January 2020 - mathfun@mailman.xmission.com

[math-fun] (Log[2]-5/8) = 0.06814718 . . .
by Brad Klee 24 Jan '20

24 Jan '20

For a horrible approximation of Pi/2, and a minor headache along the way: Ann2 ={1, -4 x, 4 (1 - x) x}; ERev1 = Hypergeometric2F1[1/2, 3/2, 2, x] x; ERev2 = ERev1*Integrate[ Series[Exp[-Integrate[Ann2[[2]]/Ann2[[3]], x] ]/ERev1^2, {x, 0, 100}], x]; Limit[D[EllipticE[1 - x] - ERev2, x], x -> 0] N[Normal[ERev2 + %*ERev1 - Pi/2] /. x -> 1] Out[] = (Log[2] - 5/8) Out[] = -0.00155653 Where did this factor (Log[2]-5/8) come from? Is asymptotic analysis the only option we have left? After a few hours today, I still have yet to come up with a convincing answer, and feel thoroughly defeated (by a computer program no less). Am I missing something? Is this (Log[2]-5/8) somehow easy to derive? --Brad See also: http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/04/01/

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[math-fun] 1/Pi^2
by françois mendzina essomba2 24 Jan '20

24 Jan '20

Hi, Dear Bill, I didn't quite understand your reccurence.It may be obvious, but not for me. here is another one I propose for 1 / Pi ^ 2. It is simpler to obtain and the method is disconcertingly simple. a[n+1]=2^(-2*n-5)/(1-sqrt(1-2^(-2*n-4)/a[n])); a[0]=1/8 n----- infinity a[n] ------- 1/Pi^2 Best regards

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Re: [math-fun] Mystery table
by Keith F. Lynch 23 Jan '20

23 Jan '20

Andres Valloud <avalloud(a)smalltalk.comcastbiz.net> wrote: > Do you know what the table represents? Yes. Let me show a little more of the table. The "all" means that all real numbers will fit there (if you write them very small). I suggest looking for patterns in where there are integers in the table. Or where a number and its negative appear in the same row. 1 2 3 4 5 6 7 8 1: all 2: -inf +inf 3: -inf 1.000000 +inf 4: -inf 0.000000 2.261406 +inf 5: -inf -0.349081 1.000000 3.074601 +inf 6: -inf -0.523305 0.523305 1.564328 3.795684 +inf 7: -inf -0.626499 0.268479 1.000000 2.000000 4.495526 +inf 8: -inf -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 +inf 9: -inf -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -inf -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 11: -inf -0.804148 -0.140255 0.265003 0.621297 1.000000 1.469217 2.158924 12: -inf -0.825509 -0.188046 0.188046 0.505587 0.825509 1.192579 1.671024 13: -inf -0.842792 -0.226700 0.126799 0.415933 0.695796 1.000000 1.366735 14: -inf -0.857046 -0.258628 0.076822 0.344232 0.595108 0.857046 1.155985 15: -inf -0.868994 -0.285461 0.035215 0.285461 0.514414 0.746112 1.000000 16: -inf -0.879146 -0.308343 0.000000 0.236333 0.448129 0.657173 0.879146 1 2 3 4 5 6 7 8

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[math-fun] Draft of essay about two proofs of a rectangle tiling theorem
by James Propp 23 Jan '20

23 Jan '20

I'm hoping to post the essay "What Proof is Best?" on my blog in a few days; comments are very welcome! The current draft is at http://jamespropp.org/mathenchant/057-draft2.pdf Thanks, Jim

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[math-fun] Mystery table
by Keith F. Lynch 23 Jan '20

23 Jan '20

Who will be the first to explain what this is a table of? (As always with tables, it's best viewed in a fixed-width font.) 2 3 4 5 6 7 8 9 3: 1.000000 4: 0.000000 2.261406 5: -0.349081 1.000000 3.074601 6: -0.523305 0.523305 1.564328 3.795684 7: -0.626499 0.268479 1.000000 2.000000 4.495526 8: -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 9: -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 6.578811

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[math-fun] Iterate, stop, restart
by Éric Angelini 21 Jan '20

