February 2019 - mathfun@mailman.xmission.com

[math-fun] surdburger
by Bill Gosper 24 Feb '19

24 Feb '19

3 (1 + 3 2^(1/3)) (2 - 3^(1/3)) (1 + 3^(1/3)) = = (1 + 2^(1/3))^3 (1 + 2^(2/3)) (-1 + 3^(1/3)) (-2^(2/3) + Sqrt[3]) (2^(2/3) + Sqrt[3]) (1 + 3^(2/3)) —rwg

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[math-fun] Fwd: Near misses
by Bill Gosper 24 Feb '19

24 Feb '19

Dang. I thought this would be about transgenders. -------- Original Message -------- Subject: [math-fun] Near misses Date: 2019-02-21 11:43 From: "Bernie Cosell" <bernie(a)fantasyfarm.com> To: math-fun(a)mailman.xmission.com Reply-To: math-fun <math-fun(a)mailman.xmission.com> < http://nautil.us/issue/69/patterns/the-impossible-mathematics-of-the-real-w… > Bogus article. There are finitely many Johnson solids only because they're restricted to properly convex. Mathematica has them all (with unit edges. Might make a nice poster): In[28]:= PolyhedronData@"Johnson" Out[28]= {"AugmentedDodecahedron", "AugmentedHexagonalPrism", \ "AugmentedPentagonalPrism", "AugmentedSphenocorona", \ "AugmentedTriangularPrism", "AugmentedTridiminishedIcosahedron", \ "AugmentedTruncatedCube", "AugmentedTruncatedDodecahedron", \ "AugmentedTruncatedTetrahedron", "BiaugmentedPentagonalPrism", \ "BiaugmentedTriangularPrism", "BiaugmentedTruncatedCube", \ "BigyrateDiminishedRhombicosidodecahedron", "Bilunabirotunda", \ "DiminishedRhombicosidodecahedron", {"Dipyramid", 3}, {"Dipyramid", 5}, "Disphenocingulum", "ElongatedPentagonalCupola", \ "ElongatedPentagonalDipyramid", "ElongatedPentagonalGyrobicupola", \ "ElongatedPentagonalGyrobirotunda", \ "ElongatedPentagonalGyrocupolarotunda", \ "ElongatedPentagonalOrthobicupola", \ "ElongatedPentagonalOrthobirotunda", \ "ElongatedPentagonalOrthocupolarotunda", \ "ElongatedPentagonalPyramid", "ElongatedPentagonalRotunda", \ "ElongatedSquareCupola", "ElongatedSquareDipyramid", \ "ElongatedSquareGyrobicupola", "ElongatedSquarePyramid", \ "ElongatedTriangularCupola", "ElongatedTriangularDipyramid", \ "ElongatedTriangularGyrobicupola", \ "ElongatedTriangularOrthobicupola", "ElongatedTriangularPyramid", \ "GyrateBidiminishedRhombicosidodecahedron", \ "GyrateRhombicosidodecahedron", "Gyrobifastigium", \ "GyroelongatedPentagonalBicupola", \ "GyroelongatedPentagonalBirotunda", "GyroelongatedPentagonalCupola", \ "GyroelongatedPentagonalCupolarotunda", \ "GyroelongatedPentagonalPyramid", "GyroelongatedPentagonalRotunda", \ "GyroelongatedSquareBicupola", "GyroelongatedSquareCupola", \ "GyroelongatedSquareDipyramid", "GyroelongatedSquarePyramid", \ "GyroelongatedTriangularBicupola", "GyroelongatedTriangularCupola", \ "Hebesphenomegacorona", "MetabiaugmentedDodecahedron", \ "MetabiaugmentedHexagonalPrism", \ "MetabiaugmentedTruncatedDodecahedron", \ "MetabidiminishedIcosahedron", \ "MetabidiminishedRhombicosidodecahedron", \ "MetabigyrateRhombicosidodecahedron", \ "MetagyrateDiminishedRhombicosidodecahedron", \ "ParabiaugmentedDodecahedron", "ParabiaugmentedHexagonalPrism", \ "ParabiaugmentedTruncatedDodecahedron", \ "ParabidiminishedRhombicosidodecahedron", \ "ParabigyrateRhombicosidodecahedron", \ "ParagyrateDiminishedRhombicosidodecahedron", "PentagonalCupola", \ "PentagonalGyrobicupola", "PentagonalGyrocupolarotunda", \ "PentagonalOrthobicupola", "PentagonalOrthobirotunda", \ "PentagonalOrthocupolarotunda", "PentagonalRotunda", {"Pyramid", 4}, {"Pyramid", 5}, "SnubDisphenoid", "SnubSquareAntiprism", "Sphenocorona", \ "Sphenomegacorona", "SquareCupola", "SquareGyrobicupola", \ "SquareOrthobicupola", "TriangularCupola", \ "TriangularHebesphenorotunda", "TriangularOrthobicupola", \ "TriaugmentedDodecahedron", "TriaugmentedHexagonalPrism", \ "TriaugmentedTriangularPrism", "TriaugmentedTruncatedDodecahedron", \ "TridiminishedIcosahedron", "TridiminishedRhombicosidodecahedron", \ "TrigyrateRhombicosidodecahedron"} (The spelling corrector just had a coronary.) Speaking of wiggle room, does anyone remember a discussion we had where I offered a bogus counterexample in the form of a fanfolded, Möbius-looped strip? Were we discussing flexible polyhedra? —rwg /b\ Bernie Cosell bernie(a)fantasyfarm.com -- Too many people; too few sheep --

