December 2017 - mathfun@mailman.xmission.com

Re: [math-fun] Draft of my December 2017 blog post
by Henry Baker 15 Dec '17

15 Dec '17

"And now that we have those computers, mod p arithmetic (where p is a huge prime) is the cornerstone of cryptographic security for internet commerce." Not so much. There have been so many problems with RSA that the most recent recommendations are to disable RSA completely, in favor of elliptic curves. "A number of the most popular websites and services online, including Facebook and *PayPal*, are vulnerable to an exploit which has resurfaced from 1998." "We believe RSA encryption modes are so risky that the only safe course of action is to disable them." http://www.zdnet.com/article/robot-exploit-from-1998-resurrected-leaves-top… https://eprint.iacr.org/2017/1189.pdf At 08:42 AM 12/15/2017, James Propp wrote: >Hi, > >I'm almost finished with an essay entitled "The Roots of Unity" and would >love to get people's feedback. I'll be publishing it in less than 48 hours, >and I don't expect many people to have a chance to read it (let alone offer >comments) before I publish it. But I'll continue to tinker with it as >comments come in, so if you find mistakes or other opportunities for >improvement, please let me know, even if it's after the 17th. > >I tried (and failed) to find a doable puzzle that makes use of the the >multiplicative structure of the complex numbers. Can you think of a good >one that'd be accessible to a bright high school student? I was going to >include the question "Let p be the probability that the sum of the numbers >you get when you roll six fair dice is a multiple of seven. Do you think p >is equal to, greater than, or less than 1/7? What if we roll seven dice >instead of six?" (There's a nice anaysis that uses 7th roots of unity.) But >this strikes me as likely to be too tricky for nearly all of my readers. >Can anyone suggest a substitute? > >I hope you'll sign your comments, even negative ones; sometimes I have a >follow-up question (like "Okay, would this way of phrasing it be less >dreadful?"), which I can't ask you if you've commented anonymously. I want >to make the essay better, and I have a fairly thick skin. > >Please leave your feedback here: > https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af > >Title: The Roots of Unity >Beginning: Two primal pleasures for me from the years of my childhood (and >maybe the years of infancy that I can't remember) are feelings that I'll >call coziness and spaciousness. The first is the feeling I'd get at night >pulling the blankets up over my head; the other is the feeling I'd get >entering a landsca... >Read more: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af >Thanks, >jamespropp

2 1

[math-fun] Draft of my December 2017 blog post
by James Propp 15 Dec '17

15 Dec '17

Hi, I'm almost finished with an essay entitled "The Roots of Unity" and would love to get people's feedback. I'll be publishing it in less than 48 hours, and I don't expect many people to have a chance to read it (let alone offer comments) before I publish it. But I'll continue to tinker with it as comments come in, so if you find mistakes or other opportunities for improvement, please let me know, even if it's after the 17th. I tried (and failed) to find a doable puzzle that makes use of the the multiplicative structure of the complex numbers. Can you think of a good one that'd be accessible to a bright high school student? I was going to include the question "Let p be the probability that the sum of the numbers you get when you roll six fair dice is a multiple of seven. Do you think p is equal to, greater than, or less than 1/7? What if we roll seven dice instead of six?" (There's a nice anaysis that uses 7th roots of unity.) But this strikes me as likely to be too tricky for nearly all of my readers. Can anyone suggest a substitute? I hope you'll sign your comments, even negative ones; sometimes I have a follow-up question (like "Okay, would this way of phrasing it be less dreadful?"), which I can't ask you if you've commented anonymously. I want to make the essay better, and I have a fairly thick skin. Please leave your feedback here: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af Title: The Roots of Unity Beginning: Two primal pleasures for me from the years of my childhood (and maybe the years of infancy that I can't remember) are feelings that I'll call coziness and spaciousness. The first is the feeling I'd get at night pulling the blankets up over my head; the other is the feeling I'd get entering a landsca... Read more: https://mathenchant.wordpress.com?p=2052&shareadraft= 5a33f99f1b1af Thanks, jamespropp

