May 2016 - mathfun@mailman.xmission.com

[math-fun] Two triangles in one (Zak Seidov)
by Warren D Smith 19 May '16

19 May '16

> See figure: > http://zak08.livejournal.com/28922.html well, obviously if the two interior triangles each are halfsize similar copies of the original, then they both will fit inside it, inscribed in one corner each, disjointly except for 1 point, and each have half the perimeter. So that is one solution. It might be feasible to describe the full set of solutions, if we knew exactly what the rules were. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

[math-fun] Quantum gravity questions from Veit Elser & blind-leading-the-blind answer (sort of)
by Warren D Smith 19 May '16

19 May '16

> 1. Re: Simple description of general relativity. Physics tries > not to waste time :) (Veit Elser) > Message: 1 > Date: Thu, 19 May 2016 16:46:07 +0000 > From: Veit Elser <ve10(a)cornell.edu> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Simple description of general relativity. > Physics tries not to waste time :) > Message-ID: <A2861FB7-5047-4610-B002-149D653C8CEC(a)cornell.edu> > Content-Type: text/plain; charset="utf-8" > > >> On May 19, 2016, at 10:27 AM, Warren D Smith <warren.wds(a)gmail.com> >> wrote: >> >> You've heard of the "principle of least time" in optics... light >> follows a path which minimizes the time it takes to get there... >> in quantum field theory there is a more general action principle, >> which is that the integral, over all paths followed by all particles, >> of exp(i*T*m*c^2/hbar) dT >> is stationarized, where T is the proper time consumed by that particle >> (of rest mass m) in following its trajectory. >> (Actually things a bit trickier for spinor particles, and there are >> certain constant factors inserted whenever two particles merge or >> bifurcate, >> but I will ignore that here.) > > Taking seriously -- in quantum mechanics -- the sum over all paths, one has > to include particle world-lines parts of which are not time-like and over > which > T is pure imaginary. With the right choice of sign, ?tachyonic? propagation > is > thereby suppressed in a direct way. > > If the Hilbert action formulation is the right way to think about classical > GR, > i.e. the Einstein equations are just a consequence, then it seems > reasonable > that action-stationarity in that setting is also the hbar->0 asymptotics of > a > sum-over-whatever (e.g. metrics) principle. Nobody has yet made sense of > that sum. Are there real-exponentially suppressed configurations? > > -Veit ---The idea you just mentioned is the Wheeler-DeWitt equation and had already been explored by Bryce DeWitt (starting in the 1950s or 1960s I think?). To put it in a nutshell, the space of all 3+1 metrics is (if I recall right) called "superspace." It is an infinite dimensional space which seems pretty damn hard to comprehend or work with. Anyhow, the Wheeler-DeWitt equation is basically Schrodinger's equation set in superspace (which they figured out how to do somehow). Then you proclaim that the universe is a quantum system whose state is a wave function on superspace. In the classical limit hbar-->0, a single trajectory in superspace gets all the probability density, just like in quantum mechanics where (allegedly) a particle in the classical limit moves along a well defined trajectory. That trajectory is the one stationarizing the action, which explains classical mechanics action principles. This "fact" is "seen" because the Feynman path-integral formulation of QM Schrodinger equation really puts in all trajectories with a complex phase arising from exp(i*action/hbar). Only the extrema-of-the-action trajectories get to interfere constructively, the rest cancel out. I am using the scare quotes to indicate that the rigor in this whole line of handwaving jive, aka theoretical physics, has pretty much never managed to come... but it is a great story. So in the Wheeler-DeWitt version of the story, the Einstein GR equations giving a unique evolution of our 3+1 metric are an approximate fiction arising in classical limit, since they arise from Hilbert action principle... and the real story is the Wheeler-DeWitt equations of quantum gravity, and the real arena of physics is not 3+1 space, it is superspace. [By the way, do not confuse this superspace with supersymmetry, which also has sometimes got something they call "superspace", totally different.] OK, sounds like a great story. As far as I understand (which is not much) they did succeed in a few cases in solving the WDW equation to get some exact solutions, but for the most part this whole theory of quantum gravity (if it is one) has been unusable. Further, I do not know if they were able to put in source terms (in fact I doubt they were able to do it in any successful way). Later so called "loop theory" [totally unrelated to abstract-algebra "loops" aka nonassociative groups] was cooked up by Abhay Ashtekar and followers as a way to try to quantize GR. And it presumably in some sense must be equivalent to the Wheeler-DeWitt equation, albeit expressed very differently... but now you are getting totally above my head. I have reason to suspect those guys do not know what they are doing. But they know more than me at least about some stuff. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

