July 2015 - mathfun@mailman.xmission.com

[math-fun] optimum wiring pattern for solar panel
by Warren D Smith 31 Jul '15

31 Jul '15

Optimum Wiring Pattern for Solar Panel =========Warren D Smith===July 2015========== (Also a highly related problem, is optimum pattern of drainage ditches for a farm field.) Consider a plane region (interior of Jordan curve, usually rectangle or circle or semicircle). It is made of silicon. On top of the silicon layer we place a wiring pattern made of sheet metal, forming a tree. The root of the tree must lie on the boundary of the region. Silicon has some constant resistivity, and metal has a much smaller constant, say K=10^4 times smaller. Metal is opaque. Imagine that the silicon is a current source (generates some constant current per exposed square cm) at constant voltage. The goal is to design the metal pattern, to get the most power out of the solar cell. The two loss mechanisms are (1) joule loss (proportional to R*I^2 where R=resistance and I=current) (2) loss due to metal covering up the silicon, reducing the exposed area. BROWNIAN MOTION REFORMULATION: It also is possible to reformulate this problem in the language of brownian motion (continuum limit of random walks). Imagine that each square-cm of exposed silicon is equally likely to spontaneously generate a randomly-walking drunk. Once generated the drunk brownian-motions until hitting metal. He also brownian motions inside metal, but with a much greater diffusion constant. The goal is to minimize the expected time before the drunk hits the tree root. ASYMPTOPIA: To make the problem easier, consider the limit of large size-scaling of the solar panel, and ignore subleading terms when considering optimality. TWO PROBLEM VERSIONS: Also, we could consider allowing the metal to have variable sheet thickness, or demand it has constant thickness. With variable thickness assume the metal casts a shadow proportional to its height. EXAMPLE: Let us first consider a fairly standard and boring kind of metal tree pattern -------*------- | -------*------- | -------*------- | -------*------- | -------*------- | -------*------- | root fitting inside a 2N wide by N high rectangle, with vertical periodicity length P. Any electron can reach the tree root by traveling at most distance P/2 in silicon then at most distance 2N in metal. With constant metal thickness and constant wire widths: it pays to make the metal wires thinner and P smaller in proportion until either (a) the wire widths become comparable to the metal's vertical (sheet) thickness, or (b) until the resistance within the metal becomes comparable to the resistance within the silicon (both as experienced by typical electron). If case b, that would suggest it is optimal to have P*K of the same order as 2*N*W where W is the wire width (assuming K is the ratio of sheet resistivities for Si and metal). You then realize that it is optimal for the left-right wires to have varying widths linearly increasing toward the center, and for the up-down wire to have width linearly increasing toward the root. More precisely, the width of each wire optimally should be proportional to the current it carries. With variable metal thickness: It is optimal for the metal thickness to be comparable to the width of a wire, i.e. all wires have fixed cross-sectional shape. In that case the wire radius should vary proportional to the 2/3 power of the current it carries (wires thicker near tree root than near leaves, as usual). BETTER PATTERN: I believe the following kind of pattern will enjoy improved efficiency compared to the pattern above. In fact I think the losses should be at least 20% smaller. Imagine rays of metal all emerging radially from center at the root, like a child's drawing of sun with rays. Equally space them angularly. So far, what we have described would involve distance between adjacent rays, that grows proportionally to distance-to-root. That in general is undesirable. We instead want the inter-ray spacing to grow proportionally to the resistance (at that distance-to-root) of the metal trace leading to the root. But actually, what we have described IS desirable, if we insist on using constant-width and constant-thickness metal wires. If we allow variable width wires (and/or variable sheet-thickness), then modify pattern as follows. The wire widths W will vary as a function of distance-to-root, so that W is proprotional to current-carried for the consant-thickness case, or to current^(2/3) for the variable thickness case (constant cross sectional shape for wires) We can vary the inter-ray spacing as a function of distance-to-root by making the rays "split" in the manner of a binary tree. E.g if we wanted constant inter-ray spacing, then every time we go out to 1+epsilon times the old distance to the root, split a fraction epsilon of the rays into two. There are some local readjustments of the pattern near each split point. Also we could terminate some rays instead of splitting them. These two maneuvers allow us get essentially any inter-ray spacing law we want, as a function of distance-to-root, with losses that will be negligible (subleading terms) versus simpler-to-think-about scenarios where all rays go exactly straight. The optimal spacing law will involve inter-ray spacing distances proportional to the metal ray's electrical resistance from there to the root. I have never seen a solar cell involving this kind of "sunburst" wiring pattern. I conclude that this is an easy way to improve solar cell efficiency. This is most natural for semicircle-shaped solar cells. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Generalized spanning trees in hypergraphs
by James Propp 31 Jul '15

31 Jul '15

Can someone give me a reference to the result of Kalai and Linial (?) generalizing Cayley's formula to spanning "trees" in hypergraphs? I mean the result in which the right-hand-side is something like n to the power of n-choose-k, counting generalized spanning trees of the complete k-hypergraph on n vertices. Thanks, Jim Propp

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[math-fun] 10-Planet Somsky-Gear System Discovered?
by William R Somsky 31 Jul '15

31 Jul '15

I've not confirmed the mesh analytically as yet, but I believe this is an exact "Somsky-Gear" system of 10 planets, all non-overlapping: Annulus: 73 teeth, Sun: 57, Planets: 2x5, 2x7, 2x8, 2x9, 2x11 Sun Displacement: 6.47941... (in units where a gear of radius n has n teeth) Static png: https://drive.google.com/open?id=0B2889vNnzpsTMjEwNDlDSnZjRWc Animated gif: https://drive.google.com/open?id=0B2889vNnzpsTNTBFUGhqVXNGTjQ -- WRSomsky <wrsomsky(a)gmail.com>

