June 2019 - mathfun@mailman.xmission.com

Re: [math-fun] What is Gibbs?
by Bill Gosper 04 Jun '19

04 Jun '19

Derp! http://www.tweedledum.com/rwg/gibbs.htm Tnx. —rwg On Tue, Jun 4, 2019 at 8:20 AM Simon Plouffe <simon.plouffe(a)gmail.com> wrote: > > > Hello, > > Gibbs is Si(Pi), is the height of a final > square wave when you sum up fourier series with sin functions, > a thing that cannot be avoided. > https://en.wikipedia.org/wiki/Gibbs_phenomenon > > The number is 1.85193705198246617036105... equal to Si(Pi), > > > Best regards, > Simon Plouffe > > > Le 2019-06-04 à 17:16, Bill Gosper a écrit : > > *wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number= <http://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=>1.828106334019465&lookup_type=simple > Your value of 1828106334019465 would be here.* > 1828106414589225 = (0001) Zeta(1/2)^Gibbs/(Zeta(1/2)^sin(Pi/12)) > —rwg > > >

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[math-fun] What is Gibbs?
by Bill Gosper 04 Jun '19

04 Jun '19

*wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number= <http://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=>1.828106334019465&lookup_type=simpleYour value of 1828106334019465 would be here.* 1828106414589225 = (0001) Zeta(1/2)^Gibbs/(Zeta(1/2)^sin(Pi/12)) —rwg

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[math-fun] Fwd: How long did it take you to determine the rule of formation for this number triangle?
by Bill Gosper 02 Jun '19

02 Jun '19

On 2019-06-02 07:49, Neil Sloane wrote: > Go to the OEIS and enter the last two rows: > 5,8,9,8,5,6,10,12,12,10,6 > The response is instantaneous I.e., the tilted times table! T(n,k)=k(n-k). Surprising(?) occurrence: superHyperfactorial (https://oeis.org/A125760) Product[Hyperfactorial@k, {k, n - 1}] == Product[k^T[n,k], {k, n}] —rwg > > On Sun, Jun 2, 2019 at 10:46 AM Fred Lunnon <fred.lunnon(a)gmail.com> > wrote: > >> Approx. 5 secs. Do I get a chocolate frog? WFL >> >> >> On 6/2/19, Mike Speciner <ms(a)alum.mit.edu> wrote: >> > Too long. >> > >> > On 02-Jun-19 07:54, Bill Gosper wrote: >> >> 1 >> >> 2 2 >> >> 3 4 3 >> >> 4 6 6 4 >> >> 5 8 9 8 5 >> >> 6 10 12 12 10 6 >> >> . . . >> >> —rwg

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[math-fun] How long did it take you to determine the rule of formation for this number triangle?
by Bill Gosper 02 Jun '19

02 Jun '19

1 2 2 3 4 3 4 6 6 4 5 8 9 8 5 6 10 12 12 10 6 . . . —rwg

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Re: [math-fun] Mathematical flimflam
by Dan Asimov 02 Jun '19

02 Jun '19

1. There is a lovely dissection proof of the fact that a rectangle of area 65 can be dissected into one of area 64. Hence 1 = 0. 2. Also, Banach-Tarski showed that a unit ball B = {p in R^3 | ||p|| <= 1} can be dissected into 5 pieces that can be reassembled to comprise a partition of *two* unit balls. A likely story! Hence 1 = 0. 3. Furthermore, the vector field on the complex plane given by V(z) = i(z^3 - z) is holomorphic, so the fact that the flow {phi_t} of V satisfies that the times t for which phi_t(z) = z for all z form a discrete subgroup G_z of the reals. Also note that phi_t(z) is jointly holomorphic in both z and t. But it's not the same subgroup for all z (!) Which clearly contradicts the principle of permanence for holomorphic functions.= —Dan

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[math-fun] Mathematical flim-flam
by Bill Gosper 02 Jun '19

