December 2019 - mathfun@mailman.xmission.com

[math-fun] Raspberry Pi and the disappearing/reappearing Mathematica
by Hilarie Orman 08 Dec '19

08 Dec '19

>From what I've been able to determine, between July and September, there was no Mathematica included in the Raspberry Pi distribution. The latest version does have Mathematica. Hilarie

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[math-fun] Rasperry Pi 4 has no Mathematica
by Hilarie Orman 08 Dec '19

08 Dec '19

Until a few months ago, Raspberry Pis came with Mathematica already installed. That is no longer true. Don't buy a Pi with Mathematica in your eye. Hilarie

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Re: [math-fun] Blankets
by Henry Baker 08 Dec '19

08 Dec '19

Depends upon what the blanket is made of. E.g., I don't think that an ordinary wool blanket would commute with one of the silvery hi-tech blankets that first responders use. Those silvery blankets *reflect* (essentially all normal) wavelengths, while a wool blanket depends upon a quiet thickness of (hopefully dry) air; the wool is merely used to keep the thickness and reduce air movement (convection). I noticed that my down sleeping bag doesn't protect me from the cold ground -- I need a separate hard foam that keeps me a certain distance from the ground. On the other hand, those silvery first responder blankets also trap moisture, so you could end up with the inner surface covered with ice -- like uninsulated windows on a winter's day. At 07:19 AM 12/5/2019, James Propp wrote: >Do blankets commute? > >Not esthetically (of course), but thermally? > >Jim Propp

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[math-fun] Blankets
by James Propp 06 Dec '19

06 Dec '19

Do blankets commute? Not esthetically (of course), but thermally? Jim Propp

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[math-fun] Oh no! More Syzygies!
by Brad Klee 05 Dec '19

05 Dec '19

Previously I claimed that the null ring of NxN matrix invariants could be constructed by observing unitary transformations of the coefficients of the characteristic equation. This is an okay method, but not the only approach. Specht's theorem [1] provides an alternative using only traces of products of a square matrix A and it's complex conjugate A*. If we assume A*=A, then the only invariants are Tr(A^n) and should be finitely generated by T(A), T(A^2), T(A^3) up to a cutoff T(A^N), which depends on the dimension of A. For dim n=2, choose diagonal A=[L1,L2], then Tr(A)=L1+L2, and Tr(A^2)=L1^2+L2^2. Alternatively, det(A-Id2*L) = L^2 - Tr(A)*L+det(A). The syzygy is that det(A) = 1/2*(Tr(A)^2-Tr(A^2)). Similarly there are syzygies for dim(A)=3 to convert Tr(A), Tr(A^2), Tr(A^3) into the coefficients of the characteristic equation. Thus we can write a set of generators for the null ring (Identities: 0 + g = g, 0*g=g, g in NR ): NR={(g1+g2)*g3: g1,g2,g3 all in NR} U{ 0, Tr(A)-Tr(B), Tr(A^2)-Tr(B^2), Tr(A^3)-Tr(B^3) } with B unitarily equivalent to diagonal A. To solve the problem of changing symbols within R=K[[b_{i,j}, L1, L2, L3]] we look for identities of the form p2 = p1 + p3*pn, with p1,p1,p3 in R and pn in NR, and say p1=p2 mod 0. Let us not forget to cite Carl Pearcy. He also did some work on 3x3 matrices, and published in transactions of the AMS , 1962 on "A complete set of unitary invariants for 3x3 complex matrices" [2]. I want to reiterate again, or to re-reiterate--In my opinion Tao et al. did not cite the Sprecht theorem of 1940 and other works thereafter in an attempt to make their work seem more original and more "unexpected". --Brad https://en.wikipedia.org/wiki/Specht%27s_theorem https://www.ams.org/journals/tran/1962-104-03/S0002-9947-1962-0144911-4/

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[math-fun] Terence Tao's Blog: The Missing Proof.
by Brad Klee 04 Dec '19

04 Dec '19

https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/ Here is the proof I wrote, in Mathematica: ZS[n_] := M[n, n] (M[1, 1] + M[2, 2] + M[3, 3]) - M[n, n] (L1 + L2 + L3) ZS[n_, Li_] := Li (-M[n, n] - t @@ Complement[Range[3], {n}] + L1 + L2 + L3) ZS0 = Plus[M[1, 1] M[2, 2] - M[1, 2] M[2, 1], M[1, 1] M[3, 3] - M[1, 3] M[3, 1], M[2, 2] M[3, 3] - M[2, 3] M[3, 2], -(L1 L2 + L1 L3 + L2 L3)]; ZSComp[n_, Li_] := Subtract[ZS[n], ZS[n, Li] + ZS0] evsq[Li_] := With[{Ann = Complement[{L1, L2, L3}, {Li}]}, Part[Dot @@ ((Array[M, {3, 3}] - # IdentityMatrix[3]) & /@ Ann), #, #] & /@ Range[3]] Expand[evsq[L1] - (ZSComp[#, L1] & /@ Range[3])] // MatrixForm Expand[evsq[L2] - (ZSComp[#, L2] & /@ Range[3])] // MatrixForm Expand[evsq[L3] - (ZSComp[#, L3] & /@ Range[3])] // MatrixForm The so-called "new identity" is just the projector formula modulo a few zero-sums. The zero-sums are well known in the theory of matrix invariants, see for example: https://en.wikipedia.org/wiki/Invariants_of_tensors#Principal_invariants --Brad