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Re: [math-fun] Linear algebra over finite fields ?
by Henry Baker 23 Feb '19

23 Feb '19

Looks to be spot on! It's extremely interesting that one of the key elements is the *quadratic extension* of the base field. This makes some sense, as the singular values are *square roots* of the eigenvalues. Interesting: posted Jan 6, 2019 -- although first version was posted May, 2018. It's possible that the last time I searched for this particular combination was prior to May, 2018, so I would have missed this paper. At 05:27 PM 2/21/2019, Victor Miller wrote: >Look here: >https://arxiv.org/pdf/1805.06999.pdf > >On Thu, Feb 21, 2019 at 20:06 Henry Baker <hbaker1(a)pipeline.com> wrote: >> Hi Gene: >> >> Over the past several years, I've probably downloaded north of 1,000 >> papers on this subject, but I still haven't found any that address this >> direct question: >> >> "So you've heard all this cool stuff about matrix factorization >> (eigenvalue decomp, SVD, polar decomp, etc.) over the reals & complex >> numbers; what happens to all of this stuff when you move over to the Galois >> fields?" >> >> There are huge numbers of papers about aspects of linear algebra over >> Galois fields in coding theory, which is all interesting, but -- so far as >> I know -- these traditional matrix decomps aren't used in coding theory >> (are they?). >> >> At 04:27 PM 2/21/2019, Eugene Salamin via math-fun wrote: >> >Lots of Google hits for linear algebra over finite fields, as well as >> p-adic linear algebra.

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Re: [math-fun] Linear algebra over finite fields ?
by Dan Asimov 22 Feb '19

22 Feb '19

The paper that Victor linked to is on the singular value decomposition (SVD) for finite fields. One reason I like the SVD, at least over the reals, is its connection to the linear geometry of Euclidean space. Suppose P and Q are each a k-dimensional linear subspace of R^n, with k <= n. How can we describe the "angle" between P and Q ? We can ask for the unit vectors u in P and v in Q whose least nonnegative angle ang_1 = Ang(u,v) is minimized over all choices of unit vectors u and v. But wait, there's more. Now consider P_1 = u^perp and Q_1 = v^perp, the orthogonal complements of u in P and v in Q, and lather, rinse, repeat. This way we get a well-defined sequence of angles: 0 <= ang_1 <= ang_2 <= ... ang_k <= π/2 such that the vector (ang_1, ang_2, ..., ang_n) uniquely characterizes the pair of k-planes P and Q in R^n up to an isometry of R^n. That is, if P' and Q' have the same angle-vector as P and Q, then there's an orthogonal matrix J in the group O(n) such that J(P) = P' and J(Q) = Q'. A convenient way to calculate these angles is with the SVD. First choose any orthonormal bases {u_j} for P and {v_j} for Q, and now let the k x k matrix M be defined as M_ij = <u_i, v_j>, the dot product of u_i and v_j. The singular values SVD(M) of M are then the *cosines* of the angles ang_k. This is independent of the choices of orthonormal bases (very short proof). * * * So, if we can get real eigenvalues from matrices over finite fields F, then maybe it makes sense to talk about angles between subspaces of a vector space over F. —Dan Victor Miller écrit: ----- Look here: https://arxiv.org/pdf/1805.06999.pdf Henry Baker <hbaker1(a)pipeline.com> écrit: ----- ... > "So you've heard all this cool stuff about matrix factorization > (eigenvalue decomp, SVD, polar decomp, etc.) over the reals & complex > numbers; what happens to all of this stuff when you move over to the Galois > fields?" ... ----- -----