1 0

Re: [math-fun] math-fun Digest, Vol 178, Issue 23
by Don Reble 15 Dec '17

15 Dec '17

> "Given a circle of n lights, exactly one of which is initially on, it > is permitted to change the state of a bulb provided that one also > changes the state of every dth bulb after it (where d is a divisor of > n strictly less than n), provided that all n/d bulbs were originally > in the same state as one another. Is it possible to turn all the bulbs > on by making a sequence of moves of this kind?" > OEIS http://oeis.org/A103314 looks very apropos. A070894 looks even better. The set of bulbs changed by a move has a center-of-light at zero. (Arrange the bulbs evenly around the circle. And then all go on or all go off.) The original center is non-zero, and a move changes it to a weighted combination of zero and the old value: but never to zero. So you can't even turn them all off. Please refute this, if you're the last one to bed. -- Don Reble djr(a)nk.ca

3 3

[math-fun] Circle of lights puzzle
by James Propp 15 Dec '17

15 Dec '17

Does anyone know the origin of the following Olympiad study problem? "Given a circle of n lights, exactly one of which is initially on, it is permitted to change the state of a bulb provided that one also changes the state of every dth bulb after it (where d is a divisor of n strictly less than n), provided that all n/d bulbs were originally in the same state as one another. Is it possible to turn all the bulbs on by making a sequence of moves of this kind?" I've found a statement of the problem at http://www.math.northwestern.edu/~mlerma/problem_solving/ putnam/training-complex and I learned from a Google search that the problem appears in the book "Mathematical Olympiad Challenges", but the excerpt I was able to view online doesn't explain who came up with the problem. (Maybe this is explained elsewhere in "Mathematical Olympiad Challenges", but I don't currently have access to a copy.) Was this ever used as an Olympiad problem, or does it come from some earlier problem book? Does it have a credited inventor? See http://somethingorotherwhatever.com/circular-lights-out/ for Christian's nice implementation. I also wonder whether anyone has ever computed, for various values of n, the number of different patterns of lights that are reachable from the all-lights-off position (or equivalently from the all-lights-on position). I tried entering "lights" and "circle of lights" into the OEIS but the former gave too many hits (141) while the latter gave none at all. Jim Propp

3 3

[math-fun] Optimized wildfire fighting ?
by Henry Baker 14 Dec '17

14 Dec '17

Since the "Thomas" fire has been threatening our local Santa Barbara for quite some time, and this fire has been discussed at length in the news, I was wondering whether any mathematics has been applied to firefighting, and whether there could be an "optimal" strategy to fight such fires? We have input data: Weather: * time of day * temperature * humidity * wind speed & direction * these data at some precision in 3D space & time * historical & forecasted values of these data Fuel load: * type and amount (to estimate ability to sustain & propagate fire) * historical data -- e.g., when last burned Objective function: * location & value of assets to be preserved; * details of value degradation vs. temperature & time We have these actuators: * fire suppression (water, retardant) over 2D surface * fire accentuation (ignition, napalm, etc.) * these mechanisms applied in time & space at some precision We need a mathematical model: * how fire burns out and/or propagates, based upon local fuel load, air temp, air humidity, wind velocity, presence of retardants, presence of accelerants, etc. ---- The idea is to figure out when & where to optimally supply fire retardants and fire accentuation to minimize the damage incorporated into the objective function. If modern AI can produce winning strategies for *Go*, then why wouldn't such an AI technology be capable of solving this fire problem?

