050302080903060109 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Yes, you are touching upon some very deep concepts with this line of thought.

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

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

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

DS

Yaser Khan wrote:

Hi David,

 

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

 

Thanks!


Take care,

 

_Yaser

--------------030405050302080903060109-- From moscicki@MIT.EDU Tue Oct 4 23:05:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9535Ww0028989 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:05:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9535R6s007650 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:05:27 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9535OhC022313 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:05:24 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9535OTN022801; Tue, 4 Oct 2005 23:05:24 -0400 Received: from NEW-THREE-SIXTY-SEVEN.MIT.EDU (NEW-THREE-SIXTY-SEVEN.MIT.EDU [18.241.6.112]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:05:24 -0400 Message-ID: <20051004230524.kwm3677bu70gwso4@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:05:24 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: week 5 comment MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_3ch3plf2khds" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 3.14 X-Spam-Level: *** (3.14) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 522 Content-Length: 1159 X-Keywords: NonJunk This message is in MIME format. --=_3ch3plf2khds Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit attached --=_3ch3plf2khds Content-Type: text/plain; charset=ISO-8859-1; name="week5comment.txt" Content-Disposition: attachment; filename="week5comment.txt" Content-Transfer-Encoding: 7bit Back in the realm of intution. All the stuff before Coloring Graphs was covered in class. It was clear and I think I understood it. Coloring- an interesting way of looking at graphs. Also, the general method of shrink-down grow-back I think would have been useful for the previous problem set. Its nice to have a general method of using induction on graphs. Before I read about that, it seemed like something that would be really hard to do. Bipartite Graphs are understandable. Planar Graphs are a bit fuzzy in my mind. I definitely appreciate the applications to PCB design though. Euler's formula is fine, I have a feeling that will be important anyway. It might have been nice to state Hall's Theorem formally first so the reader can keep it in mind as we go through the example. --=_3ch3plf2khds-- From lana@MIT.EDU Tue Oct 4 23:18:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953Icw0030888 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:18:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953Ib6s015573 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:18:37 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953IYh3025001 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:18:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j953IYHW022942; Tue, 4 Oct 2005 23:18:34 -0400 Received: from NEXT-SEVEN-THIRTEEN.MIT.EDU (NEXT-SEVEN-THIRTEEN.MIT.EDU [18.242.7.202]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:18:33 -0400 Message-ID: <20051004231833.fe7j5cx0guosgk0o@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:18:33 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.165 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 523 Content-Length: 331 X-Keywords: NonJunk I was wondering if there was a way of proving that a graph cannot be planar without Euler's Formula (for example the two graphs used in the reading: the complete graph on 5 vertices and the complete bipartite graph on 6 vertices). Is there an algorithm (other than Euler's formula) of showing that a graph cannot be planar? Lana From tonyng@MIT.EDU Tue Oct 4 23:19:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953JOw0030917 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:19:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953JM6s015897 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:19:23 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953JDqZ025072 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:19:15 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004231521.020f4678@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 23:19:28 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 524 Content-Length: 463 X-Keywords: I found Coloring Graphs (section 4, page 9) most surprising. I always thought that graphs were used to visualize relations between vertices. However, I never realized it was possible to assign colors and get other useful applications such as the finals scheduling situation (I have seen assigning numbers to edges to represent distances, but I've never seen assigning properties to the vertices themselves and getting something useful from it). - Tony Ng From aka_kame@MIT.EDU Tue Oct 4 23:21:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953Ltw0031104 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:21:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953Ls6s017471 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:21:54 -0400 (EDT) Received: from [18.234.1.87] (DP-EIGHTY-SEVEN.MIT.EDU [18.234.1.87]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953LldY025539 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:21:47 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <382973AE-959B-446E-9671-C62830B1DF67@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-1--148549695 From: Akari Kameyama Subject: Week 5 comments Date: Tue, 4 Oct 2005 23:21:45 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 0.501 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 525 Content-Length: 3422 X-Keywords: NonJunk --Apple-Mail-1--148549695 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed "We can regard the corners and edges of these polyhedra as the vertices and edges of a planar graph. (This is another logical leap based on geometric intuition.)" So far we have only discussed "graphs" from a 2-dimensional perspective. Although 2D graphs can be "flexed" (for lack of a better word) to become 3D, it feels as though there exists/should exist a whole different realm of possibilities in the 3D world which cannot be translated into 2D. For example, the dog and house situation, could be done in a 3D representation yes? --Apple-Mail-1--148549695 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
"We can regard the corners and edges of these = polyhedra as the vertices and edges of a planar=A0
graph. (This is = another logical leap based on geometric = intuition.)"

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


= --Apple-Mail-1--148549695-- From jehan@MIT.EDU Tue Oct 4 23:28:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953Suw0031624 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:28:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953Ss6s021594 for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:28:54 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953SlwX026902 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 4 Oct 2005 23:28:48 -0400 (EDT) Message-Id: <5.2.1.1.2.20051004232442.00bbaed0@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 04 Oct 2005 23:30:08 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 526 Content-Length: 66 X-Keywords: NonJunk I didn't really understand the proof for euler's formula. jehan From kjhollen@MIT.EDU Tue Oct 4 23:33:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953XMw0032282 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:33:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953XL6s024226 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:33:21 -0400 (EDT) Received: from MACGREGOR-FOUR-O-FOUR.MIT.EDU (MACGREGOR-FOUR-O-FOUR.MIT.EDU [18.239.6.149]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953XB7c027714 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:33:14 -0400 (EDT) Subject: [Hanson] clarification of Hall's Theorem From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Tue, 04 Oct 2005 23:33:11 -0400 Message-Id: <1128483191.30474.4.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 2.544 X-Spam-Level: ** (2.544) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 527 Content-Length: 418 X-Keywords: NonJunk Hello, I understand what Hall's Theorem (p. 15-17) is and it's pretty neat -- it seems pretty straight forward how to use it to find a mapping between different subsets in graphs, and I can think of how it would be useful in a computer program. However, I struggled a bit with the tutor problem on how to know when you can't apply Hall's Theorem. I think I get it, but it would be nice to see another example. Kate From meyer@imap.theory.csail.mit.edu Tue Oct 04 23:43:21 2005 Message-ID: <43434BDF.8080203@csail.mit.edu> Date: Tue, 04 Oct 2005 23:43:27 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hamidou Soumare CC: 6042-staff@theory.csail.mit.edu Subject: Re: [Sayan] Weekly Email Comment References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 813 Status: RO X-UID: 528 X-Keywords: It's a misunderstanding to say "men are more promiscuous ...because..." The relative fraction of sexual partners has nothing to do with promiscuity; it just shows how misleading averages can be. For example, suppose there was a population of 100 mean and 1000 woman, and each man was married to one woman, (leaving 900 unmarried women). Now the men have on average 10 times as many partners (man average = 0.1, woman average 0.01), but I don't think you'd want to claim anybody was promiscuous. regards, A. Hamidou Soumare wrote: > I found the passage on sex in America to be the most surprising because > I never thought that you could prove mathematically, that the only > reason men are more promiscuous than women is because of the > differences in the population size. > > Hamidou Soumare From rshroff@MIT.EDU Tue Oct 4 23:49:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953niw0001460 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:49:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953nh6s003483 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:49:43 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953nfvP000680 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 4 Oct 2005 23:49:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j953nfiq031584; Tue, 4 Oct 2005 23:49:41 -0400 Received: from NEXT-SIXTY-SEVEN.MIT.EDU (NEXT-SIXTY-SEVEN.MIT.EDU [18.242.5.67]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:49:41 -0400 Message-ID: <20051004234941.22n59d76b1xc8ckw@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:49:41 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David]E-mail Comments for assigned reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 529 Content-Length: 163 X-Keywords: NonJunk Reference: Sections 3.2 and 4.2 I was a bit confused about the idea of spanning trees and Bipartite graphs, and would like to go over it in class. -Rahul Shroff From minilek@theory.csail.mit.edu Tue Oct 4 23:51:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953pMw0001590 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:51:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953pJ6s004542; Tue, 4 Oct 2005 23:51:19 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953pCpC000975 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 4 Oct 2005 23:51:16 -0400 (EDT) Message-ID: <43434DC7.8090809@theory.csail.mit.edu> Date: Tue, 04 Oct 2005 23:51:35 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Pete Kruskall Subject: Re: [Jelani] Comments for This Week References: <4342E1BF.4010003@mit.edu> In-Reply-To: <4342E1BF.4010003@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.529 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 530 Content-Length: 1182 X-Keywords: NonJunk >> A graph can only truly be bipartite if it's chromatic number is 2 (I hope I have the vocab right). This is correct, but this is true by definition (a graph is defined to be bipartite if it's chromatic number is two). As for Euler's formula, it would be best to try to understand its proof. Try to read it over and digest it -- it may take a few times reading over. Let me know if you still have questions about it after we cover it in lecture. -Jelani Pete Kruskall wrote: > I'm having difficulty with Euler's formula and getting it into my > head. Should I memorize it? Or should I understand it's concepts? > > > Also.... > > I was going to comment on Theorem 4.2 being unsupported, until I found > a great way to think of it. A graph can only truly be bipartite if > it's chromatic number is 2 (I hope I have the vocab right). If > there's an odd cycle, some adjacent nodes would need to be colored the > same... (A B A --- cycle of length 3 has AA (no good )) > >-- >Pete Kruskall >28 The Fenway >Boston, MA 02215 > >::::::::::::::::::::::::::::: >617.536.9925 :::Sigma Nu::::: >508.843.5861 ::::Cell Phone:: >::::http://tege.mit.edu:::::: > > > From minilek@theory.csail.mit.edu Tue Oct 4 23:56:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j953uSw0001859 for <6042-probs@theory.lcs.mit.edu>; Tue, 4 Oct 2005 23:56:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j953uP6s007501; Tue, 4 Oct 2005 23:56:25 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j953uBPD001826 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 4 Oct 2005 23:56:18 -0400 (EDT) Message-ID: <43434EF1.1060008@theory.csail.mit.edu> Date: Tue, 04 Oct 2005 23:56:33 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Arup Sarma Subject: Re: [Jelani] Week 5 Reading Comments References: <6.2.3.4.2.20051004172123.047bf878@po14.mit.edu> In-Reply-To: <6.2.3.4.2.20051004172123.047bf878@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 531 Content-Length: 749 X-Keywords: NonJunk Arup, P(n+1) isn't being assumed before proving it. In the inductive step, we're looking at a graph with k vertices and n+1 edges. We take he subgraph obtained by removing one of the edges arbitrarily. This subgraph has n edges, so P(n) holds on that subgraph. Now we reason about what happens when we put back the edge that was removed and show that P(n) --> P(n+1). Does this clear things up? -Jelani Arup Sarma wrote: > Section 2.5, page 6 (and used later as well): I think I don't > completely understand the "shrink-down, grow-back" style of > induction. It seems like it assumes P(n+1) before proving it. Also, > what are examples of the logical errors that can occur if this style > of argument isn't used? > > |Arup| > From minilek@theory.csail.mit.edu Wed Oct 5 00:00:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9540Qw0003013 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:00:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9540O6s009426; Wed, 5 Oct 2005 00:00:24 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9540Lne002458 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:00:22 -0400 (EDT) Message-ID: <43434FEC.1030001@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:00:44 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Crystal Chao Subject: Re: [Jelani] week 5 reading References: <8BC5300F-BC1D-48A6-88A8-24CBAFC63466@mit.edu> In-Reply-To: <8BC5300F-BC1D-48A6-88A8-24CBAFC63466@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.564 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 532 Content-Length: 1115 X-Keywords: NonJunk The faces are the connected regions the graph's edges divide the plane into. Since there are no edges in a single-vertex graph, the entire plane is in the same connected region, and this is the "outside face" the base case is referring to. Does that clear things up? Crystal Chao wrote: > Overall, I thought the reading was pretty understandable and for once > had enough examples. I didn't get this thing in the proof of Euler's > Formula (pg. 12). In the base case, it says that a graph with nothing > but a single vertex has one face, the "outside" face. What the? How > can something with no edges have faces? > > Also, I have a comment about something on page 6, in the second > paragraph after Corollary 2.10. It talks about taking an n+1 graph, > then generalizing about an n subgraph to generalize about the n+1 one > in induction. This is something that we needed in the pset that was > due on Monday (on the last problem). Suggestion: maybe in the future, > this shrink-down/grow-back paradigm in induction with graphs can be > introduced before the pset is due. > > ~Crystal From minilek@theory.csail.mit.edu Wed Oct 5 00:04:23 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9544Nw0003268 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:04:23 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9544K6s011414; Wed, 5 Oct 2005 00:04:20 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9544Itb003041 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:04:18 -0400 (EDT) Message-ID: <434350D9.3080905@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:04:41 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Natalia N Chernenko Subject: Re: [Jelani] Week 5 Comments References: <1128463908.20774.11.camel@hayden-1.mit.edu> In-Reply-To: <1128463908.20774.11.camel@hayden-1.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.576 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 533 Content-Length: 511 X-Keywords: NonJunk The proof is somewhat tricky, and may take reading over a couple times (especially the second case). Try to read it over some more to really digest the proof. If you still find some parts of the proof confusing though, just let me know. -Jelani Natalia N Chernenko wrote: >It would be nice to discuss planar graphs (Section 5, pp. 11-13) more in >lecture, especially Euler's formula (Section 5.1). I found the >derivation of the formula a little confusing (induction proof). > >Natalia Chernenko > > From lkini@MIT.EDU Wed Oct 5 00:10:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954A2w0003645 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 00:10:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954A06s014378 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 00:10:00 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9549vq7004352 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 00:09:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9549vmP001390; Wed, 5 Oct 2005 00:09:57 -0400 Received: from MACGREGOR-TWO-FORTY.MIT.EDU (MACGREGOR-TWO-FORTY.MIT.EDU [18.239.5.240]) (User authenticated as lkini@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 00:09:57 -0400 Message-ID: <20051005000957.osf3cx3xuhog400g@webmail.mit.edu> Date: Wed, 5 Oct 2005 00:09:57 -0400 From: Lohith G Kini To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Difficulty with reading lecture notes for week 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.879 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 534 Content-Length: 254 X-Keywords: NonJunk Hi Hanson, I am having trouble understanding the proof for lemma 5.2 on page 13 of the lecture notes. I understand the mathematics that leads off from statement (1) but I don't understand the explanation behind the original inequality 2e >= 3f. Lohith From minilek@theory.csail.mit.edu Wed Oct 5 00:12:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954Cww0003776 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:12:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Cu6s015926; Wed, 5 Oct 2005 00:12:56 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954CjKT004810 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:12:49 -0400 (EDT) Message-ID: <434352D4.8060102@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:13:08 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: Re: [Jelani] LN5 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 535 Content-Length: 937 X-Keywords: NonJunk >> I was confused by the formal statement of Hall's Theorem on page 17 We have some bipartite graph, so we can partition the vertices into "the left side" (L) and "the right side" (R), such that all edges have one endpoint in L and one endpoint in R. A matching for L is a function f from vertices of L to vertices of R such that no two vertices of L get mapped to the same thing, and if f(x) = y, then the edge x--y exists. Hall's Theorem is saying a matching for L exists if and only if for every subset S of L, |N(S)| >= |S|. N(S) is the set of vertices that are not in S but are adjacent to something in S (so, since S is a subset of L, N(S) will be a subset of R -- do you see why?). Let me know if you are still confused after lecture. -Jelani Amanda Seybold wrote: > I was confused by the formal statement of Hall's Theorem on page 17. I > would also like to see algorithms for matching in a bipartite graph. From iyzhang@MIT.EDU Wed Oct 5 00:13:56 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954Dtw0003847 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:55 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j954Dsjv024507 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Ds8U005560 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:54 -0400 (EDT) Received: from [18.239.7.4] (MACGREGOR-FIVE-FIFTEEN.MIT.EDU [18.239.7.4]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j954Dmok018413 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:13:51 -0400 (EDT) Message-ID: <434352EF.7050302@mit.edu> Date: Wed, 05 Oct 2005 00:13:35 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 536 Content-Length: 56 X-Keywords: I would like planar graphs explain more. thanks, irene From minilek@theory.csail.mit.edu Wed Oct 5 00:17:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954Htw0004149 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:17:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Hr6s018748; Wed, 5 Oct 2005 00:17:54 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954HmWC005553 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 00:17:51 -0400 (EDT) Message-ID: <43435403.1050606@theory.csail.mit.edu> Date: Wed, 05 Oct 2005 00:18:11 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: r n jacobs Subject: Re: [Jelani] Reading response References: <200510040036.j940aXUv024591@white-meteo.mit.edu> In-Reply-To: <200510040036.j940aXUv024591@white-meteo.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.564 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 537 Content-Length: 547 X-Keywords: NonJunk The tutor questions are supposed to reflect the reading and provide practice applying what you just learned. Is your complaint that it isn't doing this? -Jelani r n jacobs wrote: >It took a while for me to find the definition of "matching" since it >was only boldface in a paragraph. > >Are the tutor questions specifically supposed to reflect the reading? >It seems strange that the material was half already covered and half >required doing the reading. > >Otherwise, everything seems straightforward this week. > > - Robert Jacobs > > From mpapi@MIT.EDU Wed Oct 5 00:36:28 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954aSw0008614 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:36:28 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j954aRjv016087 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:36:27 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954Yj8U005998 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:34:45 -0400 (EDT) Received: from MACGREGOR-THREE-FORTY-SIX.MIT.EDU (MACGREGOR-THREE-FORTY-SIX.MIT.EDU [18.239.6.91]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j954YgMu018462 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 00:34:42 -0400 (EDT) Subject: [Jelani] Week 5 Comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 05 Oct 2005 00:34:41 -0400 Message-Id: <1128486881.7375.18.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.425 X-Spam-Level: * (1.425) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 538 Content-Length: 383 X-Keywords: NonJunk Page 17, section 6.1 & theorem 6.2, the formal statement of Hall's theorem. The marriage analogy is pretty useful in getting to understand the theorem, and the proof is good. I think it would also be helpful to further elaborate on applications of the theorem (and perhaps some not-so-obvious applications, since they seem to arise from time to time for different things). -- Matt From dshin@MIT.EDU Wed Oct 5 00:43:47 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954hlw0010006 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:43:47 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j954hkjv023440 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:43:46 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954hj8U006169; Wed, 5 Oct 2005 00:43:46 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j954hhok019215; Wed, 5 Oct 2005 00:43:43 -0400 (EDT) Message-ID: <434359E3.8090704@mit.edu> Date: Wed, 05 Oct 2005 00:43:15 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 5 comments References: <382973AE-959B-446E-9671-C62830B1DF67@mit.edu> In-Reply-To: <382973AE-959B-446E-9671-C62830B1DF67@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 539 Content-Length: 1460 X-Keywords: Think about it this way: suppose you took a "2D" graph on a piece of paper, and placed a coin on each vertex, and glued string between coins wherever there was an edge. Just by lifting some of the coins up in the air, you'd be making this graph "3D"! Now, take some "3D" graph, "frozen in time and space", and imagine placing a floating coin at each vertex, and gluing a string between coins wherever there is an edge. Then, bam, "unfreeze" time, and gravity pulls the coins (and strings) down to the ground. And presto, you have a "2D" graph! All of the concepts we've studied so far (matchings, cycles, simple paths, connected components) stay the same in these two graphs. If you could understand this little thought experiment, you'd see that 2D graphs and 3D graphs really are the same thing - there is no such thing as a "2D" or "3D" graph! Akari Kameyama wrote: > "We can regard the corners and edges of these polyhedra as the > vertices and edges of a planar > graph. (This is another logical leap based on geometric intuition.)" > > So far we have only discussed "graphs" from a 2-dimensional > perspective. Although 2D graphs can be "flexed" (for lack of a better > word) to become 3D, it feels as though there exists/should exist a > whole different realm of possibilities in the 3D world which cannot be > translated into 2D. For example, the dog and house situation, could > be done in a 3D representation yes? > > From antonk@MIT.EDU Wed Oct 5 00:46:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954kKw0010151 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:46:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954kJ6s003055 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:46:19 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954kGUH009627 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:46:16 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 5 Comments Date: Wed, 5 Oct 2005 00:46:14 -0400 Organization: Massachusetts Institute of Technology Message-ID: <009301c5c967$b801d730$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0094_01C5C946.30F03730" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXJZ7cubGrNNXxNTNufAnHQiIk85w== X-Spam-Score: -1.046 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 540 Content-Length: 2180 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0094_01C5C946.30F03730 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Is it possible to go over planar graphs? I understand the faces and the math behind it, but I do not understand why we need it and the proof is a bit hard. Anton. ------=_NextPart_000_0094_01C5C946.30F03730 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Is it possible to go over planar = graphs?

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

 

Anton.

------=_NextPart_000_0094_01C5C946.30F03730-- From clintonb@MIT.EDU Wed Oct 5 00:54:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954s5w0011951 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:54:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954s46s006973 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:54:04 -0400 (EDT) Received: from ccbibmx30 (NEW-EIGHTY-TWO.MIT.EDU [18.241.5.82]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954rtuj010555 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:53:55 -0400 (EDT) Message-Id: <200510050453.j954rtuj010555@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Jelani: Week 5 Comments Date: Wed, 5 Oct 2005 00:53:26 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_005B_01C5C947.323BD490" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXJaLiPDplbT+1qQdiWpdm+G6fcJA== X-Spam-Score: 1.15 X-Spam-Level: * (1.15) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 541 Content-Length: 3510 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_005B_01C5C947.323BD490 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I actually understood most of the reading this time. I need a little more time to study the bipartite section. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_005B_01C5C947.323BD490 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

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

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_005B_01C5C947.323BD490-- From jstritar@MIT.EDU Wed Oct 5 01:10:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955A6w0012956 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 01:10:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955A46s014500 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 01:10:05 -0400 (EDT) Received: from [18.245.6.98] (BAKER-THREE-FIFTY-THREE.MIT.EDU [18.245.6.98]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955A18N012612 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 01:10:02 -0400 (EDT) Message-ID: <43436028.1080700@mit.edu> Date: Wed, 05 Oct 2005 01:10:00 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.054 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 542 Content-Length: 224 X-Keywords: NonJunk 5.3 Classifying Polyhedra I thought this section was really interesting. I have never thought of polyhedrans as planar graphs and it was pretty surprising. It took a few minutes of visualizing to see the connection. Jon From dshin@MIT.EDU Wed Oct 5 01:14:04 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955E4w0013177 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:14:04 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j955E3jv021497 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:14:03 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955E38U006736; Wed, 5 Oct 2005 01:14:03 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j955Dxok020061; Wed, 5 Oct 2005 01:13:59 -0400 (EDT) Message-ID: <434360FC.6030702@mit.edu> Date: Wed, 05 Oct 2005 01:13:32 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 5 Comments References: <200510050118.j951InMv011661@outgoing-legacy.mit.edu> In-Reply-To: <200510050118.j951InMv011661@outgoing-legacy.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 543 Content-Length: 1195 X-Keywords: It is a little difficult to describe those algorithms with the limited machinery we have developed in the course thus far. If you would like, I could show you how these algorithms work during OH or before/after class. As for problems that reduce to finding a matching, there are lots! One application: medical schools use (bipartite) matching algorithms to accept applicants to try to make sure everybody gets in somewhere. A common variant on the matching problem is to find a max/min-weight matching (when the edges have real-valued weights). This is often a useful subroutine for many problems, including some variants of the traveling salesman problem (TSP). In min-TSP, a salesman has to visit a bunch of cities (i.e., vertices on a graph) and wants to minimize the total amount of time he has to spend. (Matching is not used in min-TSP, but it is used in max-TSP). Hope that gives you a flavor of some of the uses. DS Chris Yang wrote: > On page 17, it says that there are very efficient algorithms for doing > the Hall’s Theorem matching. What are some examples of those? Also, > what sorts of problems can be reduced to finding a matching? > > Thanks, > > Chris Yang > From lmccart@MIT.EDU Wed Oct 5 01:25:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955Plw0014079 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:25:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955Pk6s021856 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:25:46 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955PdOG014147 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:25:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j955Pdax010348; Wed, 5 Oct 2005 01:25:39 -0400 Received: from KS-ONE-TWENTY-EIGHT.MIT.EDU (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 01:25:39 -0400 Message-ID: <20051005012539.cck5vdbdhdfvoso8@webmail.mit.edu> Date: Wed, 5 Oct 2005 01:25:39 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 5 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 544 Content-Length: 159 X-Keywords: NonJunk I am still a little confused about the actual definition of a connected component. Both his defintion in class and the reading were somewhat unclear. -lauren From ryan786@MIT.EDU Wed Oct 5 01:27:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955Row0014148 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:27:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955Rn6s022819 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:27:49 -0400 (EDT) Received: from ryoung (PSK-SIX.MIT.EDU [18.217.1.6]) (authenticated bits=0) (User authenticated as ryan786@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955Rk7x014339 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 01:27:47 -0400 (EDT) Message-Id: <200510050527.j955Rk7x014339@outgoing.mit.edu> Reply-To: From: "ryan" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 01:27:48 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C94B.FF2CBF60" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJbYWLaqr8h/Z0Ts6GE4a5JVJBJA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.375 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 545 Content-Length: 2797 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C94B.FF2CBF60 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit This was by far and away my favorite reading assignment thus far. I liked how there was more of an emphasis on visual representation and less of an emphasis on technical logic (although admittedly it is a necessary evil in this class). I especially liked the section on Hall's Marriage Theorem, as I had never seen it before, but once I read it, it made perfect sense. I guess I still have only a shaky hold on connected components, but that's probably more from our class Monday than the reading tonight. -Ryan Young ------=_NextPart_000_0000_01C5C94B.FF2CBF60 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

 

-Ryan Young

------=_NextPart_000_0000_01C5C94B.FF2CBF60-- From aeon@MIT.EDU Wed Oct 5 01:29:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j955TFw0014269 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:29:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j955TD6s023508 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:29:13 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j955TB9C014499 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 01:29:11 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j955TBuK026914; Wed, 5 Oct 2005 01:29:11 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 4 Oct 2005 23:29:11 -0600 Message-ID: <20051004232911.lw6n5nvxu4u8088o@webmail.mit.edu> Date: Tue, 4 Oct 2005 23:29:11 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: LN5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.686 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 546 Content-Length: 683 X-Keywords: NonJunk Page 7, "Tree Properties" Just about everything here can be applied to data network topology, so I found it to be particularly useful. I.e., k-connectedness describes the robustness and fault tolerance of a network, spanning trees give the cheapest setup possible, etc. In general, I just found this to be cool. Page 9, "Coloring Graphs" This idea was surprising to me. I didn't even think about "coloring" a graph. What I wonder is how a program could be written to assign colors and then make sure all nodes are different without individually checking them all... it must sort the graph somehow, but it's not immediately apparent how that sort would be........ ;) - John Marrero From sriaz@MIT.EDU Wed Oct 5 02:23:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956NGw0025950 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:23:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j956NFIs014676 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:23:15 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j956NCZb018556 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:23:13 -0400 (EDT) Message-ID: <43437152.8010903@mit.edu> Date: Wed, 05 Oct 2005 02:23:14 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.881 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 547 Content-Length: 264 X-Keywords: NonJunk One of the issues I had with this week's reading, and the tutor problems, was the questions regarding Graph Coloring. In Section 4, coloring graphs are gone over, but I would have liked to see more detail as that would have helped with the TP. Thanks Sameer From shreyes19@gmail.com Wed Oct 5 02:25:48 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.195]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956Pmw0026554 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:25:48 -0400 Received: by xproxy.gmail.com with SMTP id s9so68595wxc for <6042-probs@theory.lcs.mit.edu>; Tue, 04 Oct 2005 23:25:43 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=B7fMoyo9D6KFNBxZ6hu8asWOwiqR+DM2BJazZbCR6JAE8mFRNTpBOYHlK3Gh/HeLXcNIKsdcKNwoVCjyBB8YWruUCzeWYulaNPvGzHrC9rwXq886Kdx3NL/sXyAt61GIMudKxeFFZ1Kw7Lilqg0j0rObnuPJhXPJPTscNjGjNKg= Received: by 10.70.59.17 with SMTP id h17mr304424wxa; Tue, 04 Oct 2005 23:25:43 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Tue, 4 Oct 2005 23:25:43 -0700 (PDT) Message-ID: <3c33fe380510042325p4122fdfeqfcb19cf04f6c805d@mail.gmail.com> Date: Wed, 5 Oct 2005 02:25:43 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 5 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_23205_25177894.1128493543150" Status: RO X-UID: 548 Content-Length: 2036 X-Keywords: NonJunk ------=_Part_23205_25177894.1128493543150 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, The part in this week's reading I found most surprising/interesting was the section on polyhedra. "Thus, we can regard the corners and edges of these polyhedra as the vertices and edges of a planar graph. (This is another logical leap based on geometric intuition.) This means Euler's formula for planar graphs can help guide our search for regular polyhedra." (Page 14). = I just thought it was an interesting way to approach the problem of finding all regular polyhedra, and was surprised to see that translating the proble= m into 2D planar graphs made it much simpler to visualize and inspect. I also thought it was interesting to be able to define a regular polyhedra in term= s of its qualities v, e, and f, without regard at first to its possible shape= . Thanks, Shreyes Seshasai Group 7 ------=_Part_23205_25177894.1128493543150 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

