February 2016 - mathfun@mailman.xmission.com

[math-fun] A=bh/2
by Bill Gosper 14 Feb '16

14 Feb '16

Remember when they taught you that the area of a triangle is half the area of the parallelogram you get by gluing two copies together? And the area of the parallelogram is the area of the rectangle you get by chopping a triangle off one end and gluing it on the other? Did anybody complain that this doesn't work for sufficiently skewed parallelograms? Fix: gosper.org/A=f(bh).gif --rwg

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[math-fun] Perigal's Pythagorean proof
by Bill Gosper 13 Feb '16

13 Feb '16

gosper.org/Perigal.gif --rwg

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Re: [math-fun] Gravitational waves found
by Warren D Smith 12 Feb '16

12 Feb '16

--more papers collected here: http://www.virgo-gw.eu/papers.html

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[math-fun] Gravitational waves found
by Warren D Smith 12 Feb '16

12 Feb '16

news media now onto it: https://www.washingtonpost.com/news/speaking-of-science/wp/2016/02/11/cosmi… https://www.theguardian.com/science/2016/feb/11/gravitational-waves-discove… http://www.bbc.com/news/science-environment-35524440 "The published paper names a staggering 1,004 individual authors... It was a clear, compelling signal of two black holes coalescing, LIGO scientists said. It lasted less than half a second... These black holes were each approximately the diameter of a major metropolis. They orbited one another at a furious pace at the very end, speeding up to about 75 orbits per second -- warping the space around them like a blender cranked to infinity -- until finally the two black holes became one... LIGO scientists said they are analyzing additional data from the observational run lasting from September to early January, and that they may find other signs of black hole mergers. One candidate for such an event, picked up in October, is still being analyzed, they said... One of the two black holes had a mass about 36 times greater than our sun. The other registered at 29 solar masses. Both were rather massive as black holes go -- 10 solar masses is more typical...the resulting black hole was not the 65 solar masses you'd expect from basic arithmetic, but only 62. The rest was converted to energy that radiated across space in a grand gravitational burp...That burp first reached the LIGO facility in Louisiana, then the one in Washington state just 7 milliseconds later. The sequence is important, as it allowed physicists to chart the black-hole collision back to somewhere in the southern sky. And the incredibly brief time delay supports something that theorists have long believed about gravitational waves: They move at the speed of light...LIGO is still only about a third as sensitive as it is designed to be, and improvements in coming months should let it pick up signals from deeper regions of space, the scientists said." The paper is in PHYSICAL REVIEW LETTER and something also is in SCIENCE magazine: http://www.sciencemag.org/news/2016/02/gravitational-waves-einstein-s-rippl… https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102 >From which we find: Incredibly, these two black holes are believed to have been 1.3 billion light years away... the signal hit us on 14 September 2015... the wave stretched space around here by about 1 part in 10^21, "making the entire Earth expand and contract by 1/100,000 of a nanometer... The oscillation emerged at a frequency of 35 cycles per second, or Hertz, and sped up to 250 Hz before disappearing 0.25 seconds later. The increasing frequency, or chirp, jibes with two massive bodies spiraling into each other. The 0.007-second delay between the signals in Louisiana and Washington is the right timing for a light-speed wave zipping across both detectors... Comparison with computer simulations reveals that the wave came from two objects 29 and 36 times as massive as the sun spiraling to within 210 kilometers of each other before merging. Only a black hole... can pack so much mass into so little space, says Bruce Allen, a LIGO member at the Max Planck Institute for Gravitational Physics in Hanover, Germany. The observation provides the first evidence for black holes that does not depend on watching hot gas or stars swirl around them at far greater distances. 'Before, you could argue in principle whether or not black holes exist,' Allen says. 'Now you can't.... For a tenth of a second [the collision shined] brighter than all of the stars in all the galaxies. But only in gravitational waves [not in any other kind of radiation].' Kip Thorne: 'It is by far the most powerful explosion humans have ever detected except for the big bang'." Rainer Weiss says this is the first test of GR in highly nonlinear regime and it passed with flying colors. You can actually listen to an audio recording of the signal -- better listen carefully, it is near my perception threshold -- here: https://soundcloud.com/emily-lakdawalla

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Re: [math-fun] Can a nonconstant continuous function on the reals
by Warren D Smith 11 Feb '16

11 Feb '16

Question: Can a nonconstant continuous function on the reals assume only rational values? Answer: No, because of https://en.wikipedia.org/wiki/Intermediate_value_theorem More interesting question: Find a function F(x), which is continuous (the smoother the better) which maps rationals to rationals, but is not a rational function. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Something I figured out
by David Wilson 11 Feb '16

11 Feb '16

For 2 x 2 matrix M define F(M) = f : R->R : x => (M11 x + M12) / (M21 x + M22). Then F(AB) = F(A) o F(B) So the composition of unreduced order 1 rational functions is isomorphic to the product of 2x2 matrices. I assume this is well known, but I thought it was pretty cool. This means that finding, say, all order 1 rational functions f with f(f(x)) = x would reduce to finding all 2x2 matrices M with M^2 = I.