[math-fun] Wilson a^2+b^2-c^2 Conjecture; now proven & extended
by Warren D Smith 19 May '16

19 May '16

RCS's proof is (rewritten so I can follow it): If N>=0 then (5N)^2 < (3N-1)^2 + (4N+1)^2 = (5N)^2 + 2N+2 < (5N+1)^2 (5N)^2 < (3N+2)^2 + (4N)^2 = (5N)^2 + 12N+4 = (5N+1)^2 + 2N+3 < (5N+2)^2 Therefore any even number of form 2N+2 or odd number of form 2N+3 if N>50, is representable in the Wilson a^2+b^2-c^2 form with c=floor(sqrt(a^2+b^2)); and this is all numbers except for some finite set of small integers, which are covered by the computer search I'd done (which was large overkill for that). Actually the exceptional set is {0, 1} so if RCS and I split up the work we can probably handle it even without a computer. Brilliantly simple. (I'd actually thought of the same kind of idea but had somehow managed to confuse myself into thinking it was not adequate...) ===== Now to go a bit further, be nice to prove that not only is every number representable but in fact has an infinite set of representations. Which I shall now show. If r and s fixed integers then for all sufficiently large N (5N+12r+12s)^2 < (3N+20r)^2+(4N+15s)^2 = (5N)^2 + 120(r+s)N+(20r)^2+(15s)^2 = (5N+12r+12s)^2 + (16r-9s)^2 < (5N+12r+12s+1)^2 inspiring (5N+12r+12s)^2 < (3N+20r-1)^2+(4N+15s+1)^2 = = (5N+12r+12s)^2 + (16r-9s)^2 + 2N+2+30s-40r = (5N+12r+12s+1)^2 + (16r-9s)^2 + 1+6s-64r-8N So every even number (2N+2+30s-40r) is representable in an infinite set of Wilsonian ways, e.g. 2N+2>=0 is by considering the infinite set of (r,s) with 30s-40r=0. (5N+12r+12s)^2 < (3N+20r+2)^2+(4N+15s)^2 = = (5N+12r+12s)^2 + (16r-9s)^2 + 12N+4+80r = (5N+12r+12s+1)^2 + (16r-9s)^2 + 2N+3+56r-24s = (5N+12r+12s+2)^2 + (16r-9s)^2 + 32r-48s-8N So every odd number (2N+3+56r-24s) is representable in an infinite set of Wilsonian ways, e.g. 2N+3>=0 is by considering the infinite set of (r,s) with 56r=24s and r>=0. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

2 2

[math-fun] Simple description of general relativity. Physics tries not to waste time :)
by Warren D Smith 19 May '16

19 May '16

You've heard of the "principle of least time" in optics... light follows a path which minimizes the time it takes to get there... in quantum field theory there is a more general action principle, which is that the integral, over all paths followed by all particles, of exp(i*T*m*c^2/hbar) dT is stationarized, where T is the proper time consumed by that particle (of rest mass m) in following its trajectory. (Actually things a bit trickier for spinor particles, and there are certain constant factors inserted whenever two particles merge or bifurcate, but I will ignore that here.) So my claim (which I haven't seen before but it isn't really new in the sense it is just a physical rephrasing of the mathematical Hilbert action) is: general relativity is also viewable as a stationarized-time principle. Sit someplace. (And by "you" I mean, a particle of infinitesimal mass & diameter, equipped with clock, free will, observation and communication capabilities, etc.) Distribute your pals (similar particles) around you uniformly on the surface of a infinitesimal-radius sphere centered at you. Now wait. The clock readings of you and your pals will gradually diverge if you are not in "flat space." There will be a beginning and an end of this experiment. At the end, the total score is the difference between your clock time, and the mean among your pals' clock readings ("wasted time"). So what is general relativity? It is just this. At the beginning, imagine such experiments are happening everywhere, uniformly on the spatial metric. (The "beginning" is a spatial slice of a 3+1 dimensional metric. So is the "end.") In between the beginning and the end, the 3+1 metric evolves in some manner. There are time-slices, each a spatial section (3-metric) of the 3+1 metric, which are infinitesimally close together in time. Each adjacent time slice should be regarded as the end of an old, and beginning of a new, such experiment. My claim is: the 3+1 metric is the one which stationarizes the total score of all the experiments. Physics is weird and "paradoxical" in the sense time is not the same for me and for you. But the Einstein vacuum is sort of trying to keep everything as consistent as possible given the situation at the beginning. Trying to be the least paradoxical it can be, sort of. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