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

Thanks,
Shreyes Seshasai
Group 7
------=_Part_23205_25177894.1128493543150-- From bens@MIT.EDU Wed Oct 5 02:31:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956Vpw0026850 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 02:31:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j956VnIs018156 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 02:31:49 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j956Vh4V019107 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 02:31:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j956Vgib009113; Wed, 5 Oct 2005 02:31:42 -0400 Received: from LSLATER.MIT.EDU (LSLATER.MIT.EDU [18.221.0.171]) (User authenticated as bens@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 02:31:42 -0400 Message-ID: <20051005023142.dci3ppmec1wkcgg0@webmail.mit.edu> Date: Wed, 5 Oct 2005 02:31:42 -0400 From: Benjamin M Schwartz To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Week 5 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 549 Content-Length: 64 X-Keywords: NonJunk I cannot understand Euler's formula (p. 12) this late at night. From mracich@MIT.EDU Wed Oct 5 02:44:56 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j956iuw0029142 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:56 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j956isvr018666 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j956is25008681 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:54 -0400 (EDT) Received: from MACGREGOR-THREE-SIXTY-TWO.MIT.EDU (MACGREGOR-THREE-SIXTY-TWO.MIT.EDU [18.239.6.107]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j956ijMu022160 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 02:44:45 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 5 (Graphs) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 05 Oct 2005 02:44:42 -0400 Message-Id: <1128494682.23545.11.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 550 Content-Length: 202 X-Keywords: NonJunk I found section 5.3, Classifying Polyhedra, (starting on page 14) interesting, but also kind of confusing. I would appreciate it if this was reviewed in greater depth during lecture. Moira Racich From bakster@MIT.EDU Wed Oct 5 03:20:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j957KRw0001754 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 03:20:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j957KQIs006783 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 03:20:26 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j957KIhW021666 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 03:20:19 -0400 (EDT) Message-ID: <43437EB2.5040400@mit.edu> Date: Wed, 05 Oct 2005 03:20:18 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] Week 5 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 551 Content-Length: 134 X-Keywords: NonJunk I just wanted to make sure that we go over planar graphs in class. The tutor problem set didn't seem to cover them. Alexander Bakst From scot@MIT.EDU Wed Oct 5 04:25:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j958PJw0011207 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 04:25:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j958PIIs028941 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 04:25:18 -0400 (EDT) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j958PE7V024038 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 04:25:15 -0400 (EDT) Message-ID: <43438DD7.3030903@mit.edu> Date: Wed, 05 Oct 2005 04:24:55 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: week 5 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.903 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 552 Content-Length: 343 X-Status: A X-Keywords: NonJunk It was unclear at my table during last lecture whether the fault of the sex statistics had to do with people mis-reporting, or the statistical analysis. From page 5, and lecture, I'm still confused on what we are using for the term of connectedness, and from the in-class problem are we saying it is connected if there is any simple path? From zev@MIT.EDU Wed Oct 5 04:48:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j958mMw0013774 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 04:48:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j958mKIs007107 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 04:48:21 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j958mDh4024800 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 04:48:13 -0400 (EDT) Message-ID: <4343934C.70107@mit.edu> Date: Wed, 05 Oct 2005 04:48:12 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 5 comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.073 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 553 Content-Length: 405 X-Keywords: -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 I like the use of diagrams and examples in these notes: they made the sections much more clear. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDQ5NMlO3j8HLL0+4RAjLYAKDx7fTI1VLmaqoMms/G4TPpUFA9ywCg087+ UJLmVVexJMVbuhHB3ASi46I= =iPX7 -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Wed Oct 05 06:14:58 2005 Message-ID: <4343A7A9.2050703@csail.mit.edu> Date: Wed, 05 Oct 2005 06:15:05 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Scot Frank Subject: Re: week 5 References: <43438DD7.3030903@mit.edu> In-Reply-To: <43438DD7.3030903@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 1098 Status: RO X-UID: 554 X-Keywords: two vertices are connected iff there is a path that begins at one and ends at the other. if there's any path, there will also be a simple one. I don't know what the fault was in the sex statistics -- I just know that the claim that in the averages for the whole population, men have 74% more partners than women, is impossible. One possible explanation is that the men exaggerated their number of partners (and the researchers mistakenly believed them). Another explanation is that the researchers tested a sample of men and women, presumably including all the partners of everybody in the sample, and wound up with 74% more women in the sample. regards, A. P.S. Remember to put your TA's name in the subject. Scot Frank wrote: > It was unclear at my table during last lecture whether the fault of the > sex statistics had to do with people mis-reporting, or the statistical > analysis. From page 5, and lecture, I'm still confused on what we are > using for the term of connectedness, and from the in-class problem are > we saying it is connected if there is any simple path? From medrano@MIT.EDU Wed Oct 5 06:40:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Ae1w0026621 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 06:40:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95AdxPG015695 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 06:39:59 -0400 (EDT) Received: from grumpy-fuzzball.mit.edu (GRUMPY-FUZZBALL.MIT.EDU [18.7.16.79]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95AdvbP028997 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 06:39:57 -0400 (EDT) Received: (from medrano@localhost) by grumpy-fuzzball.mit.edu (8.12.9) id j95AdvWp004207; Wed, 5 Oct 2005 06:39:57 -0400 (EDT) Date: Wed, 5 Oct 2005 06:39:56 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]comments on reading Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 555 Content-Length: 257 X-Keywords: NonJunk I'm am a bit confused about coloring. I thought that any map of the USA (a graph) is colorable with at least 4 colors, but theorem 4.1 says that a graph with maximum degree at most k is (k+1)-colorable. Can it be less than (k+1) colorable? Jesus Medrano From nedzel@MIT.EDU Wed Oct 5 07:06:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95B6Cw0029327 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 07:06:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95B68PG026114 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 07:06:08 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SEVENTY-SEVEN.MIT.EDU [18.96.5.77]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95B64Qj000616 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 07:06:04 -0400 (EDT) Message-Id: <200510051106.j95B64Qj000616@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 07:06:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.272 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 556 Content-Length: 72 X-Keywords: NonJunk In section 5.1, what is the significance of planer embedding? - David From mdmurray@MIT.EDU Wed Oct 5 08:54:09 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Cs9w0011662 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 08:54:09 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Cs9Zq027521 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 08:54:09 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95Cs1ok005202 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 08:54:02 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-7--114245286 From: Michael Murray Subject: week 5 Date: Wed, 5 Oct 2005 08:53:29 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.098 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 557 Content-Length: 8744 X-Status: A X-Keywords: --Apple-Mail-7--114245286 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed On pg. 13: " Otherwise, G has at least one cycle. Select a spanning tree and an =20 edge u=E2=80=94v in the cycle, but not in the tree. (The spanning tree can not contain all edges in =20 the cycle, since trees are acyclic.) Removing u=E2=80=94v merges the two faces on either side = of =20 the edge and leaves a graph G0 with only e edges and some number of vertices v and faces =20 f. Graph G0 is connected, because there is a path between every pair of vertices =20 within the spanning tree. So v=E2=88=92e+f =3D 2 by the induction assumption P(e). Thus, the = original =20 graph G had v vertices, e+1 edges, and f +1 faces. Since v=E2=88=92(e+1)+(f +1) =3D v=E2=88=92e+f = =3D 2, P(e=20 +1) is again true. Otherwise, G has at least one cycle. Select a spanning tree and an =20 edge u=E2=80=94v in the cycle, but not in the tree. (The spanning tree can not contain all edges in =20 the cycle, since trees are acyclic.) Removing u=E2=80=94v merges the two faces on either side = of =20 the edge and leaves a graph G0 with only e edges and some number of vertices v and faces =20 f. Graph G0 is connected, because there is a path between every pair of vertices =20 within the spanning tree. So v=E2=88=92e+f =3D 2 by the induction assumption P(e). Thus, the = original =20 graph G had v vertices, e+1 edges, and f +1 faces. Since v=E2=88=92(e+1)+(f +1) =3D v=E2=88=92e+f = =3D 2, P(e=20 +1) is again true. " I did not understand this part of the proof or Euler's formula. Why =20 do we take a spanning tree and a cycle not in the tree? How do we =20 argue from there? This proof confused me. Michael Murray= --Apple-Mail-7--114245286 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
On pg. 13:

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


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

Michael = Murray
= --Apple-Mail-7--114245286-- From a_lopez@MIT.EDU Wed Oct 5 08:57:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Cvnw0012202 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 08:57:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Cvmcq024959 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 08:57:48 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95CvkOw019949 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 08:57:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j95CvkpG002479; Wed, 5 Oct 2005 08:57:46 -0400 Received: from MDMURRAY.MIT.EDU (MDMURRAY.MIT.EDU [18.234.0.116]) (User authenticated as a_lopez@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 08:57:46 -0400 Message-ID: <20051005085746.ikssu33okdmskc8w@webmail.mit.edu> Date: Wed, 5 Oct 2005 08:57:46 -0400 From: a_lopez@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: week 5 readings MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 1.031 X-Spam-Level: * (1.031) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 558 Content-Length: 642 X-Status: A X-Keywords: NonJunk "Suppose the graph is drawn in the plane with f faces. Every edge on a cycle in the graph borders two faces and is traversed once by each of the two cycles bounding these two faces, and not by any other face boundary. If an edge touches only one face, then it is traversed twice by the cycle that borders the face (one ?going? and the other ?coming?), and not by any other face boundary. So the total number of traversals of any edge by the face boundaries is exactly two." pg. 13 I did not understand this proof at all. I got lost after the second sentence. I've read it a couple of times but I still don't understand it. Adriana Lopez From kromer@MIT.EDU Wed Oct 5 09:14:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DEUw0014873 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:14:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DETcq006943 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:14:29 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DEQ7D025332 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:14:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j95DEQKN009922; Wed, 5 Oct 2005 09:14:26 -0400 Received: from BURTON-TWO-O-TWO.MIT.EDU (BURTON-TWO-O-TWO.MIT.EDU [18.247.5.202]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:14:26 -0400 Message-ID: <20051005091426.v8b03pef1lccoowo@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:14:26 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 559 Content-Length: 423 X-Keywords: NonJunk p. 15 "Checking the finitely-many cases that remain turns up five solutions. For each valid combination of n and m, we can compute the associated number of vertices v, edges e, and faces f. And polyhedra with these properties do actually exist." I found it surprising that there are only 5 regular polyhedra, and that all the regular polyhedra computed to be possible with Euler's formula actually exist. Katherine Romer From hejing85@MIT.EDU Wed Oct 5 09:14:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DElw0014890 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:14:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DEkcq007212 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:14:46 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DEeR0025402 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:14:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j95DEdu9003251; Wed, 5 Oct 2005 09:14:39 -0400 Received: from NEW-ONE-O-SEVEN.MIT.EDU (NEW-ONE-O-SEVEN.MIT.EDU [18.241.5.107]) (User authenticated as hejing85@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:14:39 -0400 Message-ID: <20051005091439.yfo57e4xc7k84ogs@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:14:39 -0400 From: Jing He To: 6042-probs@theory.csail.MIT.EDU Subject: [David] week 5 comments MIME-Version: 1.0 Content-Type: text/plain; charset=GB2312 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 560 Content-Length: 102 X-Keywords: NonJunk The part of the readings that talked about Euler's Formula and the polyhedra was interesting. --Jing From jeffhoff@MIT.EDU Wed Oct 5 09:16:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DGPw0015036 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 09:16:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DGOcq008647 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 09:16:24 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DG4V7025932 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 09:16:05 -0400 (EDT) Message-Id: <5.2.1.1.2.20051005090947.04f56990@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 05 Oct 2005 09:18:06 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 5 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 2.544 X-Spam-Level: ** (2.544) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 561 Content-Length: 255 X-Keywords: NonJunk Page 9 - Section 4 Coloring graphs I understand coloring graphs and sort of how to do them. Is there a trick or something to doing them easier instead of "brute force"? Can we go over it more in class? Group 3 / C Jeffrey D. Hoff jeffhoff@mit.edu From manosdefierro@gmail.com Wed Oct 5 09:18:14 2005 Return-Path: Received: from wproxy.gmail.com (wproxy.gmail.com [64.233.184.196]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DIDw0015123 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:18:13 -0400 Received: by wproxy.gmail.com with SMTP id i16so64297wra for <6042-probs@theory.lcs.mit.edu>; Wed, 05 Oct 2005 06:18:08 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=P6st3JL5FiJbWXwcDfYxLL6WtWQZFlYRi2ufPb5spfj6mO0/u31TjuucuoGFehmpzY0dtCRR7c1jnw229ho9crtx5ze9815Myl+GcOJEiAxe4y/Ad7Yt6g91EwrRYLrep/ombAnZuHjeE8/1WSXik1DnYbJ5SJ3hzwAu2J6zugk= Received: by 10.54.125.17 with SMTP id x17mr361464wrc; Wed, 05 Oct 2005 06:18:08 -0700 (PDT) Received: by 10.54.151.13 with HTTP; Wed, 5 Oct 2005 06:18:08 -0700 (PDT) Message-ID: <2bac7c5d0510050618j193063fctbed2df1b8f902008@mail.gmail.com> Date: Wed, 5 Oct 2005 09:18:08 -0400 From: Mario Marrufo Reply-To: joeyrufo@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: required e-mail comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_12635_14485148.1128518288571" Status: RO X-UID: 562 Content-Length: 1079 X-Keywords: NonJunk ------=_Part_12635_14485148.1128518288571 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found this week's reading assignment much easier to digest than last week's assignment. That said, there may be some stuff I've missed. For example, I'm not sure I remember the precise definition of what a "degree" is (mentioned in section 1, pages 1 and 2). Another section I wouldn't mind if was covered in lecture would be section 5.2 (page 13). ------=_Part_12635_14485148.1128518288571 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found this week's reading assignment much easier to digest than last week's assignment. That said, there may be some stuff I've missed. For example, I'm not sure I remember the precise definition of what a "degree" is (mentioned in section 1, pages 1 and 2). Another sect= ion I wouldn't mind if was covered in lecture would be section 5.2 (page 13). ------=_Part_12635_14485148.1128518288571-- From sil_03@MIT.EDU Wed Oct 5 09:32:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DWow0017108 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:32:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DWncq022786 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:32:49 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DWgIw002313 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:32:42 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j95DWgKK028715; Wed, 5 Oct 2005 09:32:42 -0400 Received: from NEW-TWO-THIRTY-EIGHT.MIT.EDU (NEW-TWO-THIRTY-EIGHT.MIT.EDU [18.241.5.238]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:32:42 -0400 Message-ID: <20051005093242.j5pudxwhaa0444cs@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:32:42 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 563 Content-Length: 166 X-Keywords: I thought the section on classifying plyhedra on page 14-15 was interesting, but I would like to know what other kinds of problems you can solve using faces. Silvia From zacharyozer@gmail.com Wed Oct 5 09:35:05 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.205]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DZ4w0017316 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:35:04 -0400 Received: by xproxy.gmail.com with SMTP id s8so100673wxc for <6042-probs@theory.lcs.mit.edu>; Wed, 05 Oct 2005 06:34:59 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=EIH8xV8HTuUj+zVkot9G7Or7KwEkp1vTSJ/gTJSZzekbaOXGvbPQw/LVLpEjFmDOesOfzEKiiaPNYhx7BrAg7KXJatg3Bkpg25AksHqA2wrLdzmrJPKwgefZ+hITzQ7jYjizUdTKKZ++8OeayKp5B1Vf3B2Bs2P1vI4LC+qIEzM= Received: by 10.70.43.2 with SMTP id q2mr404859wxq; Wed, 05 Oct 2005 06:34:59 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Wed, 5 Oct 2005 06:34:59 -0700 (PDT) Message-ID: <4f2613e50510050634pfe780d5o774a3a60dbf3dc6d@mail.gmail.com> Date: Wed, 5 Oct 2005 09:34:59 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j95DZ4w0017316 Status: RO X-UID: 564 Content-Length: 403 X-Keywords: My question is about k-colorable graphs: In 6.034 we learned that any map (which can be represented as a graph) could be colored with at most 4 colors. There was no rigid proof, although, various demonstrations led us to beleive that this was true. Why then is the upper limit on chromatic number k+1 for a graph with maximum degree k, and not k+1 for graphs less than 3, and 4 for all others? -zozer From lmccart@MIT.EDU Wed Oct 5 09:43:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dhlw0017786 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:43:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Dhkcq002507 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:43:46 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DhifI007547 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:43:44 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j95Dhiea007575; Wed, 5 Oct 2005 09:43:44 -0400 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:43:44 -0400 Message-ID: <20051005094344.hlc0xsqogey8o8c8@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:43:44 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] other question MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 565 Content-Length: 881 X-Keywords: NonJunk I keep reading these emails about normalizing scores for psets over tas, but I just don't really understand what is meant by that. I'm confused why when I do my psets with my friend and we have essentially the same answers, he gets 35 while I get a 23. There was one problem where I referenced a proof we did in class and got 1/10 while he got 9/10 for doing the same thing. I don't understand why you guys don't talk about acceptable answers before grading psets. I find everything in this class to be pretty ambiguous. It's always unclear to me what has been proven and what has not and what we're allowed to assume. Also, it's even more difficult for me when the professor puts up incorrect information on slides and frequently makes mistakes while explaining things. It seems like with some preplanning many of thse problems could be avoided. Thank you, Lauren McCarthy From miki_tnd@MIT.EDU Wed Oct 5 09:47:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dlhw0017970 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:47:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Dlfcq006084 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:47:42 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95DlZhG009320 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 09:47:35 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j95DlZ4u005794; Wed, 5 Oct 2005 09:47:35 -0400 Received: from PIKA-SEVENTY-NINE.MIT.EDU (PIKA-SEVENTY-NINE.MIT.EDU [18.214.1.79]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:47:35 -0400 Message-ID: <20051005094735.t6qyyb5cq4e84s44@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:47:35 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: (sayan) reading comment 5 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 2.727 X-Spam-Level: ** (2.727) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 566 Content-Length: 167 X-Keywords: NonJunk I've always found the proof of Euler's classification of polyhedrons unclear. On pg.14 of the reading, it would be nice if that was explained by someone in person. From nancyk@MIT.EDU Wed Oct 5 09:47:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dlmw0017975 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 09:47:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Dllcq006140; Wed, 5 Oct 2005 09:47:47 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95Dler9009343; Wed, 5 Oct 2005 09:47:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j95DlehD009762; Wed, 5 Oct 2005 09:47:40 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 5 Oct 2005 09:47:40 -0400 Message-ID: <20051005094740.2nrmctpbgy0owsss@webmail.mit.edu> Date: Wed, 5 Oct 2005 09:47:40 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 5 reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 567 Content-Length: 476 X-Keywords: NonJunk Hi, On page 12, under section 5.1 "Euler's Formula," the last sentence of the first paragraph says that only one point (a single, common endpoint) can appear on two of the curves. Since each vertex must be assigned to a distinct point in the plane, it seems like that last sentence should specify that the "single, common endpoint" must be the vertex from which the edge started, since the ending point cannot have 2 or more edges incident to it. Is this right? Nancy Keuss From veracarr@MIT.EDU Wed Oct 5 09:50:34 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95DoYw0018183 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:50:34 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95DoXZq029659 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:50:33 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95DoQok008931 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 09:50:26 -0400 (EDT) Message-ID: <4343F680.8090600@mit.edu> Date: Wed, 05 Oct 2005 09:51:28 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Graphs Reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 568 Content-Length: 271 X-Keywords: NonJunk Planar Graphs (pg 11) Is there an easy way to determine whether the graph can be planer. They gave two examples where if one edge was to be removed that it would make the graph planar. I understand the definitions but is there an easy way to determine this property? From mitras@drake.csail.mit.edu Wed Oct 5 09:55:46 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dtjw0019348; Wed, 5 Oct 2005 09:55:46 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j95Dtj1P009188; Wed, 5 Oct 2005 09:55:45 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j95Dtj1g009187; Wed, 5 Oct 2005 09:55:45 -0400 Date: Wed, 5 Oct 2005 09:55:45 -0400 From: Sayan Mitra Message-Id: <200510051355.j95Dtj1g009187@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, nedzel@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Cc: 6042-staff@theory.csail.mit.edu Status: RO X-UID: 569 Content-Length: 1215 X-Keywords: NonJunk I presume you are asking about the significance of the term "planar embedding" here and not the class of planar graphs. An embedding is a representation of a topological object, manifold, graph, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. A graph embedding is a particular drawing of a graph (with sometimes added constraint that the embedding be planar, i.e., has no crossing edges). While the underlying object is independent of the embedding, a clever choice of embedding can lead to particularly illuminating diagrams. I took these definitions from Mathworld. -S. >From nedzel@MIT.EDU Wed Oct 5 07:06:12 2005 From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 07:06:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.272 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 In section 5.1, what is the significance of planer embedding? - David From mitras@drake.csail.mit.edu Wed Oct 5 10:00:12 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95E0Cw0019620; Wed, 5 Oct 2005 10:00:12 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j95E0CA3009221; Wed, 5 Oct 2005 10:00:12 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j95E0CGm009220; Wed, 5 Oct 2005 10:00:12 -0400 Date: Wed, 5 Oct 2005 10:00:12 -0400 From: Sayan Mitra Message-Id: <200510051400.j95E0CGm009220@drake.csail.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU, medrano@MIT.EDU Subject: Re: [Sayan]comments on reading Cc: 6042-probs@theory.csail.mit.edu Status: RO X-UID: 570 Content-Length: 1048 X-Keywords: NonJunk Yes. Consider a star graph with a central vertex v with k adjacent vertices, and all these other (not v) vertices are adjacent to v only. This graph has max degree k, but it is 2 colorable. In general, you have seen from the lecture notes that a tree is always 2 colorable irrespective of the maximum degree. -S. >From medrano@MIT.EDU Wed Oct 5 06:40:01 2005 Date: Wed, 5 Oct 2005 06:39:56 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]comments on reading MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 X-UID: 555 Content-Length: 257 X-Keywords: I'm am a bit confused about coloring. I thought that any map of the USA (a graph) is colorable with at least 4 colors, but theorem 4.1 says that a graph with maximum degree at most k is (k+1)-colorable. Can it be less than (k+1) colorable? Jesus Medrano From cvnguyen@MIT.EDU Wed Oct 5 10:16:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EGWw0022983 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:16:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EGVcq002399 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:16:31 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-ONE-O-FOUR-HUNDRED-TWENTY.MIT.EDU [18.238.7.153]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EGNAx022038 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:16:23 -0400 (EDT) Message-ID: <001901c5c9b7$5ece8c80$9907ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 5 comments Date: Wed, 5 Oct 2005 10:16:24 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MIMEOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -3.142 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 571 Content-Length: 296 X-Keywords: NonJunk (page 12, section 5.1) I found the discussion of Euler's formula particularly interesting. I remember reading somewhere a few years ago that there were at least fifteen different proofs of it because it was such a profound result that people sought alternate elegant proofs. --Chieu Nguyen From mrivas03@MIT.EDU Wed Oct 5 10:38:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EcHw0027601 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:38:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EcGcq022707 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:38:16 -0400 (EDT) Received: from el-ternero.mit.edu (TDCIP95.MIT.EDU [18.237.0.95]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EcCXS001947 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 10:38:14 -0400 (EDT) Message-Id: <6.2.3.4.2.20051005103051.02e6aaf8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 05 Oct 2005 10:33:54 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: [Sayan] Week 5 Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 572 Content-Length: 206 X-Keywords: NonJunk pg 12. Section Euler's Formula. A rerun of Euler's Formula and creating Polyhedra would be helpful if included during lecture: It's proof by induction and how to catalog polyhedra. Thanks, Manuel Rivas From vbrobbey@MIT.EDU Wed Oct 5 10:42:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EgZw0028060 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 10:42:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EgYcq026886 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 10:42:34 -0400 (EDT) Received: from newhouse-6.mit.edu (NEWHOUSE-6.MIT.EDU [18.241.2.253]) (authenticated bits=56) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EgUow004217 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 10:42:31 -0400 (EDT) Received: (from vbrobbey@localhost) by newhouse-6.mit.edu (8.12.9) id j95EgUr0003197; Wed, 5 Oct 2005 10:42:30 -0400 Subject: [David] Week 5 Comments From: Valery K Brobbey To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 05 Oct 2005 10:42:30 -0400 Message-Id: <1128523350.3022.8.camel@newhouse-6.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: -0.529 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 573 Content-Length: 621 X-Keywords: NonJunk What the University of Chicago study probably meant was that 74% of the time it's men to look for opposite gender partners. Even by common sense their assertion that men have on average 74% more opposite gender partners is ridiculous. I was impressed about how the idea of colouring a graph can be used to solve problems such as scheduling. I guess the housing department uses Hall's Marriage Theorem, or something similar to assign houses to freshmen. Of course this is more complicated because you have to think of preferences. I understood the concept of planar graphs but I got lost in Classifying Polyhedra section. From rian@MIT.EDU Wed Oct 5 10:58:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95EwOw0031299 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 10:58:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95EwMcq012681 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 10:58:22 -0400 (EDT) Received: from m12-182-9.mit.edu (M12-182-9.MIT.EDU [18.19.0.40]) (authenticated bits=56) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95EwJjT011603 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 10:58:20 -0400 (EDT) Received: (from rian@localhost) by m12-182-9.mit.edu (8.12.9) id j95EwJQ6029727; Wed, 5 Oct 2005 10:58:19 -0400 Subject: (jelani) required reading comments From: rian To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Wed, 05 Oct 2005 10:58:19 -0400 Message-Id: <1128524299.28520.2.camel@m12-182-9.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 3.64 X-Spam-Level: *** (3.64) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 574 Content-Length: 212 X-Keywords: NonJunk i felt that i had a good grasp of the material in this week's reading. nothing was really surprising or difficult, and i'm impartial to anything being discussed more fully in class honestly. thanks. Rian Hunter From meyer@imap.theory.csail.mit.edu Wed Oct 05 11:04:49 2005 Message-ID: <4343EB96.6050802@csail.mit.edu> Date: Wed, 05 Oct 2005 11:04:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: a_lopez@MIT.EDU Subject: Re: week 5 readings References: <20051005085746.ikssu33okdmskc8w@webmail.mit.edu> In-Reply-To: <20051005085746.ikssu33okdmskc8w@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 854 X-UID: 575 X-Keywords: Take a look at the animation for this (link on course main page, also in the Appendix for Week 5 Notes). Otherwise, wait till Fri lecture. regards, A a_lopez@MIT.EDU wrote: > "Suppose the graph is drawn in the plane with f faces. Every edge on a cycle in > the graph > borders two faces and is traversed once by each of the two cycles bounding these > two faces, and > not by any other face boundary. If an edge touches only one face, then it is > traversed twice by > the cycle that borders the face (one ?going? and the other ?coming?), > and not by any other face > boundary. So the total number of traversals of any edge by the face boundaries > is exactly two." > > pg. 13 > > I did not understand this proof at all. I got lost after the second sentence. > I've read it a couple of times but I still don't understand it. > > Adriana Lopez From meyer@imap.theory.csail.mit.edu Wed Oct 05 11:08:07 2005 Message-ID: <4343EC5F.9050003@csail.mit.edu> Date: Wed, 05 Oct 2005 11:08:15 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Michael Murray Subject: Re: week 5 References: In-Reply-To: Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 1793 X-UID: 576 X-Keywords: Yeah, I don't care for it either, but doing it more carefully, as in the Appendix, may not be easier to understand. looking at the planar drawing animation (link on the course main page) may help. regards, A. Michael Murray wrote: > On pg. 13: > > " Otherwise, G has at least one cycle. Select a spanning tree and an > edge u—v in the cycle, > but not in the tree. (The spanning tree can not contain all edges in the > cycle, since trees > are acyclic.) Removing u—v merges the two faces on either side of the > edge and leaves > a graph G0 with only e edges and some number of vertices v and faces f. > Graph G0 is > connected, because there is a path between every pair of vertices within > the spanning tree. > So v−e+f = 2 by the induction assumption P(e). Thus, the original graph > G had v vertices, > e+1 edges, and f +1 faces. Since v−(e+1)+(f +1) = v−e+f = 2, P(e+1) is > again true. > Otherwise, G has at least one cycle. Select a spanning tree and an edge > u—v in the cycle, > but not in the tree. (The spanning tree can not contain all edges in the > cycle, since trees > are acyclic.) Removing u—v merges the two faces on either side of the > edge and leaves > a graph G0 with only e edges and some number of vertices v and faces f. > Graph G0 is > connected, because there is a path between every pair of vertices within > the spanning tree. > So v−e+f = 2 by the induction assumption P(e). Thus, the original graph > G had v vertices, > e+1 edges, and f +1 faces. Since v−(e+1)+(f +1) = v−e+f = 2, P(e+1) is > again true. " > > > I did not understand this part of the proof or Euler's formula. Why do > we take a spanning tree and a cycle not in the tree? How do we argue > from there? This proof confused me. > > Michael Murray From meyer@imap.theory.csail.mit.edu Wed Oct 05 09:55:46 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Dtjw0019348; Wed, 5 Oct 2005 09:55:46 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j95Dtj1P009188; Wed, 5 Oct 2005 09:55:45 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j95Dtj1g009187; Wed, 5 Oct 2005 09:55:45 -0400 Date: Wed, 5 Oct 2005 09:55:45 -0400 From: Sayan Mitra Message-Id: <200510051355.j95Dtj1g009187@drake.csail.mit.edu> To: 6042-probs@theory.csail.mit.edu, nedzel@MIT.EDU Subject: Re: [Sayan] Week 5 Comments Cc: 6042-staff@theory.csail.mit.edu Status: RO Content-Length: 1215 X-UID: 577 X-Keywords: NonJunk I presume you are asking about the significance of the term "planar embedding" here and not the class of planar graphs. An embedding is a representation of a topological object, manifold, graph, etc. in a certain space in such a way that its connectivity or algebraic properties are preserved. A graph embedding is a particular drawing of a graph (with sometimes added constraint that the embedding be planar, i.e., has no crossing edges). While the underlying object is independent of the embedding, a clever choice of embedding can lead to particularly illuminating diagrams. I took these definitions from Mathworld. -S. >From nedzel@MIT.EDU Wed Oct 5 07:06:12 2005 From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 5 Comments Date: Wed, 5 Oct 2005 07:06:00 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.272 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 In section 5.1, what is the significance of planer embedding? - David From meyer@imap.theory.csail.mit.edu Wed Oct 05 00:48:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j954lww0010245 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:47:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j954lv6s003849 for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:47:57 -0400 (EDT) Received: from [18.240.5.57] (MCCORMICK-FIFTY-SEVEN.MIT.EDU [18.240.5.57]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j954lnN2009799 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6.042-online-tutor@theory.csail.mit.edu>; Wed, 5 Oct 2005 00:47:50 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <0e70597c2530cc20e602a3471ad75a49@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6.042-online-tutor@theory.csail.mit.edu From: Rachel Shearer Subject: [Jelani] Week 5 Comment Date: Wed, 5 Oct 2005 00:47:48 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -2.099 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 288 X-UID: 578 X-Keywords: NonJunk The section on Euler's Formula gave me a little bit of difficulty. It's not that I don't understand it when I'm reading it, it's just that I don't think I'd be able to explain it to anyone else...if that makes any sense at all. I think listening to lecture will help though. Rachel From kkdb@MIT.EDU Wed Oct 5 11:23:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95FNUw0001780 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 11:23:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95FEw7p026862 for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 11:23:29 -0400 (EDT) Received: from w20-575-52.mit.edu (W20-575-52.MIT.EDU [18.187.0.71]) (authenticated bits=56) (User authenticated as kkdb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95F6UY1015314 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 5 Oct 2005 11:06:31 -0400 (EDT) Received: (from kkdb@localhost) by w20-575-52.mit.edu (8.12.9) id j95F6Ufv028763; Wed, 5 Oct 2005 11:06:30 -0400 (EDT) Subject: David: Email Comments for Graphs From: Kaustuv De Biswas To: 6042-probs@theory.csail.MIT.EDU Content-Type: multipart/alternative; boundary="=-wIewF5uZghRLutvtfPYl" Date: Wed, 05 Oct 2005 11:06:29 -0400 Message-Id: <1128524789.28702.5.camel@w20-575-52.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.077 X-Spam-Level: * (1.077) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 579 Content-Length: 918 X-Keywords: NonJunk --=-wIewF5uZghRLutvtfPYl Content-Type: text/plain Content-Transfer-Encoding: 7bit The Induction proof for K-Coloring of Graph was a little unclear. p(1) says 1 vertex graph is 1-colorable and then you are assuming p(n) as n-vertex graph is at most k-colorable. the relationship between n and k are not very clear in the proof. Kaustuv. --=-wIewF5uZghRLutvtfPYl Content-Type: text/html; charset=utf-8 The Induction proof for K-Coloring of Graph was a little unclear. p(1) says 1 vertex graph is 1-colorable and then you are assuming p(n) as n-vertex graph is at most k-colorable. the relationship between n and k are not very clear in the proof.
Kaustuv. --=-wIewF5uZghRLutvtfPYl-- From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:09:54 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95H9sw0028108; Wed, 5 Oct 2005 13:09:54 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95H9s0h026774; Wed, 5 Oct 2005 13:09:54 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95H9sxM026771; Wed, 5 Oct 2005 13:09:54 -0400 Date: Wed, 5 Oct 2005 13:09:53 -0400 (EDT) From: Hanson Zhou To: Karena Tyan cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Reading 5 comments In-Reply-To: <20051004193124.wo2hpl958jcwc84o@webmail.mit.edu> Message-ID: References: <20051004193124.wo2hpl958jcwc84o@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 580 Content-Length: 1337 X-Keywords: k-connected here refers to the connectivity of the graph, not between a specific pair. While it is true that deleting those three edges will not disconnect B and E, it DOES disconnect D from the rest of the graph. -Hanson On Tue, 4 Oct 2005, Karena Tyan wrote: > I found the part about k-connected vertices most confusing. Namely, the passage > on page 5: > > Definition 2.6. Two vertices in a graph are k-connected if they remain connected > in any subgraph obtained by deleting k-1 edges. A graph is k-connected if > every pair of its vertices are k-connected. > > How do you know what "k" is? What I mean to say is - to test for > k-connectivity, you have to delete k-1 edges and see if the two vertices in > question are still conected. I assume that this refers just to the edges that > exist along some path between the two vertices, but I still don't quite > understand how "k" is derived. Like, in Figure 1 on page 3, they say that B > and E are 2-connected, which means you could delete 1 edge between the two > vertices and still be connected, correct? But couldn't you delete the three > edges (B-C) (C-D) and (D-E) and still be connected? > > I'm sorry, this was very long. I'm just confused over the definition of > k-connectivity. > > - Karena > > -- > 410 Memorial Drive > Cambridge, MA 02139 > (585)957-5923 > From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:17:33 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HHWw0028579; Wed, 5 Oct 2005 13:17:32 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HHWwb026814; Wed, 5 Oct 2005 13:17:32 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HHWUs026811; Wed, 5 Oct 2005 13:17:32 -0400 Date: Wed, 5 Oct 2005 13:17:32 -0400 (EDT) From: Hanson Zhou To: Lohith G Kini cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Difficulty with reading lecture notes for week 5 In-Reply-To: <20051005000957.osf3cx3xuhog400g@webmail.mit.edu> Message-ID: References: <20051005000957.osf3cx3xuhog400g@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 581 Content-Length: 621 X-Keywords: Ok...it's a bit tricky. Go to office hours to get this cleared up. Essentially, we are counting edges as we traverse the boundary of a face. Since each edge is traversed twice(once for each face that it bounds), the total is 2e, and this is at least 3f since each face has at least 3 bounding edges. -Hanson On Wed, 5 Oct 2005, Lohith G Kini wrote: > Hi Hanson, > > I am having trouble understanding the proof for lemma 5.2 on page 13 of the > lecture notes. I understand the mathematics that leads off from statement (1) > but I don't understand the explanation behind the original inequality 2e >= 3f. > > Lohith > From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:22:52 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HMqw0029991; Wed, 5 Oct 2005 13:22:52 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HMq2O026846; Wed, 5 Oct 2005 13:22:52 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HMqwR026843; Wed, 5 Oct 2005 13:22:52 -0400 Date: Wed, 5 Oct 2005 13:22:52 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 5 reading assignment In-Reply-To: <20051005094740.2nrmctpbgy0owsss@webmail.mit.edu> Message-ID: References: <20051005094740.2nrmctpbgy0owsss@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 582 Content-Length: 730 X-Keywords: Not sure what technicality you are referring to...it is simply that two edges drawn in the plane may not cross except at possibly the common vertex to which they are both incident. -Hanson On Wed, 5 Oct 2005, Nancy L Keuss wrote: > Hi, > > On page 12, under section 5.1 "Euler's Formula," the last sentence of the first > paragraph says that only one point (a single, common endpoint) can appear on > two of the curves. Since each vertex must be assigned to a distinct point in > the plane, it seems like that last sentence should specify that the "single, > common endpoint" must be the vertex from which the edge started, since the > ending point cannot have 2 or more edges incident to it. Is this right? > > Nancy Keuss > From scholtz@MIT.EDU Wed Oct 5 13:39:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HdLw0032089 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 13:39:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95HdK6W013510 for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 13:39:20 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95HdGQ1019059 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 5 Oct 2005 13:39:16 -0400 (EDT) Message-Id: <200510051739.j95HdGQ1019059@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 5 Comments Date: Wed, 5 Oct 2005 13:39:17 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C9B2.2F02C700" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXJ07Vh2SxhJZJWQTWnRbkn6vWKOA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.856 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 583 Content-Length: 1714 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C9B2.2F02C700 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I would like to have Hall's Marriage Theorem (p 15-16) discussed more fully in lecture. ------=_NextPart_000_0000_01C5C9B2.2F02C700 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

