August 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Re: Kepler conjecture & Flyspeck
by asimovd＠aol.com 11 Aug '03

11 Aug '03

John McCarthy wrote: << When a long computer calculation is an important part of the proof of a mathematical theorem, we should ask for proof (preferably both computer-checked and hand checked) that the program does the advertised calculation. For example, we want a proof that if the calculation comes out, the four color conjecture is true. >> This is of course desirable, but brings up a new question: where does this stop? A typical "verification of correctness" of a sizeable computer program involves the application of another program. Then we need to know that the verification program is valid and was used properly. Etc. --Dan

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[math-fun] Bekenstein article about Thermo/Info in Scientific American
by Henry Baker 08 Aug '03

08 Aug '03

Add'l fuel for the fiery debate from a few weeks ago: http://www.sciam.com/print_version.cfm?articleID=000AF072-4891-1F0A-97AE80A… It looks like the entire article is online, and apparently doesn't require any kind of sign-in (at least not from me!).

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[math-fun] An interesting upcoming test case ...
by Thane Plambeck 08 Aug '03

08 Aug '03

... for the (surely true?) proposition that a trendy novelist couldnt possibly write anything interesting about mathematics http://www.amazon.com/exec/obidos/ASIN/0393003388 Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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[math-fun] mathematical aphrodesiac: love story from National Story Project
by Mike Stay 07 Aug '03

07 Aug '03

Quoted from "I Thought My Father Was God," a collection of stories contributed to NPR's National Story Project. MATHEMATICAL APHRODISIAC In the days when John and I used to break up all the time, we made a decision to see each other only casually. Dates were okay, but no more than once a week. We were going to lead separate lives, getting together occasionally when the spirit moved us, but without worrying about commitment. One day at the beginning of this period, we were sitting together on the floor of John's one-room apartment. He was knitting himself a sweater and I was reading Fermat's Last Theorem. Every now and then, I'd interrupt his knitting to read him passages from my book. "Did you ever hear of amicable numbers? They're like perfect numbers, but instead of being the sum of their own divisors, they're the sum of each others divisors. In the Middle Ages people used to carve amicable numbers into pieces of fruit. They'd eat the first piece themselves and then feed the other one to their lover. It was a mathematical aphrodisiac. I love that -- a mathematical aphrodisiac." John showed little interest. He doesn't like math much. Not like I do. It was one more reason for us to be casual. Christmas fell during this period, and since I hate to shop, I was glad to be able to cross John off my shopping list. We were too casual for presents. While I was shopping for my grandmother, however, I saw a cryptic crossword puzzle book and bought it for John. We had always worked on the cryptic crossword puzzles at the back of The Nation, and for five bucks I figured I could give it to him. When Christmas rolled around, I handed John the book -- unwrapped, very casual. He didn't give me anything at all. I wasn't surprised, but my feelings were a little hurt, even though I wasn't supposed to care. The next day, John invited me over to his apartment. "I have your Christmas present" he said. "Sorry it's late." He handed me an awkwardly wrapped bundle. When I pulled it open, a rectangle of hand-knit fabric fell on my lap. I picked it up and looked at it, completely confused. One side had the number 124,155 knitted into it; the other side had 100,845. When I looked up at John again, he was barely able to contain his excitement anymore. "They're amicable numbers," he said. "I wrote a computer program and let it run for twelve hours. These were the biggest ones I found, and then I double-knit them in. It's a pot holder. I couldn't give it to you last night but I still haven't figured out how to cast off. It's kind of geeky, but I thought you might like it." After that Christmas, we were a lot of things, but we weren't casual anymore. The ancient mathematical aphrodisiac had worked again. -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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RE: [[math-fun] Nim-like game]
by Christian G. Bower 06 Aug '03

06 Aug '03

"Jon Perry" <perry(a)globalnet.co.uk> wrote: > Neat. What do 1 and 2 imply? > For the purposes of a single heap game, 0 implies 2nd player win, anything else implies first player win. The different values 1 and 2 are useful if you wish to sum this game with other games (or copies of itself). I won't go into the details of what that means. A copy of Winning Ways I'm sure will explain that in great detail. I'll give this short technical description. A postion's value is zero if all the moves from that position are zero. The value is one if the current player can move to zero but not to one. Two if the current player can move to zero or one, but not to two. > If the increase range is generalised to l<=i<=r (l=1, r=2 is my original > example - generally l>=0), how can you predict which player will win? That looks easy to model. I can't say what the answer will be until I model it. I expect similar results, but I am concerned that the decision tree can get very complicated as r-l increases. > > For the multiple heaps version, if a player can only remove from 1 heap at a > time, which the additional play that if a player can exhaust a heap this > signifies the end of their turn (i.e. they can pre-empt having to take k > buttons), there are 2 options - the next player only need take i more > buttons than the previous player took, or the pre-empting player can state > how many they took and the next player must take more than this. I don't know how to use the theory for the multiple heaps version other than to consider each case individually. I suspect that for more than 3 heaps it can get very complicated. Additionally, that rule that allows a player to state how many they "took" when emptying a heap can lead to a large number of choices for that player.