1 0

[math-fun] Spooning cold quince soup from a tall, thin jar,
by Bill Gosper 14 Dec '17

14 Dec '17

I was perplexed when the spoon suddenly jammed in the neck and refused to move up or down. After minutes of flopping it back and forth, I finally realized what was happening, and finished my soup. How? --Bill

4 4

Re: [math-fun] RMS paradox
by Keith F. Lynch 12 Dec '17

12 Dec '17

Eugene Salamin <gene_salamin(a)yahoo.com> wrote: > Actually, the conjecture for sums of multiple sine waves is false. Oops, you're right. When it comes to the sum of sine waves, I'm more of a radio and electronics guy than a mathematician. As was perhaps evident from my using the term "RMS" rather than "quadratic mean." > What is true is that the mean square MS = RMS^2 of the sum equals > the sum of the mean squares of the component sine waves, with the > proviso that the components are orthogonal. Yes, that's what I was probably thinking of. The first time I learned that power was proportional to the square of the amplitude, that struck me as very strange. Two identical radio transmitters emit four times the power of either of them alone? Eventually, I realized that a similar principle applies in lots of contexts. For instance: * In quantum mechanics, the probability of finding a particle at a given point is proportional to the square of the absolute magnitude of the complex amplitude there. (Or, equivalently, the product of the amplitude with its complex conjugate.) * The power in a sound wave is proportional to the square of the RMS overpressure. * The power in a water wave is proportional to the square of the RMS height of the wave. * The power of a gravitational wave is proportional to the square of the RMS strain (i.e. of the degree to which space is stretched or squished). * The energy in an electric or magnetic field is proportional to the square of the field strength. * Bosons tend to be in the same state as their neighbors because we're adding amplitudes, then measuring the squares of those amplitudes. * The work it takes to shovel snow is proportional to the square of the snow depth. (X times the depth of snow means X times as much snow per unit area, and also means you have to lift it X times as high to get it above the surrounding snow.) * The work it takes to dig a hole is proportional to the square of the depth of the hole. My intended solution to the RMS paradox was that as you add more and more sine waves to make a square wave, the corners continue to have spikes that rise above the flat part. Adding more terms will make those spikes narrower, but never make them shorter. At the limit, they're infinitely narrow but still no less tall than they ever were. How about the radio paradox I alluded to above? How can two identical transmitters radiating the same frequency and phase radiate four times the power as either one alone? One answer is that the antennas are generally in different places, so the waves are out of phase in some places and in phase in others. But what if the antennas are an infinitesimal distance apart, e.g. two identical parallel dipoles separated by much less than one wavelength? Then the waves would be in phase everywhere. Brent Meeker <meekerdb(a)verizon.net> wrote: > If I add two sine waves of equal amplitude but with 180deg > phase difference, I get zero - clearly not an RMS of the > amplitude/sqrt(2). Actually, my mistaken claim does work there, since the peak amplitude of the sum of those two waves is zero, and if you divide that by the square root of 2, you get zero, which is indeed the RMS.

3 2

[math-fun] Deep Mind crushes Rybka
by Guy Haworth 12 Dec '17

12 Dec '17

Just as well I didn't ask Dennis H about Deep Mind and Chess last Friday! The word on the chess networks is that Stockfish 8 was v short of hash-table allowance, and had no opening book or endgame table, whereas Alpha Zero effectively had an opening book 'absorbed' into its Neural Network settings. While Deep Mind are clearly doing impressive, mould-breaking work, it would be good to see this experiment repeated under conditions approved by Stockfish's management and the ICGA (or TCEC) people. My expectation is that, at the same tempo, the contest would be much more even with Stockfish differently set up. Nevertheless, at 'Classic Tempo', I'd still expect Alpha Zero to win. Guy