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Re: [math-fun] Linear algebra over finite fields ?
by Henry Baker 22 Feb '19

22 Feb '19

Re: p-adic Not at first. I'd like to see what analogies exist for GF(p) first, then GF(p^k). Only after these are understood would I go after p-adics. It's obvious that PDQ => M, where D diagonal, P,Q orthogonal can be constructed. So which square matrices M over GP(p) can be so factored? What works as a "square root" -- or more importantly, what works as a *square* to convert eigenvalues to singular values ? It is also easy to do the usual *iterations* over finite fields; the real (pun intended) question is: do these iterations do anything interesting and/or useful ? Continued fractions with square matrix "numbers" ? Are these useful for anything? Etc., etc. At 03:18 PM 2/21/2019, Fred Lunnon wrote: > Rather over p-adic fields, perhaps? WFL > >On 2/21/19, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Are there any tutorials or cookbooks on extending some of the usual >> linear algebra results to finite fields? >> >> I'm particularly interested in these algorithms on square matrices: >> * polar decomposition >> * eigenvalue decomposition >> * singular value decomposition >> * what happens when various classic *iterations* are performed over a finite >> field -- e.g., computing polar decomposition via iteration (does it even work?) >> * Are there any linear/convex programming ideas that carry over to finite fields? >> >> I'm also interested in a cookbook of various classical Newton-style >> iterations, but performed over the non-commutative rings of square matrices >> of real & complex #'s -- in the manner of computing the polar decomposition >> by Newton's square root iteration. Since these square matrices are in >> general non-commutative, there are a lot more choices about how to extend >> the commutative Newton iterations to the non-commutative cases. >> >> I suspect that relatively little work has be done on this last issue, >> because it's only been relatively recently that PC's have been powerful >> enough to do thousands/millions of iterations on entire matrix operations >> (as opposed to row ops or column ops).

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[math-fun] Near misses
by Bernie Cosell 22 Feb '19

22 Feb '19

<http://nautil.us/issue/69/patterns/the-impossible-mathematics-of-the-real-w…> /b\ Bernie Cosell bernie(a)fantasyfarm.com -- Too many people; too few sheep --

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[math-fun] Linear algebra over finite fields ?
by Henry Baker 21 Feb '19

21 Feb '19

Are there any tutorials or cookbooks on extending some of the usual linear algebra results to finite fields? I'm particularly interested in these algorithms on square matrices: * polar decomposition * eigenvalue decomposition * singular value decomposition * what happens when various classic *iterations* are performed over a finite field -- e.g., computing polar decomposition via iteration (does it even work?) * Are there any linear/convex programming ideas that carry over to finite fields? I'm also interested in a cookbook of various classical Newton-style iterations, but performed over the non-commutative rings of square matrices of real & complex #'s -- in the manner of computing the polar decomposition by Newton's square root iteration. Since these square matrices are in general non-commutative, there are a lot more choices about how to extend the commutative Newton iterations to the non-commutative cases. I suspect that relatively little work has be done on this last issue, because it's only been relatively recently that PC's have been powerful enough to do thousands/millions of iterations on entire matrix operations (as opposed to row ops or column ops).

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[math-fun] Question about gamma products
by James Propp 19 Feb '19

19 Feb '19

My former student Kyle Nelson has been playing around with the function f(p) = the product of 1/((sin n pi / p) ^ (n|p)) as n goes from 1 to (p-1)/2 (where p is 1 mod 4 and "(n|p)" is the Legendre symbol) along the lines of the formula that appears at the end of section 13 on page 15 of Barry Mazur's "Princeton Companion to Mathematics" essay "Algebraic Numbers" ( http://www.math.harvard.edu/~mazur/preprints/algebraic.numbers.April.30.pdf) and rediscovering some identities that appear to point to a larger picture that he'd like to know more about. He knows that this stuff isn't new, and has been told that it's related to L-functions for various algebraic number fields, but he'd like some more specific suggestions for what to read. He's been able to show that f(5) = (1 + sqrt(5))/2 f(13) = (3 + sqrt(13))/2 and f(29) = (5 + sqrt(29))/2 so he suspects that when p is of the form d^2 + 4, f(p) equals (d + sqrt(p))/2 (which has the nice continued fraction d + 1/(d + 1/(d + ...). He's also shown that f(41) = 32 + 5 sqrt(41). Anyone know the story that this is part of? Thanks, Jim Propp P.S. This originally came from looking at quotients of gamma functions evaluated at rational numbers, which can be seen by applying the reflection formula to 1 / sin.

