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Ignore my previous message!!! Re: [math-fun] Probabilistic harmonic serie
by asimovd＠aol.com 24 Jun '03

24 Jun '03

I was trying to delete my initial thoughts -- which I think are wrong, anyway -- when I accidentally sent the message. Apologies. --Dan

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Re: [math-fun] Probabilistic harmonic series
by asimovd＠aol.com 24 Jun '03

24 Jun '03

Harmut Holtzwart (hartmut.holzwart(a)allianz.de) asks << Related question not treated in Schmulands article: What is known about the distribution of Z = e_1 + 1/3 e_2 + 1/5 e_3 + 1/7 e_4 + .... with e_i i.i.d. +/-1? >> If X = sum from 1 to oo of e_k (1/k) is the random variable for the sum of the random harmonic series, then clearly Z = X_1 - 2*X_2 (where X_1 and X_2 are independent and have the same distribution as X).

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Re: Re: [math-fun] Probabilistic harmonic series
by hartmut.holzwart＠allianz.de 24 Jun '03

24 Jun '03

Related question not treated in Schmulands article: What is known about the distribution of Z = e_1 + 1/3 e_2 + 1/5 e_3 + 1/7 e_4 + .... with e_i i.i.d. +/-1? Remark: If X is the random harmonic series, then X = Y * Z, where Y is equally distributed in (-1,1). Any hints would be appreciated. Cheers, Hartmut asimovd@aol.com@mailman.xmission.com on 06.05.2003 11:06:04 Bitte antworten an math-fun(a)mailman.xmission.com Gesendet von: math-fun-bounces(a)mailman.xmission.com An: math-fun(a)mailman.xmission.com Kopie: Thema: Re: [math-fun] Probabilistic harmonic series The question was, What is the distribution of the sum of the random harmonic series ±1 ± 1/2 ± 1/3 ± ... ? By amazing synchronicity, an article on this very subject, "Random harmonic series" by Byron Schmuland, appears in the May, 2003 issue of the American Mathematical Monthly and is available from the author's website. (Thanks, Neil, for pointing this out.) As E. Clark mentioned, the article's available for downloading at <http://www.stat.ualberta.ca/people/schmu/publications.html>. I haven't studied every detail of this paper, but it's very interesting, with a number of graphs of the distribution function and related functions (like the distribution function if the same question is asked about placing random signs in front of the series of *squares* of reciprocal integers). This paper rewards whatever level of perusal you feel like giving it. It uses a clever series-rearrangement trick to nail down the distribution function in question. The author proves that this distribution function is infinitely differentiable, though he wrote me in private e-mail that for 15 years before solving the question, he believed it would not even be differentiable at all. He wrote that he did not know whether it is real analytic. In case anyone is interested, I'll mention that I've proved it's in fact real analytic. --Dan ------------------------------------------------------------------------------- ------------------------------------------------------------------------------- _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -------------------------------------------------------------------------------

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[math-fun] addition-subtraction chains
by Richard Schroeppel 23 Jun '03

23 Jun '03

I noticed the sequence A003064: 2,3,5,7,11,19,29,47,71,127,191,379,607,1087,1903,3583,6271, 11231,18287,34303,65131,110591,196591,357887 Name: Smallest number with addition chain of length n. 1903 = 11*173 is the first composite, but after that it's about 50:50. It looks like the limiting ratio might gradually approach 2. I get 2 3 5 7 11 19 29 53 87? for the corresponding addition-subtraction chain. (I'm working by hand, errors all too likely.) Rich rcs(a)cs.arizona.edu

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[math-fun] Calculus I problem
by R. William Gosper 22 Jun '03

22 Jun '03

with a nice shortcut and an interesting answer: Q: A cylindrical water tank, radius 1, rests on its side, skewered by a coaxial bar of radius r. Choose r to maximize capacity if the water level may not exceed the top of the bar. A: .retal etaitnereffid od gniog tsuj er'uoy ecnis )2^x-1(trqs largetni rotces eht od ot deen t'nod uoy taht si tuctrohs ehT .)2^ip+1(trqS/1

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Re: [math-fun] Emacs Tetris & statistics (fwd)
by Erik Demaine 20 Jun '03

20 Jun '03

On Fri, 20 Jun 2003, Hugh Everett wrote: > Dean Hickerson suggested the game: > Player 1 chooses a tetromino, and player 2 then places it according to > the rules of > tetris (any sequence of rotate, left, right, down). With optimum > strategy, can > Player 1 force the board to fill? This is the usual game of Tetris, except that in normal Tetris, player 1 is "nature" in the sense that it plays stochastically. [See Christos Papadimitriou's paper "Games Against Nature".] It is known that nature can foil player 2, with some simple distributions on pieces. This result implies what you want, if you allow randomized strategies for player 1. There are also deterministic strategies, e.g., alternating between Z and S eventually dooms player 2 for boards of width 2n for odd n. Here are the references: John Brzustowski. Can you win at Tetris? Master's thesis, University of British Columbia, 1992. Heidi Burgiel. How to lose at Tetris. Mathematical Gazette, July 1997. See the bottom of page 3 of this paper: http://theory.lcs.mit.edu/~edemaine/papers/Tetris_COCOON2003/ Erik -- Erik Demaine | edemaine(a)mit.edu | http://theory.lcs.mit.edu/~edemaine/

