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[math-fun] Fwd: Re: [ROBBINS] A 7-cycle
by rcs＠xmission.com 02 Oct '17

02 Oct '17

This seems worth passing along to the Math-Fun crowd. Read the bottom half first. My apologies to Propp & Speyer. -- Rich ----- Forwarded message from speyer(a)UMICH.EDU ----- Date: Mon, 2 Oct 2017 12:34:10 -0400 From: David Speyer <speyer(a)UMICH.EDU> Reply-To: The Robbins Forum <ROBBINS(a)LISTSERV.UML.EDU> Subject: Re: [ROBBINS] A 7-cycle To: ROBBINS(a)LISTSERV.UML.EDU This provides no insight but, yes, the composition has order 7. Define the seven cones 1: 0 < x,y 2: 0 < -x < y 3: 0 < y < -x 4: x<y<0 5: y<x<0 6: 0 < x < -y 7: 0 < y < -x Then one can check that h and v are linear on each cone, with image another cone. Specifically, h(k) = 4-k mod 7 and v(k) = -k mod 7. So hv permutes the cones by k+4 mod 7 and (hv)^7 is the identity on the set of cones; it is then easy to check that it is the identity on each cone. I'm curious about the tropicalization question -- seven isn't a number which tends to come out of cluster algebras. On Mon, Oct 2, 2017 at 12:20 PM, James Propp <jamespropp(a)gmail.com> wrote: > Apologies for the double posting to the DAC listserv and Robbins listserv; > both seem equally appropriate. > > If we define piecewise-linear continuous maps h and v from R^2 to itself > with > h((x,y)) = (-x-max(y,0),y) > and > v((x,y)) = (x,-y-max(x,0)), > then the composition of h and v appears to be of order 7. > > Is this true? > > Usually the way I prove a PL identity like this is by finding a birational > identity that tropicalizes to it (and having Mathematica do the heavy > lifting), but in this case that strategy failed; I tried the obvious > de-tropicalization > H((x,y)) = (1/(x (1 + y)), y) > and > V((x,y)) = (x, 1/(y (1 + x))) > but the composition of these maps is not of order 7. > > Is the composition of h and v the tropicalization of some birational map > of order 7? > > (If you're wondering where h and v come from: they're the horizontal and > vertical fliber-flipping maps for the polygon bounded by the lines x=-1, > x=1, y=-1, y=1, and x+y=1; the composition of h and v leaves the origin > fixed and cyclically permutes the 7 lattice points on the boundary. If the > term "fiber-flipping" is unfamiliar, see the bottom of page 3 and the top > of page 4 of https://arxiv.org/pdf/1404.3455.pdf, where the > fiber-flipping maps are called piecewise-linear toggles.) > > Jim Propp > > ----- End forwarded message ----- This provides no insight but, yes, the composition has order 7. Define the seven cones 1: 0 < x,y 2: 0 < -x < y 3: 0 < y < -x 4: x<y<0 5: y<x<0 6: 0 < x < -y 7: 0 < y < -x Then one can check that h and v are linear on each cone, with image another cone. Specifically, h(k) = 4-k mod 7 and v(k) = -k mod 7. So hv permutes the cones by k+4 mod 7 and (hv)^7 is the identity on the set of cones; it is then easy to check that it is the identity on each cone. I'm curious about the tropicalization question -- seven isn't a number which tends to come out of cluster algebras. On Mon, Oct 2, 2017 at 12:20 PM, James Propp <jamespropp(a)gmail.com> wrote: > Apologies for the double posting to the DAC listserv and Robbins listserv; > both seem equally appropriate. > > If we define piecewise-linear continuous maps h and v from R^2 to itself > with > h((x,y)) = (-x-max(y,0),y) > and > v((x,y)) = (x,-y-max(x,0)), > then the composition of h and v appears to be of order 7. > > Is this true? > > Usually the way I prove a PL identity like this is by finding a birational > identity that tropicalizes to it (and having Mathematica do the heavy > lifting), but in this case that strategy failed; I tried the obvious > de-tropicalization > H((x,y)) = (1/(x (1 + y)), y) > and > V((x,y)) = (x, 1/(y (1 + x))) > but the composition of these maps is not of order 7. > > Is the composition of h and v the tropicalization of some birational map > of order 7? > > (If you're wondering where h and v come from: they're the horizontal and > vertical fliber-flipping maps for the polygon bounded by the lines x=-1, > x=1, y=-1, y=1, and x+y=1; the composition of h and v leaves the origin > fixed and cyclically permutes the 7 lattice points on the boundary. If the > term "fiber-flipping" is unfamiliar, see the bottom of page 3 and the top > of page 4 of https://arxiv.org/pdf/1404.3455.pdf, where the > fiber-flipping maps are called piecewise-linear toggles.) > > Jim Propp > >