2 1

[math-fun] Base six ??
by Henry Baker 19 May '16

19 May '16

FYI -- two-trit? bi-trit? hexit? tritter? http://scitation.aip.org/content/aip/journal/apl/108/18/10.1063/1.4948455 Towards a six-state magnetic memory element We pattern permalloyfilms into three crossing elongated ellipses with an angle of 60Â° between the major axes of any pair of ellipses. Planar Hall effect measurements show that the magnetization in the area of overlap of the ellipses has six stable magnetic orientations parallel to the major axes of the three ellipses. We determine the effective anisotropy field for small magnetic deviations from the easy axis and the switching field between the easy axes as a function of magnetic field orientation. We compare our results with micromagnetic simulations and present an effective Hamiltonian that captures the magnetic response. We show how such magnetic structures in a magnetic tunnel junction would result in a magnetic memory element with six distinct resistance states that could be written using spin-orbit torques.

3 2

[math-fun] Plot of Rational Distance Problem triples
by Ed Pegg Jr 19 May '16

19 May '16

http://i.imgur.com/YclyNMC.gif Take a unit square. Plot a few thousand points (a,b) that are a rational distance from three vertices (all but (1,1)) in black. Plot the inverse point (a,b)/(a^2+b^2) in red. Can anyone see the ellipse 3 (x^2+ y^2)-4(x+y)+2 x y=0 ? Or the cubic curve 4 (y^3+ x^3)-3 (y^2+ x^2)+2x y ( 2y+ 2x -1)=0 ? --Ed Pegg Jr Points on the ellipse so far: {{-(4/17),16/51}, {28/33,112/99}, {-(140/387),112/129}, {-(224/627),140/209}, {340/627,272/209}, {736/1107,460/369}, {-(476/1539),544/513}, {-(80/1683),28/561}, {868/2211,992/737}, {-(368/1089),3220/3267}, {3280/3363,1148/1121}, {-(372/1441),4960/4323}, {-(1904/5907),1020/1969}, {752/1649,6580/4947}, {588/1921,7840/5763}, {-(1012/2993),5152/8979}, {-(2156/9843),3920/3281}, {-(3008/10659),3948/3553}, {-(448/3569),1540/10707}, {8176/10947,4380/3649}, {3124/12483,5680/4161}, {5056/14691,6636/4897}, {-(1968/5377),12628/16131}, {-(2132/5841),13120/17523}, {-(1820/18867),672/6289}, {-(3692/19491),7952/6497}, {3916/5457,19936/16371}, {6592/7089,22660/21267}, {5044/23859,10864/7953}, {-(2480/8241),11284/24723}, {-(6320/26643),10428/8881}, {9520/34563,15708/11521}, {5424/9073,34804/27219}, {-(12848/36579),11388/12193}, {35620/37587,13152/12529}, {-(5984/37827),2380/12609}, {6188/10001,38080/30003}, {-(1940/11649),43456/34947}, {-(15980/51507),8272/17169}, {-(6052/17441),49840/52323}, {12080/15521,54964/46563}, {2540/13889,56896/41667}, {-(476/57123),160/19041}, {28336/57171,25116/19057}, {-(8636/58179),24384/19393}, {10948/67971,30912/22657}, {-(3808/18737),68068/56211}, {-(20384/70851),9996/23617}, {-(23504/72627),24860/24209}, {67936/74787,27020/24929}, {-(27740/79827),16352/26609}, {-(9044/27537),83776/82611}, {-(12236/84579),4784/28193}, {-(12236/91443),38640/30481}, {-(11536/31553),77044/94659}, {5344/23313,95524/69939}, {73780/96387,38192/32129}, {-(17360/98643),7068/32881}, {15124/105123,47760/35041}, {-(18704/105507),43420/35169}, {45136/106947,47740/35649}, {12988/26961,106960/80883}, {113764/114243,38240/38081}, {-(14420/39729),85696/119187}, {-(14240/41361),74404/124083}, {-(12880/43601),142324/130803}, {-(45440/162963),21868/54321}, {-(19328/54417),149188/163251}, {17708/40913,164032/122739}, {-(63020/172227),46032/57409}, {-(50876/173283),25280/57761}}

