February 2013 - mathfun@mailman.xmission.com

[math-fun] Revolution in physics coming?
by Warren Smith 09 Feb '13

09 Feb '13

I'd be interested to here more ranting on the topic from H.Baker. (A bit more coherent preferably, he sounds like he's getting at something interesting, but what?)

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Re: [math-fun] Attesting to Atoms
by Henry Baker 09 Feb '13

09 Feb '13

Planck's little 1905 Dover book which resolved the "ultraviolet catastrophe" of black body radiation remains (to me) one of the outstanding triumphs of pure thought. Planck extends Maxwell & Boltzmann to the point where energy _has_ to be quantized. (I don't have my copy in front of me just now, but I think this is the correct book. I highly recommend it.) http://www.amazon.com/Theory-Radiation-Dover-Books-Physics/dp/0486668118/ Maxwell's & Boltzmann's own approaches aren't to be sneezed at. Once I read Planck's book, I could no longer accept any physical theory based on real numbers & calculus. I'm still amazed at how far physics has gotten with the absurd notions of real numbers (& therefore complex numbers) & calculus, and a complete disrespect for bits and information. The field of quantum computation seems to have gotten completely stuck, because the computer scientists have allowed the field to be taken over by physicists instead of providing the physicists an alternate paradigm. A revolution in physics is coming, very much akin to the quantum revolution, wherein real numbers & calculus are finally replaced with an inherently discrete mathematics. You can't seriously believe that a quantum state is represented by _all_ of the complex numbers, no matter how many bits of information would be required to describe that complex number. At 09:36 PM 2/8/2013, Rowan Hamilton wrote: >Rutherford scattering should give the effective size of the nucleus. >(Though note that "size" is a squishy term in quantum mechanics.) >Quantum electro-dynamics predicts the orbital structure of the >electrons, which should give an effective size to any given atom. > >As to arbitrarily small structures, the Planck scale seems to give the >bottom, if you believe quantum gravitation theories. I have always >found this idea interesting, since if you quantize gravity then >space-time becomes quantized, which means (at least to this naive >observer) that the universe is countable. How cool is that? > >Rowan. > >On Fri, Feb 8, 2013 at 9:17 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> I don't know how one would guess the size of atoms from crystal structure, >> but I'd be surprised if there weren't some way to make a reasonable guess. >> >> At the opposite extreme, if all scales made equal physical sense, >> then there would be no way for nature to choose some scale. >> >> Also, Occam's Razor would probably complain that that isn't the >> simplest physical model that fits the evidence. (Not conclusive >> of course, but suggestive.) >> >> On the third hand, I wouldn't be at all surprised if as more is learned >> about physics, it becomes increasingly clear that there is some kind of >> structure at scales that are arbitrarily small. Or not. >> >> --Dan >> >> >> On 2013-02-08, at 9:05 PM, James Propp wrote: >> >>> I just watched George Hart's video >>> https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left >>> with a vexing disquiet about the fact that the macroscopic structure of >>> crystals seems to imply the existence of atoms and yet gives us no >>> information about how big atoms are. If the observed structure of >>> macroscopic crystals is compatible with an infinite range of models of >>> reality, each positing the existence of atoms but at ever-smaller scales, >>> could there be some sort of projective limit of these theories, with "cubes >>> all the way down" but no bottom level? I'm not saying it's a believable >>> physical theory, but it seems like it would give an example of a universe >>> with crystals but without atoms. Or is this idea incoherent in some way? >>> >>> Jim Propp

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[math-fun] New Mersenne Prime
by Guy Haworth 09 Feb '13

09 Feb '13

Simon Plouffe's estimate - 'more than 2 months' - for the minimum time to run a Lucas-residue test on an M(p) ... is way over the mark. http://mersenne.org/ reports that the three confirmations of the latest prime M(p) took 3.6, 4.5 and 6 days. The main reason for confirming that prime M(p) are 'likely' to be the Nth prime M(p) by size, and 'proved' to be so ... is the residue of first- and second-LR-tests to be run. http://mersenne.org/report_milestones/default.php gives notice of: a) prime M(p), b) residual first-LR-testing of prime exponents 'p', and c) residual dual sourcing of Lucas residues for M(p) Thus, 51,552 M(p) smaller than the latest record prime M(p) have not even been LR-tested once ... and 565,667 have not been tested twice. I have not heard of any M(p) being seen as composite and then later as prime ... so accuracy has apparently been good in GIMPS territory. Note that the 46th and 47th prime M(p) found were both smaller than the 45th. That was a first for GIMPS but not a first overall. Guy

