May 2020 - mathfun@mailman.xmission.com

[math-fun] Cross-My-Primes
by Éric Angelini 06 May '20

06 May '20

Hello Math-Fun, have a look here for a clearer idea developed in the hereunder lines: https://bit.ly/2KMrmHM Best, É. --------------- Hello Math-Fun, this game (hope not old hat) is played on a Scrabble board where all squares are white, and no square has any text on it. The starting "star" square, in the center of the grid, has a 7 on it (the number, not the word, as this game is played with digits only that will form numbers – instead of letters forming words). The single player must now form prime numbers at every turn on the board – at the first turn placing the integer 1, at the second turn placing 2, then 3, then 4, then 5, etc. – this is the natural order of the positive integers as they appear (and because 7 starts the game, the player will jump from 6 to 8. No other jump will be allowed during the game). Those integers, as in the traditional game of Scrabble, must be attached to the existing structure at least by a digit (one digit per square). All visible numbers, before and after any turn, must be prime: they are read horizontally from left to right or vertically from top to bottom (again, like words in a traditional Scrabble game). Can one split the digits of a term bigger than 9 – say 10 or 11 ? Yes, as long as the split digits stay on the same line, forming one or more prime numbers within the existing structure. Example here: We start the game with 7 on the star: ....................7.................... then comes 1 and we form the prime 71; ....................71.................... then comes 2 and we form the prime 271; ...................271.................... then comes 3 and we form the prime 3271; ..................3271.................... then comes 4 and we form the prime 43271; .................43271.................... then comes 5 and we form the prime 53 (5 on top of 3 on the grid); ..................5....................... .................43271.................... (etc.) Best, É.

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Re: [math-fun] Surface with single, knotted edge
by Bill Gosper 05 May '20

05 May '20

It looks, misleadingly, like the curvature remains bounded at the corner, like the three planes of a corner cube. But no, the corner is kinky <http://gosper.org/verycross2.png>. With planarizing, it could sill make a decent corner cube. —rwg "Fresnel corner cube" <http://gosper.org/martinsmarbles.png>. I've always wanted to mount a big corner cube anemometer on my car's roof to spoof police radar. "But it's my airspeed indicator!" On Tue, May 5, 2020 at 12:25 PM Howard Cohl <howard.cohl(a)nist.gov> wrote: > That's really neat. > > I wonder if it has a name. > Also, are there analogues for other knots? > > There must be, eh? > > -- > http://www.nist.gov/itl/math/msg/howard-s-cohl.cfm > > On 5/4/20 4:55 PM, Bill Gosper wrote: > > Does this thing > <https://gcc02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fgosper.org…>have > a name? —rwg > > ParametricPlot3D[{(1 - t) Cos[2 u] (3 + Cos[3 u]), (t - 1) (3 + Cos[3 u]) > Sin[2 u], > 4 t + (1 - t) Sin[3 u]}, {u, -π, π}, {t, 0, 1}, ViewPoint -> Below] > >

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Re: [math-fun] Positive eigenvalues imply positive definite?
by Dan Asimov 05 May '20

05 May '20

A positive definite matrix always means a *symmetric* matrix, and the eigenvectors of a symmetric matrix are always perpendicular to each other as long as their eigenvalues are unequal. A nice condition equivalent to positive definiteness is that all the upper-left k x k submatrices have positive determinant. —Dan

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Re: [math-fun] Positive eigenvalues imply positive definite?
by Dan Asimov 05 May '20

05 May '20

It's just a matter of definition, not truth. I've always seen positive definite defined as a condition on symmetric matrices, so that's what I take as the definition. —Dan Gene Salamin wrote: ----- > On Tuesday, May 5, 2020, 09:58:03 AM PDT, Dan Asimov <dasimov(a)earthlink.net> wrote: ----- A positive definite matrix always means a *symmetric* matrix, ----- Not true. The nonsymmetric matrix [[a,b],[-b,c]] is positive definite when a > 0 and c > 0. -----

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[math-fun] Positive eigenvalues imply positive definite?
by Eugene Salamin 05 May '20

05 May '20

I found this book review in the April 2020 Physics Today [ https://physicstoday.scitation.org/doi/10.1063/PT.3.4456 ]. The following is a quote from it. "After stating that “an arbitrary non-singular tensor T is positive definite if v·T·v>0 for all vectors v≠0,” the book goes on to spread the myth that positive eigenvalues of T are sufficient for T to be positive definite; the 2 × 2 matrix T={{4,9},{1,4}} with v={1,−1} is a counterexample." Then in reading Arthur Gelb, "Applied Optimal Estimation", I found problem 2-3, in which A is a matrix. "Show that A is positive definite if and only if all of its eigenvalues are positive." -- Gene

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[math-fun] Surface with single, knotted edge
by Bill Gosper 05 May '20

