June 2010 - mathfun@mailman.xmission.com

[math-fun] Intuitively simple trig functions
by Jason 12 Jun '10

12 Jun '10

A few years ago I was delighted to find that calculating pi to arbitrary precision can be made simple enough for even a programmer to understand. I finally got around to finding an algorithm for computing sines and cosines that similarly relies only on high school algebra and trig. I wrote it up here with pictures and source code: http://credentiality2.blogspot.com/2010/06/few-years-ago-my-brother-in-law-…

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[math-fun] Ramanujan's Hardy-stumping CFs
by Bill Gosper 11 Jun '10

11 Jun '10

I hadn't realized how directly connected they are to the Rogers-Ramanujan identities: If f(q):=(q,q^4;q^5)_oo and g(g):=(q^2,q^3;q^5)_OO, the reciprocals of the g.f.s for the +-1 (resp +-2) mod 5 partitions, then 5 5 f(q) 1 2 (q ;q )_oo ---- = -------- = f(q) --------- g(q) q (q;q)_oo 1 + -------- 2 q 1 + ------- 3 q 1 + ----- ... which appears to come out an <algebraic #>/q^(1/5) when q = e^(pi r), r a quadratic surd (in the left halfplane) "by an appeal to the theory of elliptic theta functions" (G. Andrews: The Theory of Partitions, p105 :-). Ramanujan sent Hardy the cases q = e^(-2 pi) and - e^-pi, which presumably give the nicest algebraics, f(e^-2 pi) pi 3 pi 2 pi/5 ---------- = 4 sin -- sin ---- e , g(e^-2 pi) 20 20 whereas the nicest cf is presumably f(e^-pi) sqrt(40 + 18 sqrt(5) - 25 5^(1/4) - 11 5^(3/4)) - sqrt(36 + 18 sqrt(5) - 25 5^(1/4) - 11 5^(3/4)) pi/5 -------- = ------------------------------------------------------------------------------------------------- e . g(e^-pi) 2 ~ 0.958650181596. Now we know why Ramanujan didn't send Hardy this one. I'm really interested in how to get these theta special values (other than by numerical searching), but Andrews seems to cite only combinatoric references rather than (theta) number-theoretic. In particular, where do I learn that f and g can be evaluated separately, e.g. here in terms of e^(pi/60) because -pi -pi - pi/6 10 f(e ) g(e ) = e sqrt ------------------ . 5^3/4 - 5^(1/4) +2 Other forms of f and g are n (5n-3) n/2 Sum (-) q 1 Theta_4(7 i log(q)/4, q^(5/2)) n f(q) = ----------- = - q ------------------------------ = -------------------- , n^2 (q^5;q^5)_oo (q^5;q^5)_oo q Sum ------- n>0 (q;q)_n the "gofigurate number theorem", and n (5n-1) n/2 Sum (-) q 1 2 Theta_4(9 i log(q)/4, q^(5/2)) n g(q) = ------------ = - q ------------------------------ = -------------------- . (n+1) n (q^5;q^5)_oo (q^5;q^5)_oo q Sum ------- n>0 (q;q)_n f(q) and g(q) become Theta_1(pi/5, Q) under Jacobi's transformation. If we can express these in terms of Theta_s(0,p) ("Theta constants") we'd have a nearly automatic source of special values because the Theta constants all come out as etas, at which we're pretty good. I can do Theta_1(pi/10, Q), but so far no luck with pi/5. --rwg

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Re: [math-fun] Vladimir Arnold Dies at 72; Pioneering Mathematician
by Dan Asimov 11 Jun '10

11 Jun '10

I was also saddened to learn, a few days ago, of Arnol'd's death. Although I never met him, I was first introduced to his work via the wonderfully fun book with Avez, "Ergodic Problems in Classical Mechanics" -- which is a compendium of interesting facts that, despite the intimidating title, require very little background to appreciate (more accurately, a bachelor's in math would easily do). Another beautiful book is his "Mathematical Methods in Classical Mechanics", whose first chapter shows how to make rough estimates of various aspects of an animal, like the speed it can run, in terms of its linear size S, . (For example, its volume is proportional to S^3.) One thing he mentioned is the power of S that the height an animal can jump is proportional to. Do you want to guess? --Dan << FYI -- Arnold was one of my favorite mathematicians; I really enjoyed his work on orbits. His (English translated) books are quite readable. >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Vladimir Arnold Dies at 72; Pioneering Mathematician
by Henry Baker 11 Jun '10

