January 2019 - mathfun@mailman.xmission.com

[math-fun] Fwd: Visualization puzzles - SPOILERS
by James Propp 26 Jan '19

26 Jan '19

Andy Latto wrote: On Fri, Jan 25, 2019 at 2:54 AM James Propp <jamespropp(a)gmail.com> wrote: > > Here are some visualization challenges. See if you can answer the questions > in your head without drawing anything (or if you must draw, draw in the air > rather than on paper). > > > Picture the usual tiling of the plane by unit squares, and picture a > shifted version of that tiling whose vertices all lie in the interiors of > the squares of the first tiling. How many vertices of tiling #2 lie in each > square cell of tiling #1? > > > Now try it with an infinite tiling of the plane by regular hexagons. > > > And now try it with an infinite tiling of the plane by equilateral > triangles. > > > Bonus question: Is there an easy way to see, in each case, what the AVERAGE > number of vertices-of-tiling-#2 per cell-of-tiling-#1 must be? Sure. Each square has 4 vertices, and each vertex is a vertex of 4 squares. So in a large area, the total number of vertices = number of squares * 4 / 4, so one per square. With a hex tiling, each hex has 6 vertices, and each is a vertex of 3 hexes, so total number of vertices = number of hexes * 6 / 3 = twice the number of hexagons. For triangles, it's number of triangles * 3/6, so one per two triangles. For visualizing the square case, I found it easy to visualize moving the vertices in one grid to the centers of the squares in the other grid. For hexes, imagine the hexes having horizontal sides, and translate the grid vertically a small amount, and two of the vertices of each hex move into that hex. For triangles, if you translate along a direction that makes an angle of 30 degrees with the x-axis, every right-side-up triangle contains a vertex. Andy

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[math-fun] Obituary for Michael Atiyah, 1929-2019
by Dan Asimov 25 Jan '19

25 Jan '19

The AMS has announced the death of Michael Atiyah: ----- January 11, 2019 Sir Michael Francis Atiyah, a renowned mathematician who received the Fields Medal in 1966 and the Abel Prize in 2004, which he shared with Isadore M. Singer, died January 11 at the age of 89. He did a great deal of significant work, including creating topological K-theory with Friedrich Hirzebruch and formulating the Atiyah-Singer index theorem with Singer. ----- More at <http://www.ams.org/news?news_id=4823>. —Dan

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[math-fun] Visualization puzzles
by James Propp 25 Jan '19

25 Jan '19

Here are some visualization challenges. See if you can answer the questions in your head without drawing anything (or if you must draw, draw in the air rather than on paper). Picture the usual tiling of the plane by unit squares, and picture a shifted version of that tiling whose vertices all lie in the interiors of the squares of the first tiling. How many vertices of tiling #2 lie in each square cell of tiling #1? Now try it with an infinite tiling of the plane by regular hexagons. And now try it with an infinite tiling of the plane by equilateral triangles. Bonus question: Is there an easy way to see, in each case, what the AVERAGE number of vertices-of-tiling-#2 per cell-of-tiling-#1 must be? (Warning: Do not drive, operate heavy machinery, or make love while attempting to solve these problems.) Jim Propp

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[math-fun] Foxtrot puzzle
by rcs＠xmission.com 25 Jan '19

25 Jan '19

There's a cute puzzle at https://www.foxtrot.com/2019/01/13/cell-division/ I get a unique answer. Perhaps the smallest such puzzle? Rich

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[math-fun] CF inconsequentia
by Bill Gosper 24 Jan '19

24 Jan '19

(Very good news, if no news is good news.) I once had a jugendtraum of showing the continued fraction of π to be aperiodic mod 2 by analyzing the process of converting one of its non-regular continued fractions to its regular one. I just looked at the simpler problem of converting 1+1/(2+2/(3+3/... to 1/(e-1). It appears that its rate of convergence is incommensurable with 2,6,10,14,. . . = coth(1/2). This should enable me to prove the geezertraum that e and e are algebraically independent. Furtherless, if x has CF = 0,1,4,9,16, ..., then (1+x)/(1-x) has quasiperiod 8, with 4 constant terms and 4 quadratics. (No linears.) —rwg