6 6

[math-fun] Bar bet
by Bill Gosper 12 Dec '17

12 Dec '17

On 2016-09-22 17:59, Allan Wechsler wrote: > Son of a gun. > > On Thu, Sep 22, 2016 at 8:05 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> "The volume of a regular tetrahedron (triangular pyramid with unit edges) >> is exactly half the volume of a square pyramid with unit edges." >> True or false? --rwg Easier T|F: "The height of the square pyramid is the circumradius of its base, so that all five vertices are equidistant from that circumcenter." Eh? Obviously. Put the square's points on the equator, separated by 90º of longitude. Then the apex is at the north pole, separated by 90º of latitude, dividing the hemisphere into four equilateral spherical triangles, each with three right angles. Is this why a square pyramid of oranges has the same lattice as a tetrahedral pyramid of oranges? Final T|F: "The solid ∠ at the π/3,π/3,π/3,π/3 apex is exactly twice the solid ∠ of each π/3,π/3,π/2 base point." Spoiler: the pyramid is half an octahedron. (Whose solid ∠s = 4 ArcTan[1/√8] = 2 ArcCos[7/9] = 4 ArcCos[√8/3] = 4 ArcSin[1/3] = 8 ArcTan[√(Cot[5π/24] Tan[π/24]) Tan[π/8]] =ArcTan[56√2/17] == ArcSin[56√2/81] == ArcCos[17/81].) --rwg

1 0

[math-fun] solid angles & correction
by Bill Gosper 10 Dec '17

10 Dec '17

I CCed some people a faulty formula for the solid angle at the vertices of the base of a regular n-gonal pyramid. It worked for regular tetrahedra, and perhaps nothing else. Here's one (of many) that works for a regular n-gon base: pyramidSolidAngles[n_, 𝜃_] :={2 π - 2 n ArcTan[Cos[𝜃] Tan[π/n]], 2 ArcSec[(1 + Sin[π/n] Sin[𝜃])/(Sin[π/n] + Sin[𝜃])]} (𝜃 is the ∠ between the altitude and the generator.) There are bewilderingly many formulæ for the solid angle at a trihedral vertex with angles a, b, c. Here are five: solidAngle[a_, b_, c_] := {2 ArcTan[Sqrt[1 - Cos[a]^2 - Cos[b]^2 - Cos[c]^2 + 2 Cos[a] Cos[b] Cos[c]]/(1 + Cos[a] + Cos[b] + Cos[c])], ArcCos[-1 + (1 + Cos[a] + Cos[b] + Cos[c])^2/((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c]))], 2 ArcCos[1/4 (1 + Cos[a] + Cos[b] + Cos[c]) Sec[a/2] Sec[b/2] Sec[c/2]], 2 ArcSin[1/(4 Sqrt[2]) Sqrt[-1 - Cos[2 a] - Cos[2 b] + 4 Cos[a] Cos[b] Cos[c] - Cos[2 c]] Sec[a/2] Sec[b/2] Sec[c/2]], 4 ArcTan[\[Sqrt](Tan[1/4 (a + b - c)] Tan[1/4 (a - b + c)] Tan[1/4 (-a + b + c)] Tan[1/4 (a + b + c)])]} FullSimplify can't begin to equate them. Question: For a variable apex above a fixed base-polygon, what is the locus of that apex which preserves the solid angle? Would it be spherical, analogous to the 2D case? It's hard to see how it wouldn't be at simplest ellipsoidal, in the case of an extremely elongated base. I just realized something obvious. I've only been thinking about the solid angles at the vertices of polyhedra. But you can define a (4π supplementary) pair of solid angles for any point on a piecewise smooth surface. It assigns a sort of "topological curvature", bounded between 0 and 4π, to many points where the usual curvature blows up. At a point of bounded curvature, both solid angles are 2π. For a finite isosceles cone with angle 𝜃 between its altitude and generators, the interior solid angles at the apex and and rim are (sending n→∞ in the pyramid) coneSolidAngles[𝜃_] := {2 (1 - Cos@𝜃) π, π - 2 𝜃} I.e., the solid angle at the rim has twice the magnitude of the planar angle there. Which is also true of the smooth areas, but not the apex. --rwg Because 4π ~ 12, one face of a dodecahedron subtends only a little more than a steradian. gosper.org/1steradian.png Scandal: Due to insidious infiltration by Egyptians, Mathematica's Pyramid primitive permits only quadrilateral bases.

1 0