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[math-fun] rank of a symbolic algebra matrix
by Henry Baker 07 Oct '20

07 Oct '20

Suppose you're using a symbolic algebra system & want to estimate the rank of a matrix whose entries are symbolic expressions involving many different variables. My intuition tells me that I should be able to randomly guess values for all of the variables, numerically compute the rank for that assignment of variables, and do it again several more times. The *maximum* rank achieved should be close to the actual symbolic rank of the matrix. 2 cases: integers and floating point. If the symbolic expressions enable exact integer values, then the rank could be computed using arbitrary-precision arithmetic. However, it might be more efficient to do the computation modulo a large number of different primes. If we work in floating point, the expressions could be incredibly ill-conditioned, in which case telling the difference between very small numbers and actual zeros could be impossible. Other than such ill-conditioning, is it possible that the rank could be *over* estimated by such random sampling?

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Re: [math-fun] Integer determinant question
by Dan Asimov 07 Oct '20

07 Oct '20

I recently asked the question at bottom. This is an update. >From googling I learned that an n x n matrix whose rows are successive cyclic rotated versions of each other are called "circulant" matrices (a term I'd heard but never bothered to learn about before) — and that these matrices are actually very interesting!!! (See for instance this article: <https://www.ams.org/notices/201203/rtx120300368p.pdf>. It is apparently an open problem to determine which integers can be the determinants of integer circulant matrices — see for instance this article: <https://projecteuclid.org/download/pdf_1/euclid.ijm/1256047802>. Perhaps the additional condition I somewhat arbitrarily imposed, that the row sum a(0) + ... + a(n-1) = 1, might allow solution of the problem in this special case; maybe not. A subproblem might be to reduce entries of a circulant matrix modulo a prime p and then ask what the determinant can be modulo p. Maybe that question is more tractable. —Dan ----- The lattice in R^3 generated by the symmetrical vectors { (2,-1,0), (0,2,-1), (-1,0,2) } (whose sum is (1,1,1)) has a fundamental cell whose volume = 7. I was surprised that a 3-fold symmetrical set of vectors would generate a parallelepiped whose volume had volume = 7. So I'm curious: Suppose n integers a(0), ..., a(n-1) have sum a(0) + ... + a(n-1) = 1. Then what integers can be the determinant |M| of the n x n matrix M = (m(i,j)) if m(i,j) = a(i+j) where i+j is calculated modulo n ??? -----

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Re: [math-fun] Lucas, Fibo, Phi, capital O
by Steve Witham 07 Oct '20

07 Oct '20

> My first thought seeing this was that >Â Â Â Â Lucas(n) / Fibo(n) --> sqrt(5) > and the int sequences can be done by repeated squaring, > so you could get O(2^n) digits of sqrt(5) > with O(n) int multiplications and only one division!! Ugh, using "n" for two different things.Â Lucas(n) and Fibo(n) do have O(n) digits, but the following is wrong: > And the work that Newton's method does is also about > the same because the divisions need only 1, 2, 4... > 2^(n-1), 2^n digits of precision. Sorry, instead say 2^1, 2^2, 2^4... 2^(k/2), 2^k digits in k steps. The point is that the order of work is that of the last division. Â --Steve

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[math-fun] Lucas, Fibo, Phi, capital O
by Steve Witham 07 Oct '20

07 Oct '20

Lucas(n=0, 1, 2...) = 2, 1, 3, 4, 7, 11, 18... = phi^n + (-phi)^(-n) Fibonacci(n=0, 1, 2...) = 0, 1, 1, 2, 3, 5, 8... = (phi^n - (-phi)^(-n)) / sqrt(5) (True for n < 0 also.) My first thought seeing this was that Lucas(n) / Fibo(n) --> sqrt(5) and the int sequences can be done by repeated squaring, so you could get O(2^n) digits of sqrt(5) with O(n) int multiplications and only one division!! But then I thought, continued fractions probably solve the general sqrt(int) problem just as fast. And the work that Newton's method does is also about the same because the divisions only need 1, 2, 4... 2^(n-1), 2^n digits of precision. This is math. This is my brain on math. --Steve

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[math-fun] Integer determinant question
by Dan Asimov 06 Oct '20

06 Oct '20

The lattice in R^3 generated by the symmetrical vectors { (2,-1,0), (0,2,-1), (-1,0,2) } (whose sum is (1,1,1)) has a fundamental cell whose volume = 7. I was surprised that a 3-fold symmetrical set of vectors would generate a parallelepiped whose volume had volume = 7. So I'm curious: Suppose n integers a(0), ..., a(n-1) have sum a(0) + ... + a(n-1) = 1. Then what integers can be the determinant |M| of the n x n matrix M = (m(i,j)) if m(i,j) = a(i+j) where i+j is calculated modulo n ??? —Dan