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[math-fun] Lissajous illusion
by Bill Gosper 07 Feb '13

07 Feb '13

This simple fact might make a nice math objet d'art. Labeled[Manipulate[ Labeled[ParametricPlot[{{t - 2*\[Pi] - 1, Cos[a*(t + v) + \[Phi]]}, {Sin[b*(t + v)], t - 2*\[Pi] - 1}, {Cos[t], Sin[t]}/11 + {Sin[b*v], Cos[a*v + \[Phi]]}, {Sin[b*v], t/\[Pi] - 1}, {t/\[Pi] - 1, Cos[a*v + \[Phi]]}, {Sin[b*t], Cos[a*t + \[Phi]]}}, {t, 0, 2*\[Pi]}, PlotRange -> {{-4 - \[Pi], 9/8}, {-4 - \[Pi], 9/8}}], {StringJoin[ "Vertical frequency a=", ToString[a]], StringJoin["Horizontal frequency b=", ToString[b]], StringJoin["Phase \[Phi]=", ToString[\[Phi]/\[Pi]], "\[Pi]"]}, {Top, Right, Bottom}, RotateLabel -> True], {v, -\[Pi], \[Pi]}, {\[Phi], 0, 2 \[Pi]}, {a, Range[5]}, {b, Range[5]}], "Lissajous Mechanism", Top] Click frequencies a to 3, b to 4, say. Pop open the + sign on the phase (φ) parameter and run it slowly. Imagine the precessing Lissajous figure to be drawn on a rotating transparent cylinder. Is it vertical or horizontal? So make two metal "Lissajous hoops", one for each axis and rotate them side by side on mutually perpendicular turntables. --rwg

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[math-fun] Isotope separation via bosons/fermions
by Warren Smith 07 Feb '13

07 Feb '13

> At the risk of upsetting Big Brother, what is the boson/fermion status of > U-235 and U-238 ? > > Is one a fermion and the other a boson, or are they both fermions or both > bosons ? -- U=element 92. Since 92 is even: 235 is odd, hence U235 is a fermion. 238 is even, so U238 is a boson.

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[math-fun] Isotope separation via bosons/fermions
by Henry Baker 07 Feb '13

07 Feb '13

I just listened to a lecture regarding supercold (micro/nano Kelvin) atoms. I hadn't realized it before, but two different isotopes of the same element can have vastly different properties at extremely low temperatures if one is a boson and the other is a fermion. Apparently, someone showed this happening with two different isotopes of lithium. Can these properties be utilized to _separate_/_sort_ isotopes into the part containing the bosonic isotopes and the part containing the fermionic isotopes?

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Re: [math-fun] Isotope separation via bosons/fermions
by Henry Baker 07 Feb '13

07 Feb '13

At the risk of upsetting Big Brother, what is the boson/fermion status of U-235 and U-238 ? Is one a fermion and the other a boson, or are they both fermions or both bosons ? At 06:17 AM 2/7/2013, Veit Elser wrote: >On Feb 6, 2013, at 8:21 PM, Warut Roonguthai <warut822(a)gmail.com> wrote: > >> For the same reason, superfluid helium-3 is not miscible with superfluid helium-4. >> Liquids of the same element but don't mix, amazing? >> >> Warut > >Yes, it is amazing. However, the isotope separation is not perfect, even in the >limit of zero temperature. While the He3-rich phase is pure in that limit, about >7% He3 is dissolved in the He4-rich phase. > >Now you know the origin of the phrase "7-percent solution". > >-Veit

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[math-fun] Isotope separation via bosons/fermions
by Warren Smith 07 Feb '13

07 Feb '13

http://web.mit.edu/newsoffice/2005/matter.html I presume? Even if isotope separation based on boson/fermion-ness possible, wouldn't be economical. Cooling to 50 nanokelvin gas as in the lithium experiment is extremely expensive per kilogram. In fact, a single gram would probably bankrupt the world. --- Here's something little-related to think on. Allegedly, neutron star matter is not merely a liquid, but a superfluid. Critical temperature allegedly about 10^10 kelvin. Also electrical superconductivity allegedly going on. You may ask how they can possibly know enough theoretically or experimentally to claim that. Well, I admit to a lot of such skepticism myself, but what you could naively do is just say "well, we know helium-3 goes superfluid at 0.002491 kelvin... so now attempt to scale appropriately for neutrons given their much greater density and interaction-energies and what order of magnitude estimate would you get for neutron superfluidity temperature?" I get 10^8 kelvin, disagreeing with the allege, but making the phenomenon at least plausible. But then we note that helium stops being superfluid, in fact solidifies, when 25 atm pressure applied. So might neutron-star-stuff solidify at the (enormous) pressures it is under? I suspect not. In fact, the more pressure you apply, the more "fermi confinement energy" the neutrons should gain, presumably becoming more delocalized, i.e. more pressure should make it more liquid. That re-liquification might also happen with helium, but if it does, only at pressures higher than experimenters have yet reached.