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[math-fun] Taking the real and imaginary parts solves a(n admittedly reciprocal) sextic.
by Bill Gosper 19 Feb '19

19 Feb '19

4 ArcCos[r/(1 + r)] + ArcCos[1 - 2/(1 + r)^2] - 4 ArcCos[(1 - (-2 + r) r)/(1 + r)^2] - 4 ArcSec[1 + r] /. r -> Root[-1 - 2 #1 + 3 #1^2 + 8 #1^3 &, 1, 0]; Sign@% N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating 4 ArcCos[Root[-1+Times[<<2>>]+Times[<<2>>]+Times[<<2>>]&,1,0]/(1+Root[<<1>>&,1,0])]-4 ArcCos[(1-(-2+Root[<<3>>]) Root[<<1>>&,1,0])/(1+Root[<<1>>&,1,0])^2]+ArcCos[1-2/(1+Root[<<1>>&,1,0])^2]-4 ArcSec[1+Root[-1+Times[<<2>>]+Times[<<2>>]+Times[<<2>>]&,1,0]]. Translation: It's zero. (Note how it helpfully abbreviates Root[-1 - 2 #1 + 3 #1^2 + 8 #1^3 &, 1] to Root[-1+Times[<<2>>]+Times[<<2>>]+Times[<<2>>]&,1,0]].) After several days, FullSimplify@%% returned -((3 \[Pi])/2) - 4 ArcSin[Root[4 + 9 #1 + 2 #1^2 + #1^3 &, 1]] + I Log[Root[1 - 2048 #1 + 1460739 #1^2 + 2633728 #1^3 + 1460739 #1^4 - 2048 #1^5 + #1^6 &, 1]] But hey, look at this! In[116]:= %72 == {1, I}.RootReduce@ReIm@%72 Out[116]= Root[1 - 2048 # + 1460739 #^2 + 2633728 #^3 + 1460739 #^4 - 2048 #^5 + #^6& , 1, 0] == Root[329728 + 365184 # - 1024 #^2 + #^3& , 1, 0] + I Root[159103 + 365827 # - 643 #^2 + #^3& , 1, 0] —rwg Kids: If R is a root of a palindromic (= "reciprocal") polynomial, then so is 1/R. Why? Uh oh, Mathematica can't Solve[x^6 + x^5 + x + 1 == 5 x^3] (in radicals). Help it out! —Bill

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[math-fun] NYTimes: Getting Naked with Singularities
by Henry Baker 15 Feb '19