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[math-fun] binary roulette wheels
by Veit Elser 06 Oct '20

06 Oct '20

Is there a name for cyclic sequences (necklaces) of length 2^n that contain all the integers 0, … , 2^n-1 expressed in binary in the 2^n subsequences of length n? For example, for n=4 the sequence 0000100110101111 contains 0, 1, 2, 4, 9, 3, 6, 13, 10, 5, 11, 7, 15, 14, 12, 8. -Veit

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[math-fun] Using AI to help Superseeker guess formulas for sequences
by Neil Sloane 05 Oct '20

05 Oct '20

Dear Math Fun: I keep coming across sequences that badly need a formula, possibly expressing them in terms of some known sequence. A337641 and A337655 are two recent examples. Why not use AI to strengthen Superseeker?, I keep saying There is a talk on Monday that might be relevant: This announcement just arrived in email: Springer Nature <springernature(a)newsletter.springernature.com> George Em Karniadakis (Brown University and MIT) >From PINNs to DeepOnets: Approximating functions, functionals, and operators using deep neural networks Throughout October, authors from the new journal SN Partial Differential Equations and Applications will be hosting a series of free webinars on topics related to partial differential equations. Up next is a presentation by George Em Karniadakis of Brown University and MIT on approximating functions using deep neural networks. Join us for a lively discussion on October 5th at 9am EST, 3pm CET. I signed up, but AI and PDEs are not my field. If anyone is interested, please watch too

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Re: [math-fun] binary roulette wheels
by Dan Asimov 03 Oct '20

03 Oct '20

Allan and Andy have pointed out the ridiculousness of my original suggestion. But maybe a slight change could make it possible: Could there be a continuous function on the disk F : D^2 —> R such that for every continuous function g : [a, b] —> [0, 1] on an interval, there is a curve C = C_g C : [a, b] —> D^2 such that F(C(t)) = g(t) for all a ≤ t ≤ b ??? —Dan ----- Or even simpler than that, only 1/L constant functions, out of uncountably many, are possible. Andy On Sat, Oct 3, 2020 at 1:29 PM Allan Wechsler <acwacw(a)gmail.com> wrote: > I am feeling strongly that Dan's universal circular function is impossible. > For any L, let N be ceil(1/L). Consider just the family of N+1 functions > f[i](x) = ix, i ranging from 0 to N. It seems obvious to me that R/Z isn't > big enough to fit them all. > > On Sat, Oct 3, 2020 at 12:24 PM Dan Asimov <dasimov(a)earthlink.net> wrote: > > > I wonder if analogous continuous objects can exist. > > > > For instance, can there be a continuous function on the circle R/Z > > > > f : R/Z —> R > > > > and some fixed L > 0, such that f contains all continuous functions > > > > g : [0, L] —> R > > > > as represented on an arc [c, c+L] ⊂ R/Z of the circle, via > > > > g(x) = f(x+c) > > > > (addition modulo 1) ??? > > > > Or something along these lines, maybe for a restricted class of > > functions f and g ? > > -----

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Re: [math-fun] binary roulette wheels
by Dan Asimov 03 Oct '20

03 Oct '20

Jim Propp wrote: ----- I believe there is a sense in which the Riemann zeta function on the critical line exhibits the sort of behavior Dan is looking for. If I recall correctly, for any continuous real-valued function f on any finite interval, and for any positive epsilon, there’s an interval on the critical line such that the zeta function differs from f by less than epsilon everywhere on the interval. ----- Aha, maybe I heard of something like that once, too. Googling led to this article: <https://en.wikipedia.org/wiki/Zeta_function_universality>. It is apparently a 2-dimensional approximation in the critical strip (not line), which I guess allows for more wiggle room. —Dan

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Re: [math-fun] binary roulette wheels
by Dan Asimov 03 Oct '20

03 Oct '20

I wonder if analogous continuous objects can exist. For instance, can there be a continuous function on the circle R/Z f : R/Z —> R and some fixed L > 0, such that f contains all continuous functions g : [0, L] —> R as represented on an arc [c, c+L] ⊂ R/Z of the circle, via g(x) = f(x+c) (addition modulo 1) ??? Or something along these lines, maybe for a restricted class of functions f and g ? —Dan Adam Goucher wrote: ----- de Bruijn sequences Veit Elser wrote: ----- Is there a name for cyclic sequences (necklaces) of length 2^n that contain all the integers 0, … , 2^n-1 expressed in binary in the 2^n subsequences of length n? For example, for n=4 the sequence 0000100110101111 contains 0, 1, 2, 4, 9, 3, 6, 13, 10, 5, 11, 7, 15, 14, 12, 8. ----- -----

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