April 2004 - mathfun@mailman.xmission.com

[math-fun] (no subject)
by MCKAY john 09 Apr '04

09 Apr '04

Bill's post on nets reminds me of Jean Taylor's work Sci. Am. 1976 July on foam and bubbles. She has ten trivalent graphs on a sphere. Each has edges made of great arcs. They meet at 2.pi/3. Now it is "well-known" that the Cayley graph of the modular group (a free product <x.y> of a cyclic groups C2=<x> & C3=<y>), is a free trivalent tree with oriented nodes. (Each encoding for the triangle of the C3 action.) Any finite connected trivalent graph is the Schreier coset graph of the modular group on some subgroup. Each such graph yields permutations for the images of the two generators of the free product PSL(2,Z), generating a factor group of the modular group on cosets. We have (as Bill points out) for subgroups, the platonics, Gamma(N), N=2,3,4,5 yielding the regular representations of PSL(2,N), and the great circle itself {how to interpret this one?} and the "others": For the "other" graphs I find the disjoint cycle shapes (= cusp widths of the stabilizer of a point) to be of index: Index shape xy-image order of perm gp. comments 18: 44433 648 solvable 30: 5544444 2**3.3.5**3 solvable 36: 55553333 2**23.3**4.5.7 A9 composition factor 42: 555555444 2**10.3**9.5.7 A7 composition factor 48: 5555555544 2**14.3.5 ?solvable It would be fun if some minimality can be deduced from these groups alone! John

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[math-fun] The primes contain arbitrarily long arithmetic progressions
by Richard Schroeppel 09 Apr '04

09 Apr '04

I don't imagine this will be an easy read, but the result seems neat. Rich ------ Paper: math.NT/0404188 Date: Thu, 8 Apr 2004 16:48:41 GMT (49kb) Title: The primes contain arbitrarily long arithmetic progressions Authors: Ben Green and Terence Tao Comments: 49 pages Subj-class: Number Theory; Dynamical Systems \\ We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi's theorem that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this, one may place the primes inside a pseudorandom set of ``almost primes'' with positive relative density. \\ ( http://arXiv.org/abs/math/0404188 , 49kb)

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[math-fun] FW: Math Spam from David Wilson
by Richard Schroeppel 09 Apr '04

09 Apr '04

This is the first math message in months to appear in the old Arizona Math-Fun mailbox. No doubt a fingerslip by DW. Curiously, it has its own special spam-like qualities. It seems that within each row the rule is Next = 5 * Current + 8. I can vouch for the first three primes, 7 43 223. Rich ---------------- From: "David Wilson" <davidwwilson(a)comcast.net> To: "Math Fun" <math-fun(a)CS.Arizona.EDU> Subject: simple check Date: Fri, 9 Apr 2004 01:10:36 -0400 Could any of you verify that each of the following numbers is prime? 7 43 223 1123 5623 28123 7621 38113 190573 952873 4764373 23821873 15937 79693 398473 1992373 9961873 49809373 382933 1914673 9573373 47866873 239334373 1196671873 462271 2311363 11556823 57784123 288920623 1444603123 518803 2594023 12970123 64850623 324253123 1621265623 866941 4334713 21673573 108367873 541839373 2709196873 1025407 5127043 25635223 128176123 640880623 3204403123 16022015623 1142503 5712523 28562623 142813123 714065623 3570328123 1427347 7136743 35683723 178418623 892093123 4460465623 1742473 8712373 43561873 217809373 1089046873 5445234373 1980067 9900343 49501723 247508623 1237543123 6187715623 2343619 11718103 58590523 292952623 1464763123 7323815623 2910031 14550163 72750823 363754123 1818770623 9093853123 45469265623 3015283 15076423 75382123 376910623 1884553123 9422765623 4344121 21720613 108603073 543015373 2715076873 13575384373 4352269 21761353 108806773 544033873 2720169373 13600846873 4544209 22721053 113605273 568026373 2840131873 14200659373 5081893 25409473 127047373 635236873 3176184373 15880921873 5116543 25582723 127913623 639568123 3197840623 15989203123 5127043 25635223 128176123 640880623 3204403123 16022015623 5482531 27412663 137063323 685316623 3426583123 17132915623 5484379 27421903 137109523 685547623 3427738123 17138690623 5501473 27507373 137536873 687684373 3438421873 17192109373 5648221 28241113 141205573 706027873 3530139373 17650696873 88253484373 6452899 32264503 161322523 806612623 4033063123 20165315623 7213897 36069493 180347473 901737373 4508686873 22543434373 7968469 39842353 199211773 996058873 4980294373 24901471873 8151043 40755223 203776123 1018880623 5094403123 25472015623 8162047 40810243 204051223 1020256123 5101280623 25506403123 8236219 41181103 205905523 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478244653 2391223273 11956116373 59780581873 298902909373 1494514546873 7472572734373 96463267 482316343 2411581723 12057908623 60289543123 301447715623 98380357 491901793 2459508973 12297544873 61487724373 307438621873 101220313 506101573 2530507873 12652539373 63262696873 316313484373 <end of DW message>

