December 2013 - mathfun@mailman.xmission.com

[math-fun] electron electric dipole moment, if any, is small
by Warren D Smith 20 Dec '13

20 Dec '13

http://arxiv.org/abs/1310.7534 says electron electric dipole moment is (-2.1 +- 3.7stat +- 2.5syst) * 10^(-31) * e * meter. Press descriptions claim standard model predicts exactly zero. Really? Certainly plain QED predicts zero, but the weak force being chiral, and known T and P violations in the standard model, might perhaps allow a small nonzero value for electron, proton, or neutron electric dipole moments. I saw no basis for the (bogus?) claim made or implied in press descriptions of this, that the standard model forces these to be exactly 0. Indeed, I've seen claims SM predicts roughly neutron: 10^(-34) e * meter electron: 10^(-43) e * meter proton: 10^(-33) e * meter They say extensions to the standard model such as "weak scale supersymmetry," predict 10^(-32) to 10^(-29) e * meter for the electron electric dipole moment. They hope to wring another factor of 10 out of refined future versions of their experiment, which would mostly rule this out. How does their experiment work? Should we believe it? I follow that ThO is a highly polar neutral molecule, containing internal electric fields they say of 84 GV/cm, so if its electrons had an electric dipole moment I presume that would yield detectable energy level shifts and possible new spectral excitations. Why the hell whatever they measured proves there are no such... that was not clearly explained. Anyhow, if an electron had dipole moment D in units of e*meter, then expect energy shift of order 8.4*10^12 * D * ElectronVolt, so their claim |D|<10^(-30) means they believe they can somehow can rule out energy shifts of the order 10^(-17) ElectronVolts. I pointed out to them in email that if so, they ought to be able to show, by using ordinary nitrogen N2, that the electron electric QUADrupole |moment| is below about 0.001 barns*e. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] x^2 sin 1/x and all that
by Bill Gosper 19 Dec '13

19 Dec '13

On 2013-12-19 11:13, Veit Elser wrote: > The "shuddering" singularity of Euler's disk is the best example I can think of. > > As a prototype of mechanical behavior with a "finite time singularity", a paper > by Moffatt in Nature made quite a splash in 2000. However, there were a number > of problems with both the physics and the math in that paper. A readable summary > of the issues can be found here: > > http://ruina.tam.cornell.edu/research/topics/miscellaneous/comments_on_moff… > > If more generally you are interested in examples of physical dynamics, where > regular initial conditions lead to singularities (that are resolved in > the breakdown of > the physical model in extreme limits) in finite time, one of the most > famous is the > regularity of the solutions of the Euler equations for incompressible, > inviscid fluids. > It's believed that you can get singular vortex cores in a finite time, > but last time I > checked there was no proof. > > -Veit > > On Dec 18, 2013, at 3:24 PM, James Propp <jamespropp(a)gmail.com> wrote: > >> For years I've taught my honors calculus students about functions like x^2 >> sin 1/x, and for just as many years I've told them that they won't >> encounter functions like this outside theoretical mathematics. >> >> But now I'm wondering whether simplified mathematical models of Euler's >> disk (see http://en.wikipedia.org/wiki/Euler%27s_Disk) or other idealized >> physical systems might involve functions in which the amplitude of some >> oscillatory quantity goes to zero while the frequency goes to infinity in >> finite time, and in particular, whether there might be "natural" examples >> of differentiable functions with discontinuous derivatives. >> >> What do you think? >> >> Jim Propp It reminds me that you can follow a finite rhumb line northeast to the Pole, irreversibly winding up at a point not northeast of anywhere. http://gosper.org/rhumb.png (Contributing to the irreversibility is a doubtlessly fatal case of whiplash.) --rwg

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[math-fun] x^2 sin 1/x and all that
by James Propp 19 Dec '13

19 Dec '13

For years I've taught my honors calculus students about functions like x^2 sin 1/x, and for just as many years I've told them that they won't encounter functions like this outside theoretical mathematics. But now I'm wondering whether simplified mathematical models of Euler's disk (see http://en.wikipedia.org/wiki/Euler%27s_Disk) or other idealized physical systems might involve functions in which the amplitude of some oscillatory quantity goes to zero while the frequency goes to infinity in finite time, and in particular, whether there might be "natural" examples of differentiable functions with discontinuous derivatives. What do you think? Jim Propp

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Re: [math-fun] "Aurifeuillean" factorization of Mersenne^2+8
by Bill Gosper 19 Dec '13