05 May '20

Does this thing <http://gosper.org/verycrosscap.png>have a name? —rwg ParametricPlot3D[{(1 - t) Cos[2 u] (3 + Cos[3 u]), (t - 1) (3 + Cos[3 u]) Sin[2 u], 4 t + (1 - t) Sin[3 u]}, {u, -π, π}, {t, 0, 1}, ViewPoint -> Below]

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Re: [math-fun] Moebius-like torus name?
by Steve Witham 04 May '20

04 May '20

From: Marc LeBrun <mlb(a)well.com> Date: 5/4/20, 7:04 PM > Anyway, it seems a finite square board with edges identified would be > logically equivalent to an infinite quilt of square patches in a regular > board, each of which are initialized with a suitably rotated/reflected > copy of the initial configuration. Yes, as long as with Langton's ant you have synchronized ants running in all the patches. > Indeed this avoids local snarling, but curling up a CA generally mostly > seems to introduce additional "global" phenomena that depend > sensitively only on arbitrary details, such as the exact diameter of the > universe along each axis. Yes, that's been my holy grail... well semi-holy. Right now it occurs to me you can twist the Klein bottle by any number of cells.Â Same with a torus.Â That is, shift the top left-right relative to the bottom. I guess that corresponds to brick-laying patterns. Wait, does this mean one can do those tilings with two sizes of squares? Marc, you may have widened one of the cracks in my brain. >> Does this mean that a doubly-twisted "purse" board will behave the same >> as an untwisted "torus" board? > Duh, sorry, obviously only if the initial configuration was symmetrical... And not Langton's Ant.Â Or else two or four ants always meeting and doing some kind of do-se-do whenever they go to corners, meh. That reminds me of an experiment I've never done: put mirrors in a dish of water so that waves in the water pass straight through the mirrors. I wonder whether you could set up the tiles systematically out of sync, say, by some number of clock cycles for each tile-sized step to the North.Â I guess that creates a problem setting up the initial state... Â --Steve

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Re: [math-fun] Surface with single, knotted edge
by Dan Asimov 04 May '20

04 May '20

I think the only thing it could be called is "cone on a trefoil knot". —Dan ----- Does this thing <http://gosper.org/verycrosscap.png>have a name? —rwg ParametricPlot3D[{(1 - t) Cos[2 u] (3 + Cos[3 u]), (t - 1) (3 + Cos[3 u]) Sin[2 u], 4 t + (1 - t) Sin[3 u]}, {u, -π, π}, {t, 0, 1}, ViewPoint -> Below] -----

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[math-fun] Integrals of sums of Rademacher functions
by Dan Asimov 04 May '20

04 May '20

The Rademacher function r_n : [0,1] —> {-1,1} is your basic square wave of wavelength 1/2^n on [0,1], defined conveniently as r_n(x) = sign(sin(2^n * 2πx)) I noticed that in Mark Kac's great 1959 book "Statistical Independence in Probability, Analysis, and Number Theory" he states that Integral_{0 <= t <= 1} ((r_1(t) + ... + r_n(t))^2 dt = n It's easy to see that if the exponent 2 is replaced by an odd number, this integral is 0, by oddness. But my limited experiments have not suggested what the pattern is for even exponents, other than that it's always an integer. Question: What is an explicit formula for f(n, k) = Integral_{0 <= t <= 1} ((r_1(t) + ... + r_n(t))^(2k) dt ??? —Dan

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[math-fun] The stellated dodecahedron,
by Bill Gosper 04 May '20

04 May '20

PolyhedronData@{"DodecahedronStellation", 1} has 12 pentagonal pyramid vertices (solid angle ArcTan[38/41]) and 20 monkey saddles (solid angle 2π, exactly self-complementary <http://gosper.org/saddlekiss.png>). Unfortunately, for these the new function PolyhedronAngle gets 0.) —rwg The Regular Dodecahedron vertex solid angle is π - ArcTan[2/11] == 2 ArcTan[GoldenRatio^5] Almost trivial Cheesewedge Theorem: The dihedral angle of a cheesewedge (in radians) equals the solid angle of the sharp corners at either end of the dihedral edge (in steradians). Unless it's a mild cheese. In[1825]:= solidAngles[π/2, π/2, a] Out[1825]= {2 ArcTan[Sqrt[1 - Cos[a]^2]/(1 + Cos[a])], ArcCos[Cos[a]], 2 ArcCos[1/2 (1 + Cos[a]) Sec[a/2]], 2 ArcSin[(Sqrt[1 - Cos[2 a]] Sec[a/2])/(2 Sqrt[2])], 4 ArcTan[Sqrt[Tan[a/4]^2 Tan[1/4 (-a + \[Pi])] Tan[(a + \[Pi])/4]]]} In[1830]:= Assuming[0 < a < π, FullSimplify@%1825] Out[1830]= {a, a, a, a, a}

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