21 Jan '20

Hello Math-Fun, Start with n, then let n=2n and iterate. When the string n reappears, stop the iterations and restart the procedure with the smallest integer not yet in S. S = 1,2,4,8,16,stop,3,6,12,24,48,96,192,384,stop,5,10,20,40,80,160,320,640,1280,2560,stop,7,14,28,56,112,224,448,896,1792,stop,9,18,36,72,144,288,576,1152,2304,4608,9216,stop,11,22,44,88,... Question: is there a new n that will restart S at some point and _not_ lead to a new stop? Best, É. à+ É. Catapulté de mon aPhone

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[math-fun] Original source of "XOR doubly linked list" hack?
by Henry Baker 21 Jan '20

21 Jan '20

Wikipedia is sometimes pretty lame. Their references about the (in)famous "XOR doubly linked list" hack miss the origin by perhaps 40 years. Does anyone here know who originally came up with this hack and if so, where it was first published? I seem to recall hearing about it while an undergraduate at MIT in the late '60's, but it may have become folklore by then.

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[math-fun] Dunkin Brands π day promotions?
by Bill Gosper 20 Jan '20

20 Jan '20

I tried to interest Baskin Robbins: Does Baskin Robbins have 10 π (= 31.4159) flavors? Or only π³ (= 31.00627668) ? Unfortunately, their Contact Us UX is a disaster, so I sent it via their sister company Dunkin. Night and day! So I followed up: Since you guys are so much easier to talk to, here's your own dose of Pi Day fun: A perfect donut with diameter d and hole diameter h has glazed area A = π² (d - h) (d + h)/4 and volume V = π² (d - h) (d + h)²/32 . We shall see. —rwg

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[math-fun] Geometric construction for Pi^2
by françois mendzina essomba2 20 Jan '20

20 Jan '20

Hi, a [n+1]=2^(2*n+3)*(1-(1-a [n]/2^(2*n+2))^(1/2)) a [0]=4 ; n----> infinity a [n]----> Pi^2 We see that: 1-a [n]/2^(2*n+2)=(1-a [n+1]/2^(2*n+3))^2 Best regards Envoyé depuis Yahoo Mail pour Android

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[math-fun] Major Breakthrough In Quantum Computing Shows That MIP* = RE
by Ray Tayek 19 Jan '20

19 Jan '20

https://science.slashdot.org/story/20/01/18/0554240/major-breakthrough-in-q… Posted by EditorDavid on Saturday January 18, 2020 @03:34PM from the problem-sets dept. Slashdot reader JoshuaZ <https://slashdot.org/~JoshuaZ> writes: /In a major breakthrough in quantum computing it was shown that MIP* equals RE <https://arxiv.org/abs/2001.04383>. MIP* is the set of problems that can be efficiently demonstrated to a classical computer interacting with multiple quantum computers with any amount of shared entanglement between the quantum computers. RE is the set of problems which are recursive; this is essentially all problems which can be computed. This result comes through years of deep development of understanding interactive protocols, where one entity, a verifier, has much less computing power than another set of entities, provers, who wish to convince the verifier of the truth of a claim. In 1990, a major result was that a classical computer with a polynomial amount of time could be convince of any claim in PSPACE <https://en.wikipedia.org/wiki/PSPACE> by interacting with an arbitrarily powerful <https://en.wikipedia.org/wiki/IP_(complexity)> classical computer. Here PSPACE is the set of problems solvable by a classical computer with a polynomial amount of space. Subsequent results showed that if one allowed a verifier able to interact with multiple provers, the verifier could be convinced of a solution of any problem in NEXPTIME, a class conjectured to be much larger than PSPACE. For a while, it was believed that in the quantum case, the set of problems might actually be smaller, since multiple quantum computers might be able to use their shared entangled qubits to "cheat" the verifier. However, this has turned out not just to not be the case, but the exact opposite: MIP* is not only large, it is about as large as a computable class can naturally be. This result while a very big deal from a theoretical standpoint is unlikely to have any immediate applications since it supposes quantum computers with arbitrarily large amounts of computational power and infinite amounts of entanglement. The paper in question is a 165 tour de force which includes incidentally showing that the The Connes embedding conjecture, a 50 year old major conjecture from the theory of operator algebras, is false./ -- Honesty is a very expensive gift. So, don't expect it from cheap people - Warren Buffett http://tayek.com/

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