June 2016 - mathfun@mailman.xmission.com

[math-fun] sum of binary digits
by Erich Friedman 03 Jun '16

03 Jun '16

Let n be a positive integer. Let g(n) be the sum of the binary digits of n. I am looking for interesting facts involving g(n). Here are 3 that i have so far, in increasing difficulty to prove: Theorem 1: The minimum number of integral powers of 2 to sum to n is g(n). Theorem 2: The highest power of 2 to divide n! is 2^[ n - g(n) ]. Theorem 3: The number of odd entries in the nth row of Pascal's Triangle is 2^[ g(n) ]. Anyone know any others? Erich Friedman

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[math-fun] esoterica -- meeting on Lambert function
by rcs＠xmission.com 03 Jun '16

03 Jun '16

This poster is for the upcoming meeting on the Lambert function. www.apmaths.uwo.ca/~djeffrey/LambertW/poster.pdf www.apmaths.uwo.ca/~djeffrey/LambertW/LambertW.html Rich

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[math-fun] What has info theory ever done for multiple choice exams?
by Fred Lunnon 02 Jun '16

02 Jun '16

TT in (relatively) playful mood --- https://terrytao.wordpress.com/2016/06/01/how-to-assign-partial-credit-on-a… WFL

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Re: [math-fun] Diamonds for dopes
by Keith F. Lynch 02 Jun '16

02 Jun '16

Credit where it's due. As far as I know, the idea of using C12 and C13 in diamond as bits to store information is due to Charles Stross. The earliest mention I've been able to find is in a July 2013 Usenet post from him. He later used the idea in his Hugo-award winning novella "Palimpsest," in which high-definition video recordings of literally all of history and prehistory in vast numbers of alternate timelines is recorded in a vast library which uses nearly all of Earth's carbon. (Coincidentally, I ran into him at a convention last weekend, the first time I'd seen him in years. Another coincidence is that "Palimpsest" is basically a rewrite of a novel written by the late uncle of one of the regular posters here.) Diamond is stable over at least billions of years, probably much longer. The rigid lattice means that adjacent atoms never swap. At normal temperatures and pressures it slowly turns into graphite, but that's so slow that it's never been observed at normal temperatures. So in practice, damage by cosmic rays is certainly the fastest failure mode, assuming reasonable storage conditions. (Do not store in a fire, or in molten iron, or smash it with a hammer. Pretty much anything else is okay, including storing it in water, acids, or solvents, or injecting it under your skin.) Its data density is an astonishing 10^22 bits per gram. In your shirt pocket you could store a full-lifetime video recording of every person who ever lived, with enough room left over for all books, newspapers, magazines, letters, web pages, emails, movies, and TV shows. The whole of Wikipedia and OEIS would fit into a microscopic speck of diamond dust. There's no evidence that substituting C13 for C12 in food causes any harm. There's some weak evidence that it, and other substitutions of a heavier stable isotope for a lighter one, may increase longevity. Deuterium is an exception. It's the only substitution that can kill. No non-microscopic organisms can survive total deuteration. Even so, there's weak evidence that drinking half a cup of heavy water per day might be good for you.

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[math-fun] Orthonormal abnormality
by Fred Lunnon 01 Jun '16