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[math-fun] Torus
by Richard Guy 09 Apr '04

09 Apr '04

Yesterday we watched the video Outside In. Is it possible to turn a torus inside-out without breaking or creasing it? R.

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[math-fun] rectangle arithmetic
by R. William Gosper 07 Apr '04

07 Apr '04

Very elementary (8th gradish) paper: www.ippi.com/rwg/rectarith7.rtf . (~13pp, 9.7MB (BMPy).) E.g., can you reduce a fraction without computing the gcd? What's the quickest way to bat .239? --rwg

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[math-fun] Emil Post & bipolar disorder
by Henry Baker 07 Apr '04

07 Apr '04

Mathematics Magazine 77,1 (Feb. 2004) contains an article about the logician Emil Post, who apparently discovered unsolvability and incompleteness prior to Goedel using a diagonalization argument. But I was more intrigued by the fact that he was apparently bipolar, and was also aware of his condition. His approach to his disease: work on two different problems simultaneously, and stop working on the one making the most progress, so as not to trigger the manic phase. Presumably, if he started getting too depressed on the other problem, he would return to the first problem. A very elegant solution to his problem! Unfortunately, this algorithm didn't always work, and he therefore underwent electric shock therapy. Interestingly, Post also worked later with recursively enumerable sets (which are most interesting when their complement isn't). Many of the arguments involve timesharing a machine to work on a problem and its complement simultaneously -- something Post had much practise at.

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[math-fun] dirac scissors trick + balinese cup trick
by Thane Plambeck 07 Apr '04

07 Apr '04

> (This is the > same thing that says you can hold a full cup of coffee in one hand, and > without letting go or spilling, turn > it around twice and get back in the same position. However, if you > turn it around just once, your arm will be > in a different, probably awkward, position.) I read that Dirac illustrated this fact, that a 360 degree rotation isn't the same as a 720 degree one, using strings tied to a pair of scissors, right? My question: where/when did he do it? Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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Re: [math-fun] dirac scissors trick + balinese cup trick
by MCKAY＠vax2.concordia.ca 07 Apr '04

