September 2017 - mathfun@mailman.xmission.com

[math-fun] 1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + ... yet again
by Bill Gosper 16 Sep '17

16 Sep '17

First of all, an erratulum on p66 of my old series acceleration paper https://dspace.mit.edu/handle/1721.1/6088 should read Sum[1/Fibonacci[n], {n, ∞}] == Sum[(I^((k - 1)k) ((-1)^k*Fibonacci[2k + 2] + Fibonacci[4k + 3]))/ (Fibonacci[2k + 1]*Fibonacci[2k + 2]*Product[LucasL[2j + 1], {j, k}]), {k, 0, ∞}] (with 𝜗-like convergence and a nice matrix product). Secondly, the Lambert speedup I offered Jörg below, when cleaned up, is nifty: Sum[b^n/(1 - a*q^n), {n, 0, ∞}] == Sum[(a^k*b^k*q^k^2*(1 - a*b*q^(2*k)))/((1 - a*q^k)*(1 - b*q^k)), {k, 0, ∞}] I.e., speeding it up reveals it to be symmetric in a and b! Thus Sum[1/Fibonacci@n, {n,∞}] == Sqrt[5]*Sum[1/((-1)^n 𝜙^(1 + 2n) - 1), {n, 0,∞}] ~ 3.359885666243177 . Unfortunately, this converts very clumsily to QPolyGammas, yeilding a horrendous 0 == 1/4 Sqrt[5] (-EllipticTheta[2, 0, 1/GoldenRatio^2]^2 - (1/( 2 ArcCsch[2]))(Log[5] + 2 QPolyGamma[0, 1, 1/GoldenRatio^4] - 4 QPolyGamma[0, 1, 1/GoldenRatio^2]) + ( 1/(\[Pi] - 2 I ArcCsch[2]))(-6 \[Pi] + 4 I (Log[1 + GoldenRatio^2] + QPolyGamma[0, 1/(2 + (I \[Pi])/ArcCsch[2]), -GoldenRatio^2]))) Such results are nevertheless useful in captchas. "To prove you're not a computer, what is the value of 1/4 Sqrt[5] (-EllipticTheta[2, 0, 1/GoldenRatio^2]^2 - ..."? An answer of 0 betrays the applicant to be a dangerous AI. --rwg On Tue, Aug 25, 2015 at 7:05 AM, rwg <rwg(a)sdf.org> wrote: > On 2015-08-25 00:59, Joerg Arndt wrote: > > * rwg <rwg(a)sdf.org> [Aug 25. 2015 08:14]: > > > [...] > > Ahh, here's our problem-- > I do not accept Lambert series as closed form! > > > But why is then Theta_2 acceptable? > > 'Cause Whittaker & Watson legitimize it. > > (no idea what these QPolyGamma thingies are) > > > In[451]:= D[Log[QGamma[x, q]], x] > > Out[451]= QPolyGamma[0, x, q] > > > > To do without them, > http://gosper.org/recipfib.pdf [2 <http://gosper.org/recipfib.pdf>] > > In[147]:= D[Log[QPochhammer[q, q]], q] > > Out[147]= (-((QPochhammer[q, q]*(Log[1 - q] + QPolyGamma[0, 1, > q]))/(q*Log[q])) + > Derivative[0, 1][QPochhammer][q, q])/ > QPochhammer[q, q] > > Sum[1/Fibonacci[n], {n, Infinity}] == > (1/4)*Sqrt[5]* > ((-4*QPolyGamma[0, 1, 1/GoldenRatio^2] + > 2*QPolyGamma[0, 1, 1/GoldenRatio^4] + > Log[5])/(2*Log[GoldenRatio]) + > EllipticTheta[2, 0, 1/GoldenRatio^2]^2) > > N[List @@ %, 49] > > {3.359885666243177553172011302918927179688905133732, > 3.359885666243177553172011302918927179688905133732} > > BtW, there's a very simple path-invariant q matrix system > for converting generalized Lamberts to theta convergence. > > > Let's us see! > > http://gosper.org/lambser.png > --rwg > > > --rwg > > Links: > ------ > [1] http://arxiv.org/abs/1202.6525 > [2] http://gosper.org/recipfib.pdf > _______________________________________________ > >

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[math-fun] Prove or disprove ∑𝛾 = 0?
by Bill Gosper 15 Sep '17

