October 2004 - mathfun@mailman.xmission.com

[math-fun] numerical curiosity
by Simon Plouffe 22 Oct '04

22 Oct '04

bonjour tout le monde, Here is a numerical curiosity: 588^2 + 2353^2 = 5882353 the curious thing is that 1/17 = 0.05882353 (rounded to 7 digits). I never searched beyond that, are there other known results in this direction? Simon Plouffe

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[math-fun] Dr. Kenneth E. Iverson [fowarded from the J Forum]
by Mark D. Niemiec 21 Oct '04

21 Oct '04

-----Forwarded Message----- From: Eric Iverson <eiverson(a)SYMPATICO.CA> Sent: Oct 21, 2004 9:44 AM To: JFORUM(a)PEACH.EASE.LSOFT.COM Subject: [Jforum] Dr. Kenneth E. Iverson Ken Iverson passed away Tuesday evening, October 19th at the age of 83. Ken was at his computer Saturday afternoon working on a new J lab when he had a stroke. Three days later he died quietly with his wife Jean by his side, along with other family members. Ken had a wonderful and memorable life. He enjoyed it fully and he freely shared his joy with so many others. Many people have contributed to the start, growth, and evolution of J. But Ken's role was central and inspirational. J exists because of Ken and we will always remember that. The J Forum was a particular joy for Ken. He was an avid reader and was amazed and pleased with the civility. In his last days Ken expressed confidence that the J seeds he had planted had taken root and was satisfied that the steady and healthy growth would continue. Ken has passed a torch and it is now up to us. Ken's immediate family is having a private memorial service. In lieu of flowers, please make a memorial donation to the charity of your choosing or to the Academy of Life Long Learning (www.allto.ca) Ken's life was based on the importance of education and teaching and in recent years, with Jean's serious involvement, the Academy was an important part of Ken's wider life outside of J. /Eric Iverson -------------------------------------------------------------------------- For information about the J Forum see http://www.jsoftware.com/j_forum.htm -- Mark D. Niemiec <mniemiec(a)interserv.com>

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[math-fun] recent math story on NPR about Manjul Bhargava
by Erik Neumann 21 Oct '04

21 Oct '04

There was a nice math story on NPR's Morning Edition last Monday Oct 18. The main point was about how math has aspects of beauty and creativity. Here is a link to a page where you can hear the story, and I've copied the text below. http://www.npr.org/templates/story/story.php?storyId=4111253 At 28, Manjul Bhargava has already won a coveted full professorship at Princeton University. An expert in number theory, the study of the properties and relationships of numbers, Bhargava is also a master of the tabla, a small Indian hand drum used to create music with rhythmic, precise patterns. Number theory is the type of math that describes the swirl in the head of a sunflower and the curve of a chambered nautilus. Bhargava says it's also hidden in the rhythms of classical Indian music, which is both mathematical and improvisational. He sees close links between his two loves -- both create beauty and elegance by weaving together seemingly unconnected ideas. As part of a Morning Edition series exploring the intersection of art and science, NPR's Richard Harris reports on the beauty of mathematics, its ties to art -- and the man who straddles both worlds. Erik Neumann erikn(a)MyPhysicsLab.com

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[math-fun] 7-11 curve
by R. William Gosper 18 Oct '04

18 Oct '04

Remember the 7-11 problem where Professor O'Blivet buys four items whose sum and product both total 7.11? There is only one integer (cents) solution, but a graphical method on a certain "elliptic" curve yields infinitely many rational solutions. The curve looks like a "rounded triangle" amidst three "hyperbolas" (http://www.ippi.com/rwg/711.gif) Any straight line cuts these four arcs in exactly three places (two, if one is a graze), and all three cuts represent rational solutions if any two of them do. Inventor and curve connoisseur Alan Adler recently sought some rounded triangles, so I offered a b (x - a) (x - b) (x + -----) 2 b + a y = ---------------------------. a b 3 x - ----- + b + a b + a For a<x<b, this gives an "equilateral" version of the 7-11 rounded triangle ("threelipse"?), when a<0<-a<b<-2a. As b (= the circumradius) varies between -a and -2a, the curve morphs between a circle and an equilateral triangle. The up-down symmetry is obvious from the equation. Less so the 120 degree rotational symmetry. But x = r cos(t), y = r sin(t) gives 0 = (b+a)*r^3*cos(3*t)-(b^2+a*b+a^2)*r^2+a^2*b^2, which is unperturbed by t <- t - 2pi/3. Who was the idiot who misappropriated "hyperboloid" to name a surface? --rwg Pedants depants pentads.