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Re: [math-fun] Lots of Triangle Centers
by Henry Baker 20 Jun '03

20 Jun '03

This page is dying for an animation... You can take multiple gif's and compose them into an animated gif. There may even be free software online somewhere to do this. At 02:45 PM 6/20/03 -0700, Ed Pegg Jr wrote: > I was curious about how 800 different triangle centers moved around as the >underlying triangle changed, so I generated lots of pictures. You can see them >at http://www.mathpuzzle.com/TriCent/TriangleCenter.html . On my front page, >I've added links to Kimberling's page, Weisstein's page, and a list of 2000 >centers compiled by Edward Brisse. > >--Ed Pegg Jr, http://www.mathpuzzle.com

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[math-fun] Emacs Tetris & statistics
by Henry Baker 20 Jun '03

20 Jun '03

I have become intrigued by Tetris, not as a game, but as an online packing problem. I'm not interested in the optimum solution, per se, but in the statistics of the problem. Normally, Tetris has 10 columns and 7 tetraminos -- 2x2 "block", 2 "L"'s (L, L', mirror images), 2 "Z"'s (Z, Z', mirror images), 1 "T", and a 4x1 "line". Each tetramino has an area of 4 units. These tetraminos show up randomly, but with a uniform distribution -- the 7 different types are equally distributed. I'm curious about the interaction between the packing strategies/heuristics and the probability distribution of the pieces. Everyone who has played the game has noticed how the 4x1 pieces appear critical to achieving a high score -- if they show up too rarely, you quickly run out of space as things stack up. Tetris appears to be very close to "criticality", in the sense that with the right heuristics and statistics, one might achieve an unbounded score. One way to understand this phenomenon is to embed enough quality-playing heuristics, and then play (automatically) lots of games with slightly varying statistics of the pieces to see how the average score changes. Clearly, on a Tetris board with 14 columns, one could devote every pair of columns to a single piece type. Since each piece type can efficiently pack with itself in a 2-wide column, and since each type has the same probability and the same area, each of the column pairs will fill at the same rate, on average. So only large variances will cause the game to terminate. When the game is squeezed to only 10 columns, things get much more interesting, as there does not appear to be any similar strict "ghetto" strategy. I have taken the standard Lisp tetris code distributed with Emacs and made some radical changes to allow the parameters of the game to be modified. For example, you can now change the width (# columns) of the game and choose the probability distribution (I use Walker's algorithm, see Knuth, v.II). I intend to implement backing up (so you can practise better), additional levels of lookahead, automatic placement according to certain heuristics, etc. If anyone else on this list is interested, I'll be happy to post the code on my web site. I'll go with the usual Gnu GPL, and my changes are public domain if anyone cares. I make no guarantees about quality or reliability of the code, and it will get modified sporadically with no warning. Henry Baker

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[math-fun] nome
by R. William Gosper 20 Jun '03

20 Jun '03

Someone privately requested the definition of "nome". Not to be confused with the garden pest ( http://alyssadolls.homestead.com/potterica.html http://www.angelfire.com/stars2/harrypotterland/degnome.html http://membres.lycos.fr/justpotter/studies4.html http://www.geocities.com/hogwarts_29/ww/2/2c.html http://meisetsu.topcities.com/enchantments/beasts2.html http://neocortex.by.ru/index/d1.html#Degnoming) a nome is merely a crude approximation to a pole, as in the Alaskan city whose famed beauty pageant annually crowns "Miss Nomer." It can also refer to the little dot on the bottom of a fruit opposite the stem, as in nome de plum, (in deference to Asian- Americans, no longer called "berry button.") Finally, of course, there is the dyslexic animated fish. Hmm. The guy with the funny cigarette is no longer upwind, so let's try again. For what might seem an unmotivated web of definitions, see http://mathworld.wolfram.com/Nome.html . In summary, "nome" is merely the name for the second argument of a theta function, and is usually denoted by q. This is the same q as the "base" in "basic-" or q-hypergeometric series, so they both could have been called "nome" or "base". In fact, q is just a parameter, i.e., it tends to remain fixed, undefined, or even implicit for a given application of theta functions. E.g., the tumbling of an asteroid can be described with quotients of theta functions wherein the first argument is proportional to time, while the nome remains fixed. See, e.g., www.emis.de/proceedings/7ICDGA/VI/puta.ps . No, that's the name of the author, not a piece of asteroid. So why didn't they call it "parameter" instead of "base" or "nome"? Because "parameter" was already usurped to mean two other things: modulus^2 in the case of theta, and (incompatibly) a constituent of a series coefficient in the hypergeometric case. Finally, do not be discouraged that there are four theta functions. They are just four translates of the same function, like sine vs cosine. But unlike sine and cosine, thetas almost never appear alone, but rather paired in quotients. It is as if you never needed sine, just tangent and cotangent. --rwg

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[math-fun] NYTimes: Madam, I'm Adam!
by Henry Baker 20 Jun '03

20 Jun '03

This is no joke! According to a front-page story in the NYTimes yesterday, the male human Y chromosome is (actually, contains) palindromic! By folding back upon itself, the redundancy can be used to repair bad DNA sequences. So this palindrome is now a pun!

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