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[math-fun] Fourier analysis and music
by Henry Baker 02 Mar '20

02 Mar '20

This problem has been bothering me ever since I learned about Fourier transforms and the frequency domain as an undergraduate who loves music. Unlike in electrical engineering and physics, where frequency is a *linear* scale, and an entire bounded spectrum can be translated up/down in frequency using mixing converters, 12-tone music uses a *log* frequency scale, and I'm not aware of any simple Fourier formulae that work on a log frequency scale. The ultraviolet catastrophe shows that the amount of information encoded in higher musical octaves grows enormously, so any invertible Fourier translation of standard 12-tone scales is going to have to throw away this extra information. Is anyone here aware of any interesting papers or theorems which involve *linear* time and *logarithmic* frequency? Or perhaps there are Fourier-like theorems that involve *logarithmic* time and *logarithmic* frequency? Or *exponential* time and *logarithmic* frequency?

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[math-fun] In Memoriam: Freeman Dyson
by Brad Klee 01 Mar '20

01 Mar '20

The population size of Homo sapiens has gotten so large now, it is stating the obvious to say that, despite his fame, many people never had the opportunity to meet Freeman Dyson, to attend one of his talks or lectures, or to write letters with him. A smaller segment of the population, who only encountered his works in print or video-recorded talks, probably have a similar sadness and feeling of regret for a missed opportunity, say, to have a chat about the relative merits of hooliganism and the extinction rebellion. In this situation, we have to thank science for creating such detailed systems of classification that we can have a half-way valid, rational explanation for life stories that happen, and others that do not. Personally, I can still be happy having never met Freeman Dyson, and thoroughly enjoyed studying his theory for the life of the mind. The essay "The Scientist as a Rebel" [1] has been a huge influence in my most recent years. From his book by the same title, I particularly liked "A modern Heretic", a short essay on the thinking of Thomas Gold. These works are somewhat relevant to my postings on [seqfan] and [math-fun], because I wholeheartedly bought into the idea that professional training is not a deciding factor for performance in mathematical competitions. Of course the same is true for contests in physics, and Freeman Dyson himself, is a great example and inspirational personage as to what a trained mathematician can hope to accomplish in a neighbouring discipline. I also found interest in the video interview [2] at web of stories, enjoyed hearing the stories about Bethe, and think that Gosper has a funny story about the Ramanujan conference at UIUC. If anyone else has suggested reading or interesting stories to share, perhaps consider posting to the list or sending me a personal message. The fact that he is now deceased is no reason to stop studying his works, quite the opposite. --Brad [1] https://www.jstor.org/stable/2974452?seq=1 [2] https://www.webofstories.com/play/freeman.dyson/95

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[math-fun] Euler quote (?)
by James Propp 01 Mar '20

01 Mar '20

I just Googled the quote “*My pen is a better mathematician than I am*” (which I’m sure I’ve seen attributed to Euler) in search of the original French or Latin phrasing, and was dismayed to see that the only verbatim appearances of this sentence are references to an earlier essay of mine! I’m hoping this quote isn’t an unintentional fabrication on my part. Can anyone provide a source or reference? Thanks, Jim Propp

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[math-fun] Epidemic mathematics
by Henry Baker 01 Mar '20

01 Mar '20

Given the spread of the Covid-19, here's how to explain the mathematics of epidemics. "The 'basic reproduction number' R0 is defined as the expected number of secondary cases produced by a single infection in a completely susceptible population. It is important to note that R0 is a dimensionless number and not a rate." R0 is the largest eigenvalue of the 'next generation matrix'. For example, R0=phi=(1+sqrt(5))/2 for Fibonacci's rabbit problem; R0='k' (neutron multiplication factor) in nuclear fission. Thus, R0 is the *base* of the exponentials & logarithms which occur in the various solutions instead of Euler's 'e'. Clearly, an epidemic won't go 'viral' unless |R0|>1. "Notes on R0" James Holland Jones, Stanford University, 5/1/2007 https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf

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Re: [math-fun] New essay
by Henry Baker 01 Mar '20