02 Jun '19

n 2019-06-01 18:08, Dan Asimov wrote: > 1. There is a lovely dissection proof of the fact that > a rectangle of area 65 can be dissected into one of > area 64. Hence 1 = 0. http://gosper.org/8x8=5x13.gif —rwg > > 2. Also, Banach-Tarski showed that a unit ball > B = {p in R^3 | ||p|| <= 1} can be dissected > into 5 pieces that can be reassembled to comprise > a partition of *two* unit balls. A likely story! > > Hence 1 = 0. > > 3. Furthermore, the vector field on the complex plane > given by > > V(z) = i(z^3 - z) > > is holomorphic, so the fact that the flow {phi_t} > of V satisfies that the times t for which > > phi_t(z) = z > > for all z form a discrete subgroup G_z of the reals. > > Also note that phi_t(z) is jointly holomorphic > in both z and t. > > But it's not the same subgroup for all z (!) Which > clearly contradicts the principle of permanence > for holomorphic functions.= > > —Dan

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[math-fun] Another Stirling's product
by Bill Gosper 01 Jun '19

01 Jun '19

https://www.wolframcloud.com/objects/d53b3672-659e-4984-a07d-4ae2003491e9 (misrendered, nonfatally). —rwg (I'll republish it if|when they fix the rendering.)

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[math-fun] Mathematical flimflam
by James Propp 01 Jun '19

01 Jun '19

As a teenager I loved "How to Lie With Statistics"; does it have any successors, especially ones that focus on non-statistical forms of mathematical flimflam? I'd also be interested in egregious and/or amusing examples of math abuse, and in particular, examples of misuse of Lagrange interpolation. We mathematicians all know that it should always be used for interpolation, never for extrapolation. But do practitioners of client disciplines know this? I hope to include some examples of mathematical flimflam in my June 17 Mathematical Enchantments essay. As always, all contributions will be gratefully acknowledged in the essay. Jim Propp

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[math-fun] Seven + ℤ special values of Hyperfactorial, BarnesG, Zeta'[-1,#], and Zeta'[2,#]
by Bill Gosper 01 Jun '19

01 Jun '19

{ (E^(-(1/9) + (-2 \[Pi]^2 + 3 PolyGamma[1, 1/3])/( 36 Sqrt[3] \[Pi])) Glaisher^(4/3))/3^(25/72) == Hyperfactorial[1/3], (2^(2/3) E^(-(1/9) + (2 \[Pi]^2 - 3 PolyGamma[1, 1/3])/( 36 Sqrt[3] \[Pi])) Glaisher^(4/3))/3^(49/72) == Hyperfactorial[2/3], (E^(-(3/32) + Catalan/(4 \[Pi])) Glaisher^(9/8))/Sqrt[2] == Hyperfactorial[1/4], (3^(3/4) E^(-(3/32) - Catalan/(4 \[Pi])) Glaisher^(9/8))/(2 Sqrt[2]) == Hyperfactorial[3/4], (E^(-((\[Pi] (5 + 2 Sqrt[3] \[Pi]) - 3 Sqrt[3] PolyGamma[1, 1/3])/ (72 \[Pi]))) Glaisher^(5/6))/(2^(11/72) 3^(23/144)) == Hyperfactorial[1/6], (E^(-((\[Pi] (5 - 2 Sqrt[3] \[Pi]) + 3 Sqrt[3] PolyGamma[1, 1/3])/( 72 \[Pi]))) (5 Glaisher)^(5/6))/(2^(59/72) 3^(119/144)) == Hyperfactorial[5/6]} But we have Glaisher == E^(1/12 - Zeta'[-1]) and Zeta'[-1, 1/2] == - Log[2]/24 -1/2 Zeta'[-1] and Log[Hyperfactorial[z]] == z Log[z] - Zeta'[-1] + Zeta'[-1, z] and D[Zeta[s, a_Integer] == 2^s \[Pi]^(-1 + s) (-s)! Sin[1/2 \[Pi] (4 a + s)] Zeta[1 - s],s] and BarnesG[1 + n] n!^-n Hyperfactorial[n] ==1 I betcha Adamchik left WRI all this stuff. —Bill It's weird that we have "closed forms" for Hyperfactorial[fraction] for seven different fractions and Factorial[fraction] for only ½.

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[math-fun] 170
by James Propp 01 Jun '19

01 Jun '19

I’ve agreed to host Carnival of Mathematics #170 next month, and I’m wondering whether the number 170 has any especially nice properties that might deserve mention. The number 170 has its own Wikipedia page, but none of the mathematical properties listed there appealed to me. I might say a little about writing numbers as sums of two squares, but I’d prefer something a little more exciting. Jim Propp

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