March 2011 - mathfun@mailman.xmission.com

Re: [math-fun] Factorials in non-hypergeometric sums
by James Propp 15 Mar '11

15 Mar '11

>Whenever that happened to me, I was just glad that I was on the right track! I remember the number-theorist Michael Larsen as a teen-ager (back when I was a teen-ager too), sharing with me his excitement over his graduating from rediscovering nineteenth-century theorems to rediscovering twentieth- century theorems. Jim Propp

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Re: [math-fun] Factorials in non-hypergeometric sums
by Dan Asimov 15 Mar '11

15 Mar '11

Yeah, I was buoyed by having the formula confirmed (though I hadn't really doubted it). There were a few other times that my "discovery" turned out to be known for a very long time (like when I tried to define z^w for complex z and w), and eventually it sank it that it wasn't so easy to come up with something in math that wasn't already known! --Dan Mike Stay wrote: << On Tue, Mar 15, 2011 at 12:57 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: << Still, all excited about this formula, I told it to my freshman advisor, Henry McKean, on the first day we met. He casually reached over to his bookshelf, and after flipping a few pages showed me the identical formula in Whittaker an Watson from 70 years earlier. You've never seen an MIT freshman's bubble burst so quickly. >> Whenever that happened to me, I was just glad that I was on the right track! >> ________________________________________________________________________________________ "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] symbolic algebra handling of (+/-) signs
by Henry Baker 15 Mar '11

15 Mar '11

Over the years, I've had occasion to use Macsyma/Maxima to produce formulae that were then utilized in my computer code (Lisp/C/...). While these symbolic algebra tools are extremely useful, there is one area where symbolic algebra systems could be improved quite a lot -- the handling of signs (e.g., +/-). While Maxima does a pretty good job of understanding the absolute value function ('abs'), it doesn't seem to know very much about the intimately related "sign" ('signmum') function. For example, although 'asinh' is a perfectly good odd function, Maxima doesn't know how to move 'signum' in and out of the 'asinh' function. Dealing with 'signum' in a high quality way should be considerably easier than attempting to handle 'if-then-else' expressions (aka "()?:" expressions in C/C++, etc.). But there are occasions where symbolic algebra systems would be extremely helpful if they could handle such if-then-else expressions. The alternative is pretty ugly: it may be necessary to create a software "harness" and then evaluate the alternative expressions with every conceivable boolean combination -- a strategy that quickly blows up exponentially, but it is still better if this strategy is implemented mechanically, to make sure that all combinations are checked. I haven't had a lot of experience with other symbolic algebra systems. Do Maple & Mathematica (or any others) do a better job of handling these signum and/or if-then-else expressions? Alternatively, has anyone here had any experience in teaching the Macsyma/Maxima simplifier about such expressions, so that I could expand its capabilities ?

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[math-fun] Factorials in non-hypergeometric sums
by Bill Gosper 15 Mar '11

15 Mar '11

quad> I saw this sum somewhere as a question as to whether it has a closed form. I am pretty sure it doesn't have any elementary closed form, but I'd think that this isn't a whole new class of functions (with respect to special functions). (%i1) 'sum(1/(x^k + k!), k, 0, inf); inf ==== \ 1 (%o1) > ------- / k ==== x + k! k = 0 for some real x (we might even just consider positive integral x). Maybe even just (%i2) 'sum(1/(1 + k!), k, 0, inf); inf ==== \ 1 (%o2) > ------ / k! + 1 ==== k = 0 Can we write this in any other useful forms? (cue the Gosperfunctionology) -Robert ----------------------- You gotta be kidding. I'm trying to show the kids bilateral infinite matrix products and I can't even "shew that" Limit[(E^(I N t) (-b + N)! Hypergeometric2F1[1, a - N, b - N, E^(-I t)])/(-a + N)!, N -> \[Infinity]] == -(2^(-a + b) E^(1/2 I (b (-3 Pi + t) + a (Pi + t))) Pi Csc[b Pi]/ ((-Sin[t/2])^(a - b) (-1 + E^(I t)) Gamma[-a + b])) (integer N, b>a, t>pi/2, and probably more widely). Where's the nearest home for the bewildered? --rwg

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Re: [math-fun] Factorials in non-hypergeometric sums
by Dan Asimov 15 Mar '11