15 Sep '17

Out[542]= Sum[((-1)^n*Floor[(EulerGamma + Log[n])/Log[2]])/n, {n,∞}] == 0 (%544 = %542 with terms grouped pairwise, comforting NSum.) Testing: In[547]:= tim[Out[544][[1]] /. Sum[a_, b_] :> NSum[a, b, NSumTerms -> 9999]] During evaluation of In[547]:= NIntegrate::ncvb (NSum remains uncomfortable.) During evaluation of In[547]:= 212.330211,0 Out[547]= -0.000047302360632835656 In[550]:= tim[Out[544][[1]] /. Sum[a_, b_] :> NSum[a, b, NSumTerms -> 9999999]] During evaluation of In[550]:= NIntegrate::ncvb During evaluation of In[550]:= 5.096415,0 Out[550]= -5.549175287798655*^-9 (212 sec for 9999 terms, 5 sec for 10^7 terms.) The 1st 16 terms: In[553]:= %542[[1]]/.{∞->16,Sum->Table} Out[553]= {0,1/2,-2/3,1/2,-3/5,1/2,-3/7,3/8,-4/9,2/5,-4/11,1/3,-4/13,2/7,-4/15,1/4} Repeatedly adding pairwise. In[556]:= Plus@@#&/@Partition[%553,2] Out[556]= {1/2,-1/6,-1/10,-3/56,-2/45,-1/33,-2/91,-1/60} In[557]:= Plus@@#&/@Partition[%,2] Out[557]= {1/3,-43/280,-37/495,-211/5460} In[558]:= Plus@@#&/@Partition[%,2] Out[558]= {151/840,-20431/180180} In[559]:= Plus@@#&/@Partition[%,2] Out[559]= {23917/360360} --rwg

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[math-fun] Draft of my September 2017 blog post
by James Propp 13 Sep '17

13 Sep '17

Hi, I started writing a new draft titled "How Do You Write One Hundred in Base 3/2?" and would love to get your feedback. I plan on publishing it on the 17th. One reader of last month's essay suggested that I create hyperlinks from the main text to the Endnotes. Great suggestion! (I plan to implement it between now and Sunday.) More suggestions of this kind would be welcome, along with comments on the substance of the article, stylistic suggestions, typographical corrections, etc. Please leave your feedback here: https://mathenchant.wordpress.com?p=1835&shareadraft=59b9a33fe5bf1 Title: How Do You Write One Hundred in Base 3/2? Beginning: You and your computer have a fundamental disagreement about how to represent numbers. Your computer was designed to calculate in base two (binary), while you were brought up speaking and writing in base ten (decimal). But there's something that your decimal self and your binary computer (or its desi... Read more: https://mathenchant.wordpress.com?p=1835&shareadraft= 59b9a33fe5bf1 Thanks, jamespropp

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[math-fun] Reversible computation
by Fred Lunnon 12 Sep '17

12 Sep '17

This idea has been around since I first encountered computers --- maybe not as exotic as "quantum compution", but appears to this casual sceptic a good deal more soundly based. Wot, me worry? I fully expect to be pushing up daisies long before either notion becomes at all practicable! Mind you, I once said the same about climate change ... https://spectrum.ieee.org/computing/hardware/the-future-of-computing-depend… WFL

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Re: [math-fun] 50,847,478
by Henry Baker 12 Sep '17

12 Sep '17

= 2*11*2311249. ??? At 03:53 PM 8/29/2017, Tomas Rokicki wrote: >-tom > >-- >-- http://cube20.org/ -- http://golly.sf.net/ --

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Re: [math-fun] classes of sq matrices closed under matrix multiply ?
by Henry Baker 10 Sep '17