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[math-fun] Stupid error-correction coding question
by Henry Baker 24 Jan '19

24 Jan '19

We often use *linear equations* in coding theory, although the ring/field may vary significantly. If the linear code has no redundancy, the set of equations has exactly one solution, which presumably can be found via Gaussian elimination. If the linear code has redundancy, then the system of equations is overdetermined, and only has a solution if the code hasn't been corrupted. All of the cleverness of ECC codes boils down to how to determine the *maximum likelihood* solution. Now in a field of characteristic 0 -- e.g., R or C -- we have the method of *least squares* to find solutions to sets of overdetermined equations. Are any of the standard ECC methods for other fields -- e.g., various Galois fields -- *equivalent to a least squares* method of solution? I.e., if Ax=b is the overdetermined system, and A' is the Hermitian conjugate transpose of A, are the x's found by solving (A'A)x=(A'b) the same as those computed by any other ECC schemes? What if we "lift" the original equation set Ax=b from one field into another, larger field?

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[math-fun] Approximate factorial ( Gamma(z+1))
by Bill Gosper 22 Jan '19

22 Jan '19

For modest |z| and increasing k and n, E^((-z (-z/n)^k LerchPhi[-z/n, 1, 1 + k] + n Log[(n + z)/n])/n)/(1 + z/n) respectably approximates 1. For specific positive integers k, Mathematica finds that Product[E^((-z (-z/n)^k LerchPhi[-z/n, 1, 1 + k] + n Log[(n + z)/n])/n)/(1 + z/n), {n, m,∞}] is a multiple of z! E.g., for k=4, m=9, -1/2 < z < 1/2, 1.00000015 > z! E^(EulerGamma z + 1/8 z (-1 + z/16 - z^2/192 + z^3/2048 + (8 Log[1 + z/8])/z) + 1/7 z (-1 + z/14 - z^2/147 + z^3/1372 + (7 Log[1 + z/7])/z) + 1/6 z (-1 + z/12 - z^2/108 + z^3/864 + (6 Log[1 + z/6])/z) + 1/5 z (-1 + z/10 - z^2/75 + z^3/500 + (5 Log[1 + z/5])/z) + 1/4 z (-1 + z/8 - z^2/48 + z^3/256 + (4 Log[1 + z/4])/z) + 1/3 z (-1 + z/6 - z^2/27 + z^3/108 + (3 Log[1 + z/3])/z) + 1/2 z (-1 + z/4 - z^2/12 + z^3/32 + (2 Log[1 + z/2])/z) + z (-1 + z/2 - z^2/3 + z^3/4 + Log[1 + z]/z) - 1/360 z^2 (30 \[Pi]^2 + \[Pi]^4 z^2 - 120 z Zeta[3])) > 1 - .00000015. —rwg

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[math-fun] 2. Re: math-fun Digest, Vol 191, Issue 31 (Stuart Anderson)
by Stuart Anderson 21 Jan '19