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Re: [math-fun] More Cardano: The Quantum Astrologer's Handbook
by Fred Lunnon 28 Sep '17

28 Sep '17

A source who prefers to remain anonymous writes: > FYI - out in October > > * Michael Brooks > The Quantum Astrologer's Handbook > (Scribe, Brunswick, Vic., Australia, 2017) > <https://books.google.com/books?id=AG00DwAAQBAJ> WFL

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[math-fun] polyhedral "state machines" ?
by Henry Baker 28 Sep '17

28 Sep '17

Is there a characterization of all Moore (?) state machines that can be implemented in the following fashion: Given a convex polyhedron P; it is "in the state" represented by the letter "A" iff the polyhedron is sitting stably (under normal gravity) on a face labeled with "A". The polyhedron switches state by rotating on the edge bounding the "A" face and the "B" face until it now sits on the "B" face. The label on the edge between the "A" face and the "B" face is associated with the "input" -- e.g., it might be input "b". Note that we might have to have several different faces labeled "A" in order to provide enough flexibility for a wider range of possible state machines. We can also have several different edges labeled "b". Clearly, we can "unroll" this polyhedron into an infinite 2-D network with the face labels and edge labels intact. So I guess I'm asking whether we could unroll any Moore state machine into a similar infinite network, and then re-roll it back into the form of a polyhedron.

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[math-fun] Lax Pairs and Toda Lattices
by Henry Baker 28 Sep '17

28 Sep '17

"Lax Pairs" seem to be the generalization of systems with soliton solutions. Most of the work on Lax Pairs seems to have gone into models of actual physical systems. What I'm looking for is a more methodical mathematical *catalog* of possible Lax Pairs -- whether or not they are physically realizable. I'm curious about the range of behavior that these various soliton solutions can exhibit. I've heard that the *inverse scattering transform* (IST) is a non-linear generalization of the Fourier Transform. I'd also be interested in a *catalog* of IST pairs, analogous to the f(t), FT(f(t)) pairs show in Fourier Transform catalogs. "Toda Lattices" appear to be the finite element versions of the continuous KdV soliton equation. Perhaps Toda Lattices might be capable of Conway "Life" emulation?

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[math-fun] Richard K Guy : 101 years old on sept. 30.
by Simon Plouffe 27 Sep '17

27 Sep '17

Hello, maybe it would be worth noting that Richard K. Guy will be 101 years old on september 30 this year. 101 is prime, so is Richard K. Guy too, of prime importance. Best regards, Simon Plouffe

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[math-fun] tenfold trinskys
by Bill Gosper 27 Sep '17

27 Sep '17

To some of you I recently blathered that the only known rotationally symmetric (modulo slight smushing) "recursive" trinskys were of orders 3 ,4, 5, and 6, completely forgetting the 8 and 12 found by my coauthors. But I don't think we had a 10 until gosper.org/TrinskyXYPlot[599,599,3÷2-√5÷2,1÷2,0,0,1,1,999999][[8,3]].png , which has an infinite hierarchy of nested "star loops". gosper.org/monstar.png . Empirically, they can be started at x = y = 4 + 2 GoldenRatio^(-5 - 2 n)/√5 + 2 GoldenRatio^(5 + 2 n))/√5, with period 140/3 (5 4^n - 2). --rwg PS here's an 8: gosper.org/octoplasm.png and a 12: gosper.org/dodecadence.png .

