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[math-fun] Complexity of detecting whether the intersection of two regular languages is empty?
by Mike Stay 30 Nov '17

30 Nov '17

Given two regular expressions, how hard is it to tell whether they unify, i.e. whether the intersection of the languages they denote is empty? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Lights Out online
by James Propp 30 Nov '17

30 Nov '17

Anyone know of a good website that lets you play Lights Out? I found some websites that offer playing for free, but they aren't the sort of sites I'd recommend to others, because they have offensive ad-content or require downloads or just seem sketchy in one way or another. I'd especially like to know of a site that lets you play a mod 3 version, where clicking on a cell increments it and its neighbors by 1 mod 3. (The usual version is mod 2.) Thanks, Jim Propp

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[math-fun] Parenthesizing
by James Propp 29 Nov '17

29 Nov '17

Apropos of parenthesizing exponential towers, here's an easier question (probably not new): Look at all the (Catalan-many) ways to parenthesize a_1 - a_2 - ... - a_n. Not all are distinct as linear functions of a_1, a_2, ..., a_n; e.g., (a - (b - c)) - d = a - (b - (c - d)). How many different functions can be obtained? I don't know the answer. Is it 2^(n-2)? Jim Propp

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[math-fun] Complex exponentiation
by Dan Asimov 29 Nov '17

29 Nov '17

When z is a nonzero complex number z = r exp(i*theta) and alpha is any complex number, then classically: z^alpha is defined to be the set of complex values given by (*) {exp(alpha * log(z)) where log(z) takes exactly all the values of the form {c + 2K*pi*i | K in Z} for some complex number c with exp(c) = z, e.g., c = ln(r) + i*theta . Now (*) exp(alpha * (ln(r) + i*theta). Let alpha = a + bi for a, b real. Then we have z^alpha = {exp((a + bi)*(ln(r) + i*theta)) = {exp(a*ln(r) - b*theta) + i(a*theta + b*ln(r))) = {exp(a*ln(r) - b*theta_0) i(a*theta_0 + b*ln(r))) * exp(2pi*(-b + a*i))^K where (say) -pi < theta_0 <= pi defines theta_0 uniquely, so the last equation reads: z^alpha = {u * v^K | K in Z} for complex numbers u, v with u = exp(a*ln(r) - b*theta_0) i(a*theta_0 + b*ln(r))), and v = exp(2pi*(-b + a*i)). Now, when people talk about i^i^i^i^.... they usually mean by taking the parentheses from the top down — and using the principal logarithm: i^z = exp((pi*i/2)*z) and iterating: i^...^z = i^(i^(...^(i^z)...)) = exp((pi*i/2)*(exp((pi*i/2)*...*exp((pi*i/2)z)...)) But what if we *didn't* restrict to using the principal logarithm, but instead considered *all* the determinations of i^previous stuff each time we exponentiated. Then at each stage there will be countably many determinations. Now (as usual) let the *number of exponentiations* approach oo. QUESTION: What are the limit points of the determinations: -------- Those points z of C that are *limits* of any sequence of points of the form z_n = something born at exactly n exponentiations, i.e., such that for every epsilon > 0 there exists N such that there is a point z_n that is a result of n > N exponentiations with |z - z_n| < epsilon. —Dan

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[math-fun] some biggish complex numbers
by Neil Sloane 28 Nov '17

28 Nov '17

There was a discussion here recently about a non-integral sequence where the 43d term was falsely claimed to be "too big to compute exactly" Here is a different question. Sequence https://oeis.org/A198683 is defined to be the "Number of distinct values taken by i^i^...^i (with n i's and parentheses inserted in all possible ways) where i = sqrt(-1) and ^ denotes the principal value of the exponential function." The first 11 terms are 1, 1, 2, 3, 7, 15, 34, 77, 187, 462, 1152 and there seem to be two different opinions about the 12th term, which may be either 2919 or 2926. The number of ways to insert the parens is of course a Catalan number, but the tricky part is testing for equality when the numbers get large (or small). Jon Schoenfield has been trying to settle the 12th term, and his comments (see also the "History" tab in that sequence) are very interesting. He points out that one of the values for n=11 is about 4.1007...*10^41232950809707420597749203381002924. Maybe one of experts on this list could help?