15 Mar '11

Robert wrote: << For the poor, unfortunate souls who have no idea why Gosper said "shew that": http://i.imgur.com/lmFFL.jpg >> It that by any chance Whattaker & Witson? I remember that the summer after high school I was excited by the idea of finding a natural continuous function f(x) whose value at each positive integers n was f(n) = H_n := 1 + 1/2 + . . . + 1/n. This wasn't too hard, integrating from 0 to 1 the formula 1+u+...+u^(n-1) = (1-u^n)/(1-u) and allowing the n on the RHS to take real values. Soon I generalized it to sums of real powers of the reciprocals of integers. What felt like a really neat conclusion was that, letting the n -> oo, this led to an integral formula for Gamma(x)*Zeta(x). (Of course I had no idea at age 17 of how to make such a proof rigorous, and didn't yet know about contour integrals in C.) Still, all excited about this formula, I told it to my freshman advisor, Henry McKean, on the first day we met. He casually reached over to his bookshelf, and after flipping a few pages showed me the identical formula in Whittaker an Watson from 70 years earlier. You've never seen an MIT freshman's bubble burst so quickly. --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] [seqfan] A000041, the partitions : the classical example of p(200), macmahon, hardy and ramanujan
by Simon Plouffe 14 Mar '11

14 Mar '11

Yes it is!, the expression with the GAMMA function and power of pi is not the classical formula at all, it resembles only. I made this exercise to see if it was possible to get the famous p(200) computation without using all the theory behind it and just the plain decimal expansion of 1 constant, it worked. Actually, the number C*exp(Pi*10)^200, where C is the constant contains the 205 coefficients of the partitions, that is p(1) to p(205) all at once. Once computed (with 2800 digits) it is only a matter of extracting the coefficients using the ordinary change of base formula which is y(n) = [k*x(n)] and x(n+1) = {k*x(n)}, where [ ] and { } are the floor and fractional part of a real number. y(n) are the coefficients. The value of p(200) is obtained at the 199'th evaluation of the formula above. The point is (if I may)that within the decimal expansion of certains numbers lies the coefficients of some sequences and vice-versa, take this example : when f(x) = 1/sqrt(1-4*x), evaluated at x=1/100 will give the first few coefficients of binomial(2*n,n). The coefficients can be <seen> directly since they occur in this case in base 100. I wanted to see if that idea could be applied to more interesting examples. Since the base is exp(Pi*10) then each term is separated by something like 14 decimal digits, exp(Pi*10) is about 4.4x10^13. This is why the computation of p(200) needs 2800 digits to be valid. The example of 1/sqrt(1-4*x) is obvious the case of p(n) is less obvious but equally valid and much more interesting!, I could get the value of F(16), it is an horrible expression that contains an algebraic number of degree 32, Mathematica was quite excellent to extract the algebraic number and Mr Gosper got the value of F(24) which could also be used to get more coefficients. The big drawback is of course the numerical precision needed to complete the calculation. This is an academic exercise but I find it very fruitful. The formula for F(32) is an algebraic number of degree 64, I don't think it is elegant. best regards, simon plouffe

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[math-fun] A000041, the partitions : the classical example of p(200), macmahon, hardy and ramanujan
by Simon Plouffe 14 Mar '11

14 Mar '11

Hello, I have been musing around the expansion of certain numbers in the exp(Pi) base. Some results are I think interesting. If I use F(x) = product(1/(1-x^n),n=1..infinity), then when x -> exp(-Pi*x) the value for F(10) is easily found to be 1/2 1/2 GAMMA(3/4) (5 + 5 5 ) -------------------------- = F(10). 1/4 5 Pi Pi exp(----) 12 and lprinted : F(10) = GAMMA(3/4)*(5+5*5^(1/2))^(1/2)/(Pi^(1/4)*exp(5/12*Pi)) If I expand this number in base exp(Pi*10) then the first 205 coefficients can be obtained. In other words, I can compute p(n) for up to n=205 using 1 mathematical constant and enough precision. note : all the coefficients from n=1 to n=205 are computed at once. The <enough precision> is 2800 digits, unfortunately. I can only think at the Major Percy MacMahon that took 6 months to compute p(200) by hand and later Hardy and Ramanujan, using a very clever method to get that number! The method I use is not very clever, I only use a numerical trick, but it does work. With the base exp(Pi*16) I could obtain the algebraic value also and get : F(16) = 2*2^(3/4)*(560+396*2^(1/2)+3*(69708+49291*2^(1/2))^(1/2))^(1/8)*GAMMA(3/4)/Pi^(1/4)/exp(2/3*Pi) well, by using 11000 digits, I can get the first 500 values, not bad. NOTE 2) the computation of F(x) when x is small is more suitable when I use the infinite product, it converges quite fast, it is nice to have the exact value of F(x) but faster to use the classical formula. Best regards, Simon Plouffe