15 Feb '19

In case you missed it: https://www.nytimes.com/2019/02/14/science/math-algorithm-valentine.html The Perfect Valentine? A Math Formula Nothing says 'I love you' like a customizable algebraic equation. By Siobhan Roberts Feb. 14, 2019 Meet Suss, a math widget after your own heart. (You can also visit the widget on its website here, which you might want to do if you're reading this on a smartphone.) http://love.imaginary.org/ Like many geometric figures, a heart can be captured in all its curvaceous glory by a single algebraic equation. The equation for a sphere looks simple enough: x^2+y^2+z^2=1. A heart is something more complex: (x^2+((1+b)y)^2+z^2-1)^3 - x^2 z^3 - a y^2 z^3=0 Suss -- German for "sweet" -- is an interactive widget that allows you to tweak the algebra and customize the heart to your souls's delight. It was created for Valentine's Day by Imaginary, a nonprofit organization in Berlin that designs open-source mathematics programs and exhibitions. You can stretch and squeeze the heart by moving the two left-most sliders, which change the "a" and "b" parameters; the right-most slider zooms in and out. Better yet, canoodle directly with Suss's equation and engage in the dialectical interplay between algebra and geometry. (Change that final z^3 to a z^2 to see the heart in its underwear.) In the 17th century the French mathematician and philosopher Rene Descartes built a bridge between the algebraic and geometric realms when he devised the Cartesian system of coordinates. (He also classified six primitive passions: wonder and love, hatred and desire, sadness and joy.) Suss -- occupying three-dimensional space, defined by the coordinates x, y and z -- also incorporates a more recent concept: "extreme points," or "singularities," which are their own subject of study in the field of algebraic geometry. "If you look at the shape of the heart, you will see one peak on the bottom, and another peak on the top," said Andreas Daniel Matt, a mathematician and the director at Imaginary. (The peak at the top is indented.) In general, a singularity typically corresponds to a jump in an otherwise continuous process. In physics, it's a point with an infinite value. The Big Bang singularity is the biggest singularity in the history of the universe. In general relativity, a singularity is the heart of a black hole from which one would never emerge. A "naked singularity," by contrast, would be more forgiving (although it's not thought to actually exist in nature). "Roughly, a naked singularity is one you could fall into and you could escape from," said Roger Penrose, a mathematical physicist at the University of Oxford, who has an article on singularity theorems in the forthcoming book "Topology and Physics." https://www.worldscientific.com/worldscibooks/10.1142/11217 A singularity is where things go wrong, said Dr. Matt, as also happens in love: "The super high peaks, and the super low peaks. Love is full of extreme points." Mathematically, singularities are endowed with special properties, and it's these properties that cause problems. A singularity is fragile -- it can break with very small changes to the equation; formerly joined surfaces can separate. When singularities break, "they are very difficult to grasp, to study and especially to resolve," Dr. Matt said. To see a singularity in action, replace the equation in the Suss widget with a different one: x^2+y^2-z^2-a+0.5. (But use this slightly unsightly version -- x^2+y^2-z^2-a+0.5 -- so it works in the widget.) Then play with the left-most slider (parameter a), and notice the point where the double cone breaks apart, and where the cones rejoin. Before Suss was an interactive widget, it was a member of the Gallery of Singularities, a collection of algebraic surfaces created and curated in 2006 by Herwig Hauser, a geometer at the University of Vienna, and a master at resolving singularities. https://homepage.univie.ac.at/herwig.hauser/gallery.html Other gallery specimens include "Flirt," "Daisy" -- named for Donald Duck's girlfriend -- and "Solitude." (The word "singularity" comes from the Latin "singularis," meaning alone.) Granted, some of the entities in the gallery do not seem so singular -- "SphÃ¤re," for instance. But given the right mathematical techniques, Dr. Hauser said, he could squeeze that sphere down into a point. "Or I could sit on it," he added. The process of identifying intriguing algebraic surfaces that possessed singularities sometimes took Dr. Hauser and his students days. More often than not the results were incompatible: attractive equations gave rise to shapes with scant mathematical appeal. Dr. Hauser was matchmaking, essentially. "Sometimes you play completely on the algebraic side," he said. "You plug in an equation and discover what the geometric object looks like. And sometimes you do the opposite." The opposite -- reverse engineering, having a figure in mind and then finding the equation that generates it -- is much more difficult, almost impossibly so. Nonetheless, ten years ago Valentina Galata, then a high-school student, saw a poster of the Suss heart during a math-class excursion to an exhibit featuring Surfer, rendering software made by Imaginary that allows users to experiment in creating algebraic surfaces. Inspired, Galata created a collection of still lifes such as "Cherries" and "Spoon." https://www.behance.net/gallery/65537953/Re-editing-of-Surfer2008-images. She did not find the task impossible. "Challenging is a better word for it," said Ms. Galata, now a Ph.D. student in bioinformatics at Saarland University, in Germany. "It was fun to have a tool, using formulas to approximate shapes you have in your head, rather than just drawing on paper." Dr. Hauser, a purist, is not fond of rendering real-life objects. In fact, he is not particularly enamored of Suss. Too many people loved it too much: It became the poster heart for algebraic surfaces -- on thousands of actual posters in hundreds of Germans schools, and featured in an installation for a shopping-center car park. "It's just kitsch," he said. "Mathematically, it is not so interesting." Sebastian Gann, one of Dr. Hauser's former Ph.D. students on the Gallery of Singularities project, still finds Suss more appealing. "The intrinsic fascination remains," he said. "Looking at the image of an emotionally charged symbol, and knowing that the particular shape is defined by some sober maths. How do you perceive it?" Dr. Hauser prefers "Herz" (German for "heart"), which looks a bit more like an actual human heart: Herz is alluring for what's not there, he said: "How do you create a hole in a surface when you choose an equation?" One of the bigger questions within algebraic geometry is how to "resolve" singularities -- that is, how to get rid of them. Mathematically, resolving a singularity means smoothing over the problematic peaks in a surface, but this often requires jumping to a higher dimension. Alas, in resolving matters of the heart, that strategy seldom works.

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