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[math-fun] Abhyankar: #zeros = #poles -> no gravitational pull in hollow shell
by Bill Dubuque 01 Aug '03

01 Aug '03

I'd be grateful if someone could elaborate on Abhyankar's remarks below, esp. precisely how the mathematical principles are realized physically (alas, my physics knowledge has atrophied) "On the other hand, Riemann's approach appears to have been existential: know that the formulas exist! Many of his existence arguments were based on "Dirichlet's principle," the genesis of which can be expressed by saying something like this: the real and imaginary parts of an analytic function of a complex variable satisfy the same sort of partial differential equation as a gravitational potential or an electric potential (or a fluid flow or heat conduction ...); so the existence of analytic functions or differentials with prescribed boundary behaviour or assigned singularities can be adduced from physical considerations. To carry the analogy further (or backwards), once could say that the mathematical formula "number of zeros of a function = number of its poles" corresponds to the physical fact that there is no gravitational pull inside a hollow shell, or that there is no electric intensity inside a hollow charged conductor. Riemann's approach was furthered by Hilbert who put the Dirichlet principle on firmer grounds." The above is excerpted from p.411 of Abhyankar's classic Monthly paper [1], for which he was awarded the Chauvenet Prize [0] by the MAA (for outstanding expository article on a mathematical topic). The award is well-deserved. If you haven't yet read his ramblings I highly recommend doing so. Below is its jstor link and Math Review. See also his introductory textbook Algebraic Geometry for Scientists and Engineers, AMS, 1990, where he says "I have tried to tell the story of algebraic geometry and to bring out the poetry in it". If only we had more such mathematical poets... -Bill Dubuque [0] http://www.maa.org/awards/chauvent.html ------------------------------------------------------------------------------ [1] 53 #5581 14-02 (14E15 32C45) Abhyankar, Shreeram S. Historical ramblings in algebraic geometry and related algebra. Amer. Math. Monthly 83 (1976), no. 6, 409-448. http://links.jstor.org/sici?sici=0002-9890(197606/07)83:6%3C409:HRIAGA%3E ------------------------------------------------------------------------------ The algebra that is used to formulate ideas and to prove theorems in algebraic geometry comes in several different varieties. The author distinguishes three of these, namely, "high-school algebra" (polynomials, power series), "college algebra" (rings, ideals, fields) and "university algebra" (functors, homological algebra). The fundamental thesis of this entertaining and most readable paper is, in the author's words, "The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester." The author takes as his theme the genus of a plane algebraic curve (for example over the complex numbers) and gives several interpretations in terms of the various algebras and of analysis, sketching proofs of the equivalence of the various definitions. There are also four short illustrations of the author's fundamental thesis, drawn from his own experience. (In particular he attributes the gradual progress he has made with the resolution of singularities in nonzero characteristic to a systematic weeding-out of college algebra, the better for the high-school algebra to flourish!) The definitions of genus are briefly these: (1) function-theoretic: "2g-2 = number of zeros of a differential form - number of poles"; (2) topological: "number of handles on the Riemann surface"; (3) algebraic: (N-1)(N-2)/2 - sum \d(P) for a curve of order N, where \d(P) is the number of double points concentrated at P, which varies over the curve -- four interpretations of \d(P) are used and reconciled; (4) dimension of the Jacobian variety; (5) (mentioned only) via cohomology. In the course of sketching proofs of the equivalence of the various definitions of genus -- and in particular of the various definitions of \d -- the author gives rapid introductions to a number of topics, such as abelian integrals, resultants and discriminants, conductors, blowing up singularities by means of quadratic transformations, Bezout's theorem, Puiseux expansions, valuations and the monodromy group. The bibliography of 81 items ranges from Newton ("truly the father of us all"), through Euler and Weierstrass ("a most distinguished high-school algebraist") to Salmon, Cayley, Chrystal and Macaulay ("a grand master of high-school algebra"), apart from the more predictable modern references. Reviewed by Peter Giblin

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[math-fun] cyclic polygons
by Richard Schroeppel 01 Aug '03

01 Aug '03

David Robbins has joined Math-Fun temporarily. Rich rcs(a)cs.arizona.edu

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