6 10

[math-fun] solar cell efficiency record
by Warren D Smith 19 May '16

19 May '16

35.4% energy conversion (sun->electric) efficiency achieved. The light was split into 4 spectral bands each handled with a different semiconductor: Ge: 0.67eV Si: 1.1eV GaInAs: 1.41eV GaInP: 1.88eV I don't understand their splitting scheme. Can it be used with scattered sunlight, i.e. which is not coming from just one direction? In any event, it seems to me this whole approach probably is totally uneconomical because Ge and In are rare. http://newsroom.unsw.edu.au/news/science-tech/milestone-solar-cell-efficien… (Even greater efficiency has been achieved by the same team but using concentrated sunlight.) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

3 2

Re: [math-fun] solar cell efficiency record
by Keith F. Lynch 19 May '16

19 May '16

Brent Meeker <meekerdb(a)verizon.net> wrote: > Using a different semi-conductor to capture photons of different > energy wouldn't depend on directionality. Unless each semiconductor is transparent to all wavelengths except the ones at which it's most efficient, yes it does. To direct different wavelengths to different semiconductors, whether with prisms or with diffraction gratings, the light has to be coming from more or less the same direction. If white light is equally bright in all directions, nothing can sort it into different colors. (You could, of course, color it by selectively absorbing and discarding different wavelengths, as with gels on stage lights, but that inefficient process doesn't gain you anything.) > ... capturing a photon with just enough energy to boost the electron > into the conduction band with little or no excess. Right. For those without a background in solid state physics, a better mental picture might be a vacuum tube in which light shines on the cathode causing it to emit electrons due to the photoelectric effect. This generates electricity most efficiently if the voltage on the tube is such that the electrons can just barely make it to the anode. Too low a photon energy for the voltage, and the electrons fall back to the cathode, generating no current. Too high a photon energy for the voltage and the electrons hit the anode hard, wasting energy heating it. If the tube is exposed to white light, most photons will have either too little or too much energy, and the efficiency will be low. Both the spread of wavelengths in white light and the spread of directions in diffuse light are forms of entropy. And there's no getting around that. Based on that, I conclude that no semiconductor is transparent to all wavelengths except the ones at which it's most efficient, as that would allow construction of a perpetual motion machine of the second class. The good news is that the entropy of sunlight in space is constant, i.e. you can do just as much useful work with it by capturing it at 100 AU as by capturing it at 1 AU (except for the tiny fraction which is absorbed by planets and other debris between 1 and 100 AU). Of course you need a larger collector if you're further away; that's why nobody powers anything by the light from Sirius. On the other hand, what if you're not trying to maximize energy, but computation? The amount of computation you can do per unit energy is proportional to the absolute temperature of the computer. And how cool you can keep something in space depends on its surface area. So to get as much computation as possible from the sun, build a sphere around it as far away as you can. Of course then you'll suffer from latency getting signals from one part of your immense sphere to another.

3 2

Re: [math-fun] solar cell efficiency record
by Keith F. Lynch 18 May '16

18 May '16

"Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: > The amount of computation you can do per unit energy is proportional > to the absolute temperature of the computer. Sigh. I meant *inversely* proportional. > And how cool you can keep something in space depends on its surface > area. The real reason the Death Stars kept blowing up had nothing to do with Luke Skywalker. Look up their size. Assume heat is generated inside them at about the same rate as an average office building. Use the Stefan-Boltzmann law to calculate their surface temperatures. Prepare to be surprised. :-)

1 0

Re: [math-fun] Decomposing Rotation into set of sub-rotations
by Warren D Smith 18 May '16

18 May '16

Well, as I just said to Lunnon, I think you can zero the entries of the lower triangle of an arbitrary NxN matrix in the order N-1 : 2N-3 : : 4 N+3 3 N+2 : 2 N+1 2N-5 1 N 2N-4 ... etc, i.e. raster scan, column by column, going upward within each column. Each zeroing is performed using an adjacent-coordinate 2x2 rotation affecting rows k and k-1 only and zeroing the entry in row k. Note no previously zeroed entry ever is unzeroed. If original matrix was orthogonal then final matrix is upper-triangular orthog hence is identity matrix. And by the way, every procedure of this kind is quite numerically stable as is well known (see any computational linear algebra book), unlike if non-orthogonal transformations were being used, when that often is much more of a problem. I had also earlier suggested to Lunnon this alternative ordering of the entries being zeroed 7 4 8 2 5 9 1 3 6 ... again each entry is zeroed by a 2x2 rotation affecting only rows k and k-1, where the entry in row k is the one being zeroed (and the entry above it, in row k-1, is altered too); and again each zeroing should never unzero any previously zeroed entry. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0