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Re[2]: [math-fun] beautiful equations and E.T.
by M. Stay 10 Oct '04

10 Oct '04

>1. Computability. The fact that there exists a reasonably >robust theory of computability, and that this is intimately >connected with the fact of the incompleteness of number theory >(Goedel's Theorem, et al). It turns out that computability >is even easier (in some respects) to present than arithmetic >(at least if you know Lisp & EVAL). I particularly like Barker's universal combinator \x.xSK where \ is lambda. -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] an unreasonably handy tool
by R. William Gosper 09 Oct '04

09 Oct '04

[In my integrate atan^2 msg, I should have made it clearer that Mma was refusing on license-enforcement grounds. Ed Pegg suggested integrals.wolfram.com, which gives a somewhat messy, complex indefinite integral, and apparently doesn't offer the definite option. Wouter, the upper limit wants to be Tan[n*Pi/d], not Pi*n/d, thanks.] Back in the 70's, Rich Schroeppel informally described his integer lattice algorithm to Robert Maas, who called it LINREL for LINear RELation finder. Almost trivial by comparison is my TLINREL, which seeks linear relations among Taylor expansions using polynomials, instead of integers, as coefficients. Suppose you need the formula for inf ==== 2 \ n > q , / ==== n = 1 but all you remember is that the bilateral series had something to do with the infinite products inf inf /===\ /===\ | | n | | n | | (1 - q ), | | (q + 1), and maybe | | | | n = 1 n = 1 inf /===\ | | 2 n | | (q + 1). | | n = 1 Equate the logs of all four quantities with expansions to a dozen or so terms: (c197) (log(1+2*sum(q^n^2,n,1,inf)),%% = taylor(%%,q,0,9)) Time= 10 msec. inf ==== 2 3 5 \ n 2 8 q 4 12 q (d197)/T/ log(2 > q + 1) = 2 q - 2 q + ---- - 2 q + ----- / 3 5 ==== n = 1 6 7 9 8 q 16 q 8 26 q - ---- + ----- - 2 q + ----- + . . . 3 7 9 (c198) (log(product(1-q^n,n,1,inf)),%% = taylor(%%,q,0,9)) Time= 10 msec. inf /===\ 2 3 4 5 | | n 3 q 4 q 7 q 6 q 6 (d198)/T/ log( | | (1 - q )) = - q - ---- - ---- - ---- - ---- - 2 q | | 2 3 4 5 n = 1 7 8 9 8 q 15 q 13 q - ---- - ----- - ----- + . . . 7 8 9 (c199) (log(product(1+q^n,n,1,inf)),%% = taylor(%%,q,0,9)) Time= 10 msec. inf /===\ 2 3 4 5 6 | | n q 4 q q 6 q 2 q (d199)/T/ log( | | (q + 1)) = q + -- + ---- + -- + ---- + ---- | | 2 3 4 5 3 n = 1 7 8 9 8 q q 13 q + ---- + -- + ----- + . . . 7 8 9 (c200) (log(product(1+q^(2*n),n,1,inf)),%% = taylor(%%,q,0,9)) Time= 10 msec. inf /===\ 4 6 8 | | 2 n 2 q 4 q q (d200)/T/ log( | | (q + 1)) = q + -- + ---- + -- + . . . | | 2 3 4 n = 1 Then simply (c201) tlinrel(d197,d198,d199,d200) Time= 602 msec. inf ==== 2 inf \ n /===\ log(2 > q + 1) | | 2 n / log( | | (q + 1)) ==== | | n = 1 n = 1 (d201)/T/ - -------------------- - --------------------- 4 4 2 q q inf inf /===\ /===\ | | n | | n 3 log( | | (q + 1)) log( | | (1 - q )) | | | | n = 1 n = 1 + --------------------- + ------------------- = 0 + . . . 