01 Mar '20

Fourier transforms? (This use of pi, sqrt(pi) can be forgiven, due to the *circles*/*cycles* involved in Fourier analysis.) n-dimensional volumes involve pi^(n/2). Why the "1/2" ? (These formulae also involve gamma(n+1/2)). Is there a relationship between your integration of the Gaussian and n-dimensional volumes? (Google "Gaussian integrals" and wikipedia.) At 06:58 AM 3/1/2020, James Propp wrote: >I've written a new essay, which I hope to post on March 10th (rather than >the 17th). > >http://faculty.uml.edu/jpropp/mathenchant/059-draft1.pdf > >Suggestions for improvements of all kinds are welcome, including other cool >facts about the square root of pi, and good references for writeups and >videos that evaluate the integral of exp -x^2. > >Jim Propp

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[math-fun] New essay
by James Propp 01 Mar '20

01 Mar '20

I've written a new essay, which I hope to post on March 10th (rather than the 17th). http://faculty.uml.edu/jpropp/mathenchant/059-draft1.pdf Suggestions for improvements of all kinds are welcome, including other cool facts about the square root of pi, and good references for writeups and videos that evaluate the integral of exp -x^2. Jim Propp

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[math-fun] Fibonacci, cosh, sinh
by Henry Baker 01 Mar '20

01 Mar '20

It is well-known that fib(n) = cosh(n*alpha)/cosh(alpha) (n odd) fib(n) = sinh(n*alpha)/cosh(alpha) (n even) alpha = asinh(1/2). Also, fib(n) = phi^n-(-phi)^(-n) ----------------- sqrt(5) Is there any version of these equations fib(n) = A*exp(a*n)+B*exp(b*n)+C*exp(c*n)+... that *doesn't* require splitting up into n odd, n even? (I'm willing to accept complex A,a,B,b,C,c, etc.)

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Re: [math-fun] sum of 2 triangular numbers is a triangular number
by Neil Sloane 27 Feb '20

27 Feb '20

I mentioned this on Feb 19 2020 and several people sent very helpful replies. For squares, we have the theorem that all solutions to x^2+y^2=z^2 in integers are given by Pythagorean triples, t*[u^2-v^2, 2uv, u^2+v^2]. Expanding on the replies from Michael Collins and Rich, it looks like the following is the analog for triangular numbers T_n = n(n+1)/2. ALL solutions to T_n+T_k=T_m are given by the list of what one might call Trithagorean triples: these are the triples [n,k,m] = [n, T/Q-(Q+1)/2,T/Q+(Q-1)/2] where n >= 2, T=n(n+1)/2, and Q is any odd divisor of T less than n, plus these triples with the first two coordinates swapped. I haven't found this in the literature, but it can hardly be new.

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Re: [math-fun] A + B + C = D and A, B, C substrings of D
by Tom Duff 25 Feb '20

25 Feb '20

I don't understand how you add up 1820 from 3 of its substrings. The available substrings are 1 2 8 18 20 82 182 820. one of them has to be as big as 1820/3=606 -- that's 820. One of the 2 remaining has to be as big as (1820-820)/2=500. That's also 820. The third has to be 1280-820-820=180, which isn't one of the choices. On Tue, Feb 25, 2020 at 12:10 PM Michael Greenwald <mbgreen(a)seas.upenn.edu> wrote: > > On 2020-02-25 10:28, Tom Duff wrote: > > I should say, I excluded 0 as an addend, because then you get a=a+0+0 > > for every a with a zero in it, which strikes me as not interesting. > > Adding 0's to a previous solution is also uninteresting. I assume > you meant to cut those out, too, but you (mistakenly?) included > all forms of 150* = 50* + 50* + 50* (but almost no other examples > [e.g. 182, 1820, 18200, ...], although you do include > 195000=5000+95000+95000 and 19500, and 195 = 5 + 95 + 95)

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[math-fun] a(n) and a(n+1) substrings of their product
by Éric Angelini 25 Feb '20

25 Feb '20

Hello Math-Fun, Carole Dubois and myself have tried to extend S -- but could not find a new term < 9 999 999 999. S = 1, 10, 11, 100, 21, 1000, 31, 10000, 41, 100000, 51, 1000000, 61, 10000000, 71, 100000000, 81, 2817. S should be the lexicographically earliest seq of distinct positive terms such that a(n) and a(n+1) are substrings of their product. Example: 81 * 2817 = 228177. Does S end there? Would a different a(1) produce more terms? Is there another way to exploit this idea of two terms visible in their product? (respectively addition, division) Or is this old hat? Best, É.

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