07 Apr '04

CC: Subj: RE: [math-fun] dirac scissors trick + balinese cup trick Return-path: <math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com> Received: from newclyde.concordia.ca (newclyde.Concordia.CA [132.205.1.11]) by vax2.concordia.ca (PMDF V6.2 #30759) with ESMTP id <01L8N682POJQ003UGC(a)vax2.concordia.ca> for mckay(a)vax2.concordia.ca (ORCPT mckay(a)vax2.concordia.ca) Wed, 07 Apr 2004 09:12:27 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by newclyde.concordia.ca (8.12.10/8.12.10) with ESMTP id i37DCNtq015945 for <mckay(a)vax2.concordia.ca>; Wed, 07 Apr 2004 09:12:23 -0400 Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.35 #1 (Debian)) id 1BBCXh-0003Td-09 for <mckay(a)vax2.concordia.ca>; Wed, 07 Apr 2004 06:52:33 -0600 Received: from mgr3.xmission.com ([198.60.22.203]) by mailman.xmission.com with esmtp (Exim 3.35 #1 (Debian)) id 1BBCXW-0003TW-00 for <math-fun(a)mailman.xmission.com>; Wed, 07 Apr 2004 06:52:22 -0600 Received: from [65.171.232.3] (helo=ldap.micgi.com) by mgr3.xmission.com with smtp (Exim 4.30) id 1BBCXT-0004RV-QE for math-fun(a)mailman.xmission.com; Wed, 07 Apr 2004 06:52:19 -0600 Received: (qmail 39785 invoked from network); Wed, 07 Apr 2004 12:52:17 +0000 Received: from unknown (HELO alan-oxyfzctp75.neondsl.com) (209.248.59.156) by ldap.micgi.com with SMTP; Wed, 07 Apr 2004 12:52:17 +0000 Date: Wed, 07 Apr 2004 07:52:21 -0500 From: Alan Schoen <alanschoen(a)neondsl.com> Subject: Re: [math-fun] dirac scissors trick + balinese cup trick In-reply-to: <013501c41a75$541e6370$a52efea9@TASSO> Sender: math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com X-Sender: alanschoen@neondsl.com@mail.neondsl.com (Unverified) q To: math-fun <math-fun(a)mailman.xmission.com> Errors-to: math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com Reply-to: math-fun <math-fun(a)mailman.xmission.com> Message-id: <6.0.1.1.2.20040407074303.01c3a118(a)mail.neondsl.com> MIME-version: 1.0 X-Mailer: QUALCOMM Windows Eudora Version 6.0.1.1 Content-type: text/plain; format=flowed; charset=us-ascii Precedence: list X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mgr3.xmission.com X-Spam-Level: X-Spam-Status: No, hits=0.0 required=8.0 tests=none autolearn=no version=2.63 X-SA-Exim-Mail-From: alanschoen(a)neondsl.com X-SA-Exim-Version: 3.1 (built Mon Jan 26 13:00:24 MST 2004) X-SA-Exim-Scanned: Yes X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.1 X-Scanned-By: MIMEDefang 2.35 References: <013501c41a75$541e6370$a52efea9@TASSO> List-Post: <mailto:math-fun@mailman.xmission.com> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Id: math-fun <math-fun.mailman.xmission.com> Original-recipient: rfc822;mckay(a)vax2.concordia.ca Thane: Dirac visited SIUC around 1978 for an Einstein symposium. When I met him, I mentioned to him that I recalled having seen -- around 1965 -- a brief reference to his spinor-related trick in Time magazine. The article suggested that he had discovered it around 1932. He confirmed that that was probably about the right date. I suggest that you ask my physicist colleague Jerzy Kocik (jkocik(a)math.siu.edu) to fill you in on the details of the spinor connection. He has demonstrated the trick (with flailing arms) and he has expert knowledge of quantum mechanics. I'll send a copy of this email to Jurek (Jerzy). Alan At 01:47 PM 4/4/2004, you wrote: > > (This is the > > same thing that says you can hold a full cup of coffee in one hand, and > > without letting go or spilling, turn > > it around twice and get back in the same position. However, if you > > turn it around just once, your arm will be > > in a different, probably awkward, position.) > >I read that Dirac illustrated this fact, that a 360 degree rotation isn't >the same as >a 720 degree one, using strings tied to a pair of scissors, right? > >My question: where/when did he do it? > >Thane Plambeck >650 321 4884 office >650 323 4928 fax >http://www.plambeck.org > > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] dirac scissors trick + balinese cup trick
by Jpr2718＠aol.com 04 Apr '04

04 Apr '04

Thane Plambeck wrote: > I read that Dirac illustrated this fact, that a 360 degree rotation isn't the same > as a 720 degree one, using strings tied to a pair of scissors, right? > My question: where/when did he do it? The article The Spinor Spanner, Ethan D. Bolker, The American Mathematical Monthly, Vol. 80, No. 9. (Nov., 1973), pp. 977-984 suggests that this was a regular part of certain courses he taught. (Whether it answers the question asked or not, it's a lovely article. It's easy to build one -- 3 sufficiently long cords connected to a frame and to a spanner, wrench, scissors, piece of wood cut to the shape of a wrench, whatever -- and it's fun to play with.) John Robertson

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[math-fun] random walks: self-avoiding but with max self-cuddle
by wouter meeussen 03 Apr '04

03 Apr '04

imagine a self-avoiding 3D walk on a diamond (tetrahedral) net, where randomness is minimised by choosing among non-occupied sites, the one(s) with maximum number of neighbours. In case of zero neighbours, explore one step further ahead until one or more sites with neighbours is found. This reduces the number of random choices between 'equivalent' continuations. Q: how far does the randomness (entropy per step) get reduced? Experiment gives about 50% reduction vs non-self-avoiding reference. Anyone seen this before? On the physics side, I can imagine processes of polymerisation where the newly formed polymer strand coils upon itself (plastics, proteins,..) Q: how far is this away from space-filling? W.

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