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Re: [math-fun] Miscounting in the Cheddar Gorge ...
by Dan Asimov 28 Nov '17

28 Nov '17

It seems that 98.6 is an overestimate for average human body temp. In 1992 they got a figure of 98.2: https://www.ncbi.nlm.nih.gov/pubmed/1302471 —Dan ----- From: Andy Latto <andy.latto(a)pobox.com> On Sat, Nov 18, 2017 at 11:11 AM, Leo Broukhis <leobro(a)gmail.com> wrote: >> This was converted from 37.0 Celsius. Do you have a source for the value being given to three significant figures in Celsius? I looked online, and found "Normal body temperature is considered to be 37°C (98.6°F); however, a wide variation is seen. Among normal individuals, mean daily temperature can differ by 0.5°C (0.9°F), and daily variations can be as much as 0.25 to 0.5°C" on the NIH site at https://www.ncbi.nlm.nih.gov/books/NBK331/, quoted from "Clinical Methods: The History, Physical, and Laboratory Examinations. 3rd edition.". If it varies among individuals by .5 decrees C, and daily variations can be as much as .5 degrees C, 37 seems a better description than 37.0 -----

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[math-fun] weird looking EllipticK valuation
by Bill Gosper 27 Nov '17

27 Nov '17

Stumbling through old mail I was startled by Hypergeometric2F1[1/3, 2/3, 1, 1 - GoldenRatio] == 2 Sqrt[GoldenRatio] π/(5^(7/12) Gamma[1/3] Gamma[11/15] Gamma[14/15]) I have no idea where I got it, but a fairly immediate consequence is EllipticK[1/64 (47 + 17 I Sqrt[3] - 21 Sqrt[5] - 7 I Sqrt[15])] == ((1 + Sqrt[5]) (17 + 17 I Sqrt[3] + 21 Sqrt[5] - 7 I Sqrt[15])^(1/4) π^2)/( 2 Sqrt[2] 3^(3/4) 5^(7/12) Gamma[1/3] Gamma[11/15] Gamma[14/15]) I don't have any η(i√15)s in my identity collection, but I strongly suspect Gamma[1/3] Gamma[11/15] Gamma[14/15] will be Chowla-Selberg's prediction for the transcendental part. The algebraic part was mostly E^(I ArcCot[1/8 (1 + Sqrt[3])^4 (Sqrt[3] + Sqrt[5])^2]) . (Huh? That's algebraic? Obviously!) --rwg Many of these 2F1s come out as algebraic*ElliptickK[algebraic] π^rational. MROB: Let me know if you ever add K to ries. --Bill

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[math-fun] A completely unrelated question triggered by the word "convex"
by Dan Asimov 27 Nov '17

27 Nov '17

Suppose you have a finite set of points in the plane. The process known as "convex peeling" arose as one way of defining the "median" of a finite set of 2-dimensional data, analogous to the usual one for 1-dimensional data: Let the set of points be X. Define X_0 = X and X_(n+1) = X_n - {p in X_n | p is on the boundary of the convex hull of X_n} for n >= 0. This sequential "filtration" of X is known as convex peeling. About 37 year ago someone showed me that if you start with a large number of points uniformly distributed in a square and then proceed to remove them in stages via "convex peeling" — the boundaries of the last groups of points remaining are exceedingly circular. So that there appears to be a true statement, something like: ----- With probability = 1, the shape of the normalized result of convex peeling [toward the end of the process] approaches [in some sense] a circle, as the number of points approaches oo. ----- But I've never seen a proof of this. Any ideas? —Dan

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[math-fun] square root approximating sequence ?
by Henry Baker 27 Nov '17

27 Nov '17

I'm looking for a simple approximation to sqrt(abs(b*c)) that goes something like this: b*c=b_1*c_1=b_2*c_2=...=(sign(b)*sqrt(b*c))*(sign(c)*sqrt(b*c)) where b_(i+1),c_(i+1) are rationally computed from b_i,c_i. In other words, we preserve the product while approximating the square root. What I'm really after is a sequence: [a b] [a b_1] [a b_2] [c d], [c_1 d], [c_2 d], ... such that the determinant is preserved, but each matrix is computed from the previous one by X.M.Y, where X,Y are 2x2 matrices rationally computed from the entries of M.

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[math-fun] Mobius motion
by James Propp 26 Nov '17

26 Nov '17

I'm planning to make a short video about hexaflexagons in which I'll demonstrate a kind of move I haven't seen described elsewhere: sliding the band "along" itself. This is not to be confused with flexing, which causes the band to slide "through" itself. "Along", "through" --- ordinary English prepositions aren't much help here. Is there any existing terminology for describing these kinds of motions of a band? To be clearer (or at least more techincal) about what I mean: Imagine we coordinatize the band with coordinates theta and y, where theta goes from 0 to 2pi and y goes from -1 to +1. These coordinates hide the details of the imbedding (and the three half-twists), but for present purposes that's desirable. If you imagine making each point (theta,y) travel in a circle of radius |y| lying in a plane locally perpendicular to the theta-axis, then what you're doing (more or less) is flexing the hexaflexagon. In contrast, what I'm talking about is moving the point (theta,y) along the band parallel to the theta-axis. I've gone through all Martin Gardner's writings, and watched a few YouTube videos about hexaflexagons, but haven't found anything along these lines. Jim Propp

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