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[math-fun] What the brain does when the mind does math
by James Propp 11 Feb '16

11 Feb '16

Has anyone used brain-imaging to figure out which brain-regions get repurposed when the owner of a brain decides to use that brain for off-label purposes like thinking about mathematical objects, proofs, etc.? I've often suspected that sensory and motor circuits in my brain are involved when I do math, and it would be neat if there were neurological data to back up my suspicions that (for instance) when I'm following a logical argument that binds epsilon, and then binds delta, and then unbinds them, the feeling that I'm "submerging" and then "emerging" is not just a subjective feeling but part of the anatomical substrate of my experience. Are there books about this sort of thing? I gather that George Lakoff tries to think about this stuff, but I went to a talk of his a few years ago, and I got the feeling he didn't really understand math at a deep enough level to have worthwhile insights. He was all caught up in trying to understand how the brain processes Godel's Theorem, which is a very atypical piece of mathematical cognition. He also didn't seem very interested in input from the mathematicians who attended his talk (me included), which made me skeptical. Jim Propp

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Re: [math-fun] Can a nonconstant continuous function on the reals
by Warren D Smith 11 Feb '16

11 Feb '16

> Here is my example of a continuous function which maps rationals to > rationals, but is not a rational function: > > F(x) = (-1)^floor(x) * frac(|x|) * (frac(|x|) - 1), > > where frac(x) is the fractional part of x. --well, that was too easy. I should have said "not a piecewise rational function" to make it harder. Or perhaps, as Dan Asimov suggested, demanding it be analytic, or Cinfinity. And one can similarly ask: is there an analytic or Cinfinity function that maps algebraic numbers to algebraic numbers, but is not an algebraic function? Here's one that comes close. Consider the "Klein j-invariant" https://en.wikipedia.org/wiki/J-invariant which is sort of the most fundamental "modular function." "In 1937 Theodor Schneider proved that if X is a quadratic irrational number in the upper half plane then j(X) is an algebraic integer. In addition he proved that if X is an algebraic number but not imaginary quadratic then j(X) is transcendental." Are all algebraic integers generated in this way? Here's another interesting real valued function F(X) of real X. Let [a0; a1,a2,a3,a4...] be the continued fraction expansion of X. For each a[j] which is (i) greater than 4, and (ii) both its neighbors a[j+1] and a[j-1] also are greater than 4, replace a[j] by a[j]+1. This editing process converts X into F(X). This function F(X) has the properties that (1) if X is rational, then F(X) is too. (2) F(X) is not a rational function. Nor is it a piecewise rational function (or anyhow if it were considered one then the number of pieces would be uncountably infinite and the breakpoints would be dense everywhere). (3) if X is quadratic irrational, which happens exactly when X's continued fraction is ultimately-periodic, then F(X) also is. But F(X) is not an algebraic function. (4) I cannot believe F(X) is analytic. (5) Is F(X) continuous? (I think yes. Certainly if X and Y are both rational and |X-Y|-->0, then |F(X)-F(Y)|-->0.) (6) If F(X) differentiable? Well, it certainly is possible to define a derivative F'(X) whenever X is a rational number. Indeed, I think the Kth derivative may be defined whenever X is a rational number, for any K=1,2,3,...? I'm not sure whether these are continuous functions of X; but if they all were then F(X) would be Cinfinity. There also are interesting variants of this construction, such as, a[j] is edited to become, not a[j]+1, but rather a[j]+2^(-j).

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[math-fun] alleged detection of gravity waves by LIGO?
by Warren D Smith 10 Feb '16

10 Feb '16

A press conference will be held thursday (11 Feb 2016) by the LIGO team and a paper released. Allegedly (according to leaks) both LIGO sites saw an event corresponding to a merger via inspiral with later "kerr ringdown," of two black holes of masses 36 & 29 solar masses with final mass 62. Thus about 3 solar masses worth of energy were released in the form of gravity waves in a matter of seconds. The reconstructed orbital speed is almost exactly the speed of light. This is the first direct detection of gravity waves and presumably an excellent confirmation of general relativity. It took 14 years for LIGO to detect its first signal and allegedly they believe they can and will enhance its sensitivity further by the year 2021. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Can a nonconstant continuous function on the reals
by Warren D Smith 10 Feb '16

10 Feb '16

> From: Marc LeBrun <mlb(a)well.com> > I'm curious how this applies to another sort of edit, the "rebasing" > operations, for instance > > G(X) := replace 2^n in the binary expansion of X with 3^n. > > G maps rationals into rationals, the unit interval into itself, preserves > ordering and so on. > >> (5) Is F(X) continuous? (I think yes. Certainly if X and Y are both >> rational >> and |X-Y|-->0, then |F(X)-F(Y)|-->0.) > > Also true for G, I guess? However there are an awful lot of "holes" in the > range -- every number that contains a 2 anywhere in its ternary expansion. --umm. Actually in view of the intermediate value theorem, neither F nor G can be continuous, because of these holes. Oops.

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