------=_NextPart_000_0000_01C5C9B2.2F02C700-- From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:39:59 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95Hdww0032237; Wed, 5 Oct 2005 13:39:59 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HdwYv026938; Wed, 5 Oct 2005 13:39:58 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HdwiU026935; Wed, 5 Oct 2005 13:39:58 -0400 Date: Wed, 5 Oct 2005 13:39:58 -0400 (EDT) From: Hanson Zhou To: Vera Carr cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Graphs Reading In-Reply-To: <4343F680.8090600@mit.edu> Message-ID: References: <4343F680.8090600@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 584 Content-Length: 632 X-Keywords: Good question. Somewhat beyond the scope of this course though. Kuratowski's theorem characterizes planar graphs as those without a K3,3 or a K5 minor, and there might be algorithms to test planarity along those lines. I am not aware of them though it seems like there would be research done on this. -Hanson On Wed, 5 Oct 2005, Vera Carr wrote: > Planar Graphs (pg 11) > > Is there an easy way to determine whether the graph can be planer. They > gave two examples where if one edge was to be removed that it would make > the graph planar. I understand the definitions but is there an easy way > to determine this property? > From cbossard@MIT.EDU Wed Oct 5 13:46:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HkBw0001462 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 13:46:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95Hk96W020094; Wed, 5 Oct 2005 13:46:09 -0400 (EDT) Received: from w20-575-48.mit.edu (W20-575-48.MIT.EDU [18.187.0.67]) (authenticated bits=56) (User authenticated as cbossard@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j95Hk1GM021748 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 5 Oct 2005 13:46:02 -0400 (EDT) Received: (from cbossard@localhost) by w20-575-48.mit.edu (8.12.9) id j95Hk1Nu024949; Wed, 5 Oct 2005 13:46:01 -0400 (EDT) Subject: [David] Week 5 reading comments From: Cynthia C Bossard To: 6042-probs@theory.lcs.MIT.EDU Content-Type: multipart/alternative; boundary="=-voWHfZDfmwBAnk2MDAn3" Date: Wed, 05 Oct 2005 13:46:01 -0400 Message-Id: <1128534361.24876.2.camel@w20-575-48.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.129 X-Spam-Level: * (1.129) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 585 Content-Length: 838 X-Keywords: NonJunk --=-voWHfZDfmwBAnk2MDAn3 Content-Type: text/plain Content-Transfer-Encoding: 7bit Sections 5.1 and 5.2 I am having some trouble understanding the proofs, but I think I get the concept Cynthia Sorry about the time for some reason I looked at the wrong date and thought this was due Friday. --=-voWHfZDfmwBAnk2MDAn3 Content-Type: text/html; charset=utf-8 Sections 5.1 and 5.2
I am having some trouble understanding the proofs, but I think I get the concept
Cynthia

Sorry about the time for some reason I looked at the wrong date and thought this was due Friday. --=-voWHfZDfmwBAnk2MDAn3-- From hmzhou@blackbird.csail.mit.edu Wed Oct 5 13:51:15 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95HpFw0001841; Wed, 5 Oct 2005 13:51:15 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j95HpFHZ027005; Wed, 5 Oct 2005 13:51:15 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j95HpFsv027002; Wed, 5 Oct 2005 13:51:15 -0400 Date: Wed, 5 Oct 2005 13:51:15 -0400 (EDT) From: Hanson Zhou To: Hanson Zhou cc: Vera Carr , 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Graphs Reading In-Reply-To: Message-ID: References: <4343F680.8090600@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 586 Content-Length: 804 X-Keywords: See Hopcroft and Tarjan, "Efficient Planarity Testing", 1974...and possibly many others. -Hanson On Wed, 5 Oct 2005, Hanson Zhou wrote: > Good question. Somewhat beyond the scope of this course though. > Kuratowski's theorem characterizes planar graphs as those without a K3,3 > or a K5 minor, and there might be algorithms to test planarity along those > lines. I am not aware of them though it seems like there would be > research done on this. > > -Hanson > > On Wed, 5 Oct 2005, Vera Carr wrote: > > > Planar Graphs (pg 11) > > > > Is there an easy way to determine whether the graph can be planer. They > > gave two examples where if one edge was to be removed that it would make > > the graph planar. I understand the definitions but is there an easy way > > to determine this property? > > > From dshin@MIT.EDU Wed Oct 5 17:05:51 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95L5pw0006813 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 17:05:51 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95L5nDX019163; Wed, 5 Oct 2005 17:05:49 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95L5jok008960; Wed, 5 Oct 2005 17:05:45 -0400 (EDT) Message-ID: <43444027.9090608@mit.edu> Date: Wed, 05 Oct 2005 17:05:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kaustuv De Biswas CC: 6042-probs@theory.csail.mit.edu Subject: Re: David: Email Comments for Graphs References: <1128524789.28702.5.camel@w20-575-52.mit.edu> In-Reply-To: <1128524789.28702.5.camel@w20-575-52.mit.edu> Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 587 Content-Length: 429 X-Keywords: k is considered fixed. You induct on n. For example, say k = 10. P(1): Every 1-vertex graph is 10-colorable. Is that clear now? Kaustuv De Biswas wrote: > The Induction proof for K-Coloring of Graph was a little unclear. p(1) > says 1 vertex graph is 1-colorable and then you are assuming p(n) as > n-vertex graph is at most k-colorable. the relationship between n and > k are not very clear in the proof. > Kaustuv. From dshin@MIT.EDU Wed Oct 5 17:16:20 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95LGKw0008518 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 5 Oct 2005 17:16:20 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j95LGJZf025407; Wed, 5 Oct 2005 17:16:19 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95LGIc5008023; Wed, 5 Oct 2005 17:16:18 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95LGAok009461; Wed, 5 Oct 2005 17:16:10 -0400 (EDT) Message-ID: <43444298.4020109@mit.edu> Date: Wed, 05 Oct 2005 17:16:08 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Lauren McCarthy CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [david] other question References: <20051005094344.hlc0xsqogey8o8c8@webmail.mit.edu> In-Reply-To: <20051005094344.hlc0xsqogey8o8c8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 588 Content-Length: 1212 X-Keywords: Lauren McCarthy wrote: >I keep reading these emails about normalizing scores for psets over tas, but I >just don't really understand what is meant by that. > Sorry about the confusion - what is meant is that the scores will be normalized at the END of the term, when final grades are computed. The raw scores you receive on your psets are not normalized. The staff did have a meeting and decided we would try harder to coordinate grading rubrics in the future. >I'm confused why when I do >my psets with my friend and we have essentially the same answers, he gets 35 >while I get a 23. There was one problem where I referenced a proof we did in >class and got 1/10 while he got 9/10 for doing the same thing. > If the proof you are referring to is the subset-take-away game from the in-class problems, let me assure you that there is absolutely no way to use what was proved in class to prove problem 4 from pset 2. If it is something else, please bring this problem in to office hours and I will be glad to look it over. If your friend received points he did not deserve, there is not much we can do about that, but if you did not receive points that you deserve, this will be fixed. DS From dshin@MIT.EDU Wed Oct 5 17:27:53 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j95LRrw0011101 for <6042-probs@theory.csail.mit.edu>; Wed, 5 Oct 2005 17:27:53 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j95LRoDX020268; Wed, 5 Oct 2005 17:27:50 -0400 (EDT) Received: from [192.168.1.6] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j95LRjMu009655; Wed, 5 Oct 2005 17:27:46 -0400 (EDT) Message-ID: <4344454F.9000106@mit.edu> Date: Wed, 05 Oct 2005 17:27:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Valery K Brobbey CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 5 Comments References: <1128523350.3022.8.camel@newhouse-6.mit.edu> In-Reply-To: <1128523350.3022.8.camel@newhouse-6.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 589 Content-Length: 1119 X-Keywords: There are some really interesting problems that arise with bipartite matching with preference lists (something you allude to in your comments). In this problem, you have a fully connected bipartite graph, and each vertex has a "preference list" - i.e., a ranking of the vertices on the other side. As you can imagine, this is used in all sorts of applications (perhaps housing freshmen is one of them). By the way, your TA is Hanson, not me. DS Valery K Brobbey wrote: >What the University of Chicago study probably meant was that 74% of the >time it's men to look for opposite gender partners. Even by common sense >their assertion that men have on average 74% more opposite gender >partners is ridiculous. >I was impressed about how the idea of colouring a graph can be used to >solve problems such as scheduling. I guess the housing department uses >Hall's Marriage Theorem, or something similar to assign houses to >freshmen. Of course this is more complicated because you have to think >of preferences. >I understood the concept of planar graphs but I got lost in Classifying >Polyhedra section. > > From mike_a@MIT.EDU Mon Oct 10 15:52:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9AJqdw0003484 for <6042-probs@theory.lcs.mit.edu>; Mon, 10 Oct 2005 15:52:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9AJqcIJ022651 for <6042-probs@theory.lcs.mit.edu>; Mon, 10 Oct 2005 15:52:38 -0400 (EDT) Received: from [18.238.6.106] (EASTCAMPUS-EIGHT-SEVENTY-THREE.MIT.EDU [18.238.6.106]) (authenticated bits=0) (User authenticated as mike_a@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9AJqVCY009139 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 10 Oct 2005 15:52:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <6B542EA4-FBA3-4B14-8D15-ACD34C0AF4C4@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: mike anderson Subject: weekly reading Date: Mon, 10 Oct 2005 15:52:30 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 590 Content-Length: 101 X-Keywords: NonJunk is the reading coming out soon? (i'm trying to schedule my course work for the week) thanks, mike From juang@MIT.EDU Wed Oct 12 19:19:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9CNJd6s032084 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 19:19:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9CNJbk7008911 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 19:19:38 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9CNJaBw019763 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 19:19:36 -0400 (EDT) Message-ID: <434D99FE.8060500@mit.edu> Date: Wed, 12 Oct 2005 19:19:26 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 593 Content-Length: 347 X-Keywords: NonJunk I was surprised at the seemingly obvious vulnerability in Turing's code (on page 16). (Of course, it's obvious _now_...) I would like to see some discussion of why we should be confident that RSA actually works (other than "nobody's found a vulnerability yet"), since a lot of stuff would break were a vulnerability to be found. Jason Juang. From rehughes@MIT.EDU Wed Oct 12 20:39:12 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D0dC6s008643 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:12 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9D0dBxW019097 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:12 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D0dBe0017777 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:11 -0400 (EDT) Received: from [18.244.6.55] (SENIOR-THREE-TEN.MIT.EDU [18.244.6.55]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9D0d7ok028201 for <6042-probs@theory.lcs.mit.edu>; Wed, 12 Oct 2005 20:39:07 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <6a8d0069d7ea59fe3f71e174d7f2db4a@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Richard Hughes Subject: [Jelani] Date: Wed, 12 Oct 2005 20:39:06 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 594 Content-Length: 298 X-Keywords: NonJunk I liked the famous questions in number theory. The fact that every even number less than 10^16 can be written as the sum of two primes is just crazy-amazing. Richard Hughes P.S.: The answer to the last problem in the online tutor is wrong, I'm sure of it. 3780 is definitely not 3^3*5*7^2. From brevzin@MIT.EDU Wed Oct 12 20:43:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D0hs6s009309 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 20:43:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D0hrCc010701 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 20:43:53 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D0hgUl005107 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 20:43:48 -0400 (EDT) Message-Id: <6.1.2.0.1.20051012202630.01af0880@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Wed, 12 Oct 2005 20:42:59 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: Comment to Hanson, regarding my pset Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.058 X-Spam-Level: * (1.058) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 595 Content-Length: 1081 X-Keywords: NonJunk So, on my last pset, I showed that the chain of prime numbers fits the requirement because # primes = pi(x) > x / lnx > x / log2x for all x >= 11 and then showed that for x < 11 the case still holds. Hanson said this was an open problem and was only asymptotically true and gave me 0/3. I want my 3 points back. Here is a graph of pi(x) compared to x / lnx: http://mathworld.wolfram.com/images/eps-gif/PrimePi_1001.gif You can clearly see how the red graph of x /lnx is significantly less than pi(x), and for the lower numbers where it is unclear... x pi(x) x/lnx 2 1 2.88 3 2 2.73 4 2 2.88 5 3 3.11 6 3 3.35 7 4 3.60 8 4 3.85 9 4 4.10 10 4 4.34 11 5 4.59 12 5 4.83 13 6 5.07 14 6 5.30 15 6 5.54 16 6 5.77 17 7 6.00 18 7 6.23 19 8 6.45 20 8 6.68 ... 30 10 8.82 50 15 12.78 ratio: pi(x) / (x / logx) 100 25 21.7 1.15 1000 168 145 1.15 10^4 1229 1086 1.13 10^5 9592 8684 1.10 10^10 455052511,434294482 1.05 etc. While it IS true that pi(x) / (x / logx) --> 1 as x --> infinity, pi(x) remains larger always. Can i have my 3 points back now? Thank you, Barry From rnjacobs@MIT.EDU Wed Oct 12 21:28:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D1SZ6s013952 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 21:28:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D1SVCc015033 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 21:28:31 -0400 (EDT) Received: from m38-370-17.mit.edu (M38-370-17.MIT.EDU [18.107.0.36]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D1SOmq013399 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 21:28:25 -0400 (EDT) Received: (from rnjacobs@localhost) by m38-370-17.mit.edu (8.12.9) id j9D1SOQo029380; Wed, 12 Oct 2005 21:28:24 -0400 (EDT) Message-Id: <200510130128.j9D1SOQo029380@m38-370-17.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] Reading response Date: Wed, 12 Oct 2005 21:28:24 -0400 From: r n jacobs X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 596 Content-Length: 58 X-Keywords: NonJunk Everything this week made sense to me. - Robert Jacobs From icharny@MIT.EDU Wed Oct 12 22:22:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D2Mr6s018792 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 22:22:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D2MqCc026764 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 22:22:52 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D2Mj5L024664 for <6042-probs@theory.csail.MIT.EDU>; Wed, 12 Oct 2005 22:22:45 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9D2Mj1r012334; Wed, 12 Oct 2005 22:22:45 -0400 Received: from 65-78-6-83.c3-0.ned-ubr2.sbo-ned.ma.cable.rcn.com (65-78-6-83.c3-0.ned-ubr2.sbo-ned.ma.cable.rcn.com [65.78.6.83]) (User authenticated as icharny@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 12 Oct 2005 22:22:45 -0400 Message-ID: <20051012222245.4qwrqk3coo6os0og@webmail.mit.edu> Date: Wed, 12 Oct 2005 22:22:45 -0400 From: icharny@MIT.EDU To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] Week 6 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 597 Content-Length: 117 X-Keywords: NonJunk I didn't really understand Turing's 2nd Code (page 18, section 8), and the use of modular encryption. ~Isaac Charny From vixen@MIT.EDU Wed Oct 12 22:45:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D2jY6s021944 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:45:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D2jXCc013660 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:45:33 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D2jTgP029133 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:45:31 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 12 Oct 2005 22:44:37 -0400 To: 6042-probs@theory.csail.mit.edu From: Amanda Seybold Subject: [Jelani] LN6 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.385 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 599 Content-Length: 158 X-Keywords: The explanation of how and why multiplicative inverses work to decrypt messages (pg. 20) went by a little fast. Perhaps in class it could be demonstrated. From ajshafer@MIT.EDU Wed Oct 12 22:56:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D2u46s023222 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:56:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D2tiCc021072 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:55:44 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9D2tbLj001051 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 22:55:37 -0400 (EDT) Message-ID: <000201c5cfa1$98fc4100$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 6 Comments Date: Wed, 12 Oct 2005 22:55:40 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 600 Content-Length: 397 X-Keywords: NonJunk I really liked the section on cryptography. Can we do an example of RSA cryptgraphy in class using two very small primes? Also, how does this relate to elliptic curve cryptography? Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From dshin@MIT.EDU Wed Oct 12 23:38:59 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D3cx6s025656 for <6042-probs@theory.csail.mit.edu>; Wed, 12 Oct 2005 23:38:59 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D3cvCe004644; Wed, 12 Oct 2005 23:38:57 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9D3crok004712; Wed, 12 Oct 2005 23:38:54 -0400 (EDT) Message-ID: <434DD6C3.2020401@mit.edu> Date: Wed, 12 Oct 2005 23:38:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason Juang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 6 Comments References: <434D99FE.8060500@mit.edu> In-Reply-To: <434D99FE.8060500@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 601 Content-Length: 1754 X-Keywords: A very legitimate question, and the lack of justification provided by the mathematical community can be troubling indeed. Right now, our confidence in RSA is pretty much simply, "nobody's cracked it yet". Sadly, the best justification we can give is: 1. If you can factor, you can crack RSA. 2. So, we think cracking RSA is as hard as factoring. (even though the reverse direction of 1 is what really matters) 3. We think factoring is hard. In fact, people HAVE found a slight vulnerability - the RSA function actually "leaks" information. To give you a rough idea of what this vulnerability is, note that if M is a square, so is M^d. Since RSA takes M to M^d (mod n), it "preserves squares" in some sense. So, if you can detect if a number is a square efficiently, you can gain information from M from M^d (mod n). That was a bit hand-wavish, because the real vulnerability involves some pretty advanced number theory. There's something called a Jacobi symbol (a value either +1 or -1), and the Jacobi symbol of M is always equal to the Jacobi symbol of M^d when d is odd. Also, the Jacobi symbol can be computed efficiently. So, by computing the JS of the encrypted message M^d (mod n), you can gain information about M. If you find this stuff interesting, I'd suggest thinking about course like 6.875 (Cryptography) in the future. DS Jason Juang wrote: > I was surprised at the seemingly obvious vulnerability in Turing's > code (on page 16). (Of course, it's obvious _now_...) I would like to > see some discussion of why we should be confident that RSA actually > works (other than "nobody's found a vulnerability yet"), since a lot > of stuff would break were a vulnerability to be found. > > Jason Juang. From iyzhang@MIT.EDU Thu Oct 13 02:21:58 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9D6Lw6s008486 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:58 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9D6Lvo1024183 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:57 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9D6LuOW026020 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:56 -0400 (EDT) Received: from [18.239.6.35] (MACGREGOR-TWO-NINETY.MIT.EDU [18.239.6.35]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9D6LmMu007163 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 02:21:48 -0400 (EDT) Message-ID: <434DFCF0.4000304@mit.edu> Date: Thu, 13 Oct 2005 02:21:36 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 602 Content-Length: 86 X-Keywords: I would like to hear more about interesting problems in number theory. thanks, Irene From medrano@MIT.EDU Thu Oct 13 12:20:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DGKH6s005457 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 12:20:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DGKFRV016939 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 12:20:16 -0400 (EDT) Received: from quick-statalib3.mit.edu (QUICK-STATALIB3.MIT.EDU [18.50.1.77]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DGK7OP020364 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 12:20:08 -0400 (EDT) Received: (from medrano@localhost) by quick-statalib3.mit.edu (8.12.9) id j9DGK7Zo002403; Thu, 13 Oct 2005 12:20:07 -0400 Date: Thu, 13 Oct 2005 12:20:07 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan]Number Theory Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 603 Content-Length: 375 X-Keywords: NonJunk I don't really know what to write for this section after reading. I have taken a course in number theory and I understand the material about it. This was a good refresher for me. For a moment I had forgotten how to comptuer the Euler-phi function of a composite number with powers of primes as factors. If difficulties arise during class I will let you know. Jesus From ridell@MIT.EDU Thu Oct 13 12:50:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DGoM6s011475 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 12:50:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DGoLRV020798 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 12:50:21 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DGoJvj002285 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 12:50:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9DGoJNZ013686; Thu, 13 Oct 2005 12:50:19 -0400 Received: from STRATTON-FOUR-SEVENTY-EIGHT.MIT.EDU (STRATTON-FOUR-SEVENTY-EIGHT.MIT.EDU [18.187.6.223]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 12:50:19 -0400 Message-ID: <20051013125019.zqdu835wokgk4ksg@webmail.mit.edu> Date: Thu, 13 Oct 2005 12:50:19 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: tp 6 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 604 Content-Length: 293 X-Keywords: NonJunk Can you explain more about corollary 8.3 on page 20. I am having trouble understanding the sequence. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From lye@MIT.EDU Thu Oct 13 13:07:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DH7j6s015026 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 13:07:45 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DH7hRV010607 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 13:07:43 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DH7buB009288 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 13:07:37 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9DH7b7R003408; Thu, 13 Oct 2005 13:07:37 -0400 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 13:07:37 -0400 Message-ID: <20051013130737.kug7w8m5rofks8k0@webmail.mit.edu> Date: Thu, 13 Oct 2005 13:07:37 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 6 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 605 Content-Length: 211 X-Keywords: NonJunk pg. 12, the Prime Number Theorem: "The prime number theorem gives an approximate answer [equation]" I've heard this before but never saw it proved. Could you do the proof in lecture (or outline it briefly)? From cwong08@mit.edu Thu Oct 13 13:43:49 2005 Return-Path: Received: from w20-575-35.MIT.EDU (W20-575-35.MIT.EDU [18.187.0.54]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DHhm6s022099 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 13:43:49 -0400 Received: (from cwong08@localhost) by w20-575-35.MIT.EDU (8.12.9) id j9DHhmn2006544; Thu, 13 Oct 2005 13:43:48 -0400 Subject: [TA-name] Week 6 Comments From: Christopher Wong To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 13 Oct 2005 13:43:48 -0400 Message-Id: <1129225428.6489.1.camel@w20-575-35> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 Status: RO X-UID: 606 Content-Length: 109 X-Keywords: Euler's Theorem on page 26 was what I found the most difficult passage. I don't really understand it at all. From bakster@MIT.EDU Thu Oct 13 13:52:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DHqo6s022543 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 13:52:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DHpxRV026491 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 13:51:59 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DHpv9b026594 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 13:51:57 -0400 (EDT) Message-ID: <434E9EB9.6050402@mit.edu> Date: Thu, 13 Oct 2005 13:51:53 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] TP6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 607 Content-Length: 85 X-Keywords: NonJunk What's the difference between the modulo and rem functions? Thanks. Alexander Bakst From hzhou@MIT.EDU Thu Oct 13 15:28:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DJSg6s008126 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 15:28:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DJSeSo015586 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 15:28:40 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-SEVENTY-ONE.MIT.EDU [18.247.5.71]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DJSawU007325 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 15:28:37 -0400 (EDT) Message-Id: <5.2.1.1.2.20051013152724.02a98f40@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 13 Oct 2005 15:28:36 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: reading question Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.05 X-Spam-Level: * (1.05) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 608 Content-Length: 330 X-Keywords: NonJunk Hi, I was wondering if there is an easier way to do the last problem of the tutorials. I did it be finding the factors of 3780, which are 2 3 5 and 7, and then subtracting out all numbers that are factors of 2 3 5 and 7 from 3780, which became tedious at the end when i had to account for repeat counting. thanks - Steve From petek@MIT.EDU Thu Oct 13 16:10:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DKAB6s016068 for <"6042-probs"@theory.lcs.mit.edu>; Thu, 13 Oct 2005 16:10:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DKA9So003364 for <"6042-probs"@theory.lcs.mit.edu>; Thu, 13 Oct 2005 16:10:10 -0400 (EDT) Received: from [18.51.5.140] (HAYDEN-ONE-FORTY.MIT.EDU [18.51.5.140]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DKA2rZ024255 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <"6042-probs"@theory.lcs.mit.edu>; Thu, 13 Oct 2005 16:10:07 -0400 (EDT) Message-ID: <434EBF1A.1010901@mit.edu> Date: Thu, 13 Oct 2005 16:10:02 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "6.042 Reading Comments" <"6042-probs"@theory.lcs.mit.edu> Subject: [Jelani] Comments on Number Theory LN Content-Type: multipart/alternative; boundary="------------090007000108090102010602" X-Spam-Score: -2.059 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 609 Content-Length: 1937 X-Keywords: NonJunk This is a multi-part message in MIME format. --------------090007000108090102010602 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Everything was fine up until modulus proofs, which became kinda tricky since I've always grown up with definition of modulo = rem. I have circled 8.4's 2nd step in its proof, I think for the reason that I don't understand Lemma 7.2 as well as I should. I'm going to reread the notes tonight to see if I get a better grasp, and hope that the lectures supplement the material enough to get me a good grasp on all of this. -Pete -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------090007000108090102010602 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Everything was fine up until modulus proofs, which became kinda tricky since I've always grown up with definition of modulo = rem.  I have circled 8.4's 2nd step in its proof, I think for the reason that I don't understand Lemma 7.2 as well as I should.  I'm going to reread the notes tonight to see if I get a better grasp, and hope that the lectures supplement the material enough to get me a good grasp on all of this.