11 Jun '10

FYI -- Arnold was one of my favorite mathematicians; I really enjoyed his work on orbits. His (English translated) books are quite readable. http://www.nytimes.com/2010/06/11/science/11arnold.html June 10, 2010 Vladimir Arnold Dies at 72; Pioneering Mathematician By KENNETH CHANG Vladimir I. Arnold, a Russian mathematician who discovered important theorems that found application in astronomy, mechanics and even weather forecasting, died June 3 in Paris. He was 72 and had homes in Paris and Moscow. The cause was acute pancreatitis, his son Igor said. Dr. ArnoldÂs work ranged widely across mathematics. He was a founder of singularity theory, or, as it is sometimes more ominously called, catastrophe theory. Singularity theory predicts that under certain circumstances slow, smooth changes in a system can lead to an abrupt major change, in the way that the slipping of a few small rocks can set off an avalanche. The theory has applications in physics, chemistry and biology. ÂHe was a genius and one of the greatest and most influential mathematicians of our time,Â said Boris A. Khesin, a former student of Dr. ArnoldÂs and now a professor of mathematics at the University of Toronto. One of Dr. ArnoldÂs biggest contributions was applying the methods of geometry and symmetry to the motion of particles. Dr. Arnold work on how fluids flow was applied to the dynamics of weather, providing a mathematical explanation for why it is not possible to make forecasts months in advance. Infinitesimal gaps or errors in information cause forecasts to diverge completely from reality. A similar approach can also be applied to the motion of planets. If Earth were the only planet to circle the Sun, its orbit would follow a precise elliptical path, but the gravity of the other planets disturbs the motion. Scientists found that it impossible to calculate the precise motion of the planets over very long periods of time or even prove that Earth will not one day be flung out of the solar system. Understanding the subtle and difficult-to-predict boundary between stability and instability is important not only in the study of planetary dynamics but also in other endeavors, like designing a nuclear fusion reactor. In 1954, the Russian mathematician Andrey Kolmogorov figured out a key insight to calculating whether such systems are stable. Dr. Arnold provided a rigorous proof in 1963 for one set of circumstances. Another mathematician, JÃ¼rgen Moser, provided the proof for another. The work is now collectively know at the KAM theory. ÂHis work provided a framework for studying this type of question,Â Alan D. Weinstein, a professor of mathematics at the University of California, Berkeley, said of Dr. Arnold. Dr. ArnoldÂs math explanations also exhibited playfulness. In one of his books, he explained a simple algorithm that stretches and rearranges an image, shredding it into a chaotic mess. In the book, Dr. Arnold illustrated the idea with a sketch of a catÂs head that is now known as ArnoldÂs Cat Map. Vladimir Igorevich Arnold was born June 12, 1937, in Odessa, now part of Ukraine. While a 19-year-old undergraduate at Moscow State University and supervised by Dr. Kolmogorov, he came up with the final piece of a proof answering the 13th problem on the list of 23 unsolved mathematical challenges that the mathematician David Hilbert compiled in 1900. After completing his undergraduate, masterÂs and doctoral degrees, all at Moscow State, Dr. Arnold joined the Moscow State faculty in 1961. He was able to travel abroad several times in the 1960s but then ran into trouble with the authorities by signing letters criticizing the persecution of dissidents. Dr. Arnold was unable to leave the Soviet Union again until the advent of perestroika under Mikhail S. Gorbachev in the late 1980s. He was not granted membership in the Soviet Academy of Sciences until 1990, after he had already become a member of similar academies in the United States, France and Britain. In 1986 Dr. Arnold moved to the Steklov Mathematical Institute in Moscow. He had told friends that he would not join Steklov as long as Ivan M. Vinogradov, whom he thought anti-Semitic, was director. Friends and family said that stance stemmed not from Dr. ArnoldÂs heritage Â his mother was Jewish Â but from a principled insistence that people should be judged on their merits. In 1993 he began splitting his time between Steklov and Dauphine University in Paris, spending spring and summer in Paris, fall and winter in Moscow. His awards include the Wolf Prize in 2001, which mathematicians regard as the equivalent of a Nobel, and the Shaw Prize in 2008, another prestigious math award. He also has an asteroid named after him: Vladarnolda. Dr. Arnold spoke out against what he considered the overly abstract teaching of mathematics. In a 1997 talk, translated into English, he said: ÂMathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.Â Besides his son Igor, of Jersey City, he is survived by his wife of 33 years, Eleonora; another son, Dmitry, of Moscow; a brother, Dmitry, also of Moscow; a sister, Katya Arnold, of New York City; four grandchildren; and two great-grandchildren. Like his thesis adviser Dr. Kolmogorov, Dr. Arnold had an unusual approach when he got stuck on a problem. Writing in the Russian online newspaper Gazeta.ru, his former students Askold Khovanskii and Yuli Ilyashenko recalled that Dr. Arnold would ski for 25 miles or more, wearing nothing more than swim trunks. ÂAccording to him, this practice would always lead him to a new idea,Â they wrote.

