March 2008 - mathfun@mailman.xmission.com

Re: [math-fun] Characterization of irrationals x such that {frac(n!*x)} is dense in [0, 1) ?
by Dan Asimov 06 Mar '08

06 Mar '08

Using factorial base (which I'd also thought of) I can show that for any real x whose factorial base expansion is x = integer + a_2/2! + . . . + a_n/n! + . . . (0 <= a_n <= n-1) with a *bounded* set of coefficients {a_n}, the set {frac(n!*x)} cannot be dense in ([0,1). This is an uncountable set of reals {x}, but it has measure 0. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Characterization of irrationals x such that {frac(n!*x)} is dense in [0, 1) ?
by Dan Asimov 06 Mar '08

06 Mar '08

Is there a characterization of irrationals x such that {frac(n! * x) : n = 1,2,3,...} is dense in [0,1) ? Do almost all irrationals satisfy this condition? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Re help with definite integral
by Dan Asimov 05 Mar '08

05 Mar '08

Michael Kleber wrote: << On Tue, Mar 4, 2008 at 4:26 PM, Schroeppel, Richard <rschroe(a)sandia.gov> wrote: > . . . > Consider a domino, two adjacent 1x1 squares. > What's the average distance between a random point in the left square, > and a random point in the right square? > . . . Just for the record, Mma (6.0) does fine: d[x1_,y1_,x2_,y2_] := Sqrt[ (x1-x2)^2+(y1-y2)^2 ] Integrate[ d[x1,y1,x2,y2], {x1,-1,0},{x2,0,1},{y1,0,1},{y2,0,1} ] (116 - 8 Sqrt[2] - 20 Sqrt[5] + 140 ArcCsch[2] - 40 ArcSinh[1] + 80 ArcSinh[2] + Log[32] + 10 Log[-1 + Sqrt[5]] - 15 Log[123 + 55 Sqrt[5]]) / 120 >> This answer is intriguingly random-looking. It seems to be daring us to come up with an explanation for why it should be as it is. How about a much simpler task: Can anyone suggest a reason for the denominator of 120 ? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] functions behaving badly
by James Propp 05 Mar '08

05 Mar '08

Fred Lunnon writes: >I suppose you have already consulted Konrad Knopp "Counterexamples in >Analysis", reprinted by Dover? WFL I haven't been able to track down a copy. I did find the book of the same name by Gelbaum and Olmsted, but it didn't offer anything useful. Rich Schroeppel writes: >Do you mean "have a closed form integral"? Dan Asimov has it right when he replies: >No. The issue is existence, not what form it takes. (Thanks Dan!) I should remind everybody that "has an antiderivative" is not the same thing as "is integrable". For instance, the function F that's defined as x^2 sin 1/x^2 when x is non-zero and 0 when x is zero has a derivative f that's defined everywhere, so (by definition) f has an antiderivative, but f is neither Riemann integrable nor Lebesgue integrable on [-1,1]. Jim

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Re: [math-fun] functions behaving badly
by Dan Asimov 04 Mar '08

04 Mar '08

No. The issue is existence, not what form it takes. --Dan << Do you mean "have a closed form integral"? If so, f = log(x) and g = e^x gives f'*g = (e^x)/x, which requires a special function to integrate. Rich ________________________________________ From: math-fun-bounces(a)mailman.xmission.com [math-fun-bounces(a)mailman.xmission.com] On Behalf Of James Propp [jpropp(a)cs.uml.edu] Sent: Tuesday, February 26, 2008 9:13 AM To: math-fun(a)mailman.xmission.com Subject: Re: [math-fun] functions behaving badly Here's another "dirty secrets of the calculus" question (arising from my preparing to teach integration by parts): If f and g are differentiable, must f'(x) g(x) have an antiderivative? (Note that the product rule implies that if f'(x) g(x) has an antiderivative, then so does f(x) g'(x), and vice versa But it's conceivable to me that there could be cases where neither of the two functions has an antiderivative.) Jim Propp >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] functions behaving badly
by James Propp 04 Mar '08

04 Mar '08

Here's another "dirty secrets of the calculus" question (arising from my preparing to teach integration by parts): If f and g are differentiable, must f'(x) g(x) have an antiderivative? (Note that the product rule implies that if f'(x) g(x) has an antiderivative, then so does f(x) g'(x), and vice versa But it's conceivable to me that there could be cases where neither of the two functions has an antiderivative.) Jim Propp

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[math-fun] Re help with definite integral
by N. J. A. Sloane 04 Mar '08

04 Mar '08

The other day I asked for help in evaluating a definite integral Let h = 1/2, to save space. Int_{x = 0..h} Int_{r=-h-x..h-x} Int_{y=0..h} Int_ {s=-h-y..h-y} sqrt(r^2+s^2) dx dr dy ds = ? Thanks to David Cantrell, Eric Weisstein, Steve Finch, Emeric Deutsch and others who replied. To summarize the results: Integrals like this can be found in the books Finch, Encyclop. of Math. Constants (CUP); Santalo, Integral Geometry (Addison-Wesley) There is a web page in Spanish: <http://groups.google.com/group/es.ciencia.matematicas/browse_frm/thread/efe… d222ada58ab7> Mathematica can do this integral, but you have to put the variables in the right order, and you need a machine with the right operating system (it worked on a Sun Linux 64-bit machine, but - with the same version of Mma - failed on an Intel-based Linux 64-bit machine) Best regards Neil

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[math-fun] Cute Puzzle: What Are The Two Numbers?
by Robert Baillie 01 Mar '08

01 Mar '08

The puzzle is from the Winter 2000 issue of The Bent of Tau Beta Pi: www.tbp.org/pages/Publications/Bent/BTs/W00.pdf There are two integers, A and B, which are integers greater than 1 and less than 101. Neither Sam nor Pete knows what they are, but Sam knows their sum, and Pete knows their product. The following conversation takes place. Pete: “I don’t know what the numbers are.” Sam: “I knew that you did not know what the numbers are.” Pete: “Now I know what the numbers are.” Sam: “Then, so do I.” What are the values of A and B? — Don Holden ----- The solution is here: www.tbp.org/pages/Publications/Bent/BTs/Sp00.pdf

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[math-fun] Help with polynomial calculations in Mathematica or Maple
by Dan Asimov 01 Mar '08

01 Mar '08

I find I quickly lose patience when the online "help" files in either of these packages doesn't seem to be helping no matter how I phrase my query. Can someone please tell me how to accomplish the following: I want to multiply out some polynomials (with symbolic coefficients) to get a big hairy polynomial (so far no problem) . . . and then reduce the result modulo setting certain (small) polynomials equal to 0. (It seems that in Mathematica's Simplify command, the Assumptions parameter is intended to do just this. But apparently I can't figure out the syntax, and the "help" files aren't telling.) (What I tried was Simpify[hairypolynomial, {smallpoly = 0}] but it didn't like the syntax for {smallpoly = 0}.) Thanks in advance for any assistance with this. --Dan P.S. In an alternative attempt to get Mathematica to enforce the simple condition that x^5 = 1, I tried just setting x = Exp[2 Pi i/5]. It certainly understood this, since when I typed x^5 it responded with: 2 i Pi Out[2]= E . But an attempt to Simplify that expression left it unchanged. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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