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

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------090007000108090102010602-- From benlu@MIT.EDU Thu Oct 13 17:19:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DLJ86s026974 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 17:19:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DLJ6j5012357 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 17:19:06 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DLIxKs019914 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 17:19:00 -0400 (EDT) Message-ID: <434ECF53.1080601@mit.edu> Date: Thu, 13 Oct 2005 17:19:15 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 6 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.073 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 610 Content-Length: 529 X-Keywords: "On the other hand, inverses generally do not exist over the integers. For example, 7 can not be multiplied by another integer to give 1. Surprisingly, multiplicative inverses do exist when we're working modulo a prime number. For example, if we're working modulo 5, then 3 is a multiplicative inverse of 7." ~ pg19 Whoa. Whoa. Still hurting my brain. In fact, I think most of the things about modular arithmetic are still hurting my brain. Like division only working for relatively prime factors and such. Ouch. ~Ben Lu From dowgun@MIT.EDU Thu Oct 13 17:26:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DLQU6s027652 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 17:26:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DLQTj5019449 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 17:26:29 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DLQQeW022219 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 17:26:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9DLQQw8014641; Thu, 13 Oct 2005 17:26:26 -0400 Received: from W20-575-34.MIT.EDU (W20-575-34.MIT.EDU [18.187.0.53]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 17:26:26 -0400 Message-ID: <20051013172626.mhi7ko9t63ccgk0w@webmail.mit.edu> Date: Thu, 13 Oct 2005 17:26:26 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: ln6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 611 Content-Length: 710 X-Keywords: NonJunk Wow. So... I hope we're going over Modulo operations a lot in class. My main quesiton is this (pretend the equals signs have three lines) - if 9 = 2 (mod 7) is our expression, is this expressed as "9 mod 7 equals 2" or "9 equals 2 mod 7". Both are certainly true statements, but it becomes confusing when you refer to multiplicative inverses, since 9 and 2 do not have the same multiplicative inverse ... wait they do! yikes. Is this always true? Probably. Anyway, I dont think it would be fair to put a subject that we are just learning the class before a test on the test, especially because the only applications of Number Theory seem to involve causing confusion (among people you are at war with). Neil From alisonc@MIT.EDU Thu Oct 13 19:55:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9DNtY6s016063 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 19:55:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9DNtWZl019807 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 19:55:33 -0400 (EDT) Received: from [18.58.6.141] (PIERCE-THREE-NINETY-SIX.MIT.EDU [18.58.6.141]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9DNs6Zx023250 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 19:54:07 -0400 (EDT) Message-ID: <434EF3A4.5030200@mit.edu> Date: Thu, 13 Oct 2005 19:54:12 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Comments on lecture notes #6 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 612 Content-Length: 456 X-Keywords: NonJunk It's always interesting to see direct connections between abstract math stuff and very real things like online security. It's really interesting to see the huge variety of ways that it's possible for a code to really fail: there's so many places where potential weaknesses could occur even in seemingly-secure schemes (that it's almost difficult not to be a bit paranoid about all the information floating around out there in so-far-unbroken codes). From ozcan@MIT.EDU Thu Oct 13 20:34:05 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E0Y56s018678 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:34:05 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E0Y4Rr012940 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:34:05 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E0Y2ol026993 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:34:03 -0400 (EDT) Message-Id: <5.2.1.1.2.20051013203037.029c2c10@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 13 Oct 2005 20:33:49 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson]reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 613 Content-Length: 1421 X-Keywords: The first 2 paragraphs of section 8 were amazing. I realized how theoretical math could be used in real life and this increases my motivation to study. Quotation (Section 8): In 1940 France had fallen before Hitler's army, and Britain alone stood against the Nazis in western Europe. British resistance depended on a steady flow of supplies brought across the north Atlantic from the United States by convoys of ships. These convoys were engaged in a cat-and-mouse game with German "U-boat" submarines, which prowled the Atlantic, trying to sink supply ships and starve Britain into submission. The outcome of this struggle pivoted on a balance of information: could the Germans locate convoys better than the Allies could locate U-boats or vice versa? Germany lost. But a critical reason behind Germany's loss was made public only in 1974: the British had broken Germany's naval code, Enigma. Through much of the war, the Allies were able to route convoys around German submarines by listening into German communications. The British government didn't explain how Enigma was broken until 1996. When the analysis was finally released (by the US), the author was none other than Alan Turing. In 1939 he had joined the secret British codebreaking effort at Bletchley Park. There, he played a central role in cracking the German's Enigma code and thus in preventing Britain from falling into Hitler's hands. From ereid@MIT.EDU Thu Oct 13 20:59:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E0xe6s022663 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:59:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E0xdZj002436 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:59:39 -0400 (EDT) Received: from [18.242.5.227] (NEXT-TWO-TWENTY-SEVEN.MIT.EDU [18.242.5.227]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E0xW95003663 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 20:59:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <47C17CD4-3BC1-4A64-AC64-E81B9B54EE6A@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Week 6 reading comment Date: Thu, 13 Oct 2005 20:59:24 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.088 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 614 Content-Length: 142 X-Keywords: NonJunk I'm still a little confused as to where the prime number theorem came from, and how exactly it was used for the prime-number-in-e question. From sil_03@MIT.EDU Thu Oct 13 21:06:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E1626s023103 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 21:06:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E160Zj006684 for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 21:06:01 -0400 (EDT) Received: from w20-575-32.mit.edu (W20-575-32.MIT.EDU [18.187.0.51]) (authenticated bits=56) (User authenticated as sil_03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E15wi5004702 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Thu, 13 Oct 2005 21:05:59 -0400 (EDT) Received: (from sil_03@localhost) by w20-575-32.mit.edu (8.12.9) id j9E15wNL019466; Thu, 13 Oct 2005 21:05:58 -0400 (EDT) Subject: [David] questions From: Silvia F Baptista To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 13 Oct 2005 21:05:58 -0400 Message-Id: <1129251958.19267.5.camel@w20-575-32.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 615 Content-Length: 241 X-Keywords: NonJunk I'm really confused about mod, congruence, and rem. I would like to see more examples. Also, on the bottom of page 19 it says p|(1-tk) and therefore tk=1 (mod P). Shouldn't it be 1=tk (mod p) since a=b (mod n) implies n|(a-b)??? ~Silvia From mdmurray@MIT.EDU Thu Oct 13 21:53:14 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E1rE6s030902 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:14 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9E1rDBs018463 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:13 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E1rCjw003124 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:12 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E1r5ok029747 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 21:53:05 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <5F83572A-10B3-43DE-AF8F-D4E174313546@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Michael Murray Subject: [Hanson] Reading assignment - week 6 Date: Thu, 13 Oct 2005 21:52:53 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 616 Content-Length: 316 X-Keywords: NonJunk I found the reading assignment very enjoyable and interesting this week, although after reading though it I am still unclear as to the difference between rem and mod, they seem to be able to be used interchangeably and yet there is a distinction made at the beginning of the reading. Thanks, Michael Murray From xiaoranz@MIT.EDU Thu Oct 13 21:58:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E1wL6s031413 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 21:58:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E1wKZj015258 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 21:58:20 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E1wIwT014280 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 13 Oct 2005 21:58:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9E1wIwp028728; Thu, 13 Oct 2005 21:58:18 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 13 Oct 2005 21:58:18 -0400 Message-ID: <20051013215818.mpamimfkb6hc8g0g@webmail.mit.edu> Date: Thu, 13 Oct 2005 21:58:18 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 6 Comments References: In-Reply-To: MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 617 Content-Length: 350 X-Keywords: NonJunk I found section 8 (page 18-22) about the Turing code most interesting/surprising because in highschool I did a project on cryptography and came across RSA/Turing code but didn't fully understand the concept behind it before. Reading about it in association with what we learnt in number theory is refreshing and interesting. Xiaoran (Sharon) Zhang From a_lopez@MIT.EDU Thu Oct 13 22:13:45 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2Dj6s001686 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:13:45 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2DgZj027294 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:13:43 -0400 (EDT) Received: from [18.239.6.154] (MACGREGOR-FOUR-O-NINE.MIT.EDU [18.239.6.154]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2DZYt017120 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:13:36 -0400 (EDT) Message-ID: <434F1455.1070401@mit.edu> Date: Thu, 13 Oct 2005 22:13:41 -0400 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: week 6 reading [revised] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 619 Content-Length: 673 X-Keywords: NonJunk On pg. 27: "We can find a multiplicative inverses using Euler's theorem as we did with Fermat's theorem: if k is relatively prime to n, then k^(phi(n) -1) is a multiplicative inverse of k modulo n." Can we also find an inverse for k modulo n by finding a linear combination of k and n that equals 1, that is, find integers s and t such that ks + nt = 1? We can find such integers using "the pulverizer" (we know they exist because gcd(k, n) = 1). (let a =c b (mod n) denot a is congruent to b modulo n) Since ks + nt = 1, then ks + nt =c 1 (mod n) so ks =c 1 (mod n) and s is a multiplicative inverse of k modulo n. Which method is most efficient? Adriana From jjmonzon@MIT.EDU Thu Oct 13 22:20:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2KC6s002724 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:20:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2KAZj002475 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:20:10 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-TWO-TWELVE.MIT.EDU [18.247.5.212]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2K3ee018367 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 22:20:04 -0400 (EDT) Message-Id: <6.2.3.4.2.20051013221456.01e81438@po9.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 13 Oct 2005 22:20:10 -0400 To: 6042-probs@theory.csail.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 620 Content-Length: 317 X-Keywords: NonJunk I found the RSA to be very cool because I finally saw a connection between 6.042 and computer science. It would be really great if RSA is explained well in class. Josh Monzon Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From jstritar@MIT.EDU Thu Oct 13 22:26:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2Q06s003738 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:26:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2PxZj006801 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:25:59 -0400 (EDT) Received: from [18.245.6.110] (BAKER-THREE-SIXTY-FIVE.MIT.EDU [18.245.6.110]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2Pop7019447 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:25:52 -0400 (EDT) Message-ID: <434F172C.7060001@mit.edu> Date: Thu, 13 Oct 2005 22:25:48 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 621 Content-Length: 230 X-Keywords: NonJunk 7. Modular Arithmetic I thought the section on Modular Arithmetic was pretty hard to understand, probably because it was not very familiar. I also thought the sections on encoding/decoding were really interesting. Jon Stritar From hsoumare@MIT.EDU Thu Oct 13 22:27:37 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2Rb6s003902 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:37 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2RaBs001466 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:36 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2RZjw003996 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:35 -0400 (EDT) Received: from [18.241.6.193] (NEW-FOUR-FORTY-EIGHT.MIT.EDU [18.241.6.193]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E2RSok001174 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:27:28 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <41C83F1A-5B1D-4F20-9690-9141156DA9F3@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: {Sayan} Weekly Email Comment Date: Thu, 13 Oct 2005 22:36:10 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 622 Content-Length: 221 X-Keywords: NonJunk Hi, I would like the passage at the top of pg 20 to be explained in lecture because I don't totally see how the inverse of the key and the remainder function are used to decode the message. Thanks. Hamidou Soumare From bens@MIT.EDU Thu Oct 13 22:30:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E2UB6s004414 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:30:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E2U9Zj009825 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:30:09 -0400 (EDT) Received: from [18.221.0.117] ([18.221.0.117]) (authenticated bits=0) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E2U2Xb020183 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 22:30:07 -0400 (EDT) Message-ID: <434F1812.7010508@mit.edu> Date: Thu, 13 Oct 2005 22:29:38 -0400 From: "Benjamin M. Schwartz" User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050723) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] reading Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 623 Content-Length: 88 X-Keywords: page 27: So what do phi() and inverses have to do with RSA? Will I find out in lecture? From hectorb@MIT.EDU Thu Oct 13 23:06:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E36E6s010492 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:06:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E36B4S004678; Thu, 13 Oct 2005 23:06:11 -0400 (EDT) Received: from [18.246.7.127] (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (authenticated bits=0) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E369is026104 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 13 Oct 2005 23:06:10 -0400 (EDT) Message-ID: <434F209F.4050400@mit.edu> Date: Thu, 13 Oct 2005 23:06:07 -0400 From: Hector Beltran User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 624 Content-Length: 272 X-Keywords: NonJunk "In 1977 at MIT, Ronald Rivest, Adi Shamir, and Leonard Adleman proposed a highly secure cryptosystem (called RSA) based on number theory. Despite decades of attack, no significant weakness has been found." - pg 23 Amazing, would like to hear more about this in class. From fluff@MIT.EDU Thu Oct 13 23:27:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3RA6s014737 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:27:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3R94S019937 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:27:09 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E3R2Mo029605 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:27:02 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <1514E291-B632-485C-AEE8-E0AE5F84BAFE@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [jelani] week 6 reading Date: Thu, 13 Oct 2005 23:27:38 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 625 Content-Length: 378 X-Keywords: NonJunk Um, holy crap. This reading was SO FRICKIN' LONG. I thought the narrative parts and the cryptographic applications were extremely interesting, and were definitely a nice fast-read break in between the proofs (since following proofs is slow reading). However, I am almost certain I did not retain anything past page 20, i.e. all the modular arithmetic. Geez. ~Crystal From clintonb@MIT.EDU Thu Oct 13 23:34:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3YI6s015448 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:34:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3YG4S024996 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:34:17 -0400 (EDT) Received: from ccbibmx30 (NEW-ONE-FORTY-EIGHT.MIT.EDU [18.241.5.148]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E3YDnQ000806 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:34:14 -0400 (EDT) Message-Id: <200510140334.j9E3YDnQ000806@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Jelani: Week 6 Comments Date: Thu, 13 Oct 2005 23:34:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0057_01C5D04E.9798D7D0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXQcB3U7NONQDfxRs20YLXwKo89KQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.344 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 626 Content-Length: 3370 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0057_01C5D04E.9798D7D0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The Die Hard jug problem is pretty neat. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0057_01C5D04E.9798D7D0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

The Die Hard jug problem is pretty neat. =

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_0057_01C5D04E.9798D7D0-- From brevzin@MIT.EDU Thu Oct 13 23:38:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3cJ6s015797 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:38:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3cI4S027790 for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:38:18 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E3cB9O001389 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 13 Oct 2005 23:38:15 -0400 (EDT) Message-Id: <6.1.2.0.1.20051013233355.01aac540@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Thu, 13 Oct 2005 23:38:14 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.057 X-Spam-Level: * (1.057) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 627 Content-Length: 908 X-Keywords: NonJunk I don't understand the necessity of the difference between rem and mod. They mean preciesly the same thing, don't they ? I think it's much more confusing to have to write proofs out with the rem's, such as with the proof of Euler's thm. I mean, clearly A rem N = A mod N... always, so why the difference ? And I actually used the AKS algorithm over the summer in dealing with encryption software, it is a surprisingly simple algorithm that takes a surprising amount of time to actually get to work right. I was especially a fan of the step toward the beginning that is something to the effect of If N is some a^b, return composite. Took me a while to figure out how to write that one efficiently. Also used RSA, which might become obsolete with quantum computing since it's security is solely based on the fact that factoring takes AGES. Might also happen if somebody solves Riemann... ? Barry From mracich@MIT.EDU Thu Oct 13 23:48:28 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E3mS6s017197 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:48:28 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E3mSqd018519 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:48:28 -0400 (EDT) Received: from MACGREGOR-THREE-THIRTY-NINE.MIT.EDU (MACGREGOR-THREE-THIRTY-NINE.MIT.EDU [18.239.6.84]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E3mKMu000282 for <6042-probs@theory.lcs.mit.edu>; Thu, 13 Oct 2005 23:48:20 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 6 (Introduction to Number Theory) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Thu, 13 Oct 2005 23:48:19 -0400 Message-Id: <1129261699.5576.38.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 628 Content-Length: 225 X-Keywords: I found section 7, Modular Arithmetic, (starting on page 16) kind of confusing. I would appreciate it if this was reviewed in greater depth during lecture (and I would appreciate seeing some more examples). Moira Racich From mpapi@MIT.EDU Fri Oct 14 00:03:39 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E43c6s019965 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 00:03:38 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E43cqd018840 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 00:03:38 -0400 (EDT) Received: from MACGREGOR-TWO-O-FOUR.MIT.EDU (MACGREGOR-TWO-O-FOUR.MIT.EDU [18.239.5.204]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E43UMu000842 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 00:03:30 -0400 (EDT) Subject: [Jelani] Week 6 Comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 14 Oct 2005 00:03:29 -0400 Message-Id: <1129262610.7478.41.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 629 Content-Length: 278 X-Keywords: NonJunk Section 8.3 on page 21, on Fermat's Theorem; I found that I had to read it over a few times for it to click. Since the theorem has some relevant and interesting applications, I'd like to see it explained further in class, particularly the section on finding inverses. -- Matt From aston@MIT.EDU Fri Oct 14 00:32:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E4W26s022605 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 00:32:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E4W04S007434 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 00:32:00 -0400 (EDT) Received: from aston (ASTON.MIT.EDU [18.241.3.49]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E4VvJk009896 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 00:31:58 -0400 (EDT) Message-Id: <200510140431.j9E4VvJk009896@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: [TA-name] Week 6 Comments Date: Fri, 14 Oct 2005 00:31:51 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_003F_01C5D056.AD2A69D0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXQeDKi4kK9s/8iTNO18FdWkiaDYA== X-Spam-Score: 0.275 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 630 Content-Length: 2536 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_003F_01C5D056.AD2A69D0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The multiplicative inverse modulo primes (page 19) is a concept I've not heard of before, and seems pretty cool. I missed, however, how we dropped the rem on page 20 in proving that m* x k^-1 is congruent to m (mod p) - Aston ------=_NextPart_000_003F_01C5D056.AD2A69D0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

 

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

 

         =    - Aston

------=_NextPart_000_003F_01C5D056.AD2A69D0-- From dangut@MIT.EDU Fri Oct 14 00:36:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E4a56s022870 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 00:36:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E4a44S010054 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 00:36:04 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E4ZwWG010394 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 00:35:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9E4Zwfn005476; Fri, 14 Oct 2005 00:35:58 -0400 Received: from MACGREGOR-ONE-O-NINE.MIT.EDU (MACGREGOR-ONE-O-NINE.MIT.EDU [18.239.5.109]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 00:35:58 -0400 Message-ID: <20051014003558.c1ovs3i5ff7ooc4w@webmail.mit.edu> Date: Fri, 14 Oct 2005 00:35:58 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: TP6 response R13 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 631 Content-Length: 135 X-Keywords: NonJunk I think the pulverizing method for getting the coefficients is neat, other than that, most of this isn't too unfamiliar for this week. From sergiob@MIT.EDU Fri Oct 14 01:08:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E58B6s027053 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:08:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E58A4S002243 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:08:10 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E57tsT014543 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:08:04 -0400 (EDT) Message-Id: <5.2.1.1.2.20051013230108.03181ef0@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 13 Oct 2005 23:07:50 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 1.045 X-Spam-Level: * (1.045) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9E58B6s027053 Status: RO X-UID: 632 Content-Length: 330 X-Keywords: I found Turing´s life the most interesting part of the lecture notes. His contributions were incredibly seminal, and the way that his life ended is very sad :( . He devoted his life to making theories about computing, when the most efficient computers were still mechanical. The impact of his work is mind-boggling. Sergio From dshin@MIT.EDU Fri Oct 14 01:20:48 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E5Km6s029914 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 01:20:48 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5Kkqd020391; Fri, 14 Oct 2005 01:20:47 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E5KhMu003327; Fri, 14 Oct 2005 01:20:43 -0400 (EDT) Message-ID: <434F401E.7070301@mit.edu> Date: Fri, 14 Oct 2005 01:20:30 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Silvia F Baptista CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] questions References: <1129251958.19267.5.camel@w20-575-32.mit.edu> In-Reply-To: <1129251958.19267.5.camel@w20-575-32.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 633 Content-Length: 424 X-Keywords: Are you confused about their definitions, or their applications? tk=1 (mod p) is the same as 1=tk (mod p). This is because p|n iff p|-n. DS Silvia F Baptista wrote: >I'm really confused about mod, congruence, and rem. I would like to see >more examples. > >Also, on the bottom of page 19 it says p|(1-tk) and therefore tk=1 (mod >P). Shouldn't it be 1=tk (mod p) since a=b (mod n) implies n|(a-b)??? > >~Silvia > > From dshin@MIT.EDU Fri Oct 14 01:33:25 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E5XP6s030633 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 01:33:25 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5XOH2022787 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 01:33:24 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5XMWh009048; Fri, 14 Oct 2005 01:33:22 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E5XJok007436; Fri, 14 Oct 2005 01:33:19 -0400 (EDT) Message-ID: <434F4312.3080009@mit.edu> Date: Fri, 14 Oct 2005 01:33:06 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Alexander Bakst CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] TP6 Comments References: <434E9EB9.6050402@mit.edu> In-Reply-To: <434E9EB9.6050402@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 634 Content-Length: 439 X-Keywords: a rem b is the remainder when dividing a by b. Note that rem is a function (maps integers to integers) In this course, mod is not a function. We write a = b (mod n) if n divides a-b. Note that 2 = 7 (mod 5), but 2 rem 5 does not equal 7. On the other hand, 7 = 2 (mod 5), and 7 rem 5 = 2. Hope that helps. DS Alexander Bakst wrote: > What's the difference between the modulo and rem functions? > > Thanks. > Alexander Bakst From dshin@MIT.EDU Fri Oct 14 01:35:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E5ZZ6s031621 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 01:35:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E5ZXqd020657; Fri, 14 Oct 2005 01:35:33 -0400 (EDT) Received: from [192.168.1.7] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E5ZPMu003994; Fri, 14 Oct 2005 01:35:25 -0400 (EDT) Message-ID: <434F4391.7090202@mit.edu> Date: Fri, 14 Oct 2005 01:35:13 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Steven Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: reading question References: <5.2.1.1.2.20051013152724.02a98f40@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20051013152724.02a98f40@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 635 Content-Length: 670 X-Keywords: Yes, there is a function called Euler's phi function defined in the notes. phi(n) = # of integers from 1 to n relatively prime to n. The notes give an efficient way of computing phi. Go back to the tutor and click "Check Answers" to see what the computation looks like for the tutor problem. DS Steven Zhou wrote: > Hi, > > I was wondering if there is an easier way to do the last problem of > the tutorials. I did it be finding the factors of 3780, which are 2 3 > 5 and 7, and then subtracting out all numbers that are factors of 2 3 > 5 and 7 from 3780, which became tedious at the end when i had to > account for repeat counting. thanks > > - Steve > From shauni@MIT.EDU Fri Oct 14 02:09:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E69g6s004472 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 02:09:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E69fs7008843 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 02:09:41 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E69dCp020113 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 02:09:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9E69da6007774; Fri, 14 Oct 2005 02:09:39 -0400 Received: from SIMMONS-ONE-SEVENTY-NINE.MIT.EDU (SIMMONS-ONE-SEVENTY-NINE.MIT.EDU [18.96.5.179]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 02:09:39 -0400 Message-ID: <20051014020939.rjsfhf0dacf4kcwg@webmail.mit.edu> Date: Fri, 14 Oct 2005 02:09:39 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] week 6 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.287 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 640 Content-Length: 133 X-Keywords: NonJunk I thought the most interesting part of this week's reading was the history of Alan Turing and codebreaking during WWII (p.18, 23) :) From sriaz@MIT.EDU Fri Oct 14 02:11:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6Bq6s004811 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:11:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6Bps7010124 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:11:51 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6BnEk020285 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:11:50 -0400 (EDT) Message-ID: <434F4C28.6030108@mit.edu> Date: Fri, 14 Oct 2005 02:11:52 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Readings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 641 Content-Length: 253 X-Keywords: NonJunk One of the things I found interesting on this week's reading was the discussion of the RSA algorithm (Section 10, page 23). This is something I had covered while taking 18.310, but the lecture notes here provided a greater breakground to the issue. From lkini@MIT.EDU Fri Oct 14 02:13:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6DX6s005001 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:13:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6DVs7011267 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:13:32 -0400 (EDT) Received: from [18.239.6.97] (MACGREGOR-THREE-FIFTY-TWO.MIT.EDU [18.239.6.97]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6DSMr020392 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:13:30 -0400 (EDT) Message-ID: <434F4C87.6050906@mit.edu> Date: Fri, 14 Oct 2005 02:13:27 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 6 Lecture Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 642 Content-Length: 274 X-Keywords: NonJunk Hi Hanson, The only question I had after reading the lecture notes was the proof of Corollary 8.3. I don't understand why the first sequence must contain all of the numbers from 0 to p-1. I would appreciate it if you could go explain that a little more. Thanks, Lohith From cvnguyen@MIT.EDU Fri Oct 14 02:16:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6Gs6s005201 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:16:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6Grs7013263 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:16:53 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-FIVE-TWENTY-EIGHT.MIT.EDU [18.238.5.17]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6Gox8020635 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:16:51 -0400 (EDT) Message-ID: <010b01c5d086$e026c740$9907ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 6 comments Date: Fri, 14 Oct 2005 02:16:54 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MIMEOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.627 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 643 Content-Length: 236 X-Keywords: I enjoyed the discussion of the role of Turing's code during World War II and of Turing's life. It's always beneficial to see how the subject material relates to real life and how people have applied it in the past. --Chieu Nguyen From lana@MIT.EDU Fri Oct 14 02:19:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6Jm6s005272 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:19:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6Jls7014883 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:19:47 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6JeRt020815 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:19:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9E6JeS6028416; Fri, 14 Oct 2005 02:19:40 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 02:19:40 -0400 Message-ID: <20051014021940.zfum59m9ktw80wo0@webmail.mit.edu> Date: Fri, 14 Oct 2005 02:19:40 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 644 Content-Length: 253 X-Keywords: NonJunk The part that I found most interesting was Euler's theorem and everything that was related to Phi. The function phi in general seems to be a somewhat unusual function because there does not seem to be an empirical way of directly deriving it. Lana 6 From ryan786@MIT.EDU Fri Oct 14 02:40:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6em6s006559 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:40:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6eks7027729 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:40:47 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6ehLP022252 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 02:40:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9E6ehM5020489; Fri, 14 Oct 2005 02:40:43 -0400 Received: from PSK-NINETY-ONE.MIT.EDU (PSK-NINETY-ONE.MIT.EDU [18.217.1.91]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 02:40:43 -0400 Message-ID: <20051014024043.hf6627c5xps8cg8c@webmail.mit.edu> Date: Fri, 14 Oct 2005 02:40:43 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [sayan] email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 645 Content-Length: 392 X-Keywords: NonJunk I really enjoyed the brief background on Alan Turing. Stuff like that helps liven up the more boring parts of the reading. One question: what is the large, bold pi shape on the right hand side of Riemann Hypothesis equation? The same as a summation, but a multiplication? Sorry I didn't pick up on it. No real complaints, aside from the fact that it was really really long. -Ryan Young From tonyng@MIT.EDU Fri Oct 14 02:46:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E6k16s006800 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:46:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E6jxs7001047 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:45:59 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E6ju4A022605 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 02:45:57 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014024111.02103400@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 02:45:59 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 646 Content-Length: 302 X-Keywords: NonJunk I found the RSA encryption method (starting in pg 23 section 10 on Arithmetic with an Arbitrary Modulus) very interesting. It surprised me that it is possible to use a public key and still have the system be completely secure. I am hoping lecture will give more details on how and why this works. From yangc@MIT.EDU Fri Oct 14 03:06:57 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E76v6s008430 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:06:57 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E76vUP022777 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:06:57 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9E76jMv007659 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:06:50 -0400 (EDT) Message-Id: <200510140706.j9E76jMv007659@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] TP6 Comments Date: Fri, 14 Oct 2005 03:06:39 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00CA_01C5D06C.4BE52140" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXQjdKZZVEjLPehSJyNAcPKA/OeKg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 647 Content-Length: 2215 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_00CA_01C5D06C.4BE52140 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit You mention this amazingly simple algorithm for primality testing (twice mentioned, in fact) - but neglect to state how this amazing algorithm works. I guess I'll just ask now - how does this amazingly simple algorithm work? -Chris Yang ------=_NextPart_000_00CA_01C5D06C.4BE52140 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

 

-Chris Yang

------=_NextPart_000_00CA_01C5D06C.4BE52140-- From scot@MIT.EDU Fri Oct 14 03:12:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7CY6s009380 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:12:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7CXs7016301 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:12:33 -0400 (EDT) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7CLba024081 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:12:22 -0400 (EDT) Message-ID: <434F5A55.9030505@mit.edu> Date: Fri, 14 Oct 2005 03:12:21 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: this week's comments (Jelani) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.186 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 648 Content-Length: 357 X-Keywords: Hi, I had a question about page 12. This may be simple but when I read it I don't directly see why primes gradually taper off. Are you saying that there become fewer and fewer as the number of digits increase? I thought that the number of primes was infinite. I feel so far the class up to this point in the reading has been sufficient. thanks, scot From zev@MIT.EDU Fri Oct 14 03:26:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7Qu6s010106 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:26:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7Qts7024147 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:26:55 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7QlMN024822 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:26:52 -0400 (EDT) Message-ID: <434F5DB6.10207@mit.edu> Date: Fri, 14 Oct 2005 03:26:46 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 6 comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.239 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 649 Content-Length: 676 X-Keywords: NonJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 The solution to 6.5 has a typo. It claims (4-2) * (27-9) * (5-1) * (49-7) = 864 it means (4-2) * (27-9) * (5-1) * (7-1) = 864 In general, the notes were much more entertaining than those in the past. However, there was a lot of information thrown at us. I didn't really understand the more advanced modular athithmetic sections or the statements about sequences. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDT122lO3j8HLL0+4RAnqRAJ4ilWz8pdIJAun63SgqrKU1RsOUOwCfXAsD 3ofMK3UG1deLlXzVO3ZmdG8= =1NpS -----END PGP SIGNATURE----- From tylerw@MIT.EDU Fri Oct 14 03:30:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7Uv6s010601 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:30:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7Uts7026434 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:30:56 -0400 (EDT) Received: from [18.246.6.207] (BEXLEY-FOUR-SIXTY-TWO.MIT.EDU [18.246.6.207]) (authenticated bits=0) (User authenticated as tylerw@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7UnE3025025 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 03:30:50 -0400 (EDT) Message-ID: <434F5EAE.4000303@mit.edu> Date: Fri, 14 Oct 2005 03:30:54 -0400 From: Tyler Williams User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 650 Content-Length: 331 X-Keywords: NonJunk Hey, so my comment for this week about the reading was that I really enjoyed how the reading was interspersed with history about Turing, and RSA, etc. This made the reading much more interesting to read, and the whole thing about RSA was cool because it was practical. I hope there are future readings also like this... Tyler From scholtz@MIT.EDU Fri Oct 14 03:32:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7WX6s010715 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:32:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7WVs7027200 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:32:32 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7WPsY025080 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:32:26 -0400 (EDT) Message-Id: <200510140732.j9E7WPsY025080@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 6 Comments Date: Fri, 14 Oct 2005 03:32:34 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5D06F.EAEC9E50" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXQkXFj9rSBNhdSQE2K7O58PL35sw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.593 X-Spam-Level: ** (2.593) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 651 Content-Length: 1810 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5D06F.EAEC9E50 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I would like to have the Riemann Hypothesis discussed more fully in lecture. I don't understand what it means or why it's so famous. ------=_NextPart_000_0000_01C5D06F.EAEC9E50 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

------=_NextPart_000_0000_01C5D06F.EAEC9E50-- From kjhollen@MIT.EDU Fri Oct 14 03:35:02 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7Z26s011063 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:35:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7Z1s7028558 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:35:01 -0400 (EDT) Received: from MACGREGOR-TWO-NINETY-NINE.MIT.EDU (MACGREGOR-TWO-NINETY-NINE.MIT.EDU [18.239.6.44]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7Ysa0025168 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:34:55 -0400 (EDT) Subject: [Hanson] about rsa and number theory From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 14 Oct 2005 03:34:54 -0400 Message-Id: <1129275294.29833.3.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 652 Content-Length: 184 X-Keywords: NonJunk I am interested to see how the RSA decoding works in class tomorrow. It might also be nice to quickly go over how it works, because I think I understand but I am not completely sure. From aeon@MIT.EDU Fri Oct 14 03:40:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7eE6s011605 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 03:40:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7eDs7001548 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 03:40:13 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7eB3W025439 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 03:40:11 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9E7eBds031296; Fri, 14 Oct 2005 03:40:11 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 01:40:11 -0600 Message-ID: <20051014014011.y61phbs76w0g8oww@webmail.mit.edu> Date: Fri, 14 Oct 2005 01:40:11 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: LN6 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.671 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 653 Content-Length: 403 X-Keywords: NonJunk Page 11, "The Fundamental Theorem of Arithmetic" This makes sense and all, but I fail to see where it was applied after it was introduced, exactly.. how does it involve the material afterwards? Page 23, "Arithmetic with an Arbitrary Modulus Great, but very abstract. A simple example with actual values and strings would help reinforce the ideas behind the RSA protocol, if possible. - John Marrero From dnreshef@MIT.EDU Fri Oct 14 03:42:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7gC6s011741 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 03:42:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7gBs7002633 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 03:42:11 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7g9ir025534 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 03:42:09 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9E7g9eE019651; Fri, 14 Oct 2005 03:42:09 -0400 Received: from MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU (MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU [18.239.7.33]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 03:42:09 -0400 Message-ID: <20051014034209.hkert02mh9kokgsk@webmail.mit.edu> Date: Fri, 14 Oct 2005 03:42:09 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: HANSON: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 3.14 X-Spam-Level: *** (3.14) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 654 Content-Length: 182 X-Keywords: NonJunk Hey, This week's reading was awesome! I found some of the things regarding GCDs a little confusing. Things like question 3 on the tutor. Also, RSA was a little confusing. -Dave From rshearer@MIT.EDU Fri Oct 14 03:46:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E7kd6s012249 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:46:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E7kcs7005245 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:46:38 -0400 (EDT) Received: from [18.240.5.57] (MCCORMICK-FIFTY-SEVEN.MIT.EDU [18.240.5.57]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E7kZSa025765 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 03:46:36 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) In-Reply-To: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> References: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Rachel Shearer Subject: [Jelani] Week 6 Comment Date: Fri, 14 Oct 2005 03:46:28 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: -1.346 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 655 Content-Length: 299 X-Keywords: NonJunk I didn't really understand how to use multiplicative inverses in decryption using Turing's code (with modular arithmetic). I mean, supposedly you can "recover the original message" by multiplying by the inverse of the key but I don't understand how that math works (page 20). Rachel Shearer From yaser@MIT.EDU Fri Oct 14 04:11:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E8BD6s014876 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 04:11:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E8BCs7018851 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 04:11:12 -0400 (EDT) Received: from Charizard (NEW-ONE-THIRTY-SEVEN.MIT.EDU [18.241.5.137]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E8B970026828 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 04:11:09 -0400 (EDT) Message-Id: <200510140811.j9E8B970026828@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: reading email for 0ct 14 Date: Fri, 14 Oct 2005 04:11:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0015_01C5D075.49CF98A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXQltBO6EZ2T2xxSsCKqEl3jRJAKg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.178 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 656 Content-Length: 5494 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0015_01C5D075.49CF98A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi David, I think I get most of the reading (that is, understand it enough to follow lecture), but I would definitely like it if we could go over in detail how the RSA encryption technique actually works. It looks nice & dandy on paper, but seems almost a bit too simple to be true. Perhaps a real-life example would be useful. Thanks! _Yaser ------=_NextPart_000_0015_01C5D075.49CF98A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

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

 

Thanks!