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[math-fun] Martin Gardner & humour
by Eric Angelini 07 Jun '10

07 Jun '10

Hello Math-Fun and SeqFan, I'm asked to write a 10,000-sign paper on Martin Gardner's sense of humour (the man &/or his articles, columns, books). Do some of you have nice stories or souvenirs? Best, É.

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[math-fun] Fwd: DDLab version June 2010
by Fred lunnon 06 Jun '10

06 Jun '10

Nice-looking site for cellular automata which I didn't know about earlier. [But no Genaro, I'm NOT going to look any further just now --- it's easy enough to distract me as things are!] WFL ---------- Forwarded message ---------- From: Genaro <genaro.juarez.martinez(a)googlemail.com> Date: Jun 6, 2010 5:15 PM Subject: DDLab version June 2010 To: Genaro Juarez Martinez <genarojm(a)googlemail.com> Dear friends and colleagues, The latest June 2010 Andy Wuensche update his DDLab software (for Linux, Cygwin, Mac and DOS) and its respective manual as well. That is available at www.ddlab.org Feedback is appreciated to andy AT ddlab DOT org All best, Genaro -- Genaro Juarez Martinez Unconventional Computing Centre, UWE, UK http://uncomp.uwe.ac.uk/genaro/

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Re: [math-fun] Is this possible?
by Dan Asimov 06 Jun '10

06 Jun '10

Fred wrote: << A simple-minded search for solutions modulo primes of form p = 12 k + 1 yields 2640 solutions when p = 157 , whence the density of solutions cannot exceed 0.00000435 . As the number of solutions looks to be something of order very roughly p ^ (3/2) , the density is very likely actually zero. I haven't checked whether any of these Z_p solutions is actually a solution over Z. On 6/5/10, Hans Havermann <pxp(a)rogers.com> wrote: << Quadruples {a,b,c,d}, a>b>c>d>0, where a+b, a+c, a+d, b+c, b+d, c+d are all squares, a+b+c, a+b+d, a+c+d, b+c+d are all cubes, and a+b+c+d is a fourth power. >> >> Very interesting! I was thinking the number of conditions (11) is a lot bigger than the number of unknowns (4), so was guessing there might be no solution. Clearly if modulo any number at all there are no solutions, then there are none over Z, so I was wondering about p = 7, which looked like the first plausible case where no solutions might exist. So, Fred, can you run your program on primes of other forms than 12k + 1, like 8k + 7, or just all primes up to some size ? Or are primes 12k + 1 easier to solve Hans's equations for ? --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] a problem from martin gardner
by Dan Asimov 05 Jun '10

05 Jun '10

I inaccurately wrote: << See this web page: < http://faculty.plattsburgh.edu/don.west/tiling/ >. The short version: It can be done if and only if n == 0, 2, 9, or 11 (mod 12). >> Correction: I read Thane's question too fast. The statement above addresses only the question of whether a triangle with n dots on a side can be tiled by copies of the 3-dot triangle (2 dots on a side). On the other hand, copies of the 3-dot straight line can never tile a dotty triangle. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] a problem from martin gardner
by Dan Asimov 05 Jun '10

05 Jun '10

See this web page: < http://faculty.plattsburgh.edu/don.west/tiling/ >. The short version: It can be done if and only if n == 0, 2, 9, or 11 (mod 12). --Dan --------------- Thane wrote: << I found a short note from Martin Gardner to David A Klarner in my garage (long story; anyway...) The letter is dated 4 Oct 1983. This is what it says (in part) * * * *begin quotation * * * Do you know whether an n-rowed triangle-shaped array of dots like this * * * * * * * * * * ..... can be tiled by single triangles shaped like this * * * or by 1x3 straight lines like this * * * It's been driving me nuts for two weeks! Is there a coloring argument? Martin ******* end quotation ***** >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] a problem from martin gardner
by James Propp 05 Jun '10

05 Jun '10

>It's been driving me nuts for two weeks! Is there a coloring argument? For the full story, see J.H. Conway and J.C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), no. 2, pp. 183-208. Jim

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