4 4 2 q 2 q Solve for the desired sum: (c202) solve(lhs(%),sum(q^n^2,n,1,inf)) Time= 50 msec. inf ==== 2 \ n (d202) [ > q = / ==== n = 1 inf inf inf /===\ /===\ /===\ | | 2 n 2 | | n | | n 3 ( | | (q + 1)) - ( | | (1 - q )) ( | | (q + 1)) | | | | | | n = 1 n = 1 n = 1 - --------------------------------------------------------] inf /===\ | | 2 n 2 2 ( | | (q + 1)) | | n = 1 For brevity, combine the products: (c203) prodcontract(intoprod(expand(%[1]))) Time= 30 msec. inf /===\ n n 3 | | (1 - q ) (q + 1) inf | | ------------------ ==== 2 | | 2 n 2 \ n n = 1 (q + 1) 1 (d203) > q = ------------------------ - - / 2 2 ==== n = 1 Test: (c204) taylor(rhs(%),q,0,100) Time= 2283 msec. 4 9 16 25 36 49 64 81 100 (d204)/T/ q + q + q + q + q + q + q + q + q + q + . . . One more example. The series for log(1-x)^2 is "triangular hypergeometric", i.e., it requires a double sum, though only a single matrix product: [ 2 x ] inf [ x --- 0 ] /===\ [ n ] [ 2 ] | | [ ] [ 0 0 log (1 - x) ] | | [ x ] = [ ] . | | [ 0 x - ] [ 0 0 - log(1 - x) ] n = 1 [ n ] [ ] [ ] [ 0 0 1 ] [ 0 0 1 ] But I wanted [ n x 1 ] inf [ ----- - 0 ] /===\ [ n + 2 2 ] [ 0 0 z(x) ] | | [ ] [ ] | | [ n x ] = [ 0 0 y(x) ], | | [ 0 ----- 1 ] [ ] n = 1 [ n + 2 ] [ 0 0 1 ] [ ] [ 0 0 1 ] which is sort of "contiguous" to the previous product, so maybe z(x) and y(x) likewise to come out in terms of log(1-x) and log(1-x)^2. Taking the right column of the above product to 15 terms: (c209) taylor(col(prud(%,n,1,15),3),x,0,14) Time= 481 msec. 2 3 4 5 6 1 5 x x 49 x 41 x 109 x 853 x (d209)/T/ matrix([- + --- + -- + ----- + ----- + ------ + ------ 2 12 3 180 180 560 5040 7 8 9 10 11 12 209 x 1679 x 19981 x 18167 x 252341 x 3488333 x + ------ + ------- + -------- + --------- + ---------- + ----------- 1400 12600 166320 166320 2522520 37837800 13 2 3 4 5 6 7 8 3694253 x x x x x x x x x + ----------- + . . .], [1 + - + -- + -- + -- + -- + -- + -- + -- 43243200 3 6 10 15 21 28 36 45 9 10 11 12 13 14 x x x x x x + -- + --- + --- + --- + --- + --- + . . .], [1 + . . .]) 55 66 78 91 105 120 Expanding the log (over x, a non-essential micro-optimization): (c210) (log(1-x)/x,%% = taylor(%%,x,0,14)) Time= 0 msec. 2 3 4 5 6 7 8 9 log(1 - x) x x x x x x x x x (d210)/T/ ---------- = - 1 - - - -- - -- - -- - -- - -- - -- - -- - -- x 2 3 4 5 6 7 8 9 10 10 11 12 13 14 x x x x x - --- - --- - --- - --- - --- + . . . 11 12 13 14 15 Finally, as a stunt, solve for both z znd y simultaneously: (c211) tlinrel(%,%^2,a*(z = d209[1,1])+b*(y = d209[2,1])) Time= 27139 msec. 3 2 2 3 2520 (128 b - 216 a b + 3 a b + 4 a ) (a z + b y) (d211)/T/ ---------------------------------------------------- 2 4 b (2 b + a) (16 b - 27 a) x 3 2 2 3 + 2520 (128 b - 216 a b + 3 a b + 4 a ) log(1 - x) 2 6 (2 b x - 2 b + a)/(b (2 b + a) (16 b - 27 a) x ) 3 2 2 3 5040 (128 b - 216 a b + 3 a b + 4 a ) - ---------------------------------------- 2 5 (2 b + a) (16 b - 27 a) x 3 2 2 3 2 2520 a (128 b - 216 a b + 3 a b + 4 a ) log (1 - x) (x - 1) - -------------------------------------------------------------- = 2 7 b (2 b + a) (16 b - 27 a) x 0 + . . . (c212) solve(lhs(%),a*z+b*y)[1] Time= 130 msec. 2 (d212) a z + b y = - ((2 b log(1 - x) - 2 b) x 2 2 3 + ((a - 2 b) log(1 - x) - a log (1 - x)) x + a log (1 - x))/x The function is strikingly short: /* -*- Mode: Macsyma -*- */ tlinrel([eqns]):=catch(block([len:length(eqns),mat,rot], push(1=1,eqns), mat:addcol(transpose(map('rhs,eqns)), apply('taylor,cons(ident(1+len),first(taylorinfo(rhs(eqns[2])))))), rot:addrow(ematrix(1,len+1,1,1,len+1),addcol(ident(len),ematrix(len,1,0,1,1)-1)), do (for k:2 thru len+1 do (mat[k], if %%[1]=0 then throw(eqns.sqfr(rest(%%))) else mat[k]:%%/first(%%[1])), mat:rot.mat)))$ --rwg Any insufficiently advanced technology is indistinguishable from Microsoft.