_Yaser

------=_NextPart_000_0015_01C5D075.49CF98A0-- From adnaan@MIT.EDU Fri Oct 14 04:25:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E8PA6s015731 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 04:25:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E8P9s7026409 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 04:25:09 -0400 (EDT) Received: from AMMAR (NEW-FOUR-SIXTY-SEVEN.MIT.EDU [18.241.6.212]) (authenticated bits=0) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E8P2Mp027350 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 04:25:02 -0400 (EDT) Message-ID: <000d01c5d098$c7284a40$d406f112@AMMAR> From: "Adnaan" To: <6042-probs@theory.csail.mit.edu> Subject: week reading - hanson Date: Fri, 14 Oct 2005 04:25:03 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000A_01C5D077.3F985280" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.952 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 657 Content-Length: 790 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_000A_01C5D077.3F985280 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable the reading section on turning code is confusing ------=_NextPart_000_000A_01C5D077.3F985280 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
the reading section on turning code is=20 confusing
------=_NextPart_000_000A_01C5D077.3F985280-- From kevin08@MIT.EDU Fri Oct 14 05:05:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E95U6s019975 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:05:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E95TJD017528 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:05:29 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-THREE-SEVENTY-EIGHT.MIT.EDU [18.242.6.123]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E95Mlk028796 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:05:23 -0400 (EDT) Message-Id: <6.2.3.4.2.20051014050241.01ea38b8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 14 Oct 2005 05:05:20 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Number Theory Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 658 Content-Length: 305 X-Keywords: I had a question about Section 8.4, page 22 where we are computing the key k. I don't see how the second line of the congruence (which uses "part 2 of Lemma 7.2") follows from the first line and part 2 of Lemma 7.2: specifically how (mk rem p) becomes just mk. Does it use part 1 of Lemma 2? -Kevin From mrivas03@MIT.EDU Fri Oct 14 05:14:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E9EV6s020967 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:14:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E9ETJD022286 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:14:29 -0400 (EDT) Received: from el-ternero.mit.edu (TDCIP95.MIT.EDU [18.237.0.95]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E9EMVw029093 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 05:14:22 -0400 (EDT) Message-Id: <6.2.3.4.2.20051014051147.02e2cb08@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 14 Oct 2005 05:14:00 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: [Sayan] Week 6 Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.726 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 659 Content-Length: 397 X-Keywords: NonJunk Hello Sayan, Just wanted to point out that from the reading Section 6.1 page 15 was extremely interesting to me (Turing's Code). If possible, it would be great to go over this in class. The question of ensuring that two numbers are prime and the security of turing's code and how it was possible to break it makes the concepts in the rest of the reading very interesting. Thanks, Manuel From jeffhoff@MIT.EDU Fri Oct 14 05:16:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9E9Gn6s021161 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 05:16:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9E9GlJD023533 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 05:16:47 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9E9GdYp029192 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 05:16:41 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014051503.0281c650@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 05:16:38 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 660 Content-Length: 235 X-Keywords: NonJunk Section 3.4 The Pulverizer This section cleared up what we went over in class. In class several of us were confused as to how the pulverizer actually helped and gave us the final answer, but now it's clear with the substitution. From cbossard@MIT.EDU Fri Oct 14 07:12:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EBCS6s003923 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 07:12:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EBCRJD027180 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 07:12:27 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EBCPSO004418 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 07:12:25 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9EBCPRe010817; Fri, 14 Oct 2005 07:12:25 -0400 Received: from MCCORMICK-TWO-FOURTEEN.MIT.EDU (MCCORMICK-TWO-FOURTEEN.MIT.EDU [18.240.5.214]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 07:12:25 -0400 Message-ID: <20051014071225.mzg9o7npi1sgwgws@webmail.mit.edu> Date: Fri, 14 Oct 2005 07:12:25 -0400 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 6 Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.839 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 661 Content-Length: 165 X-Keywords: NonJunk I found this whole thing a little confusing. Probably the biggest issue I have is with mod. I keep forgetting how to find mod and how it differs from rem. Cynthia From kromer@MIT.EDU Fri Oct 14 09:10:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDAx6s018844 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:10:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDAvbg020232 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:10:57 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDAuVH024065 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:10:56 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9EDAtqM018446; Fri, 14 Oct 2005 09:10:55 -0400 Received: from BURTON-TWO-O-TWO.MIT.EDU (BURTON-TWO-O-TWO.MIT.EDU [18.247.5.202]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 09:10:55 -0400 Message-ID: <20051014091055.bqpu4dsbamo8s4kc@webmail.mit.edu> Date: Fri, 14 Oct 2005 09:10:55 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.486 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 662 Content-Length: 259 X-Keywords: p. 12 "As a rule of thumb, about 1 integer out of every ln x in the vicinity of x is prime." I find it strange (and vert interesting) that it's possible to find and prove an average distribution of primes, even though primes are very erratically distributed. From letrec@MIT.EDU Fri Oct 14 09:39:50 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDdo6s021485 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:39:50 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDdoDi001797 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:39:50 -0400 (EDT) Received: from [18.237.1.57] (TDC-FIFTY-SEVEN.MIT.EDU [18.237.1.57]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9EDdhok025686 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:39:43 -0400 (EDT) Message-ID: <434FB51E.3010202@mit.edu> Date: Fri, 14 Oct 2005 09:39:42 -0400 From: Alton Torregano User-Agent: Debian Thunderbird 1.0.7 (X11/20051010) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: 6042-student: Hanson, Week 6 Comments Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 663 Content-Length: 303 X-Keywords: NonJunk Notes are very thorough, I found the section on the Nazi's/Turing to be a useful historical review/mathematical refresher as I was presented with the material. However, I felt the notes could use slightly more information on RSA encryption (though I assume it will mostly be covered in slides). ~alton From veracarr@MIT.EDU Fri Oct 14 09:47:42 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDlg6s022167 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:47:42 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDlgDi002104 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:47:42 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9EDldMu023512 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 09:47:40 -0400 (EDT) Message-ID: <434FD35A.2010003@mit.edu> Date: Fri, 14 Oct 2005 09:48:42 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Intro to Number Theory References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 664 Content-Length: 513 X-Keywords: NonJunk Pg. 25 - Theorem 10.1 What i don't understand is if you have, for example, 27 as the number you want to find the relative primes of by part 1 of Theorem 10.1 you can way f(27)=f(3*3*3)=f(3)*f(3)*f(3) and we know for 3 that is has 2 relative primes (2 and 1) so we can say f(3)*f(3)*f(3)=2*2*2? I realize that by using part 2 (where p is a prime...) of Theorm 10.1 you get the right answer but I don't understand why you can't multiply the number of relative primes that each factor has as noted in part 1? From fgreen@MIT.EDU Fri Oct 14 09:55:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDto6s023053 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:55:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDtnbg004367 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:55:49 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDtksM010327 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:55:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9EDtkNN005705; Fri, 14 Oct 2005 09:55:46 -0400 Received: from EASTCAMPUS-SIX-FIFTY-FOUR.MIT.EDU (EASTCAMPUS-SIX-FIFTY-FOUR.MIT.EDU [18.238.5.143]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 09:55:46 -0400 Message-ID: <20051014095546.ijfr3ppvzqgo8wc8@webmail.mit.edu> Date: Fri, 14 Oct 2005 09:55:46 -0400 From: Forrest O Green To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Reading Comment Forrest Green MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 665 Content-Length: 323 X-Keywords: NonJunk "An amazingly simple, yet efficient method was finally discovered in 2002 by Agrawal, Kayal, and Saxena." >From page 3. I would very much like to hear an explination of the AKS primality test algorithm that would qualify as simple. I will not be in lecture on Friday, but I may ask in person next Wen. -Forrest Green From nedzel@MIT.EDU Fri Oct 14 09:59:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDx66s023215 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 09:59:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDx5bg007617 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 09:59:05 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-FOUR-NINETY-SEVEN.MIT.EDU [18.96.6.242]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDx3dW011658 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 09:59:03 -0400 (EDT) Message-Id: <200510141359.j9EDx3dW011658@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 6 Comments Date: Fri, 14 Oct 2005 09:58:58 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.909 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 666 Content-Length: 220 X-Keywords: NonJunk I don't have any reading questions at the moment; I enjoyed the story behind the lecture notes. Are there any sample exams that I can study for Monday, or should I just review class problems and problem sets? - David From miki_tnd@MIT.EDU Fri Oct 14 09:59:09 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EDx96s023220 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:59:09 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EDx8bg007633 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:59:08 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EDx6Tb011673 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 09:59:06 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9EDx6oq023829; Fri, 14 Oct 2005 09:59:06 -0400 Received: from PIKA-ONE-O-NINE.MIT.EDU (PIKA-ONE-O-NINE.MIT.EDU [18.214.1.109]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 09:59:06 -0400 Message-ID: <20051014095906.7bb0av9wmrv48scw@webmail.mit.edu> Date: Fri, 14 Oct 2005 09:59:06 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: sayan reading assignment 6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 667 Content-Length: 289 X-Keywords: Wow, so that's what the billboard down mass ave is about.... Anyway, I found most of the number theory topics interesting, and would like to know the proof for the phi function described on pg 25. I'm also interested in the significance of the rienmann hypothesis - what's the big deal? From zacharyozer@gmail.com Fri Oct 14 10:20:16 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.207]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEKF6s027343 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:20:15 -0400 Received: by xproxy.gmail.com with SMTP id t10so452165wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 07:20:15 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=Q/qeh8cXNSxUHnMDMGTHihyqOdBVBUolrmvpk04oQbjN9K1iWwiZbWvWiHaANVg4G1rAwAE8wVevb1BIlrUkYQku3BDtpPmhrvdR9R3uTVnfFE05pIstpQXPjl2xKPgZCpXYgzRWquzqwId3zxsyFEiryer7HOAE+KqR1WhtUuk= Received: by 10.70.16.16 with SMTP id 16mr1179043wxp; Fri, 14 Oct 2005 07:20:15 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Fri, 14 Oct 2005 07:20:15 -0700 (PDT) Message-ID: <4f2613e50510140720w68babb78q28dda12d6b5c7545@mail.gmail.com> Date: Fri, 14 Oct 2005 10:20:15 -0400 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Proof for Theorem 3.1 Page 6 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9EEKF6s027343 Status: RO X-UID: 668 Content-Length: 208 X-Keywords: NonJunk I found the first convoluted statement to be "Thus, in particular, gdc(a,b) divides m and so gcd(a,b) <= m." It is not entirely clear to me why this is true based on the previous steps of the proof. -zozer From ctsims@MIT.EDU Fri Oct 14 10:31:05 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEV56s029484 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:31:05 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEV4bg011337 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:31:04 -0400 (EDT) Received: from HowardRoark.mit.edu ([18.90.7.228]) (authenticated bits=0) (User authenticated as ctsims@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEV2Uv024685 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 10:31:03 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014101900.0280bd00@po12.mit.edu> X-Sender: ctsims@po12.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 10:29:33 -0400 To: 6042-probs@theory.lcs.mit.edu From: Clayton Sims Subject: Reading Comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 669 Content-Length: 274 X-Keywords: NonJunk I would like The implications of the Riemann Hypothesis (page 24) explained in further detail in lecture. The Hypothesis seems to have some implications to the phi function, but I could not determine which connection was supposed to be drawn from the reading. -Clayton From aka_kame@MIT.EDU Fri Oct 14 10:41:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEfh6s031032 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 10:41:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEfgbg022261 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 10:41:42 -0400 (EDT) Received: from [18.58.6.191] (PIERCE-FOUR-FORTY-SIX.MIT.EDU [18.58.6.191]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEfWJE029199 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 10:41:32 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-10-669836657 From: Akari Kameyama Subject: Week 6 Comment Date: Fri, 14 Oct 2005 10:41:31 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.331 X-Spam-Level: * (1.331) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 670 Content-Length: 7085 X-Keywords: NonJunk --Apple-Mail-10-669836657 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed In the RSA scheme 3. Select an integer e such that gcd(e, (p =E2=88=92 1)(q =E2=88=92 1)) = =3D 1. The public key is the pair (e, n). This should be distributed widely. 4. Compute d such that de =E2=89=A1 1 (mod (p =E2=88=92 1)(q =E2=88=92 = 1)). The secret key is the pair (d,n). This should be kept hidden! If we know that d is derived from de =E2=89=A1 1, then since e is =20 distributed widely, doesnt this give away d? I guess I am missing something here... --Apple-Mail-10-669836657 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
In the RSA scheme

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

If we know that d = is derived from=C2=A0de=C2=A0=E2=89=A1=C2=A01, then since e is distributed widely, doesnt = this give away d?=C2=A0=C2=A0
I guess I am missing something = here...
= --Apple-Mail-10-669836657-- From mukkala@MIT.EDU Fri Oct 14 10:47:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EElY6s000361 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:47:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EElXbg028740 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:47:33 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EElRi1001713 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:47:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EElRs6022235; Fri, 14 Oct 2005 10:47:27 -0400 Received: from AP-SIXTY-FOUR.MIT.EDU (AP-SIXTY-FOUR.MIT.EDU [18.153.1.64]) (User authenticated as mukkala@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:47:27 -0400 Message-ID: <20051014104727.ommfax2zre3pk4ww@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:47:27 -0400 From: Praveen Pamidimukkala To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 6 Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 671 Content-Length: 490 X-Keywords: In section 10.3 entitled "Euler's Theorem," Lemma 10.4 states, "Suppose n is a positive integer and k is relatively prime to n. Let k1,...,kr denote all the integers relatively prime to n in the range 0<=ki Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEnK6s000618 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:49:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEnIbg000424 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:49:18 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEnG12002439 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:49:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EEnGCD022396; Fri, 14 Oct 2005 10:49:16 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:49:16 -0400 Message-ID: <20051014104916.1zga4x3jni80o0ks@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:49:16 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 672 Content-Length: 288 X-Keywords: As I'm sure will be convered in friday's lecture, I am having a bit of trouble getting my mind around congruences (page 16 section 7). I understand the definition, but I probably just need more practice working with them in order to develop an intuitive grasp on them. -Rob Crowell From kktyan@MIT.EDU Fri Oct 14 10:52:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEqo6s001359 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:52:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEqmbg004188 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:52:48 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEqkV0003905 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:52:47 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9EEqk6G030042; Fri, 14 Oct 2005 10:52:46 -0400 Received: from BURTON-THREE-SIXTY-SEVEN.MIT.EDU (BURTON-THREE-SIXTY-SEVEN.MIT.EDU [18.247.6.112]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:52:46 -0400 Message-ID: <20051014105246.gjpieggicl1c8wk4@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:52:46 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading 6 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.542 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 673 Content-Length: 714 X-Status: A X-Keywords: I actually found most of the concepts in the reading understandable after rereading a few times. I only have a question which seems trivial in the grand scheme of things: What exactly are the subtle but profound shades of difference between modulo and remainder? i.e. y(mod x) and y rem x? (if I put the first term in correct form). On page 2, it says that "n rem d is the remainder when n is divided by d". But isn't modulo also the remainder when n is divided by d? I'm sorry; it's really a small, trite question, but it's bugging me, as we often see rem and mod used in similar contexts and I'm not really sure what the difference is. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From moscicki@MIT.EDU Fri Oct 14 10:56:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EEuK6s002454 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:56:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EEuJbg008069 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:56:19 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EEuGqK005418 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 10:56:17 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EEuGW4023077; Fri, 14 Oct 2005 10:56:16 -0400 Received: from NONE-FOUR-SEVENTEEN.MIT.EDU (NONE-FOUR-SEVENTEEN.MIT.EDU [18.54.6.162]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 10:56:16 -0400 Message-ID: <20051014105616.mxoxcp0b3pwss0wo@webmail.mit.edu> Date: Fri, 14 Oct 2005 10:56:16 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: week 6 comments MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_736bs5v5j668" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 674 Content-Length: 1716 X-Keywords: This message is in MIME format. --=_736bs5v5j668 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit wrote them last night and then realized i never e-mailed them. --=_736bs5v5j668 Content-Type: text/plain; charset=ISO-8859-1; name="week6comment.txt" Content-Disposition: attachment; filename="week6comment.txt" Content-Transfer-Encoding: 7bit Number theory... I have always been weak in it. Since I know someone actually reads this stuff, I can comment on an unrelated thing: A lot of the time on my homework I get the comment "this is the right reasoning for the proof its just poorly written". I'm not the only one that gets that. Might be something to address. Divisors, divison algorithm: clear enough. Euclids algorithm: clear after a little thought. I think this stuff will require a lot of practice, but it will be fine after that, and I think I don't speak only for myself. Its a little different way of thinking. The Pulverizer: extension of Euclid's Algorithm. Fundamental Theorem of Arithmetic: knew it before but took it for granted. Reading about it is pretty cool :) Cryptography- extremely interesting. Modulo arithmetic- had the good fortune to have dealt with this stuff before in the context of building divisibility tests for some arbitrary number. Turing 2.0- really cool! i think this is how such encryption programs as PGP works? Multiplicative Inverse: It is followable so it seems clear but it remains to be seen whether it is repeatable or not. To answer my own pgp question- i guess not. The part about permuting the sequence is even more cool. This has been the most interesting reading so far! --=_736bs5v5j668-- From lmccart@MIT.EDU Fri Oct 14 11:06:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EF676s004115 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 11:06:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EF5hbg018549 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 11:05:43 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EF5a8m010083 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 11:05:37 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9EF5aVY021021; Fri, 14 Oct 2005 11:05:36 -0400 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 11:05:36 -0400 Message-ID: <20051014110536.5kvnz9esi8g8gco8@webmail.mit.edu> Date: Fri, 14 Oct 2005 11:05:36 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.401 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 675 Content-Length: 106 X-Keywords: I didn't find anything in the reading very confusing but I found the code stuff most interesting. -lauren From antonk@MIT.EDU Fri Oct 14 11:12:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFCs6s004898 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:12:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EFCrHx023273 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:12:53 -0400 (EDT) Received: from silencer ([18.234.0.138]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EFCk1A013184 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:12:46 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] wouldlike to be discussed Date: Fri, 14 Oct 2005 11:12:45 -0400 Organization: Massachusetts Institute of Technology Message-ID: <000701c5d0d1$bb9e6770$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0008_01C5D0B0.348CC770" X-Mailer: Microsoft Office Outlook 11 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXQ0bsBOz7io9UORdSB2Ng4/ugNDQ== X-Spam-Score: -1.214 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 677 Content-Length: 2343 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0008_01C5D0B0.348CC770 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit The prime number theorem and the solution to the google problem. I understand the approximation but does this mean that the first X consecutive numbers will be prime? Anton. ------=_NextPart_000_0008_01C5D0B0.348CC770 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

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

 

Anton.

 

------=_NextPart_000_0008_01C5D0B0.348CC770-- From nancyk@MIT.EDU Fri Oct 14 11:16:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFGe6s005752 for <6042-probs@theory.csail.MIT.EDU>; Fri, 14 Oct 2005 11:16:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EFAYQ7021317; Fri, 14 Oct 2005 11:16:39 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EF6pxc010672; Fri, 14 Oct 2005 11:06:52 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EF6pmt024262; Fri, 14 Oct 2005 11:06:51 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 11:06:51 -0400 Message-ID: <20051014110651.ld5tvwsmnf4848og@webmail.mit.edu> Date: Fri, 14 Oct 2005 11:06:51 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 6 reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 678 Content-Length: 265 X-Keywords: NonJunk Hi, I was confused about Theorem 3.1 under the section "Linear Combinations and the GCD." I don't know if I understand why it refers to a "positive linear combination." Does this not mean that each of the coefficients of a and b are positive? Thanks. Nancy Keuss From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:29:30 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFTU6s009549; Fri, 14 Oct 2005 11:29:30 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFTUk0003210; Fri, 14 Oct 2005 11:29:30 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFTTND003207; Fri, 14 Oct 2005 11:29:30 -0400 Date: Fri, 14 Oct 2005 11:29:29 -0400 (EDT) From: Hanson Zhou To: Aston Motes cc: 6042-probs@theory.csail.mit.edu Subject: Re: [TA-name] Week 6 Comments In-Reply-To: <200510140431.j9E4VvJk009896@outgoing.mit.edu> Message-ID: References: <200510140431.j9E4VvJk009896@outgoing.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 679 Content-Length: 365 X-Keywords: See Lemma 7.2. By TA-name, you mean "Hanson" :). -Hanson On Fri, 14 Oct 2005, Aston Motes wrote: > The multiplicative inverse modulo primes (page 19) is a concept I've not > heard of before, and seems pretty cool. > > > > I missed, however, how we dropped the rem on page 20 in proving that m* x > k^-1 is congruent to m (mod p) > > > > - Aston > > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:32:45 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFWj6s010136; Fri, 14 Oct 2005 11:32:45 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFWj4e003247; Fri, 14 Oct 2005 11:32:45 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFWj2L003244; Fri, 14 Oct 2005 11:32:45 -0400 Date: Fri, 14 Oct 2005 11:32:45 -0400 (EDT) From: Hanson Zhou To: Lohith Kini cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 6 Lecture Notes In-Reply-To: <434F4C87.6050906@mit.edu> Message-ID: References: <434F4C87.6050906@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 680 Content-Length: 636 X-Keywords: The first sequence has all those numbers because 1. there are p numbers in the sequence 2. they are all different mod p The proof shows how to prove the latter. Essentially, you use cancellation and proof by contradiction to show distinctness. If this is still confusing, go to office hours. -Hanson On Fri, 14 Oct 2005, Lohith Kini wrote: > Hi Hanson, > > The only question I had after reading the lecture notes was the proof of > Corollary 8.3. I don't understand why the first sequence must contain > all of the numbers from 0 to p-1. I would appreciate it if you could go > explain that a little more. > > Thanks, > Lohith > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:36:13 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFaC6s010736; Fri, 14 Oct 2005 11:36:12 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFaCsB003283; Fri, 14 Oct 2005 11:36:12 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFaCF5003280; Fri, 14 Oct 2005 11:36:12 -0400 Date: Fri, 14 Oct 2005 11:36:12 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Number Theory Reading Comments In-Reply-To: <6.2.3.4.2.20051014050241.01ea38b8@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051014050241.01ea38b8@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 681 Content-Length: 372 X-Keywords: yes, part 1. On Fri, 14 Oct 2005, Kevin Wang wrote: > I had a question about Section 8.4, page 22 where we are computing > the key k. I don't see how the second line of the congruence (which > uses "part 2 of Lemma 7.2") follows from the first line and part 2 of > Lemma 7.2: specifically how (mk rem p) becomes just mk. Does it use > part 1 of Lemma 2? > > -Kevin > > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:40:53 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFer6s011430; Fri, 14 Oct 2005 11:40:53 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFerx5003331; Fri, 14 Oct 2005 11:40:53 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFerC1003328; Fri, 14 Oct 2005 11:40:53 -0400 Date: Fri, 14 Oct 2005 11:40:53 -0400 (EDT) From: Hanson Zhou To: Vera Carr cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Intro to Number Theory In-Reply-To: <434FD35A.2010003@mit.edu> Message-ID: References: <434FD35A.2010003@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 682 Content-Length: 825 X-Keywords: I'm not sure I understand this question...f(27) cannot be decomposed in the way you have written. f(27)=f(3^3), which we know how to compute. f(ab)=f(a)f(b) for a and b that are relatively prime. You are not meant to understand why this works yet. -Hanson On Fri, 14 Oct 2005, Vera Carr wrote: > Pg. 25 - Theorem 10.1 > > What i don't understand is if you have, for example, 27 as the number > you want to find the relative primes of by part 1 of Theorem 10.1 you > can way f(27)=f(3*3*3)=f(3)*f(3)*f(3) and we know for 3 that is has 2 > relative primes (2 and 1) so we can say f(3)*f(3)*f(3)=2*2*2? I realize > that by using part 2 (where p is a prime...) of Theorm 10.1 you get the > right answer but I don't understand why you can't multiply the number of > relative primes that each factor has as noted in part 1? > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 11:42:47 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFgl6s011868; Fri, 14 Oct 2005 11:42:47 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EFglPA003352; Fri, 14 Oct 2005 11:42:47 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EFgl4Z003349; Fri, 14 Oct 2005 11:42:47 -0400 Date: Fri, 14 Oct 2005 11:42:47 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 6 reading assignment In-Reply-To: <20051014110651.ld5tvwsmnf4848og@webmail.mit.edu> Message-ID: References: <20051014110651.ld5tvwsmnf4848og@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 683 Content-Length: 430 X-Keywords: Positive linear combination refers to a linear combination that evaluates to a positive number. -Hanson On Fri, 14 Oct 2005, Nancy L Keuss wrote: > Hi, > > I was confused about Theorem 3.1 under the section "Linear Combinations and the > GCD." I don't know if I understand why it refers to a "positive linear > combination." Does this not mean that each of the coefficients of a and b are > positive? Thanks. > > Nancy Keuss > From hmzhou@blackbird.csail.mit.edu Fri Oct 14 12:10:22 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EGAM6s021410; Fri, 14 Oct 2005 12:10:22 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9EGAMqZ003746; Fri, 14 Oct 2005 12:10:22 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9EGAMAd003743; Fri, 14 Oct 2005 12:10:22 -0400 Date: Fri, 14 Oct 2005 12:10:22 -0400 (EDT) From: Hanson Zhou To: Zachary Adam Ozer cc: 6042-probs@theory.lcs.mit.edu Subject: Re: Proof for Theorem 3.1 Page 6 In-Reply-To: <4f2613e50510140720w68babb78q28dda12d6b5c7545@mail.gmail.com> Message-ID: References: <4f2613e50510140720w68babb78q28dda12d6b5c7545@mail.gmail.com> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 684 Content-Length: 367 X-Keywords: What is confusing here? Note that m is a positive number, so if a|m then clearly a<=m. -Hanson On Fri, 14 Oct 2005, Zachary Adam Ozer wrote: > I found the first convoluted statement to be "Thus, in particular, > gdc(a,b) divides m and so gcd(a,b) <= m." It is not entirely clear to > me why this is true based on the previous steps of the proof. > > -zozer > > From dshin@MIT.EDU Fri Oct 14 13:37:54 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EHbs6s009543 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 13:37:54 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EHbq3w012175; Fri, 14 Oct 2005 13:37:52 -0400 (EDT) Received: from [192.168.1.3] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9EHbjMu008437; Fri, 14 Oct 2005 13:37:45 -0400 (EDT) Message-ID: <434FECDA.5060305@mit.edu> Date: Fri, 14 Oct 2005 13:37:30 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 6 Comment References: In-Reply-To: Content-Type: multipart/alternative; boundary="------------010406080202030305080709" X-Spam-Score: 2.82 X-Spam-Level: ** (2.82) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 685 Content-Length: 6401 X-Status: A X-Keywords: This is a multi-part message in MIME format. --------------010406080202030305080709 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Yes, we know de = 1 (mod (p-1)(q-1)), but we don't know what (p-1)(q-1) is. DS Akari Kameyama wrote: > In the RSA scheme > > 3. Select an integer e such that gcd(e, (p − 1)(q − 1)) = 1. > The public key is the pair (e, n). This should be distributed widely. > 4. Compute d such that de ≡ 1 (mod (p − 1)(q − 1)). > The secret key is the pair (d,n). This should be kept hidden! > > If we know that d is derived from de ≡ 1, then since e is distributed > widely, doesnt this give away d? > I guess I am missing something here... --------------010406080202030305080709 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: 8bit Yes, we know de = 1 (mod (p-1)(q-1)), but we don't know what (p-1)(q-1) is.

DS

Akari Kameyama wrote:
In the RSA scheme

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

If we know that d is derived from de ≡ 1, then since e is distributed widely, doesnt this give away d?  
I guess I am missing something here...
--------------010406080202030305080709-- From meyer@imap.theory.csail.mit.edu Fri Oct 14 14:06:54 2005 Message-ID: <434FF3BF.1040206@csail.mit.edu> Date: Fri, 14 Oct 2005 14:06:55 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Karena Tyan Subject: Re: [Hanson] Reading 6 Comments References: <20051014105246.gjpieggicl1c8wk4@webmail.mit.edu> In-Reply-To: <20051014105246.gjpieggicl1c8wk4@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 787 Status: RO X-UID: 686 X-Keywords: you're not alone on remainder vs mod; will be clarified in lecture today. regards, A. Karena Tyan wrote: >I actually found most of the concepts in the reading understandable after >rereading a few times. I only have a question which seems trivial in the grand >scheme of things: > >What exactly are the subtle but profound shades of difference between modulo and >remainder? i.e. y(mod x) and y rem x? (if I put the first term in correct >form). On page 2, it says that "n rem d is the remainder when n is divided by >d". But isn't modulo also the remainder when n is divided by d? I'm sorry; >it's really a small, trite question, but it's bugging me, as we often see rem >and mod used in similar contexts and I'm not really sure what the difference >is. > >- Karena > > > > From shreyes19@gmail.com Fri Oct 14 14:11:39 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.195]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIBd6s016764 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 14:11:39 -0400 Received: by xproxy.gmail.com with SMTP id s7so498091wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 11:11:38 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=SylmuW63EZoyMu4FP8Cs2/riDBmplVyHEdsjzHK5Cqa0AVUYUJt8+d91U1gdaPf5aiBJLB6vo7p/1Xk0VnP1QT5ximzAZQCrbc4gZydI43jEybNwRzWgyiupnaE7wolsECqyaQp0BNIGgfsRdoZiIE64gdUwqd/+LxQrios9Lnc= Received: by 10.70.129.10 with SMTP id b10mr1340399wxd; Fri, 14 Oct 2005 11:11:38 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Fri, 14 Oct 2005 11:11:38 -0700 (PDT) Message-ID: <3c33fe380510141111w556d188ag6c919e8a014e5ab6@mail.gmail.com> Date: Fri, 14 Oct 2005 14:11:38 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 6 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_25948_28406409.1129313498754" Status: RO X-UID: 687 Content-Length: 1523 X-Keywords: NonJunk ------=_Part_25948_28406409.1129313498754 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, Here are my comments for the week 6 reading: The part that still remains a little unclear to me is the difference in use between rem and mod. On page 18, it makes the following assertions: 1. a = =3D (a rem n) (mod n) 2. a =3D b (mod n) if and only if (a rem n) =3D (b rem n) These make sense to me, but I'm still confused on when to use mod and when to use rem. For example, Lemma 8.2 uses mod, while the corrollary 8.3 you got from it uses rem. Are there specific cases where one or the other is preferred? Thanks, Shreyes Seshasai Group 7 ------=_Part_25948_28406409.1129313498754 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

Here are my comments for the week 6 reading:

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

Thanks,
Shreyes Seshasai
Group 7
------=_Part_25948_28406409.1129313498754-- From hkhall@MIT.EDU Fri Oct 14 14:26:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIQQ6s020089 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 14:26:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EIQOYa029226 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 14:26:24 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EIQH5r027396 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 14 Oct 2005 14:26:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9EIQHoN010631; Fri, 14 Oct 2005 14:26:17 -0400 Received: from PLP-TWELVE.MIT.EDU (PLP-TWELVE.MIT.EDU [18.218.1.12]) (User authenticated as hkhall@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 14 Oct 2005 14:26:17 -0400 Message-ID: <20051014142617.7qezvz5mrcgs400c@webmail.mit.edu> Date: Fri, 14 Oct 2005 14:26:17 -0400 From: Harrison K Hall To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 6 comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 688 Content-Length: 428 X-Keywords: NonJunk David- Sorry this has taken me so long to send. My parents came into town and I just lost track of time. I am doing the TP now as well. Hopefully I can still get some credit out of it. The part of the reading that I found most difficult is Fermat's Theorem. Even though I have seen it before I still Don't really understand its proof. I understand how it can be useful, but not its workings. Thanks for understanding, Harrison From minilek@theory.csail.mit.edu Fri Oct 14 14:47:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIl06s023597 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 14:47:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EIkvYa020966; Fri, 14 Oct 2005 14:46:57 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EIkk07006053 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Fri, 14 Oct 2005 14:46:50 -0400 (EDT) Message-ID: <434FFD27.5010401@theory.csail.mit.edu> Date: Fri, 14 Oct 2005 14:47:03 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051011) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Jelani] LN6 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 689 Content-Length: 634 X-Keywords: NonJunk The idea was that if I wanted to send a message to you, I'd represent the message as a number, m, and send you mk (mod p). p would be public knowledge, and k would be our secret. You could then decrypt mk by multiplying by k^{-1} (the inverse of k). Later a vulnerability was discussed where if an adversary knew m mod p and mk mod p, they could figure out the secret, k using Fermat's Little Theorem. Are things clearer now? -Jelani Amanda Seybold wrote: > The explanation of how and why multiplicative inverses work to decrypt > messages (pg. 20) went by a little fast. Perhaps in class it could be > demonstrated. From minilek@theory.csail.mit.edu Fri Oct 14 14:50:40 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EIoe6s023992 for <6042-probs@theory.lcs.mit.edu>; Fri, 14 Oct 2005 14:50:40 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EIobYa024525; Fri, 14 Oct 2005 14:50:37 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EIoVMU007431 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Fri, 14 Oct 2005 14:50:31 -0400 (EDT) Message-ID: <434FFE09.3050604@theory.csail.mit.edu> Date: Fri, 14 Oct 2005 14:50:49 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051011) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Benjamin M. Schwartz" CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Jelani] reading References: <434F1812.7010508@mit.edu> In-Reply-To: <434F1812.7010508@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.11 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 690 Content-Length: 317 X-Keywords: NonJunk Look at step 4 where the receiver computes d such that de = 1 (mod (p-1)(q-1)) (note (p-1)(q-1) is phi(n)). Think about how the receiver would compute such a d given that he knows phi(n). Benjamin M. Schwartz wrote: > page 27: > So what do phi() and inverses have to do with RSA? > Will I find out in lecture? From meyer@imap.theory.csail.mit.edu Fri Oct 14 11:10:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9EFAb6s004728 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 11:10:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9EFAYI3021317 for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 11:10:36 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9EF9TAF011792 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 14 Oct 2005 11:09:30 -0400 (EDT) Message-Id: <5.2.1.1.2.20051014110901.00bfedd8@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 14 Oct 2005 11:09:29 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 6 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.923 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 67 X-UID: 691 X-Keywords: NonJunk I found the description of RSA encoding very interesting. Jehan From meyer@imap.theory.csail.mit.edu Mon Oct 17 08:30:24 2005 Message-ID: <43539962.3010305@csail.mit.edu> Date: Mon, 17 Oct 2005 08:30:26 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez Subject: Re: Wed Oct 5 Reading assignment reply, R13 References: <20051004155212.soy36pqy00gss00k@webmail.mit.edu> In-Reply-To: <20051004155212.soy36pqy00gss00k@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 660 Status: RO X-UID: 692 X-Keywords: In case you haven't gotten this straight already: A "Lemma," like a Theorem or Corollary, asserts the truth of some mathematical statement. A "proof" is an explanation of why the asserted statement is true. Important statements are generally labeled "Theorems." An important statement that is a special case of, or follows directly from, a Theorem is called a "Corollary" of that Theorem. A statement whose importance comes mainly from its later use in proving a Theorem is called a "Lemma." Regards, A. Daniel A Gutierrez wrote: > I'm okay with this material, a sidenote question though is what is the > difference between a Lemma and a proof? From juang@MIT.EDU Fri Oct 21 22:12:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9M2Ctd0002364 for <6042-probs@theory.csail.mit.edu>; Fri, 21 Oct 2005 22:12:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9M2Csow000709 for <6042-probs@theory.csail.mit.edu>; Fri, 21 Oct 2005 22:12:54 -0400 (EDT) Received: from [18.238.5.31] (EASTCAMPUS-FIVE-FORTY-TWO.MIT.EDU [18.238.5.31]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9M2Cp2M025157 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 21 Oct 2005 22:12:52 -0400 (EDT) Message-ID: <4359A010.6030908@mit.edu> Date: Fri, 21 Oct 2005 22:12:32 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [David] Week 7 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 694 Content-Length: 342 X-Keywords: NonJunk On Page 14 of the reading: "Theorem 5.8. The Mating Algorithm marries every boy to his optimal mate and every girl to her pessimal mate." This is surprising. (Well, it's not _that_ surprising, and it's only surprising before the proof is presented. But still, mildly surprising.) Is there a practical "gender-equal" algorithm? Jason. From lmccart@MIT.EDU Fri Oct 21 23:47:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9M3lrd0013679 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 21 Oct 2005 23:47:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9M3lqow015804 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 21 Oct 2005 23:47:52 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9M3loF8003938 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 21 Oct 2005 23:47:50 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9M3loHF002287; Fri, 21 Oct 2005 23:47:50 -0400 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 21 Oct 2005 23:47:50 -0400 Message-ID: <20051021234750.c0zpnoc756lw84oo@webmail.mit.edu> Date: Fri, 21 Oct 2005 23:47:50 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 7 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 695 Content-Length: 459 X-Status: A X-Keywords: The part of the reading that I found most confusing was the 'extended euclidean algorithm'. I think I understand the concept but a brief review of the idea in class would be helpful. Also, would it be possible to switch groups? It's really difficult to work with that asian kid with braces, I'm not sure of his name, because he talks to me like I'm stupid. When I have the correct answer and he does not, he refuses to listen to my ideas. thanks, lauren From dshin@MIT.EDU Sat Oct 22 01:11:20 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9M5BKd0025151; Sat, 22 Oct 2005 01:11:20 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9M5BJaP018400; Sat, 22 Oct 2005 01:11:19 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9M5BJKr000781; Sat, 22 Oct 2005 01:11:19 -0400 (EDT) Received: from [192.168.1.5] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9M5BCok012576; Sat, 22 Oct 2005 01:11:12 -0400 (EDT) Message-ID: <4359C9EE.7070505@mit.edu> Date: Sat, 22 Oct 2005 01:11:10 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason Juang CC: 6042-probs@theory.csail.mit.edu, 6042 staff <6042-staff@theory.csail.mit.edu> Subject: Re: [David] Week 7 Comments References: <4359A010.6030908@mit.edu> In-Reply-To: <4359A010.6030908@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 696 Content-Length: 1705 X-Keywords: A very good question. I do not know of algorithms that "naturally" induce a "gender-equal" marriage. However, it is usually possible to construct marriages that are neither male-optimal nor female-optimal. There is a nice result that the set of ALL stable marriages form what is called a "distribute lattice": roughly, this means that given any two stable marriages M and M', you can generate two other stable marriages via some operations. The word "lattice" should hint at this idea - the geometric analogue would be like taking two vectors v and v' and generating two other vectors (v+v' and v-v'). In particular, if we have M: male-optimal marriage M': female-optimal marriage we can generate: M ^ M': matching obtainined by giving each woman the worse of her partners in the two matchings M v M': matching obtainined by giving each woman the better of her partners in the two matchings Usually, M ^ M' and M v M' will neither be male-optimal nor female-optimal. (You might want to prove to yourself that these are still stable marriages) For even more of a challege, you might want to show that these stable marriage operations are distributive (hence, "distributive lattice"): 1. A ^ (B v C) = (A ^ B) v (A ^ C) 2. A v (B ^ C) = (A v B) ^ (A v C) for all stable marriages A, B, C. DS Jason Juang wrote: > On Page 14 of the reading: > > "Theorem 5.8. The Mating Algorithm marries every boy to his optimal > mate and every girl to her pessimal mate." > > This is surprising. (Well, it's not _that_ surprising, and it's only > surprising before the proof is presented. But still, mildly > surprising.) Is there a practical "gender-equal" algorithm? > > Jason. From lye@MIT.EDU Sat Oct 22 10:02:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ME2hd0015759 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 10:02:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ME2fow012421 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 10:02:42 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ME2eBN002968 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 10:02:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9ME2e3R003006; Sat, 22 Oct 2005 10:02:40 -0400 Received: from RANDOM-FOUR-FORTY.MIT.EDU (RANDOM-FOUR-FORTY.MIT.EDU [18.243.6.185]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sat, 22 Oct 2005 10:02:40 -0400 Message-ID: <20051022100240.iwx7ot6u5ykosowo@webmail.mit.edu> Date: Sat, 22 Oct 2005 10:02:40 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 7 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 697 Content-Length: 430 X-Keywords: NonJunk Section 5.5, Theorem 5.4: Everyone is married by the Mating Algorithm I found this most difficult to follow because it seems like a boy could wind up not serenading anyone, if he and another boy always happened to serenade the same girl. (Of course this isn't true because there are no ties in ranking -- there's always at least 1 girl for every boy left over.) Maybe an example of the algorithm at work would have been helpful? From ereid@MIT.EDU Sat Oct 22 16:04:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MK4Gd0001161 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 16:04:16 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MK4Fow018168 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 16:04:15 -0400 (EDT) Received: from [18.242.5.109] (NEXT-ONE-O-NINE.MIT.EDU [18.242.5.109]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MK48w4014517 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 16:04:08 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <6D0B2A24-A01B-46D0-8083-DAA7C23F25F5@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] reading comments Date: Sat, 22 Oct 2005 16:04:00 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 698 Content-Length: 130 X-Keywords: I found understanding why the mating algorithm always worked the most difficult, but it makes sense if I think about it enough. From cwong08@MIT.EDU Sat Oct 22 16:29:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MKTZd0003794 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 16:29:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MKTY5Y006296 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 16:29:35 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9MKTJol017880 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 16:29:32 -0400 (EDT) Message-ID: <435AA108.2040306@mit.edu> Date: Sat, 22 Oct 2005 16:28:56 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 7 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 699 Content-Length: 275 X-Keywords: I found the stable marriage problem a little depressing. In some cases, people were forced to marry their least preferred guy/girl, yet all the marriages were stable. It was very interesting, though, about how true it applies to the world. I really enjoyed that example. From bens@MIT.EDU Sat Oct 22 17:41:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MLfQd0018386 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 17:41:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MLfPXO005939 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 17:41:25 -0400 (EDT) Received: from [18.221.0.117] ([18.221.0.117]) (authenticated bits=0) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MLfMcD025334 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 17:41:23 -0400 (EDT) Message-ID: <435AB201.40703@mit.edu> Date: Sat, 22 Oct 2005 17:41:21 -0400 From: "Benjamin M. Schwartz" User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051014) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Stable buddy problem Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 700 Content-Length: 254 X-Keywords: I was surprised that the stable marriage problem is always solvable. I would like to see any connection there is between it and antichains of anded partial orders (the girl's order and the boy's order, like the height and age thing in class). -Ben From scholtz@MIT.EDU Sat Oct 22 17:51:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MLpBd0018900 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 17:51:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MLpAXO010605 for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 17:51:10 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MLp8Av026304 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 22 Oct 2005 17:51:09 -0400 (EDT) Message-Id: <200510222151.j9MLp8Av026304@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 7 Comments Date: Sat, 22 Oct 2005 17:51:10 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0003_01C5D731.3068A1A0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXXUrb0wTDpSYL1RYi8RRJI/fJDlQ== X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 701 Content-Length: 2145 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0003_01C5D731.3068A1A0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I wasn't sure how to create the invariant predicate for Problem 2 (page 4). It would help if this example was reviewed in lecture. Eddie Scholtz ------=_NextPart_000_0003_01C5D731.3068A1A0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

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

 

Eddie Scholtz

------=_NextPart_000_0003_01C5D731.3068A1A0-- From ajshafer@MIT.EDU Sat Oct 22 18:25:56 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MMPud0024823 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 18:25:56 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9MMPtXO027912 for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 18:25:55 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9MMPrFk000289 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sat, 22 Oct 2005 18:25:54 -0400 (EDT) Message-ID: <000201c5d757$8fbaae70$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 7 Comments Date: Sat, 22 Oct 2005 18:25:51 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -1.215 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 702 Content-Length: 545 X-Keywords: For the Stable Marriage problem: This pattern showed how to make two possible lists (start with boys on balconies or girls on balconies). Are there more possible matches? Is the only way to find all posible stable marriages to do some type of deeper search, or does this algorithm generate the only two possibilities? Thanks, Andrew -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From vixen@MIT.EDU Sat Oct 22 19:02:41 2005 Return-Path: Received: from mrow.mit.edu (MROW.MIT.EDU [18.238.3.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9MN2fd0028306 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 22 Oct 2005 19:02:41 -0400 Received: (from vixen@localhost) by mrow.mit.edu (8.12.9) id j9MN2eeJ024582; Sat, 22 Oct 2005 19:02:40 -0400 (EDT) Date: Sat, 22 Oct 2005 19:02:39 -0400 (EDT) From: Amanda L Seybold To: 6042-probs@theory.lcs.MIT.EDU Subject: LN7 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Status: RO X-UID: 703 Content-Length: 197 X-Keywords: I found the section on derived variables (pg 8) confusing. I didn't understand what they were or what role they played. There also didn't seem to be much done with state machines in the notes. From hectorb@MIT.EDU Sat Oct 22 20:23:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9N0NBd0003299 for <6042-probs@theory.csail.MIT.EDU>; Sat, 22 Oct 2005 20:23:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9N0N85w019744; Sat, 22 Oct 2005 20:23:08 -0400 (EDT) Received: from besko.mit.edu (BESKO.MIT.EDU [18.246.0.69]) (authenticated bits=56) (User authenticated as hectorb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9N0N1Bs010900 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sat, 22 Oct 2005 20:23:02 -0400 (EDT) Received: (from hectorb@localhost) by besko.mit.edu (8.12.9) id j9N0N15x020818; Sat, 22 Oct 2005 20:23:01 -0400 Subject: [Hanson] Week 7 Comments From: Hector Beltran To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Sat, 22 Oct 2005 20:23:00 -0400 Message-Id: <1130026981.20607.2.camel@besko.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 704 Content-Length: 339 X-Keywords: "Similarly, a boy keeps serenading the girl he most prefers among those on his list until he must cross her off, at which point he serenades the next most preferred girl on his list..." p.14 I would find it useful to see a live demonstration of the mating algorithm by some 6.042 students, with boys actually serenading girls they like. From aston@MIT.EDU Sun Oct 23 01:49:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9N5n2d0018967 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 01:49:02 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9N5mvIJ005292 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 01:48:57 -0400 (EDT) Received: from astonlaptop (NEW-ONE-SIXTY-NINE.MIT.EDU [18.241.5.169]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9N5mtMn005481 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 01:48:55 -0400 (EDT) Message-Id: <200510230548.j9N5mtMn005481@outgoing.mit.edu> Reply-To: From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 7 Comments Date: Sun, 23 Oct 2005 01:48:54 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00AC_01C5D773.ED65F4A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXXlXQSLZg3BliaQdq0m3apvENY7Q== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.038 X-Spam-Level: ** (2.038) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 705 Content-Length: 1246 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_00AC_01C5D773.ED65F4A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Page 9's "weakly decreasing" variable is an interesting concept. I'd like to see more in lecture about the termination guarantee when it's on a variable like this. - Aston ------=_NextPart_000_00AC_01C5D773.ED65F4A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable
Page = 9's "weakly=20 decreasing" variable is an interesting concept. I'd like to see more in = lecture=20 about the termination guarantee when it's on a variable like=20 this.
 
    -=20 Aston
------=_NextPart_000_00AC_01C5D773.ED65F4A0-- From dowgun@MIT.EDU Sun Oct 23 14:09:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NI9Ld0010442 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:09:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NI9J2F029062 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:09:20 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NI9IXt024411 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:09:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9NI9IoX012999; Sun, 23 Oct 2005 14:09:18 -0400 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 14:09:18 -0400 Message-ID: <20051023140918.57wc546kxj5s0k4g@webmail.mit.edu> Date: Sun, 23 Oct 2005 14:09:18 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] state machines MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 706 Content-Length: 446 X-Keywords: NonJunk It seems to me that proving termination is the most important use of the Well Ordered Principle. My question for the week is - are there any other ways to prove termination? Every example given here, including the Stable Marraige problem, seems to have termination proven by a strictly decreasing variable over the set of natural numbers. In fact, are there any examples of state machines that arent operating solely on the natural numbers? Neil From rnjacobs@MIT.EDU Sun Oct 23 14:17:38 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NIHcd0011468 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:38 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9NIHbDT026506 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:37 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NIHbpH028951 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:37 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9NIHUok011879 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 14:17:31 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j9NIHUeP022329; Sun, 23 Oct 2005 14:17:30 -0400 Message-Id: <200510231817.j9NIHUeP022329@white-meteo.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU Subject: [Jelani] Reading response Date: Sun, 23 Oct 2005 14:17:30 -0400 From: r n jacobs X-Spam-Score: 3.631 X-Spam-Level: *** (3.631) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 707 Content-Length: 65 X-Keywords: NonJunk Everything seemed straightforward this week. - Robert Jacobs From bilodeau@MIT.EDU Sun Oct 23 15:30:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NJUsd0021775 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 15:30:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NJUq2F008926 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 15:30:53 -0400 (EDT) Received: from pbilodeau (MACGREGOR-TWO-TWENTY-ONE.MIT.EDU [18.239.5.221]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NJUplG005476 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 15:30:51 -0400 (EDT) Message-Id: <200510231930.j9NJUplG005476@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [jelani] state machine reading comments Date: Sun, 23 Oct 2005 15:30:49 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5D7E6.BF760580" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXYCEYSmyGnqkMsRHWg5+naEpCGTQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.779 X-Spam-Level: ** (2.779) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 708 Content-Length: 2413 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5D7E6.BF760580 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit The hardest part of this reading to understand came at the end, when, in the discussion of the mating algorithm, names were replaces by variable, and similar variables at that, B B' G etc. yet the language of the mating problem was retained. (B prefers to marry G.) I guess the problem for me was there being no distinction between discussion using mathematical language, sets and functions, state machines, and the vocabulary of the simple, intuitive real world application. ------=_NextPart_000_0000_01C5D7E6.BF760580 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

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

------=_NextPart_000_0000_01C5D7E6.BF760580-- From ridell@MIT.EDU Sun Oct 23 17:01:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NL1gd0003165 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 17:01:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NL1eKA022030 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 17:01:41 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NL1YH2017754 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 17:01:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9NL1Ynr031880; Sun, 23 Oct 2005 17:01:34 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 17:01:34 -0400 Message-ID: <20051023170134.3bvzlnukri80kcc8@webmail.mit.edu> Date: Sun, 23 Oct 2005 17:01:34 -0400 From: Rebecca Idell To: 6042-probs@theory.lcs.MIT.EDU Subject: idell week 7 reading [hanson] MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 709 Content-Length: 351 X-Keywords: NonJunk I think that the marriage problem is very interesting. In class can we go over more thoroughly how to make it into a state machine like on p12 of the reading? -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From avalys@MIT.EDU Sun Oct 23 17:07:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NL7od0003390 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:07:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NL7nKA025021 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:07:49 -0400 (EDT) Received: from [18.96.6.192] (SIMMONS-FOUR-FORTY-SEVEN.MIT.EDU [18.96.6.192]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NL7mJ5018683 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:07:48 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Reading comments Date: Sun, 23 Oct 2005 17:07:46 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 710 Content-Length: 115 X-Keywords: NonJunk I found section 5.6 ("...And the Boys Live Especially Happily", page 14) confusing (and surprising). Alex Valys From jstritar@MIT.EDU Sun Oct 23 17:38:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NLcXd0007966 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:38:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NLcVKA010885 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:38:31 -0400 (EDT) Received: from [18.245.6.249] (BAKER-FIVE-O-FOUR.MIT.EDU [18.245.6.249]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NLcRjl023412 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 17:38:29 -0400 (EDT) Message-ID: <435C02D3.9020203@mit.edu> Date: Sun, 23 Oct 2005 17:38:27 -0400 From: Jon Stritar User-Agent: Thunderbird 1.4.1 (Windows/20051006) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 711 Content-Length: 230 X-Keywords: NonJunk 3.1 Proving Correctness I thought this part was pretty interesting because I have taken both 6.041 and 6.004 which talk about state machines, but I have never used them in this context or had to prove partial correctness. Jon From aka_kame@MIT.EDU Sun Oct 23 17:40:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NLevd0008114 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 17:40:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NLeuKA012009 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 17:40:56 -0400 (EDT) Received: from [18.187.7.125] (STRATTON-SIX-THIRTY-SIX.MIT.EDU [18.187.7.125]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NLettU023771 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 17:40:55 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <4A29788B-A474-44C9-801C-4B35EC9E12DB@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-8--674883877 From: Akari Kameyama Subject: Week 7 Comments Date: Sun, 23 Oct 2005 17:40:55 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.218 X-Spam-Level: * (1.218) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 712 Content-Length: 4611 X-Status: A X-Keywords: NonJunk --Apple-Mail-8--674883877 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed "Acting on behalf of a consortium of major hospitals (playing the =20 role of the girls), the NRMP has, since the turn of the twentieth century, =20 assigned each year =92s pool of medical school graduates (playing the role of boys) to hospital residencies =20 (formerly called =93internships=94)" "Theorem 5.8. The Mating Algorithm marries every boy to his optimal =20 mate and every girl to her pessimal mate" So if every "girl" is married to her pessimal mate, according to the =20 Mating Algorithm, when this algorithm was applied to the intern-=20 hospital situation, every hospital was getting paired with their =20 least prefered student? I would have thought that the hospitals (who =20= are the "more important" side, i think) would have been the "boys" =20 and gotten their optimal choice. --Apple-Mail-8--674883877 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=WINDOWS-1252
"Acting on behalf of a consortium of major = hospitals (playing the role of the=A0
girls), the NRMP = has, since the turn of the twentieth century, assigned each year =92s = pool of medical=A0
school graduates (playing the role of boys) = to hospital residencies (formerly called = =93internships=94)"

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

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

= --Apple-Mail-8--674883877-- From minilek@theory.csail.mit.edu Sun Oct 23 19:00:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NN0Cd0021529 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 19:00:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NN0BKA024655; Sun, 23 Oct 2005 19:00:11 -0400 (EDT) Received: from [18.250.5.115] (ASHDOWN-ONE-FIFTEEN.MIT.EDU [18.250.5.115]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NN03oK006173 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sun, 23 Oct 2005 19:00:06 -0400 (EDT) Message-ID: <435C15FB.3010303@theory.csail.mit.edu> Date: Sun, 23 Oct 2005 19:00:11 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.7 (X11/20051011) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda L Seybold CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: LN7 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 713 Content-Length: 864 X-Status: A X-Keywords: NonJunk Derived variables are really functions that map states to numbers. If you recall in our proof of termination of the gcd algorithm (page 6 of the lecture notes), we argued termination by reasoning that the variable y was strictly decreasing. y is actually a derived variable, where the states are the (x,y) pairs in our execution of the Euclidean algorithm. Derived variables are useful because if we can come up with a derived variable that is a natural number and strictly decreases on state transitions, then we can invoke the well-ordering principle to prove that the state machine terminates. Does this help? -Jelani Amanda L Seybold wrote: > I found the section on derived variables (pg 8) confusing. I didn't > understand what they were or what role they played. There also didn't > seem to be much done with state machines in the notes. From meyer@imap.theory.csail.mit.edu Sun Oct 23 19:29:28 2005 Message-ID: <435C1CDD.9020803@csail.mit.edu> Date: Sun, 23 Oct 2005 19:29:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama Subject: Re: Week 7 Comments References: <4A29788B-A474-44C9-801C-4B35EC9E12DB@mit.edu> In-Reply-To: <4A29788B-A474-44C9-801C-4B35EC9E12DB@mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Content-Length: 1014 Status: RO X-UID: 714 X-Keywords: They did originally, but there was an uproar among the interns, and the case was made that since any stable set was defensible, the interns should be given precedence. Regards, A. Akari Kameyama wrote: > "Acting on behalf of a consortium of major hospitals (playing the role > of the > girls), the NRMP has, since the turn of the twentieth century, > assigned each year ’s pool of medical > school graduates (playing the role of boys) to hospital residencies > (formerly called “internships”)" > > "Theorem 5.8. The Mating Algorithm marries every boy to his optimal > mate and every girl to her pessimal > mate" > > So if every "girl" is married to her pessimal mate, according to the > Mating Algorithm, when this algorithm was applied to the > intern-hospital situation, every hospital was getting paired with > their least prefered student? I would have thought that the hospitals > (who are the "more important" side, i think) would have been the > "boys" and gotten their optimal choice. > From kjhollen@MIT.EDU Sun Oct 23 19:47:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NNlCd0027218 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 19:47:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NNlBKA021952 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 19:47:11 -0400 (EDT) Received: from MACGREGOR-TWO-SIXTY-ONE.MIT.EDU (MACGREGOR-TWO-SIXTY-ONE.MIT.EDU [18.239.6.6]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9NNl9II014252 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 19:47:10 -0400 (EDT) Subject: [Hanson] marriage state problem From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Sun, 23 Oct 2005 19:47:09 -0400 Message-Id: <1130111229.4793.9.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 2.706 X-Spam-Level: ** (2.706) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 715 Content-Length: 180 X-Keywords: NonJunk I think I understand the marriage state problem/algorithm (p. 9-15) pretty well, but it might be nice to go over it tomorrow in class with more strictly mathematical terms. Kate From meyer@imap.theory.csail.mit.edu Sun Oct 23 18:55:58 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9NMtvd0021128 for <6042-staff@theory.csail.mit.edu>; Sun, 23 Oct 2005 18:55:57 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9NMtuBL019501; Sun, 23 Oct 2005 18:55:56 -0400 (EDT) Received: from [192.168.1.4] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9NMtmok025075; Sun, 23 Oct 2005 18:55:48 -0400 (EDT) Message-ID: <435C14E8.1050805@mit.edu> Date: Sun, 23 Oct 2005 18:55:36 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-staff@theory.csail.mit.edu Subject: Re: [David] Week 7 E-Mail Comments References: <200510232245.j9NMjkbs003903@outgoing.mit.edu> In-Reply-To: <200510232245.j9NMjkbs003903@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------060607010908010909000502" X-Spam-Score: 1.352 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,HTML_MESSAGE autolearn=ham version=2.63 Status: RO Content-Length: 4057 X-UID: 716 X-Keywords: This is a multi-part message in MIME format. --------------060607010908010909000502 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit First of all, please send these emails to 6042-probs! Second, yes, choosing a good invariant is somewhat of a black art. In fact, it's funny that you specifically used the word "deterministic method" - it is possible to prove that there is NO deterministic method (i.e., "algorithm") to find an invariant capable of proving termination! More formally, given a state machine description, there is no algorithm that is able to read it in and output "HALT" exactly when the state machine terminates and output "DOESN'T HALT" exactly when it doesn't terminate. This is called the Halting Problem - we may come across this later in the course. DS Chris Yang wrote: > Selecting a correct invariant predicate for a state machine seems > really important to be able to prove a given machine is terminable. > In the examples, they seem like sort of arbitrary selections - is > there a systematic or at least deterministic method for selecting a > useful invariant predicate, or is this choice what makes state > machines sort of an art? > > > > Thanks, > > Chris Yang > --------------060607010908010909000502 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit First of all, please send these emails to 6042-probs!

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

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

DS

Chris Yang wrote:

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

 

Thanks,

Chris Yang

--------------060607010908010909000502-- From petek@MIT.EDU Sun Oct 23 20:56:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O0uJd0004805 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 20:56:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O0uIeQ000272 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 20:56:18 -0400 (EDT) Received: from [18.194.1.37] (SN-THIRTY-SEVEN.MIT.EDU [18.194.1.37]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O0uGMI024890 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 20:56:17 -0400 (EDT) Message-ID: <435C3130.6070305@mit.edu> Date: Sun, 23 Oct 2005 20:56:16 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Comments LN7 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.929 X-Spam-Level: * (1.929) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 717 Content-Length: 936 X-Keywords: NonJunk
   State Machines I think are going to end up being one of the most useful topics I study in this class, especially in its relevance to artificial intelligence.  Anyway, most of the notes are fairly straight forward.  I guess I'd like to see some lecturing on derived variables, just so I have the chance to talk them over in the class' forum.  Thx