19 Dec '13

Quoting Bill Gosper <billgosper(a)gmail.com>: <clip> On Tue, Dec 17, 2013 at 6:20 PM, Bill Gosper <billgosper(a)gmail.com> wrote: Neil[B] and I recently noticed (2^(2*k + 1) - 1)^2 + 8 == (2^(2*k + 1) + 2^(k + 2) + 3) * (2^(2*k + 1) - 2^(k + 2) + 3) Old? --rwg This is pretty dull, since Factor[...] can simply do it. Slightly more interesting is In[182]:= Factor[(2^(k + 1) - 3)^2 - 8] Out[182]= 1 - 3 2^(2 + k) + 2^(2 + 2 k) (Fails) but both the "odd" and "even" subcases work: In[185]:= Factor[(2^(2*k + 1) - 3)^2 - 8] Out[185]= (1 - 2^(2 + k) + 2^(1 + 2 k)) (1 + 2^(2 + k) + 2^(1 + 2 k)) In[186]:= Factor[(2^(2*k) - 3)^2 - 8] Out[186]= (-1 + 2^(2 k) - 2^(1 + k)) (-1 + 2^(2 k) + 2^(1 + k)) --rwg In the even case, we can generalize 2^_ to x^_: In[208]:= Factor[(x^(2*k) - 3)^2 - 8] Out[208]= (-1 - 2 x^k + x^(2 k)) (-1 + 2 x^k + x^(2 k)) In[210]:= Expand[%] Out[210]= 1 - 6 x^(2 k) + x^(4 k) But In[209]:= Factor[(x^(2*k + 1) - 3)^2 - 8] Out[209]= 1 - 6 x^(1 + 2 k) + x^(2 + 4 k) In fact, In[210]:= PrimeQ[% /. x -> 6 /. k -> 16] Out[210]= True --rwg rcs> Out209 can be morphed to Y^4 - 6 Y^2 + 1 = (Y^2 - 1)^2 - 4 Y^2 = (Y^2 - 2 Y - 1) * (Y^2 + 2 Y - 1). This is similar to the canonical Aurif factorizations for 2^(4K+2)+1 and 3^(6K+3)+1, which can be morphed to factorizations of 4 Y^4 + 1 = (2Y^2+1)^2 - 4Y^2 and 27 Y^6 + 1 = (3Y^2+1) (9Y^4 - 3Y^2 + 1), with the final factor splitting as (3Y^2+1)^2 - 9Y^2. I like to think of these as [1 -6 1] = [1 -2 1] - [4 0] and [4 0 1] = [4 4 1] - [4 0] and [9 -3 1] = [9 6 1] - [9 0], followed by making the subtrahends into squares. Rich ------ Rich, then how would you explain the primality of (6^33-3)^2-8? Your "morphing" requires Factor[_,Extension->√2] . --rwg

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[math-fun] "Aurifeuillean" factorization of Mersenne^2+8
by Bill Gosper 19 Dec '13

19 Dec '13

Neil and I recently noticed > (2^(2*k + 1) - 1)^2 + 8 == > (2^(2*k + 1) + 2^(k + 2) + 3) * (2^(2*k + 1) - 2^(k + 2) + 3) > Old? > --rwg >

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[math-fun] What the Milky Way looks like (Warren D Smith)
by Warren D Smith 18 Dec '13

18 Dec '13

another picture is http://images.sciencedaily.com/2013/12/131217085054-large.jpg which has a circled + indicating location of our sun. We appear to have defied probability a bit by not being in either the nucleus or a major arm, sort of off in a minor arm, boondocks area. The red blobs indicate especially bright stars.

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Re: [math-fun] math-fun Digest, Vol 130, Issue 16
by Warren D Smith 18 Dec '13

18 Dec '13

> 7. What the Milky Way looks like (Warren D Smith) another picture is http://images.sciencedaily.com/2013/12/131217085054-large.jpg which has a circled + indicating location of our sun. We appear to have defied probability a bit by not being in either the nucleus or a major arm, sort of off in a minor arm, boondocks area. The red blobs indicate especially bright stars.

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[math-fun] What the Milky Way looks like
by Warren D Smith 18 Dec '13

18 Dec '13

http://mnras.oxfordjournals.org/content/437/2/1791/F6.large.jpg based on some thousands of star distance & angle measurements. Picture seems to be missing the little arrow saying "you are here."

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Re: [math-fun] "life" generalized CAs classified ala Wolfram?
by Schroeppel, Richard 17 Dec '13

17 Dec '13

One detail-level problem with the 4-cell neighborhood is: Consider the options for a 0 cell with three 0 neighbors and one 1 neighbor: If the successor state is 0, then no pattern can grow beyond its bounding box, preventing replication and unbounded computations. If the successor state is 1, most patterns grow at the speed of light and fill space with 1s. There are several ways to get around this problem, but they all involve complications: More states, or more history, or varying the geometry, etc. Rich ________________________________________ From: math-fun-bounces(a)mailman.xmission.com [math-fun-bounces(a)mailman.xmission.com] on behalf of Adam P. Goucher [apgoucher(a)gmx.com] Sent: Tuesday, December 17, 2013 9:22 AM To: math-fun Subject: [EXTERNAL] Re: [math-fun] "life" generalized CAs classified ala Wolfram? > > Marc LeBrun wrote: > >Alternatively we might start with the simpler 4 cell neighborhood, thus only > 2^10 cases. > > --the checkerboard neighborhood graph is bipartite, so this probably > not a good idea. A cell is connected to itself, so I fail to see what the problem is. Anyway, I seem to recall that Edgar F. Codd already looked at all of these 1024 rules and found that none were interesting. > Using the equilateral triangle lattice, 6 neighbors, might be better, though. Yes, the hexagonal lattice A_2 is awesome. The next time that a single lattice simultaneously wins the packing, covering and quantising problems is the Leech lattice in 24 dimensions (rather unfairly, A_8* beats E_8 in terms of covering). Sincerely, Adam P. Goucher _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] "life" generalized CAs classified ala Wolfram?
by Adam P. Goucher 17 Dec '13

17 Dec '13

> > Marc LeBrun wrote: > >Alternatively we might start with the simpler 4 cell neighborhood, thus only > 2^10 cases. > > --the checkerboard neighborhood graph is bipartite, so this probably > not a good idea. A cell is connected to itself, so I fail to see what the problem is. Anyway, I seem to recall that Edgar F. Codd already looked at all of these 1024 rules and found that none were interesting. > Using the equilateral triangle lattice, 6 neighbors, might be better, though. Yes, the hexagonal lattice A_2 is awesome. The next time that a single lattice simultaneously wins the packing, covering and quantising problems is the Leech lattice in 24 dimensions (rather unfairly, A_8* beats E_8 in terms of covering). Sincerely, Adam P. Goucher

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