01 Jun '16

Despite generous previous assistance from all, I am still puzzling over the mysterious behaviour of factorisation into Givens' rotations. My latest head-scratchers are embedded below at (6) -- (8) within in a potted summary of developments to date. I cling to the expectation that these may also be trivial when viewed from the correct perspective ... WFL ______________________________________________________________________ (1): R_pq(u) denotes "Givens'" rotation through angle u in the (x_p, x_q)-coordinate plane about the origin of Euclidean n-space, represented by an order-n orthonormal matrix taking an elementary form: see https://en.wikipedia.org/wiki/Givens_rotation (2): Given(!) natural n , a "schedule" is a permutation of distinct [p, q] with 1 <= q < p <= n , specifying some reduction of a general order-n orthonormal matrix to diagonal form via that sequence of "moves", ie. premultiplications by R_pq(u_k) for k in [1..m] . Note that the matrix remains orthonormal throughout. (3): During reduction of a specific numerical matrix, angles u_k assume individual numerical values, assigned to clear matrix entry [p, q]_k . At each stage k there are two suitable assignments: if u is one, the other is u + pi . For the present angles are unimportant and can be ignored. Sample reduction from orthonormal to diagonal, showing values p, q, A at each stage k for random numerical matrix A : https://www.dropbox.com/s/uwsmq2habm4tfgp/given_p1.tiff (4): Given a schedule along with natural r in [1..n] , its r-th "sub-schedule" is the subsequence excluding moves for which p = r or q = r , and with indices p, q > r decremented by unity. (5): Examples --- Of 6 permutation sequences for n = 3 , just 3 constitute valid schedules: [ [2, 1], [3, 1], [3, 2] ] , [ [3, 1], [2, 1], [3, 2] ] , [ [3, 2], [2, 1], [3, 1] ] . For n = 4 , one schedule and its sub-schedule with r = 3 : [ [4, 3], [3, 1], [2, 1], [4, 1], [3, 2], [4, 2] ] -> [ [2, 1], [4, 1], [4, 2] ] -> [ [2, 1], [3, 1], [3, 2] ] , first in the list above. (6): Claim --- Every sub-schedule of a schedule is a schedule. (7): Claim --- Every schedule is a sub-schedule of some larger. (8): Puzzle --- *** Are these claims obvious? *** (9): Problem --- *** Are these claims true? *** There are numerous implications: in particular, that for each schedule there exists an angle selection strategy guaranteed to reduce every special orthonormal matrix (orientation-preserving isometry) to the identity. (10): A matrix entry [i, j] is "cleared" after move k when partial reduction by the first k moves has guaranteed that it is zero (and will remain so subsequently). (11): Remark --- [p, q]_k is a move of a valid schedule iff after move k-1 , matrix entries [p, j], [q,j] are either both cleared or both uncleared for all j in [1..n] . (12): Remark --- Move [p, q] affects only matrix rows p, q . Suppose clearing entry [r, j] causes columns j, j' to retain a single pair of uncleared entries: say [r', j], [r', j'] , where {r, r'} = {p, q} . Via orthogonality entry [r', j'] in the upper triangle j' > r' also becomes cleared. Iterate ad nauseam? For example, if move [2, 1] reduces column 1 to [1, 0, ..., 0]' , then any uncleared entries on the rest of row 1 must also be cleared in consequence. (13): Via (11) and (12) distinct schedules of order n can in principle be enumerated combinatorially, and their cardinal f(n) calculated; currently known are values n 0 1 2 3 4 5 6 f 1 1 1 3 50 6525 8515479 f/h - 1.00 1.00 0.89 0.78 0.68 0.59 Plainly f(n) <= m! where m = binom(n, 2) == n(n-1)/2 equals the number of distinct moves [p, q] . From the data it seems faintly plausible that asymptotically f(n) ~ c h(n) , where h(n) == (n/2)^m , with constant c < 1/2 by some substantial margin. Fred Lunnon 01/06/16 ______________________________________________________________________

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Re: [math-fun] highly enriched uranium (HEU) improvised nuclear bomb
by Henry Baker 01 Jun '16

01 Jun '16

"Little Boy" had 64kg of HEU. A basketball is 25cm in diameter, and has a volume of 8181 cm^3. U has a density of 19.1 g/cm^3, so this U basketball weighs 156kg (~350#), or about 2.5x Little Boy. Little Boy's yield was ~15ktons, so Perry's basketball might yield 35-40 ktons. https://en.wikipedia.org/wiki/Little_Boy It is my understanding that the "problem" with the Little Boy design is that it is incredibly inefficient in its use of fissile material. On the other hand, it is also my understanding that the design is almost trivial. Perry obviously gave himself a much larger mass to enable an even more incredibly inefficient design to still "work". However, if you're in a position to make your own material, you will have plenty of time (while it's being processed) to investigate more efficient designs. At 05:42 PM 5/19/2016, Thane Plambeck wrote: >From an email I just received that was signed by the governor of California >and William J Perry > >"A quantity of HEU the size of a basketball would be sufficient to >make an improvised nuclear bomb that had the explosive power of the >Hiroshima bomb and was small enough to fit into a delivery van. Such a >bomb, delivered by van (or fishing boat) and detonated in one of our >cities, could essentially destroy that city, causing hundreds of thousands >of casualties, as well as major social, political, and economic >disruptions." > >My question: is this really true? I mean, supposing a small group of >people accomplished getting that much HEU, is it really just a matter of >"improvisation" to detonate it? I was under the perhaps mistaken >impression that detonating it involved some serious engineering chops, >especially if you dont want to kill or fatally irradiate yourself in the >attempt. > >-- >Thane Plambeck >tplambeck(a)gmail.com >http://counterwave.com/

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Re: [math-fun] Chinese tea puzzle
by Bill Gosper 31 May '16

31 May '16

On 2016-05-31 04:55, Veit Elser wrote: > Your tea is mainly cooled by evaporation (breaking hydrogen bonds), > rather than conduction (to the air) or radiation (into space). You > therefore want to maximize the surface area accessible to the tea > — your second option. Have you tried pouring the tea onto a > cookie sheet (with rim)? > > -Veit Have you any idea the look such a request would elicit from the Chinese waiter? They already mistrust me for asking how to use a fork. I suspect the actual surface area of the teabag might be comparably huge. --rwg > >> On May 28, 2016, at 3:11 AM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> I pour a cup. It's too hot and too weak. With chopsticks I remove the bag >> from the pot >> and dunk it repeatedly in my cup. Will my tea cool faster if I dunk up and >> down rapidly, >> or pause several seconds between immersions? --rwg

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