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

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From mracich@MIT.EDU Sun Oct 23 21:36:56 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O1aud0011283 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:56 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9O1atkZ015101 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O1atLL011983 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:55 -0400 (EDT) Received: from MACGREGOR-TWO-SIXTY-TWO.MIT.EDU (MACGREGOR-TWO-SIXTY-TWO.MIT.EDU [18.239.6.7]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O1aqMu025813 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:36:53 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 7 (State Machines: Invariants and Termination) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Sun, 23 Oct 2005 21:36:51 -0400 Message-Id: <1130117811.8358.8.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 718 Content-Length: 277 X-Keywords: I found section 5.6, "...And the Boys Live Especially Happily", (starting on page 14) to be the most surprising section of the lecture notes. I was especially surprised by Theorem 5.8. and the fact that every girl gets her worst pick in the Mating Algorithm. Moira Racich From benlu@MIT.EDU Sun Oct 23 21:38:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O1c5d0011311 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:38:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O1c4eQ023055 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:38:04 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O1bwtL001593 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 21:37:58 -0400 (EDT) Message-ID: <435C3B16.9050806@mit.edu> Date: Sun, 23 Oct 2005 21:38:30 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani][6.042] comments on reading 7 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 719 Content-Length: 225 X-Keywords: Hey Jelani, I liked the whole set up with stable marriages. It's comforting to know that even though boys do all the work and get turned down over and over again, they're the ones who end up with the optimal mates. ~Ben From mpapi@MIT.EDU Sun Oct 23 22:24:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2OLd0017895 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:24:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2OJeQ019569 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:24:19 -0400 (EDT) Received: from mpapi.MIT.EDU (MPAPI.MIT.EDU [18.239.4.219]) (authenticated bits=0) (User authenticated as mpapi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O2ODTi009009 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:24:13 -0400 (EDT) Subject: [Jelani] Week 7 comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Sun, 23 Oct 2005 22:24:16 -0400 Message-Id: <1130120656.12289.13.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.817 X-Spam-Level: *** (3.817) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 720 Content-Length: 368 X-Keywords: NonJunk I found that Theorem 5.8 in section 5.6 on page 14, "The Mating Algorithm marries every boy to his optimal mate and every girl to her pessimal mate" was surprising. It became a little less so, though, after realizing that there were probably many cases where a girl's optimal and pessimal mates were the same, or at least the "pessimal" mate wasn't the last choice. From pgroudas@MIT.EDU Sun Oct 23 22:26:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2Q8d0018021 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 22:26:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2Q7eQ020818 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 22:26:07 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O2PujN009344 for <6042-probs@theory.lcs.MIT.EDU>; Sun, 23 Oct 2005 22:25:56 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9O2PuS6004218; Sun, 23 Oct 2005 22:25:56 -0400 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 22:25:56 -0400 Message-ID: <20051023222556.n2otxnkb15sg8sos@webmail.mit.edu> Date: Sun, 23 Oct 2005 22:25:56 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: [HANSON] Week 7 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 721 Content-Length: 213 X-Keywords: I feel that after reading the introduction to Derived variables on page 8 that I'm not totally getting them. They seem a simple concept and I'm sure they'll probably make more sense after lecture. -Paul Groudas From lkini@MIT.EDU Sun Oct 23 22:29:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2TId0018244 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:29:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2THeQ022778 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:29:17 -0400 (EDT) Received: from [18.239.6.97] (MACGREGOR-THREE-FIFTY-TWO.MIT.EDU [18.239.6.97]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O2TEqa009905 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 22:29:16 -0400 (EDT) Message-ID: <435C46F5.6040204@mit.edu> Date: Sun, 23 Oct 2005 22:29:09 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 7 Notes Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 722 Content-Length: 307 X-Keywords: NonJunk Hi Hanson, I would like it, if possible, if the professors could go over the inductive step of Theorem 4.2 on page 8. I understand the base case and the intuitiveness behind the induction but I would like a more formal mathematical derivation. I ask because I couldn't derive it myself. Thanks, Lohith From icharny@MIT.EDU Sun Oct 23 22:54:49 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O2snd0020643 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 22:54:49 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O2smxN026616 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 22:54:48 -0400 (EDT) Received: from [127.0.0.1] (EASTCAMPUS-SIX-TWENTY-THREE.MIT.EDU [18.238.5.112]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O2sWMw029144 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 22:54:46 -0400 (EDT) Message-ID: <435C4CE8.9080302@mit.edu> Date: Sun, 23 Oct 2005 22:54:32 -0400 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Sayan] Week 7 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 723 Content-Length: 107 X-Keywords: NonJunk I don't understand derived variables (page 8, section 4). Please discuss this in lecture. ~Isaac Charny From jeffhoff@MIT.EDU Sun Oct 23 23:01:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O31Hd0021148 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 23:01:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O31Fnj011509 for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 23:01:16 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O31DWe015450 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sun, 23 Oct 2005 23:01:14 -0400 (EDT) Message-Id: <5.2.1.1.2.20051023223910.05e09f48@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Sun, 23 Oct 2005 23:01:12 -0400 To: 6042-probs@theory.csail.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 724 Content-Length: 280 X-Keywords: NonJunk Reading section 5.6 on page 14 If in class we could explain further the proof of why boys are optimal and girls are pessimal it would be appreciated. The notes only get me to understand so far, but I feel I don't fully understand it...or maybe i'm just still a little doubtful. From ksindi@MIT.EDU Sun Oct 23 23:15:19 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3FJd0023599 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:15:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3FHnj020748 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:15:17 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3FGA3017873 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:15:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9O3FG8C008950; Sun, 23 Oct 2005 23:15:16 -0400 Received: from c-24-218-110-155.hsd1.ma.comcast.net (c-24-218-110-155.hsd1.ma.comcast.net [24.218.110.155]) (User authenticated as ksindi@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 23:15:16 -0400 Message-ID: <20051023231516.kafh91q4locgo0wg@webmail.mit.edu> Date: Sun, 23 Oct 2005 23:15:16 -0400 From: Kamil Y Sindi To: 6042-probs@theory.csail.MIT.EDU Subject: [David] Week 7 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 725 Content-Length: 143 X-Status: A X-Keywords: I found the notes clear but still don't understand the practical use of these state concepts. It seems to me that they are mere formilizations From kevin08@MIT.EDU Sun Oct 23 23:27:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3Rmd0026267 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:27:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3Rknj028341 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:27:47 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-THREE-SIXTEEN.MIT.EDU [18.242.6.61]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3Rd0C020024 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:27:40 -0400 (EDT) Message-Id: <6.2.3.4.2.20051023232523.01e7cef8@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Sun, 23 Oct 2005 23:27:33 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: [Hanson] Week 7 Readings Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 726 Content-Length: 270 X-Keywords: Hi Hanson, I was confused about the concept of a derived variable (Pg. 8, section 4), especially how derived functions help us to find the maximum run time of an algorithm and how it helps prove termination. An example of this would be very helpful. Thanks, Kevin From meyer@imap.theory.csail.mit.edu Sun Oct 23 23:32:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3WYd0026704 for <6042-webmaster@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:32:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3WXnj001203 for <6042-webmaster@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:32:33 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3WQP1020816 for <6042-webmaster@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:32:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9O3WQsG001185; Sun, 23 Oct 2005 23:32:26 -0400 Received: from PSK-NINETY-ONE.MIT.EDU (PSK-NINETY-ONE.MIT.EDU [18.217.1.91]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 23:32:26 -0400 Message-ID: <20051023233226.ldor0tnsxds08s8g@webmail.mit.edu> Date: Sun, 23 Oct 2005 23:32:26 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-webmaster@theory.csail.MIT.EDU Subject: [Sayan] reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-3.9 required=3.7 tests=BAYES_00,FROM_ENDS_IN_NUMS autolearn=no version=2.63 Status: RO Content-Length: 317 X-UID: 727 X-Keywords: NonJunk I am fairly comfortable with state machines, as we just finished covering them in depth in 6.004, which I am also taking. The fuzziest parts of the reading were partial correctness (I understand what you do, just maybe the terminology in which it is described makes it confusing) and derived variables. -Ryan Young From shauni@MIT.EDU Sun Oct 23 23:42:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3g0d0028324 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:42:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3fwnj007533 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:41:58 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3fslb022437 for <6042-probs@theory.csail.MIT.EDU>; Sun, 23 Oct 2005 23:41:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9O3da6j021990; Sun, 23 Oct 2005 23:39:36 -0400 Received: from SIMMONS-ONE-SEVENTY-NINE.MIT.EDU (SIMMONS-ONE-SEVENTY-NINE.MIT.EDU [18.96.5.179]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 23:39:36 -0400 Message-ID: <20051023233936.ir6twmolfdr4kw4g@webmail.mit.edu> Date: Sun, 23 Oct 2005 23:39:36 -0400 From: "I. Shauni Deshmukh" To: 6042-probs@theory.csail.MIT.EDU Subject: [Sayan] week 7 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 728 Content-Length: 321 X-Keywords: NonJunk I found Theorem 5.8 (p.14) interesting: At first, I would have expected that there is only one set of stable marriages and that both girls and boys do equally well. If the algorithm is reversed the girls get their optimal mates instad of the boys, but does a stable marriage set exist where girls and boys fare equally? From rshearer@MIT.EDU Sun Oct 23 23:54:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O3sAd0029931 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:54:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O3s9nj015881 for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:54:09 -0400 (EDT) Received: from [18.240.5.57] (MCCORMICK-FIFTY-SEVEN.MIT.EDU [18.240.5.57]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O3s7fM024469 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sun, 23 Oct 2005 23:54:08 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <6826b5843ef05f2637fba6c0fccd784f@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: [Jelani] Week 7 Comment Date: Sun, 23 Oct 2005 23:54:06 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 729 Content-Length: 195 X-Keywords: NonJunk I had a lot of problems with Tutor Problem 7.2 part 3. I guess I don't understand why H isn't a tree, if the edges are only marked if there isn't a marked path between the endpoints. Rachel From brevzin@MIT.EDU Mon Oct 24 00:04:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O44Zd0031387 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 00:04:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O44Ynj022820 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 00:04:34 -0400 (EDT) Received: from BARRY.mit.edu (PLP-TWELVE.MIT.EDU [18.218.1.12]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O44UXf026430 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 00:04:32 -0400 (EDT) Message-Id: <6.1.2.0.1.20041024000012.019ddea8@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Mon, 24 Oct 2005 00:04:29 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 730 Content-Length: 184 X-Keywords: NonJunk I really enjoyed the Stable mating problem, and I certainly was not expected that it was so favored toward the boys. That theorem at the end was pretty cool and interesting. Barry From hsoumare@MIT.EDU Mon Oct 24 00:06:32 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O46Vd0032673 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:06:31 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O46VxN028942 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:06:31 -0400 (EDT) Received: from [18.241.5.187] (NEW-ONE-EIGHTY-SEVEN.MIT.EDU [18.241.5.187]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O46OMu002708 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:06:24 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) In-Reply-To: <20051023144946.99856.qmail@web50214.mail.yahoo.com> References: <20051023144946.99856.qmail@web50214.mail.yahoo.com> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <3190F9DA-B292-44BC-A1EC-82FD9AFBF25D@mit.edu> Content-Transfer-Encoding: 7bit From: Hamidou Soumare Subject: {SAYAN} Weekly Email Comment Date: Mon, 24 Oct 2005 00:15:16 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 731 Content-Length: 240 X-Keywords: NonJunk I found theorem 5.8 on page 14 to be the most interesting because it was counter intuitive for me. I thought that the mating algorithm would match each girl to her optimal mate because the girls picked their favorite suitor. Hamidou From hzhou@MIT.EDU Mon Oct 24 00:21:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4LBd0003445 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:21:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4LAnj003339 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:21:10 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4L4IP029842 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:21:04 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9O4L4NA015276; Mon, 24 Oct 2005 00:21:04 -0400 Received: from BURTON-TWO-TWENTY-SIX.MIT.EDU (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (User authenticated as hzhou@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 00:21:04 -0400 Message-ID: <20051024002104.pj08nk8euo34sswo@webmail.mit.edu> Date: Mon, 24 Oct 2005 00:21:04 -0400 From: Hao Zhou To: 6042-probs@theory.lcs.MIT.EDU Subject: David TP7 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 732 Content-Length: 367 X-Keywords: NonJunk Hi Staff, I found using the state invarient is a very clever way to solve problems. However, I find it is often times hard to find an invarient for all the states. For example, for the tiling problem we did in class, the invarient was not a very obvious choice. Other than through experience, how should we look for these invarients? Thanks - Steven (Hao) Zhou From cvnguyen@MIT.EDU Mon Oct 24 00:26:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4Qhd0004632 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:26:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4Qfnj006688 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:26:41 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-FIVE-TWENTY-EIGHT.MIT.EDU [18.238.5.17]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4Qew3000660 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 00:26:40 -0400 (EDT) Message-ID: <001f01c5d853$21563470$1105ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 8 comments Date: Mon, 24 Oct 2005 00:26:39 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.274 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 733 Content-Length: 143 X-Keywords: NonJunk I found the discussion of the Mating Algorithm entertaining and interesting. I'm sure it will be covered in depth in class. --Chieu Nguyen From aeon@MIT.EDU Mon Oct 24 00:47:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4lkd0005819 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:47:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4ljnj019111 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:47:45 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4ld7t003378 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:47:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9O4ldBt029883; Mon, 24 Oct 2005 00:47:39 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 23 Oct 2005 22:47:39 -0600 Message-ID: <20051023224739.neeth8xk01y8k8w8@webmail.mit.edu> Date: Sun, 23 Oct 2005 22:47:39 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 7 Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 734 Content-Length: 528 X-Keywords: NonJunk Page 9, "The Stable Marriage Problem" Just wanted to note that this is interesting, since it's implicitly included in TONS of algorithms. I've kind of used this in many programs, never actually stating it anywhere in code, only in design. Page 14, "... and the Boys... " About the "shocking truth"... does that mean there is no possible algorithm where both sets, the-boys AND the-girls can find their optimal spouses? The proof doesn't disprove this idea, and it seems like it's pretty possible. Just a thought. John Marrero From rshroff@MIT.EDU Mon Oct 24 00:53:38 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O4rcd0006266 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:53:38 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O4rbnj025031 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:53:37 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O4rZhh006520 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 00:53:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9O4rZrq009942; Mon, 24 Oct 2005 00:53:35 -0400 Received: from NEXT-THREE-FORTY-EIGHT.MIT.EDU (NEXT-THREE-FORTY-EIGHT.MIT.EDU [18.242.6.93]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 00:53:35 -0400 Message-ID: <20051024005335.w4u6tpzivq74cs0w@webmail.mit.edu> Date: Mon, 24 Oct 2005 00:53:35 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Reading Assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 735 Content-Length: 116 X-Keywords: NonJunk I found the marriage problem extremely interesting. I hope we spend some more time over it in class. -Rahul Shroff From alisonc@MIT.EDU Mon Oct 24 01:00:16 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O50Fd0007230 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:00:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O50Enj004582 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:00:14 -0400 (EDT) Received: from [18.238.6.107] (EASTCAMPUS-EIGHT-SEVENTY-FOUR.MIT.EDU [18.238.6.107]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O5078p012853 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:00:08 -0400 (EDT) Message-ID: <435C6A59.4060904@mit.edu> Date: Mon, 24 Oct 2005 01:00:09 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.7 (Windows/20050923) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Comments on course notes #7 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 736 Content-Length: 496 X-Keywords: NonJunk I found the Stable Marriage problem to be the most interesting part of this week's set of notes -- as usual, the very real application and the math behind it are very neat. I'm curious about what other sorts of pairing algorithms might have insightful graphs or proofs behind them: the ones that MIT uses for housing, the readjustment lottery, the HASS lottery, the phys ed lottery, for instance? They're different and simpler in that preferences don't go both ways, but still interesting. From clintonb@MIT.EDU Mon Oct 24 01:07:34 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O57Yd0008464 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:07:34 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O57Xnj010000 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:07:33 -0400 (EDT) Received: from ccbibmx30 (BLACKBURN.MIT.EDU [18.241.3.107]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O57U74015101 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:07:31 -0400 (EDT) Message-Id: <200510240507.j9O57U74015101@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.lcs.mit.edu> Subject: Jelani: TP 7 Comments Date: Mon, 24 Oct 2005 01:06:14 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0075_01C5D837.21C867F0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXYWKflowW9hhfEQWGWgoP0m0vgOA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.668 X-Spam-Level: * (1.668) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 737 Content-Length: 4749 X-Keywords: NonJunk $Forwarded This is a multi-part message in MIME format. ------=_NextPart_000_0075_01C5D837.21C867F0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I'm totally digging these problems with real-world relations/examples (ie. RSA, stable marriage). I learn best through these types of examples. I get the methods; although, I'm sure I need to go back over the different theorems and proofs for the stable marriage problem. Also, what happened to the pre-filled email links? I'm referring to the mailto links with the email address and subject line. I kind of liked not having to think of a subject line (the hardest part of writing emails for me). Good morning! --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0075_01C5D837.21C867F0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

 

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

 

Good morning!

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_0075_01C5D837.21C867F0-- From hkhall@MIT.EDU Mon Oct 24 01:22:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O5Mtd0010863 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 01:22:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O5Mqnj025635; Mon, 24 Oct 2005 01:22:52 -0400 (EDT) Received: from [18.96.6.125] (SIMMONS-THREE-EIGHTY.MIT.EDU [18.96.6.125]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O5Mj2v023342 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Mon, 24 Oct 2005 01:22:46 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: [David] Reading Comments on LN7 Date: Mon, 24 Oct 2005 01:24:47 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 738 Content-Length: 362 X-Keywords: NonJunk David- The coolest thing I learned is that some optimal matching occurs and that it is provable that it occurs. I tried it myself and I couldn't come up with a set that I couldn't match given the rules laid out. Is there an algorithm for matching more optimal than the the mating algorithm that can give one group their optimal spouse? Thanks. -Harrison From sergiob@MIT.EDU Mon Oct 24 01:58:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O5w3d0016199 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 01:58:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O5w1LQ013746 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 01:58:01 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O5vw3K026544 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 01:57:59 -0400 (EDT) Message-Id: <5.2.1.1.2.20051023235417.00c4e850@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Sun, 23 Oct 2005 23:57:57 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 739 Content-Length: 283 X-Keywords: NonJunk Now, I can understand why Floyd was so excited about the invariants theorem. I would like to know more about the types of algortihms there are, and if there are any systematic ways of establishing derived variables to prove their partial correctness. Sergio Bacallado (Group 1) From scot@MIT.EDU Mon Oct 24 02:05:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O650d0016739 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 02:05:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O64xLQ017228 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 02:04:59 -0400 (EDT) Received: from [18.227.1.175] ([18.227.1.175]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O64qF0027517 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 02:04:56 -0400 (EDT) Message-ID: <435C7949.5000708@mit.edu> Date: Mon, 24 Oct 2005 02:03:53 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: comments: Jelani Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 740 Content-Length: 342 X-Keywords: NonJunk Hi, I found the notes to be clear, interesting, and good this week. The only part I found confusing was the last, during the proof of the pessimal choice for the girls. I don't see how it can stand as a valid proof by contradition. How do we know G prefers B to her mate B' by assumption? Maybe this will be cleared up in lecture. Scot From bakster@MIT.EDU Mon Oct 24 02:14:39 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O6Edd0017246 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:14:39 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O6EcLQ022451 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:14:38 -0400 (EDT) Received: from [18.246.5.237] (BEXLEY-TWO-THIRTY-SEVEN.MIT.EDU [18.246.5.237]) (authenticated bits=0) (User authenticated as bakster@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O6EWLv028341 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:14:32 -0400 (EDT) Message-ID: <435C7BC5.10201@mit.edu> Date: Mon, 24 Oct 2005 02:14:29 -0400 From: Alexander Bakst User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [David] week 7 comments Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 741 Content-Length: 193 X-Keywords: NonJunk I found this week's reading to be particularly homophobic and sexist: "... for the “boy-girl” marriage problem, a stable set of marriages does always exist." and Theorem 5.8 Alexander Bakst From arup@MIT.EDU Mon Oct 24 02:37:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O6bBd0019985 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:37:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O6bALQ004065 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:37:10 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-THREE-EIGHTEEN.MIT.EDU [18.247.6.63]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O6ax6T029898 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 02:37:04 -0400 (EDT) Message-Id: <6.2.3.4.2.20051024023520.0352d078@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Mon, 24 Oct 2005 02:36:19 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 742 Content-Length: 172 X-Keywords: NonJunk Section 5.6, pp 14-15, theorem 5.8 & proofs: I would like to have this discussed more fully in lecture, since I didn't really follow the proofs in the reading. |Arup| From a_lopez@MIT.EDU Mon Oct 24 03:07:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O77Wd0022457 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 03:07:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O77ULQ018850 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 03:07:30 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O77Twd001477 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 03:07:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j9O77TvL025340; Mon, 24 Oct 2005 03:07:29 -0400 Received: from MDMURRAY.MIT.EDU (MDMURRAY.MIT.EDU [18.234.0.116]) (User authenticated as a_lopez@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 03:07:29 -0400 Message-ID: <20051024030729.kd5dlqyyd0kggsw4@webmail.mit.edu> Date: Mon, 24 Oct 2005 03:07:29 -0400 From: a_lopez@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [Jelani] week 7 readings MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 743 Content-Length: 522 X-Keywords: NonJunk Reading the notes on the Marriage Algorithm, got me thinking... is this the algorithm currently used to match people to their preferences? For example, is this the algorithm that MIT uses in the freshman housing lottery? It seems like a powerful algorithm, but the fact that it favors one group over another is a bit troubling (is optimal for one group and pessimal for the other). Are there other marriage algorithms that balance the marriage choices better, so that one group is not favored over the other? Adriana From yaser@MIT.EDU Mon Oct 24 03:15:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7FTd0025119 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:15:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7FSLQ022604 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:15:28 -0400 (EDT) Received: from Charizard (NEW-ONE-THIRTY-SEVEN.MIT.EDU [18.241.5.137]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7FKJR001860 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:15:21 -0400 (EDT) Message-Id: <200510240715.j9O7FKJR001860@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: David: week7 reading response Date: Mon, 24 Oct 2005 03:15:16 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_002E_01C5D849.2826F0A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXYaq6TnPEQ9BHjR8CIhZ1fbTDILg== X-Spam-Score: 2.891 X-Spam-Level: ** (2.891) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 744 Content-Length: 5320 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_002E_01C5D849.2826F0A0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Hi David, Out of curiosity: does the 2-domain marriage algorithm readily scale to 3-domain? I.e., say you are matching manufactories, retailers and distributors, and each has a certain preference of the other two. does the algorithm still hold? Thanks! _Yaser ------=_NextPart_000_002E_01C5D849.2826F0A0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

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

 

Thanks!

_Yaser

------=_NextPart_000_002E_01C5D849.2826F0A0-- From mdmurray@MIT.EDU Mon Oct 24 03:24:03 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7O3d0027814 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:24:03 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7O3VH004155 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:24:03 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9O7NtMu010244 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:23:55 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Michael Murray Subject: [Hanson] E-mail question Date: Mon, 24 Oct 2005 03:23:31 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 745 Content-Length: 485 X-Keywords: NonJunk As a general comment I like this topic of state machines and wish it were more than 2 lectures long. Anyway, as regards to the reading, I like the format of the stable marriage problem and the example was very easy to follow and understand. However, I would like to have a more formal and generic description of the algorithm than the one presented in the notes. Maybe, however, the reason there is no such description is that it is not required for this class? Michael From antonk@MIT.EDU Mon Oct 24 03:24:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7OAd0027824 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:24:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7O9LQ026503 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:24:09 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7O6Sr002270 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:24:07 -0400 (EDT) From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] need clarification Date: Mon, 24 Oct 2005 03:24:04 -0400 Organization: Massachusetts Institute of Technology Message-ID: <003201c5d86b$ea70d2d0$0100a8c0@silencer> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0033_01C5D84A.6360B970" X-Mailer: Microsoft Office Outlook 11 Thread-Index: AcXYa+k6R6FJwhRIQ2aAtb37KIfIjQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.214 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 746 Content-Length: 3227 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0033_01C5D84A.6360B970 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit On the first line of partial correctness of GCD I don't believe that I fully understand the definition of the proof statement. Something is not clear about the general concept of terminology when you say that if the process terminates then (the GCD(a,b)) is equal to x. what is the y then? Is it just a helper variable that we use along the way during the run of the machine? Thank you, Anton. ------=_NextPart_000_0033_01C5D84A.6360B970 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

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

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

 

Thank you,

 

Anton.

 

------=_NextPart_000_0033_01C5D84A.6360B970-- From vbrobbey@MIT.EDU Mon Oct 24 03:53:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7rZd0031676 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:53:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7rYLQ010285 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:53:34 -0400 (EDT) Received: from TheSlate (NEW-FOUR-TEN.MIT.EDU [18.241.6.155]) (authenticated bits=0) (User authenticated as vbrobbey@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7rRKe003537 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 03:53:28 -0400 (EDT) Message-ID: <004b01c5d870$0a4bbe40$9b06f112@TheSlate> From: "Valery Kwasi Brobbey" To: <6042-probs@theory.csail.mit.edu> Subject: [Hanson] Week 7 Comments Date: Mon, 24 Oct 2005 03:53:35 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0048_01C5D84E.820E9690" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.214 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 747 Content-Length: 2385 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0048_01C5D84E.820E9690 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable This week's material is very similar to material we've already seen. = State Machine is basically a graph but then edges exist between vertices = based on certain conditions/rules. When it comes to derived variables, = the concept of weak/strict is like the concept of weak/strict partial = orders. Halls marriage theorem is similar to the Stable marriage problem = except that the stable marriage problem takes rank into consideration. I = realized that this was a weakness in Hall's marriage theorem and = expected there to be some improved version that looks at rank/order. I = hope there's another marriage theorem that takes into account the fact = that opinion and interests change. For instance when the housing office = assigns dorms to students, it has to do a readjustment lottery to = account for change in preferences. ------=_NextPart_000_0048_01C5D84E.820E9690 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
This week's material is very similar to = material=20 we've already seen. State Machine is basically a graph but then edges = exist=20 between vertices based on certain conditions/rules. When it comes to = derived=20 variables, the concept of weak/strict is like the concept of weak/strict = partial=20 orders. Halls marriage theorem is similar to the Stable marriage problem = except=20 that the stable marriage problem takes rank into consideration. I = realized that=20 this was a weakness in Hall's marriage theorem and expected there to be = some=20 improved version that looks at rank/order. I hope there's another = marriage=20 theorem that takes into account the fact that opinion and interests=20 change. For instance when the housing office assigns dorms to = students, it=20 has to do a readjustment lottery to account for change in=20 preferences.
------=_NextPart_000_0048_01C5D84E.820E9690-- From kkdb@MIT.EDU Mon Oct 24 03:59:01 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O7x1d0031862 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:59:01 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O7x0LQ012773 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:59:00 -0400 (EDT) Received: from KKDB (MACLAURIN-FOURTEEN-O-FOUR.MIT.EDU [18.80.6.125]) (authenticated bits=0) (User authenticated as kkdb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O7wwJb003758 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 03:58:58 -0400 (EDT) Message-ID: <001001c5d870$cbe00480$7d065012@KKDB> From: "Kaustuv DeBiswas" To: <6042-probs@theory.lcs.mit.edu> References: Subject: [David] Week 7 comment Date: Mon, 24 Oct 2005 03:59:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000D_01C5D84F.446C9570" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.52 X-Spam-Level: * (1.52) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 748 Content-Length: 1870 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_000D_01C5D84F.446C9570 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Hi David, I was confused by section 21./pg3 (the waterjug proof) where is it says = "proof divides into cases, according to which transition rule is used. = For example, suppose the transition followed from the "fill the little = jug" rule. This means (b, l) -> (b, 3). But P((b, l) impiles that b is = an integer multiple of 3, and of course 3 is an integer multiple of 3, = so P still holds for the new state (b, 3)." how does P(b,l) imply that b | 3? thanks, Kaustuv. ------=_NextPart_000_000D_01C5D84F.446C9570 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable

Hi David,

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

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

thanks,=20 Kaustuv.

------=_NextPart_000_000D_01C5D84F.446C9570-- From jacques@MIT.EDU Mon Oct 24 04:02:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O82nd0032550 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:02:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O82mLQ014779 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:02:48 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O82k8L003930 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:02:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9O82kDx008910; Mon, 24 Oct 2005 04:02:46 -0400 Received: from EASTCAMPUS-SEVEN-O-SEVEN.MIT.EDU (EASTCAMPUS-SEVEN-O-SEVEN.MIT.EDU [18.238.5.196]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 04:02:46 -0400 Message-ID: <20051024040246.sx741hh16p0kkw8k@webmail.mit.edu> Date: Mon, 24 Oct 2005 04:02:46 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 7 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 749 Content-Length: 260 X-Keywords: NonJunk On page 14, it surprised me that the Mating Algorithm actually did better for the boys than the girls. I'm glad those last two pages are there, because I probably wouldn't have given it much thought and just continued to assume the girls were the better off. From fluff@MIT.EDU Mon Oct 24 04:26:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O8QRd0003025 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:26:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O8QPLQ025320 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:26:25 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O8QJM1004891 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:26:19 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 7 reading Date: Mon, 24 Oct 2005 04:24:03 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 750 Content-Length: 423 X-Keywords: NonJunk I thought this week's reading was way friendlier than last week's. It was nice that the state machine stuff was introduced in lecture first (the counter, jug problem, and diagonal robot), which made the reading a lot easier to understand. I also thought the "mating algorithm" was very interesting, and the fact that a stable set of marriages always exists. And also how all the girls get screwed over. ~Crystal From tonyng@MIT.EDU Mon Oct 24 04:39:18 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O8dId0004729 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:39:18 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O8dGLQ001290 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:39:17 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O8dAmk005387 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 04:39:11 -0400 (EDT) Message-Id: <5.2.1.1.2.20051024043123.0211eb40@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 24 Oct 2005 04:39:21 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: [Jelani] Reading Comments: Week 7 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 751 Content-Length: 482 X-Keywords: NonJunk I found the fact that the Marriage Algorithm produces optimal "spouses" pretty interesting (Section 5.6 page 14-15). I can already see many applications of this, such as a lottery to give people their preferred choices of classes, dorms, etc (people can enter their choices as ratings, and classes can prefer seniors over juniors, etc). It can maximize the happiness of people with their results because it is unlikely people will get paired off with a worst choice. - Tony From medrano@MIT.EDU Mon Oct 24 04:55:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O8tWd0006408 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:55:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O8tNLQ008352 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:55:23 -0400 (EDT) Received: from scrubbing-bubbles.mit.edu (SCRUBBING-BUBBLES.MIT.EDU [18.7.16.68]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O8tLIY005942 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 04:55:22 -0400 (EDT) Received: (from medrano@localhost) by scrubbing-bubbles.mit.edu (8.12.9) id j9O8tLnf016530; Mon, 24 Oct 2005 04:55:21 -0400 (EDT) Date: Mon, 24 Oct 2005 04:55:21 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] weekly reading question Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 752 Content-Length: 267 X-Keywords: NonJunk I was wondering if the mating ritual had any other applications than the one used in the reading? Even though it's funny to have theorems about people being happy who they are matched with, what are the actual computer applications of these theorems. Jesus Medrano From zev@MIT.EDU Mon Oct 24 05:38:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9O9csd0011971 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 05:38:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9O9crqJ026300 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 05:38:53 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9O9cme8007565 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 05:38:51 -0400 (EDT) Message-ID: <435CABA8.5040302@mit.edu> Date: Mon, 24 Oct 2005 05:38:48 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 7 comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 753 Content-Length: 462 X-Keywords: NonJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 Overall the lecture notes were quite straight forward. I would have liked an example of the mating algorithm, but the tutor problems took care of that. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDXKuolO3j8HLL0+4RAl5TAJ4+9I39fhy684FWZDGkFo4/DGiFoQCgmc/8 ZWAJDJwz0qcpaNXO3IZzcOk= =l0uH -----END PGP SIGNATURE----- From ozcan@MIT.EDU Mon Oct 24 08:55:37 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OCtbd0014035 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 08:55:37 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OCtbSi013766 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 08:55:37 -0400 (EDT) Received: from ebrum.mit.edu (NEW-TWO-O-FIVE.MIT.EDU [18.241.5.205]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OCtYol000938 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 08:55:35 -0400 (EDT) Message-Id: <5.2.1.1.2.20051024084808.026a4ca0@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 24 Oct 2005 08:55:32 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson] reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 754 Content-Length: 453 X-Keywords: NonJunk I am pretty amazed with the fact that there is an entire section on marriage problem, which in fact describes a matching algorithm. I also had a class in which we tried to maximize the probability of finding the "best woman" by setting a number of dates a man should go out, and the probability we ended up with was a very high one, 1/e. These kind of approaches to math is quite fun and intriguing. Quote: Entire Section 5, The marriage problem From mukkala@MIT.EDU Mon Oct 24 09:11:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODBBd0018635 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 09:11:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODBAmj025895 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 09:11:10 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-FIFTY-TWO.MIT.EDU [18.235.1.152]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODB0YP005056 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 09:11:04 -0400 (EDT) Message-Id: <6.2.1.2.2.20051024081344.04304e20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Mon, 24 Oct 2005 09:11:01 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 7 Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 755 Content-Length: 311 X-Keywords: NonJunk In section 3.2.1, entitled "Partial Correctness of GCD," I don't understand how partial correctness is proven. I understand that the gcd(x,y) = gcd(x1,0) = x1 = d. However, why is it that this proves only partial correctness, instead of the entire correctness of the claim? Thanks, Praveen Pamidimukkala From nancyk@MIT.EDU Mon Oct 24 09:29:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODTsd0020958 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:29:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODTpmj014254; Mon, 24 Oct 2005 09:29:51 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODTnuj012096; Mon, 24 Oct 2005 09:29:50 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j9ODTnHR011619; Mon, 24 Oct 2005 09:29:49 -0400 Received: from WAREHOUSE-FORTY-SIX.MIT.EDU (WAREHOUSE-FORTY-SIX.MIT.EDU [18.139.5.46]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 09:29:49 -0400 Message-ID: <20051024092949.b7oj7majk68kgkk4@webmail.mit.edu> Date: Mon, 24 Oct 2005 09:29:49 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] week 7 reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 756 Content-Length: 330 X-Keywords: NonJunk Hi, I thought the stable marriage problem (detailed on pages 9-12) was interesting. I don't have any specific questions about it, but it is interesting that without the gender constraint, we're not guaranteed to find a stable matching. I enjoyed reading about the concepts in terms of the marriage scenario. Thanks, Nancy Keuss From nedzel@MIT.EDU Mon Oct 24 09:43:50 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODhod0021834 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 09:43:50 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODhmmj028760 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 09:43:48 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-TWO-TWENTY-FOUR.MIT.EDU [18.96.5.224]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODhfVf017723 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 09:43:42 -0400 (EDT) Message-Id: <200510241343.j9ODhfVf017723@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 7 Comments Date: Mon, 24 Oct 2005 09:43:39 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.592 X-Spam-Level: ** (2.592) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 757 Content-Length: 111 X-Keywords: NonJunk How are the state machines described in the reading similar or different to Markov state machines? - David From dnreshef@MIT.EDU Mon Oct 24 09:44:12 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODiCd0021864 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:44:12 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODiBmj029088 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:44:11 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODi9ah017903 for <6042-probs@theory.csail.MIT.EDU>; Mon, 24 Oct 2005 09:44:09 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9ODi9Uk014692; Mon, 24 Oct 2005 09:44:09 -0400 Received: from MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU (MACGREGOR-FIVE-FORTY-FOUR.MIT.EDU [18.239.7.33]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 09:44:09 -0400 Message-ID: <20051024094409.cm7xdsqpbu968s4k@webmail.mit.edu> Date: Mon, 24 Oct 2005 09:44:09 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] week 7 Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 758 Content-Length: 183 X-Keywords: NonJunk In this week's reading on state machines, I found the topics of invariants on page 2 and derived variables on page 8 the hardest. I would appreciate gonig over these in class. -Dave From xiaoranz@MIT.EDU Mon Oct 24 09:58:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9ODwld0025295 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 09:58:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9ODwkmj014566 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 09:58:46 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9ODwiiO024062 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 09:58:45 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9ODwgxE023527; Mon, 24 Oct 2005 09:58:42 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 09:58:42 -0400 Message-ID: <20051024095842.um0z31procasg4g0@webmail.mit.edu> Date: Mon, 24 Oct 2005 09:58:42 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Weekly Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 759 Content-Length: 638 X-Keywords: NonJunk I think 3.3 about Extended Euclidean Algorithm is interesting because it ties the state machine and mathematical concept to computer programming. I'm' a little confused as to how to determine the invariant for a state machine, if there's a method or a trick to determine what values are invariant or not. Sharon ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From hmzhou@blackbird.csail.mit.edu Mon Oct 24 10:12:21 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OECLd0029370; Mon, 24 Oct 2005 10:12:21 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OECLbv022762; Mon, 24 Oct 2005 10:12:21 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OECL93022759; Mon, 24 Oct 2005 10:12:21 -0400 Date: Mon, 24 Oct 2005 10:12:21 -0400 (EDT) From: Hanson Zhou To: "Xiaoran (Sharon) Zhang" cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Weekly Comments In-Reply-To: <20051024095842.um0z31procasg4g0@webmail.mit.edu> Message-ID: References: <20051024095842.um0z31procasg4g0@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 760 Content-Length: 1064 X-Keywords: NonJunk Careful...the invariant need not be tied to any specific variable in the state machine, as the reading often demonstrates. The invariant is often a function of those variables or some other derived value. If you are asking how one comes up with an invariant, there is no good answer. That's an art, though with enough problems and examples, you get a feel for what tends to work. -Hanson > I think 3.3 about Extended Euclidean Algorithm is interesting because it ties > the state machine and mathematical concept to computer programming. I'm' a > little confused as to how to determine the invariant for a state machine, if > there's a method or a trick to determine what values are invariant or not. > > Sharon > > ************************************************** > Xiaoran (Sharon) Zhang > Class of 2008 > Department of Biology & > Department of Electrical Engineering and Computer Science > Massachusetts Institute of Technology > 320 Memorial Drive, > Cambridge, MA 02139 > E-mail: xiaoranz@mit.edu > ************************************************** > From kromer@MIT.EDU Mon Oct 24 10:16:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEGLd0029706 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:16:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEGKmj004492 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:16:20 -0400 (EDT) Received: from m12-182-9.mit.edu (M12-182-9.MIT.EDU [18.19.0.40]) (authenticated bits=56) (User authenticated as kromer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEGHD4002350 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:16:18 -0400 (EDT) Received: (from kromer@localhost) by m12-182-9.mit.edu (8.12.9) id j9OEGH3Q030056; Mon, 24 Oct 2005 10:16:17 -0400 Subject: [David] Email comments for reading From: Katherine Romer To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 24 Oct 2005 10:16:17 -0400 Message-Id: <1130163377.30018.4.camel@m12-182-9.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 761 Content-Length: 502 X-Keywords: NonJunk p. 7: "We claim that invariant properties that can be used to prove partial correctness are: gcd(X,Y)=gcd(a,b) Sa+Tb=Y, and Ua+Vb=X" Given these invariant properties, I can verify them and see how they can be used to show the partial correctness of the extended Euclidean Algorithm, but I don't think I could come up with them on my own--it would be nice to see an example of the thinking process used to come up with the invarant properties to use in a proof of partial correctness. Katherine Romer From cbossard@MIT.EDU Mon Oct 24 10:23:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OENMd0031536 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:23:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OENLmj012064 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:23:21 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OENE6V005297 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:23:15 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j9OENEu6026006; Mon, 24 Oct 2005 10:23:14 -0400 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:23:14 -0400 Message-ID: <20051024102314.baisrkhigpw0kw0s@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:23:14 -0400 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 7 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 762 Content-Length: 117 X-Keywords: NonJunk Section 5.2 I get the general jist of the mating algorithm and I found it to be a very interesting example. Cynthia From kushan@MIT.EDU Mon Oct 24 10:37:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEbXd0003179 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:37:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEbVmj028281 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:37:32 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEbP5r011635 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:37:25 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9OEbP1d006943; Mon, 24 Oct 2005 10:37:25 -0400 Received: from RLE-13-099.MIT.EDU (RLE-13-099.MIT.EDU [18.62.13.99]) (User authenticated as kushan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:37:25 -0400 Message-ID: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:37:25 -0400 From: Kushan K Surana To: 6042-probs@theory.lcs.MIT.EDU Subject: TA - Hanson: 6.042 Week 7 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 763 Content-Length: 165 X-Status: A X-Keywords: NonJunk - On page 6, the proof of the partial correctness of GCD is unclear. Why is the predicate P(x,y) true for every state? The GCD is found only at the very last state. From mitras@theory.csail.mit.edu Mon Oct 24 10:41:49 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEfnd0003428 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:41:49 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEfmls015818 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:41:48 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEfkZf007028; Mon, 24 Oct 2005 10:41:46 -0400 (EDT) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OEfiMu001318; Mon, 24 Oct 2005 10:41:44 -0400 (EDT) Message-ID: <435CF2A7.3090206@theory.csail.mit.edu> Date: Mon, 24 Oct 2005 10:41:43 -0400 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala , 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 7 Reading Comments References: <6.2.1.2.2.20051024081344.04304e20@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20051024081344.04304e20@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 764 Content-Length: 662 X-Keywords: Its partial correctness because it proves that if the algorithm terminates, then it gives the correct answer; however, it does not prove that the algorithm indeed terminates. This termination part is done in 3.2.2. Together these two theorems imply the (complete) correctness of the algorithm. -sayan Praveen Pamidimukkala wrote: > In section 3.2.1, entitled "Partial Correctness of GCD," I don't > understand how partial correctness is proven. I understand that the > gcd(x,y) = gcd(x1,0) = x1 = d. However, why is it that this proves > only partial correctness, instead of the entire correctness of the claim? > > Thanks, > Praveen Pamidimukkala > From shreyes19@gmail.com Mon Oct 24 10:43:41 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEhfd0003573 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:43:41 -0400 Received: by xproxy.gmail.com with SMTP id r21so192641wxc for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 07:43:40 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type; b=j0rMZXeBFhs6Tzr72BpQ6JsTujSKfj4FcxX0z1C+R2YsGL5ql34rhGeW57vAEMMxCzcZhk4+uESkK6gCa91V0XcvciP1MCSRrrTo+WNIHlmlpckgKEMD950wV9opP7uu0Sp9K6ApJ21RDeWDeKozBq553BKHt65k25aVoEdGWmo= Received: by 10.70.73.1 with SMTP id v1mr3647223wxa; Mon, 24 Oct 2005 07:43:40 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Mon, 24 Oct 2005 07:43:40 -0700 (PDT) Message-ID: <3c33fe380510240743x51a711ei2a8ef63b3637436f@mail.gmail.com> Date: Mon, 24 Oct 2005 10:43:40 -0400 From: Shreyes Seshasai Sender: shreyes19@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 7 Reading comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_22743_4688988.1130165020768" Status: RO X-UID: 765 Content-Length: 1558 X-Keywords: NonJunk ------=_Part_22743_4688988.1130165020768 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, After reading the lecture notes for week 7, one thing I'd like to have discussed more in class is more examples of how exactly to use derived variables. I follow the definitions fine on the end of page 8 and beginning of page 9, but it's not clear to me how knowing if something is decreasing can help in an invariants problem. The only application of it that's clear to me is proving termination, but that is clear by the Well Ordering Principle. Are there other uses of knowing when state machines are weakly o= r strictly decreasing? Thanks, Shreyes Seshasai Group 7 ------=_Part_22743_4688988.1130165020768 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