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[math-fun] Re New Mersenne Prime
by Guy Haworth 06 Feb '13

06 Feb '13

Why is the new prime M(p) not the 48th known? As Charles Greathouse says, GIMPS is a distributed project. So, inevitably, any unfinished Lucas-residue computations on smaller M(p) will have to be completed, and verified as correct, before it is 'certain' where this new prime is in the sequence of prime M(p). It was precisely this systematic twin-sourcing that was missing before my work with the ICL DAP team in the 1980s set that standard. Slowinski 'aimed' the early CRAYs at the next prime M(p). Second-sourcing known-factors and zero Lucas-residues indicating prime M(p) is obviously not a problem. The latest prime find has been sourced four independent ways already. Naturally, people want to find the next record M(p) or even the first 100m-digit prime M(p) - so there is a bit of a resource-blight behind the largest M(p). Fortunately, older machines with less memory cache tend to be given this second-sourcing task. http://mersenne.org/primenet/ (Primenet Summary, Work Distribution Map) indicates that: - there's quite a bit of LR second-sourcing to be done in the 'p in (26m, 44m)' range - below 58m, over 67,000 M(p) have not had a single LR-test, let alone two, - curiously, some validated LR-testing effort is being put in on a few M(p) with p = ~195m Some analysis of the results-distribution would indicate what the balance of effort is on second-sourcing LRs, 1st-LR-testing below the largest prime M(p) and 1st LR-testing above the new record. Suffice it to say that the 6-year time-lag (in placing the 42nd prime M(p) AS the 42nd prime M(p)) is a measure of the lack of second-sourcing effort. Guy http://www.reading.ac.uk/sse/about/staff/g-haworth.aspx http://centaur.reading.ac.uk/view/creators/90000763.default.html

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Re: [math-fun] New Mersenne Prime
by Guy Haworth 06 Feb '13

06 Feb '13

[ I think 'Mersenne Primes' is why I'm on the maths-fun list. Thanks Richard. ] The record M(57,885,161), discovered by Prof. Cooper et al, is a prime with 17,425,170 decimal digits. It is the 48th known prime M(p). It may yet not be the 48th in size: GIMPS have only just confirmed the '42nd by size' position of the prime M(p) discovered in 2005. http://www.mersenne.org/report_milestones/ Its discovery ends the longest wait - nearly 4 years - for the 'largest prime' record to be broken since the Great Internet Mersenne Prime Search (GIMPS) began. Curt Noll kindly credited my Mersenne work 30 years ago as being part of the reason GIMPS began, hence Chris Caldwell's note at http://primes.utm.edu/glossary/xpage/GIMPS.html Dan Shanks, whose wonderful book first interested me in Mersenne Numbers in 1967, included an amusing piece about that work in the 3rd edition. GIMPS is the oldest continuously running crowd-sourced programme running on the web and has clocked over 100 TFLOPS. The prime-indexes, circa 100,000, that my ICL colleagues were working with then seem so small now! The back-history to 1987, and our modest results of the 1980s, can be consulted here: Back-history: http://centaur.reading.ac.uk/4571/1/1987_H_Mersenne_Numbers.pdf ICL DAP results: http://centaur.reading.ac.uk/5948/1/1986_H_Mp_Consolidated_Results.pdf AMS Abstracts note 1: http://centaur.reading.ac.uk/4555/1/1983_Abstracts_of_the_AMS_v4.2_p196.pdf AMS Abstracts note 2: http://centaur.reading.ac.uk/4572/1/1986_Abstracts_of_the_AMS_v7.2_pp224-5.… Huge congratulations to Prof. Cooper and team. It is the third prime M(p) that they have found. M(M(31)) remains the largest Double Mersenne Number whose primality status is known and that record may stand for 31 years (1983-2014) ... see 'Back History' above ... :-) Guy

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