10 Sep '17

Thanks, Edwin! It would appear that if matrix multiplication is closed in a set S of square matrices, then (M^^-1).S.M = S' is another closed set of square matrices, where M is any invertible matrix whose elements come from an extension ring of the element ring of S. Indeed, if S1, S2 are elements of S, then ((M^^-1).S1.M).((M^^-1).S2.M) = (M^^-1).S1.(M.(M^^-1)).S2.M = (M^^-1).(S1.S2).M Thus, S1,S2 should be considered equivalent. This means that there are a number of classes of matrices that are "equivalent" to upper triangular matrices, even though they aren't upper triangular themselves; ditto for other classes. "Upper triangular" -- by itself -- doesn't completely specify a unique representative of the class; other restrictions are needed. At 09:08 AM 9/10/2017, W. Edwin Clark wrote: >These "Pattern algebras" or "incidence algebras" as I recall can be constructed >by taking any finite poset (P,<=) and then considering the algebra spanned by the matrix >units e_{i,j} where i <= j Barry Mitchell (of category fame) also studied these >in a slightly more general setting. He called this homological tic tac toe since he >was interested (as was I at one time) in relating the "global dimension" of such >an algebra to properties of the corresponding poset. > >If you play around a bit with matrix units semigroups (that is set of matrix units, >e_{i,j} that are closed under multiplication) you will quickly convince yourself of >the plethora of such "pattern algebra". > >Back to Victor's suggestion concerning "algebraic" groups much work has also been >done on "algebraic" semigroups (algebraic varieties of matrices closed >under multip.)--mostly as I recall by Mohan Putcha. Before him I >played around with "affine semigroups" which are linear varieties of >matrices closed under multiplication. > >Of course most "classical" groups are most easily described as sets of matrices >that are closed under multiplication (and inversion). > >Note also that any finite semigroup can be faithfully represented as a semigroup >of matrices--as can every finite group. Generally restricting semigroups to semigroups >of matrices doesn't buy very much without further hypotheses. > >On 9/7/17, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Thanks, Victor! >> >> The article looks quite interesting, but also quite tough sledding... >> >> At the topmost level, are there any classes of matrices other than >> upper triangular, lower triangular, both/diagonal, and/or block >> versions of same, that preserve the overall pattern of zeros? >> >> From a directed graph adjacency matrix perspective, triangular >> matrices are partial orders forced into a linear order; block >> matrices have subsets of vertices with cyclic orders (equivalence >> classes). >> >> It seems to me that these might exhaust the classes of matrices >> with patterns of zeros. >> >> At 06:25 AM 9/7/2017, Victor Miller wrote: >>>Henry, This may be more than you bargained for, but this is a well-studied >>>problem. To make your question more precise, you can ask for algebraic >>>subgroups of matrices -- i.e. subsets of matrices that are the set of >>>solutions to a bunch of polynomials in their coordinates (think >>>determinant = 1, for example). These polynomials are said to "cut out" >>>the group, The following paper surveys the classification of such: >>>https://www.dpmms.cam.ac.uk/~nd332/alg_gps.pdf >>> >>>Besides the ones that you mentioned, you can also have the set of matrices >>>that preserve a quadratic form: x --> x^T Q x. So you'd want all matrices, >>>A, such that A^T Q A = Q. >>>If A is the identity you get the orthogonal group, If A has dimension 2n >>>and has n +1, and n -1 on the diagonal and 0 everywhere else, you get the >>>Symplectic group. >>> >>>The nice thing about algebraic groups, is that you can restrict their >>>coordinates to lie in a field which is a subfield of the coefficients used >>>to define the polynomials that cut out the group, you get a subgroup. >>> >>>Victor >>> >>>On Wed, Sep 6, 2017 at 9:12 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>> >>>> Is there an encyclopedia/catalog somewhere of classes of square matrices >>>> closed under matrix multiply? >>>> >>>> I'm interested in generic classes -- e.g., things that work for arbitrary >>>> n, e.g., diagonal matrices, orthogonal matrices, upper triangular, lower >>>> triangular, etc.

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Re: [math-fun] classes of sq matrices closed under matrix multiply ?
by Henry Baker 10 Sep '17

10 Sep '17

Thanks, Victor! The article looks quite interesting, but also quite tough sledding... At the topmost level, are there any classes of matrices other than upper triangular, lower triangular, both/diagonal, and/or block versions of same, that preserve the overall pattern of zeros? From a directed graph adjacency matrix perspective, triangular matrices are partial orders forced into a linear order; block matrices have subsets of vertices with cyclic orders (equivalence classes). It seems to me that these might exhaust the classes of matrices with patterns of zeros. At 06:25 AM 9/7/2017, Victor Miller wrote: >Henry, This may be more than you bargained for, but this is a well-studied >problem. To make your question more precise, you can ask for algebraic >subgroups of matrices -- i.e. subsets of matrices that are the set of >solutions to a bunch of polynomials in their coordinates (think >determinant = 1, for example). These polynomials are said to "cut out" >the group, The following paper surveys the classification of such: >https://www.dpmms.cam.ac.uk/~nd332/alg_gps.pdf > >Besides the ones that you mentioned, you can also have the set of matrices >that preserve a quadratic form: x --> x^T Q x. So you'd want all matrices, >A, such that A^T Q A = Q. >If A is the identity you get the orthogonal group, If A has dimension 2n >and has n +1, and n -1 on the diagonal and 0 everywhere else, you get the >Symplectic group. > >The nice thing about algebraic groups, is that you can restrict their >coordinates to lie in a field which is a subfield of the coefficients used >to define the polynomials that cut out the group, you get a subgroup. > >Victor > >On Wed, Sep 6, 2017 at 9:12 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Is there an encyclopedia/catalog somewhere of classes of square matrices >> closed under matrix multiply? >> >> I'm interested in generic classes -- e.g., things that work for arbitrary >> n, e.g., diagonal matrices, orthogonal matrices, upper triangular, lower >> triangular, etc.