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[math-fun] beautiful equations and E.T.
by Simon Plouffe 09 Oct '04

09 Oct '04

Hello, they made a poll asking what are the most beautiful equations to a group of people. http://news.bbc.co.uk/1/hi/magazine/3721406.stm Euler's equation came second : exp(Pi * i) + 1 = 0. Maxwell equation came first. I think that these people are in physics. Nevertheless, I am wondering what if we would make a kind of message to extra-terrestrial intelligence and we would like to show them that we know all these things. What would be the best way to write them down in a universal language because for ET the equation exp(Pi*i)+1=0 means nothing at all if we can't explain what is Pi, e, 1, 0 and the equal sign, isn't? Any clue? ps : E = Mc^2 is only the 6'th. Simon Plouffe

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RE: [math-fun] beautiful equations and E.T.
by Daniel Asimov 09 Oct '04

09 Oct '04

Simon asks: << What would be the best way to write them down in a universal language because for ET the equation exp(Pi*i)+1=0 means nothing at all if we can't explain what is Pi, e, 1, 0 and the equal sign, isn't? >> Cool question! It may be self-evident that multiple vertical strokes signify 1,2,3,...: |, ||, |||, . . . A few simple equations like | + | = || and || + ||| = ||||| should reveal the meanings of + and =. A circle with one diameter drawn in, labeled | could reveal pi by showing a sequence of increasingly unrolled circles, culminating in a straight line, all the while keeping its length constant. This straight line can be labeled with the symbol pi. (The meaning of labeling could be explained by showing two diameters end to end labeled ||, three labeled |||, etc.) If E.T. knows anything about exp, then by this point they will recognize the series definition for exp(z) (for simplicity, call it e(z)) without further explanation: e(z) = | + z/(|) + zz/(|| |) + zzz/(||| || |) + ... Now writing | + -| = -| + | = should explain the minus sign, and finally i i = -| should yield the (ambiguous) meaning of i. Finally, e(pi i) + | = ought to be clear to any being who's familiar with it, irrespective of their symbology for it (assuming they use symbols!). --Dan

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[math-fun] integrate atan^2
by R. William Gosper 08 Oct '04

08 Oct '04

Mma refuses to try on my "new" laptop, but Derive, Macsyma, and Maple all fail. After a good deal of hand polishing (or maybe hand wringing), / 2 | 2 log(x + 1) | atan (x) dx = atan(x) (- ----------- + x atan(x) + log(2)) | 2 / %i x 1 - imagpart(li (---- + -)). 2 2 2 Note that when x is tan(pi*rational), the dilog dehisces, and d ==== %pi n \ j j 2 %pi j n tan(-----) > (- 1) Psi (-) sin(---------) d / 1 d d / ==== | 2 j = 1 | atan (x) dx = ----------------------------------- | 2 / d 0 %pi n %pi n sec(-----) %pi n tan(-----) d d %pi n (2 log(----------) - ----------------) 2 d - --------------------------------------------, d -d/2 < n < d/2, where Psi_1 is the trigamma function. In particular, 1 / 2 | 2 %pi log(2) %pi | atan (x) dx = ---------- + ---- - %catalan ~ 0.2452812034667; | 4 16 / 0 1 ------- sqrt(3) 1 / sqrt(3) Psi (-) 2 | 2 %pi log(3) 1 3 sqrt(3) %pi | atan (x) dx = ---------- - --------------- + ------------ | 6 9 12 / 0 ~ .0568881659798, --rwg

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[math-fun] "Sterreal" numbers
by Mike Stay 08 Oct '04

08 Oct '04

I was messing around with different addition rules for the surreal numbers; the simplest one gives all rationals as finite-birthday elements, and the mediant as the sum. This gives rise to the Stern-Brocot tree, thus "Sterreal" numbers. {|} = 1 {1|} = 2 {|1} = 1/2 mediant(A,B) = {A|B} if A<=B, {B|A} if B<=A, A if A=B. 1/A = {1/A_R , 1/A_L}; 1/{}={} I'm pretty sure that true addition and multiplication rules will look rather like Gosper's HAKMEM entry on continued fractions. Has anyone else looked at alternate addition rules for surreals? I didn't see any online. -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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