21 Jan '19

to: math-fun-owner(a)mailman.xmission.com date: Jan 21, 2019, 12:44 PM subject: Sorry last email from me sent in error Please ignore the last email I sent to the group, it was sent in error. On Tue, Jan 22, 2019 at 6:00 AM <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: Perfect nonlinear functions n=2m (Fred Lunnon) > 2. Re: math-fun Digest, Vol 191, Issue 31 (Stuart Anderson) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Sun, 20 Jan 2019 21:14:43 +0000 > From: Fred Lunnon <fred.lunnon(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Perfect nonlinear functions n=2m > Message-ID: > <CAN57Yqs9yN9OyX-LreSgDZM5j-sZj9kbXFzC= > HUs57p1wiKxOg(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > Kaisa Nyberg (1991) on DES attacks is freely on available at > https://link.springer.com/content/pdf/10.1007/3-540-46416-6_32.pdf > > WFL > > > > On 1/20/19, Henry Baker <hbaker1(a)pipeline.com> wrote: > > In Nyberg's 1991 paper, she proves that a perfect > > nonlinear binary S-box must have an input dimension > > at least twice that of the output dimension. > > > > I'm trying to construct the most trivial versions > > of her theorem: i.e., n=2*m, m=1, 2, 3, etc. > > > > Has anyone here done such a thing? > > > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > ------------------------------ > > Message: 2 > Date: Mon, 21 Jan 2019 12:41:21 +1100 > From: Stuart Anderson <stuart.errol.anderson(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Cc: adam <rainbowflat(a)gmail.com> > Subject: Re: [math-fun] math-fun Digest, Vol 191, Issue 31 > Message-ID: > <CA+3-r9O=HO= > o-Ly_gQAQVi4fH0L-RiKfAstHDK36826Y8OSsdA(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > On Mon, Jan 21, 2019, 06:00 <math-fun-request(a)mailman.xmission.com wrote: > > > Send math-fun mailing list submissions to > > math-fun(a)mailman.xmission.com > > > > To subscribe or unsubscribe via the World Wide Web, visit > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > or, via email, send a message with subject or body 'help' to > > math-fun-request(a)mailman.xmission.com > > > > You can reach the person managing the list at > > math-fun-owner(a)mailman.xmission.com > > > > When replying, please edit your Subject line so it is more specific > > than "Re: Contents of math-fun digest..." > > > > > > Today's Topics: > > > > 1. Perfect nonlinear functions n=2m (Henry Baker) > > > > > > ---------------------------------------------------------------------- > > > > Message: 1 > > Date: Sun, 20 Jan 2019 09:25:05 -0800 > > From: Henry Baker <hbaker1(a)pipeline.com> > > To: math-fun(a)mailman.xmission.com > > Subject: [math-fun] Perfect nonlinear functions n=2m > > Message-ID: <E1glGqe-000DPR-UN(a)elasmtp-curtail.atl.sa.earthlink.net> > > Content-Type: text/plain; charset="us-ascii" > > > > In Nyberg's 1991 paper, she proves that a perfect > > nonlinear binary S-box must have an input dimension > > at least twice that of the output dimension. > > > > I'm trying to construct the most trivial versions > > of her theorem: i.e., n=2*m, m=1, 2, 3, etc. > > > > Has anyone here done such a thing? > > > > > > > > > > ------------------------------ > > > > Subject: Digest Footer > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > ------------------------------ > > > > End of math-fun Digest, Vol 191, Issue 31 > > ***************************************** > > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 191, Issue 32 > ***************************************** >

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[math-fun] e^(iπ)
by Bill Gosper 21 Jan '19

21 Jan '19

At the risk of a protracted and rancorous political dispute, I'd like an informal vote on which is most "beautiful": e^(iπ) = -1 or e^(iπ) + 1 = 0 ? —rwg I get polarized just thinking about it.

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Re: [math-fun] math-fun Digest, Vol 191, Issue 31
by Stuart Anderson 21 Jan '19

21 Jan '19

On Mon, Jan 21, 2019, 06:00 <math-fun-request(a)mailman.xmission.com wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Perfect nonlinear functions n=2m (Henry Baker) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Sun, 20 Jan 2019 09:25:05 -0800 > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Perfect nonlinear functions n=2m > Message-ID: <E1glGqe-000DPR-UN(a)elasmtp-curtail.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > In Nyberg's 1991 paper, she proves that a perfect > nonlinear binary S-box must have an input dimension > at least twice that of the output dimension. > > I'm trying to construct the most trivial versions > of her theorem: i.e., n=2*m, m=1, 2, 3, etc. > > Has anyone here done such a thing? > > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 191, Issue 31 > ***************************************** >

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