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[math-fun] classes of sq matrices closed under matrix multiply ?
by Henry Baker 07 Sep '17

07 Sep '17

Is there an encyclopedia/catalog somewhere of classes of square matrices closed under matrix multiply? I'm interested in generic classes -- e.g., things that work for arbitrary n, e.g., diagonal matrices, orthogonal matrices, upper triangular, lower triangular, etc.

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[math-fun] Cool things I forgot
by Henry Baker 07 Sep '17

07 Sep '17

I had forgotten that Vandermonde matrices diagonalize Companion matrices. (Probably because of all the barnacles that came with undergrad linear algebra.) Oh, and determinant(Vandermonde) = sqrt(discriminant of polynomial) (sort of ... modulo signs) Yet another way to see that solving the polynomial requires a sqrt. (%i1) C:polytocompanion((x-r1)*(x-r2)*(x-r3),x),factor; [ 0 0 r1 r2 r3 ] [ ] (%o1) [ 1 0 - (r2 r3 + r1 r3 + r1 r2) ] [ ] [ 0 1 r3 + r2 + r1 ] (%i2) V:vandermonde_matrix([r1,r2,r3]); [ 2 ] [ 1 r1 r1 ] [ ] (%o2) [ 2 ] [ 1 r2 r2 ] [ ] [ 2 ] [ 1 r3 r3 ] (%i3) D:diag_matrix(r1,r2,r3); [ r1 0 0 ] [ ] (%o3) [ 0 r2 0 ] [ ] [ 0 0 r3 ] (%i4) (V^^-1).D.V,factor; [ 0 0 r1 r2 r3 ] [ ] (%o4) [ 1 0 - (r2 r3 + r1 r3 + r1 r2) ] [ ] [ 0 1 r3 + r2 + r1 ] (%i5) %-C,expand; [ 0 0 0 ] [ ] (%o5) [ 0 0 0 ] [ ] [ 0 0 0 ] (%i6)

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Re: [math-fun] "quadratic residue" division ??
by Henry Baker 06 Sep '17

06 Sep '17

FWIW, this wikipedia article makes the same claim for p=4k+3, which uses the term "modulus" instead of "representative": https://en.wikipedia.org/wiki/Trigonometry_in_finite_fields The "modulus" of an element a+jb in {a+jb; a,b in GF(q); j^2=-1}, where p=4k+3, is a+jb = |a^2+b^2| = +sqrt(a^2+b^2). At 08:28 AM 8/26/2017, Henry Baker wrote: >How about this weird form of division (in maxima): > >qdivide(m,n):=block([q,r,q1,r1], q:floor(m/n), r:m-q*n, > /* r is non-negative */ > if is(jacobi(r,n)=-1) then (q1:q+1,r1:r-n) > else (q1:q,r1:r), > [q1,r1]); > >e.g., > >(%i25) qdivide(1,19); >(%o25) [0, 1] >(%i26) qdivide(2,19); >(%o26) [1, - 17] >(%i27) qdivide(3,19); >(%o27) [1, - 16] >(%i28) qdivide(4,19); >(%o28) [0, 4] >(%i29) qdivide(5,19); >(%o29) [0, 5] >(%i30) qdivide(6,19); >(%o30) [0, 6] >(%i31) qdivide(7,19); >(%o31) [0, 7] >(%i32) qdivide(8,19); >(%o32) [1, - 11] >(%i33) qdivide(9,19); >(%o33) [0, 9] >(%i34) qdivide(10,19); >(%o34) [1, - 9] >(%i35) qdivide(11,19); >(%o35) [0, 11] >(%i36) qdivide(12,19); >(%o36) [1, - 7] >(%i37) qdivide(13,19); >(%o37) [1, - 6] >(%i38) qdivide(14,19); >(%o38) [1, - 5] >(%i39) qdivide(15,19); >(%o39) [1, - 4] >(%i40) qdivide(16,19); >(%o40) [0, 16] >(%i41) qdivide(17,19); >(%o41) [0, 17] >(%i42) qdivide(18,19); >(%o42) [1, - 1] >(%i43) qdivide(19,19); >(%o43) [1, 0] > >If we utilize this form of division for base conversion to >a prime base p=3(mod 4), we automatically get a representation >whose digits are +-r, where r is a quadratic residue (mod p). > >One obvious property: perfect +- symmetry: negation means >negating every digit. > >Questions: > >Does anything interesting happen when this form of division is >used in other places -- e.g., gcd algorithms? > >Does this type of division induce a slightly different "norm" >s.t. the norm of the remainder is strictly < the norm of the >divisor? > >I suspect that there is a close relationship with division in >the Gaussian integers, but I haven't checked this yet.

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