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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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[math-fun] Rough Ellipso-Spectrometry of a "Gruppenpest"
by Brad Klee 19 Jan '20

19 Jan '20

Subtitle: Stranger Than Fiction. Fayetteville, AR, Jan. 19. A green June bug was found dead and dismembered near a sewer at the University of Arkansas on the afternoon of Friday January 17. The untimeliness of the occurrence raised enough suspicions for students and faculty to initially avoid any dealings with the mess of ill-strewn body parts. Later on Sunday evening, an intrepid reporter decided to return to the scene of the crime, where he then proceeded to discretely collect remnant fragments of the beetle's shell--the pronotum and a few abdominal sections. The beetle (possibly beetles, plural) appears to have been of the local variety Cotinis Nitida. However, if the remains were, in fact, dumped inappropriately by a university researcher, they might have come from Cotinis Mutabilis. No missing beetle reports have been filed recently, and the relevant authorities do not care to help with the investigation. The plight of the insects has largely been ignored by humans, and many species of Coleoptera have recently been derided for their role in ruining crops. Despite growing inter-species tensions, the unexpected, serendipitous appearance of a chiral-reflecting corpse can do no less than to inspire! Opportunist scientists, anyways, are usually not patient enough to just wait until the June emergence. Recent efforts have reconstructed a 100-200Ma family tree fo the Scarabaeidae, which also correlates development of anthophagy following the angiosperm takeover [1]. Other scientists have taken a narrower view of the Pleurosticti radiation, especially around the sub-family Cetoniinae [2]. As first noticed by Michelson in 1911, many of these beetles have iridescent shells which selectively reflect circular polarized light. One hypothesis is that this adaptation evolved concurrently with the rise of predatory animal species, as a new method of crypsis. In Scarab-shell ellipsometry, researchers today measure reflective change of polarization in terms of Mueller matrix elements, as functions of wavelength and scattering angle. Emphasis continues to fall on the element M_{1,4}, which determines conversion of unpolarized light into left or right circular polarization. For qualitative measurement, a full scientific setup is not necessary. Two opposed CPL camera filters or a pair of 3D glasses (see also [3]) are sufficient to verify chiral selectivity in the visible spectrum. For example, on the recently collected "specimen", we find the following: https://0x0.st/zF0V.jpg with row 1: ambient, row 2: CPL, row 3 opposite CPL. One of the circular polarizing filters eliminates almost all of the green reflection, while the other does not. This indicates that the species--probably a Cotinis Scarab--does have chiral-layered shell. A quick search of the literature did not reveal if full ellipsometery of Cotinis Nitida is known. Such an experiment may be of some interest during the summer months when specimens are readily available. Ultimately scientists would like to have a complete picture as to when and why chiral selectivity evolved, and this effort needs to include all relevant species, even the green June bugs from Arkansas. --Brad [1] https://royalsocietypublishing.org/doi/pdf/10.1098/rspb.2014.1470 [2] https://www.diva-portal.org/smash/get/diva2:800022/FULLTEXT01.pdf [3] https://www.itp.uni-hannover.de/fileadmin/arbeitsgruppen/zawischa/static_ht…

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[math-fun] Would someone with a clearer head please explain why
by Bill Gosper 18 Jan '20

18 Jan '20

these are unequal: {Sin[InverseFunction[(π + #1) + Sin[π + #1] &][t]], Sin[InverseFunction[#1 + Sin[#1] &][π + t]]} (real t) while these aren't? {Sin[InverseFunction[(π + #1) + Sin[π + #1] &][π - t]], Sin[InverseFunction[#1 + Sin[#1] &][π + t]]} —rwg

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[math-fun] Mathematicians Catch a Pattern By Figuring Out How To Avoid It
by Ray Tayek 16 Jan '20

16 Jan '20

https://science.slashdot.org/story/19/11/28/180213/mathematicians-catch-a-p… We finally know how big a set of numbers can get before it has to contain a pattern known as a "polynomial progression." <https://www.quantamagazine.org/mathematicians-catch-a-pattern-by-figuring-o…> From a report:/A new proof <https://arxiv.org/abs/1909.00309> by Sarah Peluse of the University of Oxford establishes that one particularly important type of numerical sequence is, ultimately, unavoidable: It's guaranteed to show up in every single sufficiently large collection of numbers, regardless of how the numbers are chosen. "There's a sort of indestructibility to these patterns," said Terence Tao of the University of California, Los Angeles. Peluse's proof concerns sequences of numbers called "polynomial progressions." They are easy to generate -- you could create one yourself in short order -- and they touch on the interplay between addition and multiplication among the numbers. For several decades, mathematicians have known that when a collection, or set, of numbers is small (meaning it contains relatively few numbers), the set might not contain any polynomial progressions. They also knew that as a set grows it eventually crosses a threshold, after which it has so many numbers that one of these patterns has to be there, somewhere. It's like a bowl of alphabet soup -- the more letters you have, the more likely it is that the bowl will contain words. But prior to Peluse's work, mathematicians didn't know what that critical threshold was. Her proof provides an answer -- a precise formula for determining how big a set needs to be in order to guarantee that it contains certain polynomial progressions. Previously, mathematicians had only a vague understanding that polynomial progressions are embedded among the whole numbers (1, 2, 3 and so on). Now they know exactly how to find them./ -- Honesty is a very expensive gift. So, don't expect it from cheap people - Warren Buffett http://tayek.com/

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[math-fun] Need help identifying a fractal
by Neil Sloane 14 Jan '20

14 Jan '20

I've been trying to understand the Conant Gasket https://oeis.org/A328078 but without much success. The pictures are gorgeous. It is a kind of fractal. (The definition is clear, but what are the numbers?) By accident I found a similar-looking genuine fractal in an old pile of papers - see https://oeis.org/A328078/a328078_2.pdf - does anyone recognize it?

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[math-fun] RandomPartition[n]
by James Propp 14 Jan '20

14 Jan '20

Has RandomPartition[n] been dropped from Mathematica? The documentation says that it's part of the Combinatorica package, which has now been folded into the Wolfram Language, so in theory RandomPartition[n] should be part of Mathematica. I figured out on my own (no thanks to the documentation) that Partitions[n], which appeared to have been dropped, is still available as IntegerPartitions[n], so when RandomPartition[n] didn't work I tried RandomIntegerPartition[n], but that doesn't work either. Jim Propp

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