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[math-fun] Prime factors of 10^N + 1
by Bill Gosper 27 Sep '17

27 Sep '17

On 2017-09-25 15:36, Dan Asimov wrote: > Decided to list these just for the heck of it. When's the next prime after 101? Mathematica says its ≥ 10^2^20+1. --rwg > Some amusing almost-patterns for 10^0 + 1 through 10^19 + 1: > > 1 = 1 > > 11 = 11 > > 101 = 101 > > 1001 = 7 * 11 * 13 > > 10001 = 73 * 137 > > 100001 = 11 * 9091 > > 1000001 = 101 * 9901 > > 10000001 = 11 * 909091 > > 100000001 = 17 * 5882353 > > 1000000001 = 7 * 11 * 13 * 19 * 52579 > > 10000000001 = 101 * 3541 * 27961 > > 100000000001 = 11 * 11 * 23 * 4093 * 8779 > > 1000000000001 = 73 * 137 * 99990001 > > 10000000000001 = 11 * 859 * 1058313049 > > 100000000000001 = 29 * 101 * 281 * 121499449 > > 1000000000000001 = 7 * 11 * 13 * 211 * 241 * 2161 * 9091 > > 10000000000000001 = 353 * 449 * 641 * 1409 * 69857 > > 100000000000000001 = 11 * 103 * 4013 * 21993833369 > > 1000000000000000001 = 101*9901*999999000001 > > 10000000000000000001 = 11*909090909090909091 > > 10^20 + 1 is too large for my software. > > —Dan

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[math-fun] "Hamiltonian", a derivative musical?
by Henry Baker 27 Sep '17

27 Sep '17

I saw "Hamilton", the musical, last night and was disappointed because they never mentioned quaternions, not even once! Seriously, if someone out there wants to do a mathematical parody -- or more appropriately, a mathematical "variation" -- this one is ripe for the picking! The words of the Miranda's Broadway musical are easily Googleable; I can email you my file if you're interested. You won't even have to commute a Hamiltonian path on the NY subway system...

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[math-fun] weird soliton question
by Henry Baker 27 Sep '17

27 Sep '17

Solitons use a combination of nonlinearity and dispersion to propagate pulses along a 1-D channel whose *shape is preserved*. Standard solitons "simply" pass through one another "unaltered in shape, amplitude, or velocity" (see link below). Are there weird equations and soliton solutions that attempt to preserve the "holes" between them -- i.e., in addition to preserving the shape of the pulse itself, the medium would also attempt to preserve the *space between pulses*, so that they would settle down into an orderly pulse train? ---- https://arxiv.org/abs/1407.5087 Collisions of matter-wave solitons "Solitons are localised wave disturbances that propagate without changing shape, a result of a nonlinear interaction which compensates for wave packet dispersion. Individual solitons may collide, but a defining feature is that they pass through one another and emerge from the collision unaltered in shape, amplitude, or velocity. This remarkable property is mathematically a consequence of the underlying integrability of the one-dimensional (1D) equations, such as the nonlinear Schr\"odinger equation, that describe solitons in a variety of wave contexts, including matter-waves$^{1,2}$. Here we explore the nature of soliton collisions using Bose-Einstein condensates of atoms with attractive interactions confined to a quasi-one-dimensional waveguide. ..."

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Re: [math-fun] math-fun Digest, Vol 175, Issue 18
by Stuart Anderson 26 Sep '17

26 Sep '17

Message: 1 Date: Sun, 24 Sep 2017 23:28:01 -0400 From: James Propp Mathematica-using iPhone users: Have any you tried the WolframAlpha app? Does it let you run real Mathematica code? I should probably just spend the $2.99 and find out for myself what the app can do, but I thought I'd ask first. Jim Propp I have it on my phone, when I tried to square a matrix, it squared the matrix entries rather than multiplying the matrix by itself.

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