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[math-fun] High school calculus?
by rcs＠xmission.com 13 Mar '11

13 Mar '11

This note is from Amy Johnston, a math-fun lurker. The funsters undoubtedly have many different viewpoints to offer. I've done a little editing. Rich ------ From: Anna Johnston [jannaston(a)gmail.com] Sent: Friday, March 11, 2011 8:43 AM To: Schroeppel, Richard Subject: math fun and calculus Hi Rich, I wanted to ask you a question (and perhaps get a gage through mathfun) -- old though it may be -- that's come up in my current circle: calculus in high school. I've read MAA articles dating back 11 years on the questionable nature of teaching calculus in high school, talked to high school teachers, mathematician parents, as well as my former colleagues at WSU. The basic thread of these conversations is that calculus should not be part of the high school curriculum. Instead there should be more breadth with a stronger emphasis on discrete concepts (combinatorics, number theory, probability, set theory, logic, proofs, etc). The reasons are: (1) First and foremost, there are other areas of mathematics that would help students think logically while giving them knowledge far more useful in everyday life. HS Calculus, because most students don't quite have the maturity and many teachers don't have the in-depth background needed, tend to be taught in cookbook style, with more memorization and formula plugging and less understanding. Discrete math is far more concrete and useful, with great everyday examples. A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening. (2) Secondly, society's need for calculus has been surpassed by the need for discrete math. Most kids graduating from HS don't know what a hexadecimal number is or how to read one, even though they see them regularly on product codes. Besides the screaming need for better computer literacy, most scientific fields are finding they need more discrete math than they realized. The more we learn about the universe, the more discrete it seems, from quantum physics to DNA. (3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization. Though this idea seems to be old, the only math AP exams are Calculus and stats. The HS teachers at Sam's school (Park HS -- private, progressive and with both abstract and linear algebra courses offered) commented that though they'd like to emphasize other areas of mathematics, it isn't possible due to their requirements from university admissions policy. Many parents view calculus as the apex of mathematics (leading back to university admissions and the AP tests), so there's pressure on them from parents as well. The question is: Why is calculus still the perceived linear end point to HS math and what is the best way to change perceptions and curriculum? Cheers! Amy

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Re: [math-fun] High school calculus?
by Dan Asimov 12 Mar '11

12 Mar '11

P.S. One more comment to what Anna wrote: << A solid understanding of discrete math concepts (for example, more familiarity with summations than the brief introduction they get in calculus, or the binomial expansion) would make calculus far less threatening. >> The fact that in the past many students came to calculus without adequate preparation has been appreciated years ago, and for this reason the oddly named "pre-calculus" course has appeared in many, many H.S. and college curricula. It includes trigonometry and other subjects, and has replaced what used to be a full year of trig in many H.S. curricula. --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] Affine manifolds
by Dan Asimov 12 Mar '11

12 Mar '11

That Wikipedia article mentions two other conjectures beside the one I cited. The first one refers to affine manifolds "of constant volume" -- a notion I don't understand.* And which as far as I can tell is found nowhere in the reference cited (M. Hirsch & W. Thurston, Annals, 1975) The third conjecture strikes me as highly technical. But a manifold's having Euler characteristic = 0 is equivalent to its possessing a non-singular vector field. --Dan ________________________________________________________________ * If I had to guess, I might venture that this "constant volume" means that for all Riemannian metrics on the manifold that are compatible with the affine structure -- if this even makes sense -- the corresponding volume forms are just constant positive multiples of each other. Fred wrote: << See http://en.wikipedia.org/wiki/Affine_manifold for a number of other such --- "Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases." On 3/11/11, Dan Asimov <dasimov(a)earthlink.net> wrote: << Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern): Define an *affine manifold* to be one having an atlas all of whose transition functions are affine. For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0. This has remained unresolved for over 50 years. >> >> ________________________________________________________________________________________ "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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