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[math-fun] crab megastep
by Bill Gosper 04 Dec '19

04 Dec '19

2²⁰ steps of one of the six regions of a semihex grazing the point 1+i√3/2 (which I mistook for a vertex), crab megastep <http://gosper.org/crabmega.png>. This chain is the (closed loop) image of the (short) interval [13/40, 53/160]. Both endpoints map continuously to the tip of its tail. The other four tail tip preimages are 21/80, 43/160, 77/160, and 39/80. (Only three of the grazing regions have this shape.) I can still only find sextuple points by brute force guessing, until Julian mercifully creates his inversesemihex magic. Only the vertices of the texture represent visited points. The line segments just indicate order of visitation. But the "curve" solidly fills in the *all* the points in there, a dense subset of them six times over, where ever-smaller crab tails congregate. Such are the fairy tales logically coerced by belief in the continuum. —rwg

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[math-fun] Brain hemisphere spacefills likely with sextuple points.
by Bill Gosper 02 Dec '19

02 Dec '19

Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.) Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10] Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg

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[math-fun] Did René Descartes say "Cogito, ergo sum"?
by Bill Gosper 02 Dec '19

02 Dec '19

Wikipedia: *Cogito, ergo sum*[a] <https://en.wikipedia.org/wiki/Cogito,_ergo_sum#cite_note-capitalization_and…> is a Latin <https://en.wikipedia.org/wiki/Latin> philosophical <https://en.wikipedia.org/wiki/Philosophy> proposition <https://en.wikipedia.org/wiki/Proposition> by René Descartes <https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes> usually translated into English as "*I think, therefore I am*".[b] <https://en.wikipedia.org/wiki/Cogito,_ergo_sum#cite_note-better_translation…> The phrase originally appeared in French <https://en.wikipedia.org/wiki/French_language> as *je pense**, donc je suis* in his *Discourse on the Method <https://en.wikipedia.org/wiki/Discourse_on_the_Method>*, so as to reach a wider audience than Latin would have allowed. I heard that the real reason was that Descartes was lousy at Latin and paid someone for the translation! Using Google Translate for the French pronunciation, I am baffled by the sound of mysterious consonants in "je suis". ACW, Simon, is that authentic? How is it written phonetically? I have no confidence in Google Translate's pronunciations after hearing Joe, The Pizza Guy say "Cogito, ergo sum" in "Latin". —rwg

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[math-fun] History: A memoir on the theory of matrices (1858).
by Brad Klee 01 Dec '19

01 Dec '19

Arthur Cayley (1821-1895) was a pioneer of invariant theory and among the first to work out the details of the characteristic equation for 3x3 matrices in general. The Cayley-Hamilton theorem is given on page 24 of "A memoir on the theory of matrices" [1], and quickly followed by the great sensibility of an old-style mathematician: "I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree." ( though nowadays we automate the next few cases on a computer ) Coefficients of the characteristic equation are homogeneous polynomials in the matrix elements, and may be obtained by summing over matrix minor determinants. The characteristic equation itself holds for any similar matrix, thus we may construct a zero sum for every coefficient by subtracting the most general polynomial from a polynomial in a set of distinct variables, with all off-diagonal symbols set to zero. If the dimension of the matrix is N, then we obtain N distinct polynomials z_i, such that z_i = 0. Over field K, the null ring RZ=K[[z_1,z_2,...,z_N]] is a subring of R = K[[a,b,...,L1,L2...,L_N]] with lowercase alphabet for generic matrix elements and L_i for eigenvalues. Two expressions p1 and p2 from R are equivalent whenever a third expression p3 from RZ exists such that p1 + p3 = p2. Without the normalizing factors in the denominator, the squared elements of the eigenvectors are obtained from the diagonal of projector matrices, also guaranteed to exist as corollary to the Cayley-Hamilton theorem. The real problem with Terence Tao's proposal to obtain "Eigenvectors from Eigenvalues" is that it engages in revisionist history. The authors expect to find proof an old, fundamental results in a post-Hilbert language that didn't exist during the days of Cayley and Hamilton, when the question of finding "Eigenvectors from Eigenvalues" was originally settled. Were the old masters alive today, we think that they would be more impressed with the performance of modern computer algebra systems, which easily find and prove identities within well-defined polynomial rings. Some applause is deserved by subsequent authors for finding particularly nice polynomials in R; however, the amount of success needs to be gauged relative to other theorems in the field. In our perspective, a "theorem" of the sort "p1 + p3 = p2", is not really that great compared to the discovery of the null ring KZ and the original proof of the Cayley-Hamilton theorem. In fact, infinitely many equivalent functions p2 are guaranteed to exist as a corollary. In our opinion, that is how they should be viewed. All the preceding text was written as notes for an article titled "Inside the quantum bikeshed", which planned to discuss Parkinson's law of triviality, as it often manifests in the sciences. However, Predrag C. convinced me that it would be rude and inconsiderate to write such an article. Counterpoint: If instead, we were living in a world of science fiction, and suffering from hard-scrabble existence after the fall of the empire, of course we would : * * demand the return of the Beskar Iron to its rightful owners! * * --Brad [1] https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1858.0002

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