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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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[math-fun] polygons
by Richard Schroeppel 31 Jul '03

31 Jul '03

Carrying Dan Asimov's analysis a litle further, the edges Ek fit a circumcircle of radius R (diameter D=2R) when the product Prod ( sqrt(D^2-Ek^2) + i Ek ) has 0 imaginary part, i.e. is real. This gives a messy algebraic expression for D, and hints that there might not be a closed form when the number of edges is increased, because we run afoul of unsolvable quintics etc. The full permutability of the edges is another hint in that direction. Of course there's nothing that says finding D is equivalent to finding the area of the polygon. Knowing D, we can indeed get the area as the sum of triangles of sides R,R,Ek; but perhaps the reverse direction is more difficult. Rich rcs(a)cs.arizona.edu

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Re: [math-fun] inscribed polygon area
by asimovd＠aol.com 31 Jul '03

31 Jul '03

Andy writes: << I wrote: > Wait -- consider any 3 consecutive (ccw) vertices A,B,C of a > convex n-gon. If the triangle ABC is cut from the n-gon and > re-glued so that they now appear in ccw order C,B,A, then > it's easy to see that doesn't change the circumcircle (by its > symmetry about the perpendicular bisector of AC). Since all > permutations of the cyclic ordeer of the edges are generated > by ones of this sort, Actually, they aren't, if the polygon has an even number of sides. Color the sides alternately red and blue, and note that your permutations always switch edges of the same color. >> What I intended to convey above is the operation of cutting off from the polygon the triangle whose vertices are 3 consecutive vertices A,B,C of the polygon (and therefore containing two consecutive edges AB and BC of the polygon), and gluing ABC back with AC reversed. The net result is that two adjacent edges of the polyong hav e been switched. These operations must generate the group of all permutations of the polygon edges. --Dan

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RE: [math-fun] inscribed polygon area
by Andy Latto 31 Jul '03

31 Jul '03

> > Wait -- consider any 3 consecutive (ccw) vertices A,B,C of a > convex n-gon. If the triangle ABC is cut from the n-gon and > re-glued so that they now appear in ccw order C,B,A, then > it's easy to see that doesn't change the circumcircle (by its > symmetry about the perpendicular bisector of AC). Since all > permutations of the cyclic ordeer of the edges are generated > by ones of this sort, Actually, they aren't, if the polygon has an even number of sides. Color the sides alternately red and blue, and note that your permutations always switch edges of the same color. >that shows the circumcircle and > circumradius are functions purely of the multiset of edge > lengths. But the proof you posted later, implicit in the equation that relates the circumcircle radius to the sequence of edge lengths (which, if you examine it, doesn't depend on edge order), is fine. > (If true, I never realized this independence before.) Think of cutting a circular disk into pieces; one is the polygon, and the others are each bounded by a side and an arc. Discard the polygonal piece. No matter how you rearrange the other pieces, they include a total arc length of 2piR, so they will fit together to form a full circle of the same radius no matter what order you line them up in. Andy >

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Re: [[math-fun] Nim-like game]
by Christian G. Bower 31 Jul '03

31 Jul '03

"Jon Perry" <perry(a)globalnet.co.uk> wrote: > Let there be n buttons. A player may take 1 or 2 buttons MORE than his/her > opponent took. Player 1 can take either 1 or 2 buttons. > > e.g. 28->27->25->21->15->7->0 > > (the last player must take either 8 or 9, as only 7 remain, they must take > them all). I threw this together quickly, so I apologize for any errors: In the case where winner takes the last button: I can model the Nim values as follows: f(n,k) is the Nim value for n buttons remaining, last player took k buttons. If n=0, value is 0 if 1 <= n <= k+1 value is 1 otherwise value is mex( f(n-k-1,k+1), f(n-k-2,k+2) ) where mex is the smallest nonegative integer not in the set [i.e. mex(0,0) = 1, mex(0,1) = 2, mex(1,1) = mex(1,2) = mex(2,2) = 0 With this I get for f(n,0) 1 : 1 2 : 2 3 : 0 4 : 0 5 : 2 6 : 1 7 : 1 8 : 1 9 : 2 10 : 2 11 : 0 12 : 0 13 : 0 14 : 0 15 : 2 16 : 2 17 : 1 18 : 1 19 : 1 20 : 1 21 : 1 22 : 2 23 : 2 24 : 2 25 : 0 26 : 0 27 : 0 28 : 0 29 : 0 30 : 0 For the zeroes of f(n,0) (those n from which the second player can win) I get: 3-4 11-14 25-30 45-52 71-80 103-114 141-154 185-200 235-252... The lower limits seem to follow the pattern: 3 3+8=11 11+14=25 25+20=45 ... with the increment going up by 6 each time. The size of the ranges: 2, 4, 6, 8... If player who takes the last piece loses: the analysis is similar: g(n,k): If n=0, value is 1 if 1 <= n <= k+1 value is 0 otherwise value is mex( g(n-k-1,k+1), g(n-k-2,k+2) ) 1 : 0 2 : 2 3 : 1 4 : 1 5 : 2 6 : 0 7 : 0 8 : 0 9 : 2 10 : 2 11 : 1 12 : 1 13 : 1 14 : 1 15 : 2 16 : 2 17 : 0 18 : 0 19 : 0 20 : 0 21 : 0 22 : 2 23 : 2 24 : 2 25 : 1 26 : 1 27 : 1 28 : 1 29 : 1 30 : 1 with zeroes at 1 6-8 17-21 34-40 57-65 86-96 121-133 162-176 Lower limits 1 1+5=6 6+11=17 17+17=34 ... again the increments incrementing by 6 the ranges being 1, 3, 5, 7... this time > > What are the strategies for various n, with either last button wins/loses? Well, I didn't go the extra distance to detail the strategy, but it doesn't look too difficult. Christian

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