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

Thanks,
Shreyes Seshasai
Group 7
------=_Part_22743_4688988.1130165020768-- From crowell@MIT.EDU Mon Oct 24 10:46:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEkMd0004769 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:46:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEkLmj009400 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:46:21 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEkAF0015475 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:46:10 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9OEkA2O021354; Mon, 24 Oct 2005 10:46:10 -0400 Received: from SHYGUY.MIT.EDU (SHYGUY.MIT.EDU [18.238.4.89]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:46:10 -0400 Message-ID: <20051024104610.hvux4wbsduv48g44@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:46:10 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 766 Content-Length: 199 X-Keywords: NonJunk I would like to have the State Machine model of the Marraige Problem explained again in class (5.3 page 12). This section of the notes was a little confusing to me when I read it. -Rob Crowell From lana@MIT.EDU Mon Oct 24 10:49:15 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEnFd0004964 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:49:15 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEnEmj012282 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:49:14 -0400 (EDT) Received: from w92-130-webmail-3.mit.edu (W92-130-WEBMAIL-3.MIT.EDU [18.7.22.133]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OEn5Ug016715 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 10:49:05 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-3.mit.edu (8.12.4) id j9OEn55E021654; Mon, 24 Oct 2005 10:49:05 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 10:49:05 -0400 Message-ID: <20051024104905.em36hoohqzj4cgc8@webmail.mit.edu> Date: Mon, 24 Oct 2005 10:49:05 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP7 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 767 Content-Length: 262 X-Keywords: NonJunk The part of the reading I found particularly interesting was the marriage algorithm. I am curious if it necessarily for avery single person involved to rank all possible mates or if it can still operate in the minimal conditions of Hall's marriage theorem. Lana From zacharyozer@gmail.com Mon Oct 24 10:55:20 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.201]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEtKd0006849 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 10:55:20 -0400 Received: by xproxy.gmail.com with SMTP id t12so847075wxc for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 07:55:19 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=jJZl0Qn7GgebSFcepb5dFwtGSSJKz194K8mD3+C3qqAyKsMe1YiwUbXI4Zzi7AcELh535GFUs4YW1sku5pqX4HqDuEHba4R/v6+zA4CeeCBptoWfdOK8r6V0A+4BCA5cLtM0ok0sTuznuJle7CbEh1fYDBxgQ5WXf6sKLFHih3I= Received: by 10.70.7.8 with SMTP id 8mr2127843wxg; Mon, 24 Oct 2005 07:55:19 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Mon, 24 Oct 2005 07:55:19 -0700 (PDT) Message-ID: <4f2613e50510240755p13289032q8cfa2b48b4fb04af@mail.gmail.com> Date: Mon, 24 Oct 2005 10:55:19 -0400 From: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: 6042 comments: Request for notes and lecture MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j9OEtKd0006849 Status: RO X-UID: 768 Content-Length: 432 X-Keywords: NonJunk First, I'm a bit confused as to how invarients are being used in the reading in general; it seems as though at one moment there's a theorem, but that they become some sort of equivalence statement. What determines the equivalence relation or that it's true? More importantly however, get rid of Die Hard, it's getting old at this point. Also, the bio type info on people is always awesome (vis Turing, Flyod, Meyer, etc). -zozer From veracarr@MIT.EDU Mon Oct 24 10:56:33 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEuXd0006934 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:56:33 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEuTSi020573 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:56:29 -0400 (EDT) Received: from [18.245.7.47] (BAKER-FIVE-FIFTY-EIGHT.MIT.EDU [18.245.7.47]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OEuMMu002578 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:56:22 -0400 (EDT) Message-ID: <435D1275.1050405@mit.edu> Date: Mon, 24 Oct 2005 10:57:25 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Hanson] State Machines Reading Assignment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 769 Content-Length: 140 X-Keywords: NonJunk I found it difficult to understand graph algorithm works for state machines. The first problem in 7.2 with invariants was a bit confusing. From moscicki@MIT.EDU Mon Oct 24 11:30:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OFUid0011413 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 11:30:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OFUgfe025402 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 11:30:43 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OFUfnY005792 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 11:30:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j9OFUe0l012954; Mon, 24 Oct 2005 11:30:40 -0400 Received: from NEW-THIRTY-TWO.MIT.EDU (NEW-THIRTY-TWO.MIT.EDU [18.241.5.32]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 11:30:40 -0400 Message-ID: <20051024113040.labrgilwhc04sg44@webmail.mit.edu> Date: Mon, 24 Oct 2005 11:30:40 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 7 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 770 Content-Length: 1195 X-Keywords: NonJunk State Machines: The explanation of what they are is simple, but it will require a little practice to get the hang of actually coming up with invariants for them. The ususal approach is when we know what we're trying to prove, this time we must come up with what we're trying to prove and then prove it. This is the hardest part. Reachability and all that sort of thing is fine once I have some invariant to think about. Partial Correctness and Termination make sense; its just making sure your processor does the right thing and that it can't get stuck in an infinite loop. Derived Variables seem just to be whatever variable your state machine is using; all the registers in the processor. I'm not entirely sure what this f states -> R means. The Mating Algorighm: at first I did not want to believe that one set gets matched with their pessimal mate, but thinking about it, it makes sense that if one set is matched with their optimal mates and those mates prefer them better than the rest of the bunch (so the losing set gets their mate in order of least-pessimal), it will at least form a stable system. Other than that, it reminds me of non-linear system stability from 18.03. From meyer@imap.theory.csail.mit.edu Mon Oct 24 10:33:06 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OEX6d0001551 for <6042-staff@theory.csail.mit.edu>; Mon, 24 Oct 2005 10:33:06 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OEX4Si018787; Mon, 24 Oct 2005 10:33:04 -0400 (EDT) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OEX1Mu000612; Mon, 24 Oct 2005 10:33:02 -0400 (EDT) Message-ID: <435CF09C.3060509@theory.csail.mit.edu> Date: Mon, 24 Oct 2005 10:33:00 -0400 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "David A. Nedzel" CC: 6042-staff@theory.csail.mit.edu Subject: Re: [Sayan] Week 7 Comments References: <200510241343.j9ODhfVf017723@outgoing.mit.edu> In-Reply-To: <200510241343.j9ODhfVf017723@outgoing.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 817 X-UID: 771 X-Keywords: They are quite similar actually. As you have noticed their descriptions---states, starting states, and transitions---are same as Markov chains, modulo the probabilities associated with the transitions in the case of MCs. In fact, all the concepts that we have seen in class for state machines (finite automata) have their probabilistic analogues in the MC world. For example: Invariant property of the states ----- Invariant distribution over states Reachability of a state ----- Hitting time or expected time to reach a state Traditionally, MCs have been used for different type of analysis than what finite automata have been used for. -Sayan David A. Nedzel wrote: >How are the state machines described in the reading similar or different to >Markov state machines? > > > >- David > > > From meyer@imap.theory.csail.mit.edu Mon Oct 24 12:14:27 2005 Message-ID: <435D0866.8010009@csail.mit.edu> Date: Mon, 24 Oct 2005 12:14:30 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: David Shin CC: clintonb@MIT.EDU Subject: [Fwd: Jelani: TP 7 Comments] Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Content-Length: 830 Status: RO X-UID: 772 X-Keywords: Let's keep these links updated. regards, A. -------- Original Message -------- Subject: Jelani: TP 7 Comments Date: Mon, 24 Oct 2005 01:06:14 -0400 From: Clinton Blackburn To: <6042-probs@theory.lcs.mit.edu> I’m totally digging these problems with real-world relations/examples (ie. RSA, stable marriage). I learn best through these types of examples. I get the methods; although, I’m sure I need to go back over the different theorems and proofs for the stable marriage problem. Also, what happened to the pre-filled email links? I’m referring to the mailto links with the email address and subject line. I kind of liked not having to think of a subject line (the hardest part of writing emails for me). Good morning! --- Clinton Blackburn DLP – Have you seen it? From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:15:03 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGF2d0023070; Mon, 24 Oct 2005 12:15:02 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGF2bL023870; Mon, 24 Oct 2005 12:15:02 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGF2QX023867; Mon, 24 Oct 2005 12:15:02 -0400 Date: Mon, 24 Oct 2005 12:15:02 -0400 (EDT) From: Hanson Zhou To: Kevin Wang cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 7 Readings In-Reply-To: <6.2.3.4.2.20051023232523.01e7cef8@po10.mit.edu> Message-ID: References: <6.2.3.4.2.20051023232523.01e7cef8@po10.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 773 Content-Length: 600 X-Keywords: NonJunk For example, if you have a derived variable X of the algorithm at each step, with termination corresponding to a value of X=0, and each iteration lowers the value of X by at least 1, then you know that the algorithm can take no longer than the initial value of X. -Hanson On Sun, 23 Oct 2005, Kevin Wang wrote: > Hi Hanson, > > I was confused about the concept of a derived variable (Pg. 8, > section 4), especially how derived functions help us to find the > maximum run time of an algorithm and how it helps prove termination. > An example of this would be very helpful. > > Thanks, > Kevin > > From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:25:41 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGPfd0026281; Mon, 24 Oct 2005 12:25:41 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGPfiE023968; Mon, 24 Oct 2005 12:25:41 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGPfTH023965; Mon, 24 Oct 2005 12:25:41 -0400 Date: Mon, 24 Oct 2005 12:25:41 -0400 (EDT) From: Hanson Zhou To: Kushan K Surana cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: TA - Hanson: 6.042 Week 7 comments In-Reply-To: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> Message-ID: References: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 774 Content-Length: 322 X-Keywords: NonJunk Review the number theory notes: gcd(x,y) = gcd(y, remainder(x,y)). This gives the invariant. -Hanson On Mon, 24 Oct 2005, Kushan K Surana wrote: > - On page 6, the proof of the partial correctness of GCD is unclear. Why is the > predicate P(x,y) true for every state? The GCD is found only at the very last > state. > From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:26:29 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGQTd0026321; Mon, 24 Oct 2005 12:26:29 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGQTP5023981; Mon, 24 Oct 2005 12:26:29 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGQTSm023978; Mon, 24 Oct 2005 12:26:29 -0400 Date: Mon, 24 Oct 2005 12:26:29 -0400 (EDT) From: Hanson Zhou To: Zachary Adam Ozer cc: 6042-probs@theory.lcs.mit.edu Subject: Re: 6042 comments: Request for notes and lecture In-Reply-To: <4f2613e50510240755p13289032q8cfa2b48b4fb04af@mail.gmail.com> Message-ID: References: <4f2613e50510240755p13289032q8cfa2b48b4fb04af@mail.gmail.com> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 775 Content-Length: 556 X-Keywords: NonJunk I am thoroughly confused by this question :). -Hanson On Mon, 24 Oct 2005, Zachary Adam Ozer wrote: > First, I'm a bit confused as to how invarients are being used in the > reading in general; it seems as though at one moment there's a > theorem, but that they become some sort of equivalence statement. What > determines the equivalence relation or that it's true? > > More importantly however, get rid of Die Hard, it's getting old at > this point. Also, the bio type info on people is always awesome (vis > Turing, Flyod, Meyer, etc). > > -zozer > > From tylerw@MIT.EDU Mon Oct 24 12:28:13 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGSDd0026797 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:28:13 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OGSCgE025416 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:28:12 -0400 (EDT) Received: from [18.246.1.160] (TYLERW.MIT.EDU [18.246.1.160]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OGS9ok015200 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:28:09 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <11F6B62F-0919-428E-A71F-368926B5FDC4@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Tyler Williams Subject: [SAYAN] Week 7 Comments Date: Mon, 24 Oct 2005 12:28:08 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 3.816 X-Spam-Level: *** (3.816) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 776 Content-Length: 164 X-Keywords: NonJunk The part of this weeks reading that I didn't really understand the best was the stuff about derived variables. I hope we see these in lecture. Tyler Williams From hmzhou@blackbird.csail.mit.edu Mon Oct 24 12:34:29 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGYTd0028213; Mon, 24 Oct 2005 12:34:29 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.13.4/8.13.4) with ESMTP id j9OGYSmZ024056; Mon, 24 Oct 2005 12:34:28 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.13.4/8.13.4/Submit) with ESMTP id j9OGYS0v024053; Mon, 24 Oct 2005 12:34:28 -0400 Date: Mon, 24 Oct 2005 12:34:28 -0400 (EDT) From: Hanson Zhou To: Angelique E Moscicki cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Week 7 comments In-Reply-To: <20051024113040.labrgilwhc04sg44@webmail.mit.edu> Message-ID: References: <20051024113040.labrgilwhc04sg44@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 777 Content-Length: 1817 X-Keywords: NonJunk Careful...derived variables need not be the actual variables specified in the problem. They can be functions of them, as the notes demonstrate. For example, your invariant may well be A+B rather than either A or B. The notation f: states->R simply says that a derived variable function maps a state of the state machine to a real number value. For example, if the derived variable is being used as a counter towards termination, then the associated value of a state indicates how close the state is to a finishing one. -Hanson On Mon, 24 Oct 2005, Angelique E Moscicki wrote: > State Machines: The explanation of what they are is simple, but it will require > a little practice to get the hang of actually coming up with invariants for > them. The ususal approach is when we know what we're trying to prove, this > time we must come up with what we're trying to prove and then prove it. This > is the hardest part. Reachability and all that sort of thing is fine once I > have some invariant to think about. Partial Correctness and Termination make > sense; its just making sure your processor does the right thing and that it > can't get stuck in an infinite loop. > > Derived Variables seem just to be whatever variable your state machine is using; > all the registers in the processor. I'm not entirely sure what this f states -> > R means. > > The Mating Algorighm: at first I did not want to believe that one set gets > matched with their pessimal mate, but thinking about it, it makes sense that if > one set is matched with their optimal mates and those mates prefer them better > than the rest of the bunch (so the losing set gets their mate in order of > least-pessimal), it will at least form a stable system. Other than that, it > reminds me of non-linear system stability from 18.03. > > > > From jehan@MIT.EDU Mon Oct 24 12:52:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGqvd0032551 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:52:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OGqtfe024755 for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:52:56 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OGqmQx011697 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 24 Oct 2005 12:52:49 -0400 (EDT) Message-Id: <5.2.1.1.2.20051024124927.00c0cd50@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 24 Oct 2005 12:50:50 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 7 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 778 Content-Length: 90 X-Keywords: NonJunk I'm confused as to why state machines are useful at all. What benefits do they provide? From kktyan@MIT.EDU Mon Oct 24 12:55:14 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OGtEd0032702 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 12:55:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OGtDfe026987 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 12:55:13 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OGtBIO012513 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 12:55:11 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j9OGtB1w020642; Mon, 24 Oct 2005 12:55:11 -0400 Received: from BURTON-FOUR-SIXTY-SEVEN.MIT.EDU (BURTON-FOUR-SIXTY-SEVEN.MIT.EDU [18.247.6.212]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 24 Oct 2005 12:55:11 -0400 Message-ID: <20051024125511.vpt9an0gshs0gggw@webmail.mit.edu> Date: Mon, 24 Oct 2005 12:55:11 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 779 Content-Length: 280 X-Keywords: NonJunk I had little trouble understanding the reading in general. I just have a point that I'd like to clarify a little: a derived variable (page 8) is a function, essentially? Like, a stored function? Not a value? - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From rian@MIT.EDU Mon Oct 24 13:24:30 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OHOUd0007793 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 13:24:30 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OHOSfe028871 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 13:24:29 -0400 (EDT) Received: from m56-129-21.mit.edu (M56-129-21.MIT.EDU [18.56.0.50]) (authenticated bits=56) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OHOQ6H025235 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Mon, 24 Oct 2005 13:24:27 -0400 (EDT) Received: (from rian@localhost) by m56-129-21.mit.edu (8.12.9) id j9OHOQdu004259; Mon, 24 Oct 2005 13:24:26 -0400 Subject: (jelani) required reading From: rian To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Mon, 24 Oct 2005 13:24:26 -0400 Message-Id: <1130174666.4226.1.camel@m56-129-21.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 780 Content-Length: 75 X-Status: A X-Keywords: NonJunk I honestly found no difficulty with this weeks lecture notes. Rian Hunter From meyer@imap.theory.csail.mit.edu Mon Oct 24 13:25:06 2005 Message-ID: <435D18F4.8030704@csail.mit.edu> Date: Mon, 24 Oct 2005 13:25:08 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kushan K Surana CC: Hanson M Zhou Subject: Re: TA - Hanson: 6.042 Week 7 comments References: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> In-Reply-To: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Content-Length: 867 Status: RO X-UID: 781 X-Keywords: I'm guessing you're confused by the fact that the invariant mentions the unknown quantity gcd(a,b), which is exactly what the algorithm is trying to find. But there's nothing wrong with reasoning about an unknown quantity based on the ([possibly limited) info you may have about it. For example, suppose I let Q(d) be the predicate "d is a nonnegative integer". Then I know that Q(gcd(a,b,)) is true, even though I haven't found gcd(a,b) yet. Similarly, if at the start of a procedure I set x:=a and y:=b, then I know immediately that "gcd(x,y)=gcd(a,b)" holds in the start state, even though I haven't calculated anything yet. Does that help? regards, A. Kushan K Surana wrote: > - On page 6, the proof of the partial correctness of GCD is unclear. Why is the > predicate P(x,y) true for every state? The GCD is found only at the very last > state. From meyer@csail.mit.edu Mon Oct 24 13:33:47 2005 Return-Path: Received: from [128.30.49.31] (theory-dhcp-31.csail.mit.edu [128.30.49.31]) (authenticated bits=0) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OHXUd0008368; Mon, 24 Oct 2005 13:33:46 -0400 Message-ID: <435D1AEA.2080104@csail.mit.edu> Date: Mon, 24 Oct 2005 13:33:30 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: rian CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: (jelani) required reading References: <1130174666.4226.1.camel@m56-129-21.mit.edu> In-Reply-To: <1130174666.4226.1.camel@m56-129-21.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 782 Content-Length: 423 X-Keywords: I believe you, but your email doesn't have to identify parts you found hard: you could indicate what you found most (or least) interesting, surprising, poorly (or specially well) explained, relevant to your professional interests (or not),.... I'll look forward to more specific & responsive comments next week. regards, A. rian wrote: > I honestly found no difficulty with this weeks lecture notes. > > Rian Hunter From meyer@imap.theory.csail.mit.edu Mon Oct 24 14:34:29 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OIYSd0024112 for ; Mon, 24 Oct 2005 14:34:28 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EU79Q-0007r9-KK for meyer@csail.MIT.EDU; Mon, 24 Oct 2005 14:34:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OIYQp4012779; Mon, 24 Oct 2005 14:34:26 -0400 (EDT) Received: from ccbibmx30 (BLACKBURN.MIT.EDU [18.241.3.107]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9OIYHdh024826 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT); Mon, 24 Oct 2005 14:34:17 -0400 (EDT) Message-Id: <200510241834.j9OIYHdh024826@outgoing.mit.edu> From: "Clinton Blackburn" To: "'David Shin'" , "'Albert R. Meyer'" Subject: RE: [Fwd: Jelani: TP 7 Comments] Date: Mon, 24 Oct 2005 14:32:14 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 In-Reply-To: <20051024125045.s69nn6imizkg4kow@webmail.mit.edu> Thread-Index: AcXYuzUGgdrPXXw+QISUKvJ+Tw273QADerKg X-Spam-Score: 2.037 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1478 X-UID: 783 X-Keywords: Oops. I must have been clicking and scrolling too fast. My apologies. --- Clinton Blackburn DLP - Have you seen it? http://www.dlp.com -----Original Message----- From: David Shin [mailto:dshin@MIT.EDU] Sent: Monday, October 24, 2005 12:51 PM To: Albert R. Meyer Cc: clintonb@mit.edu Subject: Re: [Fwd: Jelani: TP 7 Comments] Clinton, When I log on to the tutor and select TP7, the first page I come across has a prefilled email link ("subject=[TA-name] Week 7 Comments"). Is there another link somewhere? Please let me know. DS Quoting "Albert R. Meyer" : > Let's keep these links updated. > regards, A. > > -------- Original Message -------- > Subject: Jelani: TP 7 Comments > Date: Mon, 24 Oct 2005 01:06:14 -0400 > From: Clinton Blackburn > To: <6042-probs@theory.lcs.mit.edu> > > > > I'm totally digging these problems with real-world relations/examples > (ie. RSA, stable marriage). I learn best through these types of > examples. I get the methods; although, I'm sure I need to go back over > the different theorems and proofs for the stable marriage problem. > > > > Also, what happened to the pre-filled email links? I'm referring to the > mailto links with the email address and subject line. I kind of liked > not having to think of a subject line (the hardest part of writing > emails for me). > > > > Good morning! > > --- > > Clinton Blackburn > > DLP - Have you seen it? > > > > > > From mitras@theory.csail.mit.edu Mon Oct 24 18:27:38 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9OMRcd0008261 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 18:27:38 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9OMRaXA012523; Mon, 24 Oct 2005 18:27:37 -0400 (EDT) Received: from [128.30.51.97] (drake.csail.mit.edu [128.30.51.97]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9OMRTok007535; Mon, 24 Oct 2005 18:27:29 -0400 (EDT) Message-ID: <435D5FD1.2050807@theory.csail.mit.edu> Date: Mon, 24 Oct 2005 18:27:29 -0400 From: Sayan Mitra User-Agent: Mozilla Thunderbird 1.0.7-1.1.fc3 (X11/20050929) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jehan deFonseka CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 7 Comments References: <5.2.1.1.2.20051024124927.00c0cd50@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20051024124927.00c0cd50@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 784 Content-Length: 1613 X-Keywords: Finite state machines (FSM) serve as a natural and often tractable model for systems. For example (from Wikipedia): In a digital circuit , a FSM may be built using a programmable logic device , a programmable logic controller , logic gates and flip flops or relays . More specifically, a hardware implementation requires a register to store state variables, a block of combinational logic which determines the state transition, and a second block of combinational logic that determines the output of a FSM. There are innumerable other applications both in hardware and software. The state machine model is attractive because it comes with a bunch of proof methods, like the Invariant Theorem that we have studied in class. These mathematical proof methods can often be incorporated into analysis tools for verifying correctness of systems. For example, there are Model checking tools that can automatically check if a given state machine model with more than 10^80 states satisfies a certain invariant property! These tools are useful for catching bugs that would be hard to find otherwise. The Pentium FDIV bug was found using such tools. -Sayan Jehan deFonseka wrote: > I'm confused as to why state machines are useful at all. What benefits > do they provide? > From dshin@MIT.EDU Mon Oct 24 20:11:09 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9P0B9d0019063 for <6042-probs@theory.lcs.mit.edu>; Mon, 24 Oct 2005 20:11:09 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9P0B5sh016012; Mon, 24 Oct 2005 20:11:05 -0400 (EDT) Received: from [192.168.1.3] (c-71-192-58-238.hsd1.ma.comcast.net [71.192.58.238]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j9P0AvMu005438; Mon, 24 Oct 2005 20:10:57 -0400 (EDT) Message-ID: <435D7810.8010806@mit.edu> Date: Mon, 24 Oct 2005 20:10:56 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hao Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: David TP7 Reading Comment References: <20051024002104.pj08nk8euo34sswo@webmail.mit.edu> In-Reply-To: <20051024002104.pj08nk8euo34sswo@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 1.217 X-Spam-Level: * (1.217) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 785 Content-Length: 754 X-Keywords: Unfortunately, experience and intuition is the best answer we can really give. One can actually prove that there is no preset method that will find the right invariant! When you are designing your own state machines (i.e., "algorithms"), I think you will find that finding that invariant often goes hand-in-hand with designing the algorithm. DS Hao Zhou wrote: >Hi Staff, > >I found using the state invarient is a very clever way to solve problems. >However, I find it is often times hard to find an invarient for all the states. > For example, for the tiling problem we did in class, the invarient was not a >very obvious choice. Other than through experience, how should we look for >these invarients? Thanks > >- Steven (Hao) Zhou > > From meyer@imap.theory.csail.mit.edu Tue Oct 25 10:01:11 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j9PCqqd0002059 for ; Tue, 25 Oct 2005 08:52:52 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EUOIO-0007p4-Cl for meyer@csail.MIT.EDU; Tue, 25 Oct 2005 08:52:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j9PCqnwN008390; Tue, 25 Oct 2005 08:52:50 -0400 (EDT) Received: from cs37.mit.edu (CS37.MIT.EDU [18.62.4.237]) (authenticated bits=56) (User authenticated as kushan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j9PCqgG6015504 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Tue, 25 Oct 2005 08:52:43 -0400 (EDT) Received: (from kushan@localhost) by cs37.mit.edu (8.12.9) id j9PCqgC9028746; Tue, 25 Oct 2005 08:52:42 -0400 Date: Tue, 25 Oct 2005 08:52:42 -0400 (EDT) From: Kushan K Surana To: "Albert R. Meyer" cc: Hanson M Zhou Subject: Re: TA - Hanson: 6.042 Week 7 comments In-Reply-To: <435D18F4.8030704@csail.mit.edu> Message-ID: References: <20051024103725.421xcxdsyp448k4s@webmail.mit.edu> <435D18F4.8030704@csail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Spam-Score: 1.217 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 1043 X-UID: 786 X-Keywords: NonJunk I think so. I have to review the notes again and clear it in my mind. Thanks, Kushan On Mon, 24 Oct 2005, Albert R. Meyer wrote: > I'm guessing you're confused by the fact that the invariant mentions the > unknown quantity gcd(a,b), which is exactly what the algorithm is trying to > find. > > But there's nothing wrong with reasoning about an unknown quantity based on > the ([possibly limited) info you may have about it. For example, suppose I > let Q(d) be the predicate "d is a nonnegative integer". Then I know that > Q(gcd(a,b,)) is true, even though I haven't found gcd(a,b) yet. Similarly, > if at the start of a procedure I set x:=a and y:=b, then I know immediately > that "gcd(x,y)=gcd(a,b)" holds in the start state, even though I haven't > calculated anything yet. > > Does that help? > > regards, A. > > Kushan K Surana wrote: >> - On page 6, the proof of the partial correctness of GCD is unclear. Why is >> the >> predicate P(x,y) true for every state? The GCD is found only at the very >> last >> state. > From meyer@imap.theory.csail.mit.edu Mon Oct 24 13:33:47 2005 Message-ID: <435D1AEA.2080104@csail.mit.edu> Date: Mon, 24 Oct 2005 13:33:30 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: rian CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: (jelani) required reading References: <1130174666.4226.1.camel@m56-129-21.mit.edu> In-Reply-To: <1130174666.4226.1.camel@m56-129-21.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 424 X-UID: 787 X-Keywords: I believe you, but your email doesn't have to identify parts you found hard: you could indicate what you found most (or least) interesting, surprising, poorly (or specially well) explained, relevant to your professional interests (or not),.... I'll look forward to more specific & responsive comments next week. regards, A. rian wrote: > I honestly found no difficulty with this weeks lecture notes. > > Rian Hunter From meyer@imap.theory.csail.mit.edu Mon Oct 24 13:47:05 2005 Message-ID: <435D1E19.4070902@csail.mit.edu> Date: Mon, 24 Oct 2005 13:47:05 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kamil Y Sindi Subject: Re: [David] Week 7 Comments References: <20051023231516.kafh91q4locgo0wg@webmail.mit.edu> In-Reply-To: <20051023231516.kafh91q4locgo0wg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 331 X-UID: 788 X-Keywords: Didn't you think verification of the extended GCD was a practical application? Or did you feel it really didn't use the state machine formalism? Regards, A. Kamil Y Sindi wrote: > I found the notes clear but still don't understand the practical use of these > state concepts. It seems to me that they are mere formilizations