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[math-fun] OEIS Dragon Curve triple point drafts
by rwg 31 Jul '15

31 Jul '15

https://oeis.org/draft/A260747 et seq. Additions, corrections solicited. --rwg

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Re: [math-fun] nuclide map
by Eugene Salamin 30 Jul '15

30 Jul '15

I looked at the paper, and it is quite overwhelming. I'm saying the choice of metric is merely a convention since different authors use different choices. Any transformation that preserves one metric automatically preserves the other. So Fred, perhaps you can provide a brief explanation of what I fail to see. -- Gene From: Fred Lunnon <fred.lunnon(a)gmail.com> To: Eugene Salamin <gene_salamin(a)yahoo.com> Sent: Wednesday, July 29, 2015 10:04 AM Subject: Re: [math-fun] nuclide map << and not decidable by experimental observation >> I have to say that such a snap judgement seems rather facile to me. Just exactly what makes you so confident? Have you looked at the paper? WFL On 7/29/15, Eugene Salamin <gene_salamin(a)yahoo.com> wrote: > The (-1,+1,+1,+1) vs. (+1,-1,-1,-1) choice is one of convenience, and not > decidable by experimental observation. The former is convenient when one > prefers a positive metric on spacelike surfaces. The latter is preferred to > make E^2 - p^2 = +m^2 rather than -m^2. > > -- Gene > > > From: Fred Lunnon <fred.lunnon(a)gmail.com> > To: Eugene Salamin <gene_salamin(a)yahoo.com>; math-fun > <math-fun(a)mailman.xmission.com> > Sent: Wednesday, July 29, 2015 9:10 AM > Subject: Re: [math-fun] nuclide map > > Somewhat off-topic I know, but observation of neutrinoless double beta > decay > would apparently decide a question which has long intrigued me: is there > any > fundamental physical procedure for discriminating between relativistic > physicists' > x^2 + y^2 + z^2 - t^2 and particle physicists' t^2 - x^2 - y^2 - z^2 ? > > A long paper > http://arxiv.org/abs/math-ph/0012006 > much of which goes over my head --- and is further obscured by a > crucial misprint > on page 50, where "Pin(1,3)" should read "Pin(3,1)" --- claims that a > positive result > would support the former convention (much to my personal satisfaction). > > Sadly, more recent experimental estimates of the lifetime have grown > unfeasibly > large --- see eg. > http://arxiv.org/pdf/1402.1170.pdf > passim ... > > Fred Lunnon > > > > > > On 7/29/15, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> > wrote: >> Amongst isobars (nuclei of constant mass number) with odd mass, there is >> one >> possessing minimum mass and is the most stable. Nuclei with more >> neutrons >> beta decay by electron emission toward that stable isotope, while nuclei >> with more protons decay by electron capture or positron emission. In the >> Chart, such nuclei lie on a line with 135 degree slope. If you can find >> an >> example with more than one beta stable odd mass isotope along such a >> line, >> I'd like to know about it. >> >> For even mass isobars, the situation is more complicated. Even-even >> isotopes are more stable than odd-odd isotopes. It is possible to have >> more >> than one local minimum. For example, Sn-124, Te-124, and Xe-124 are all >> stable. In such cases, double beta decay towards the global minimum by >> emission of 2 electrons together with 2 antineutrinos is allowed, but >> highly >> suppressed by the need for two simultaneous weak interactions. Some >> favorable cases have half-lives short enough to be measurable: Se-82 >> (1e20 >> y), Cd-116 (3e19 y), Mo-100 (7e18 y). See the Wiki for a more complete >> list. >> >> https://en.wikipedia.org/wiki/Double_beta_decay >> >> The spin of a neutrino is antiparallel to its momentum, but for an >> antineutrino the spin is parallel to the momentum. (I hope I got that >> that >> right. In any case, one is parallel, the other antiparallel.) If >> neutrinos >> were massless, that relation would be Lorentz invariant. But since >> neutrinos do have mass, it is possible to go fast enough to overtake one, >> and then the momentum changes sign. Does the, normally left-handed, >> neutrino become a right-handed neutrino, with interactions even weaker >> than >> the usual weak interaction, or does it turn into an antineutrino? We >> don't >> know. In the latter case, the neutrino is its own antiparticle, and >> neutrinoless double beta decay is possible, and is being sought for >> experimentally. >> >> >> -- Gene >> >> >> >> >> From: Bill Gosper <billgosper(a)gmail.com> >> To: math-fun(a)mailman.xmission.com >> Sent: Tuesday, July 28, 2015 4:32 PM >> Subject: Re: [math-fun] nuclide map >> >> Another selenium oddity: Both Se74 and Se82 are stable (and uncommon), >> but >> are unstable under the addition or removal of a proton, neutron, or both, >> or swap. >> (Eight possibilities apiece) There are *three* isotopes of erbium with >> this property. >> --rwg >> >> On Tue, Jul 28, 2015 at 4:08 PM, Bill Gosper <billgosper(a)gmail.com> >> wrote: >> >>> Oddity: Both the Brookhaven tables list the prevalent isotope Se80 as >>> STABLE, >>> but its box, instead of black, is "beta decay magenta" with a decay mode >>> of 2β- . >>> Japan Atomic Energy Agency just says >.5 Gy. Wikipedia flatly says >>> stable, >>> which is interesting because they list no stable isotopes of tungsten, >>> but >>> ascribe >>> quintillion year half-lives to five of them. >>> >>> What gives? >>> >> With the Brookhaven tables. >> >> >> >>> --rwg >>> Isn't it odd of JAEA to lump U235 with C12, stabilitywise? >>> Hmm, my dandruff shampoo may leave me with a radiation deficiency. >>> >>> On Tue, Jul 28, 2015 at 9:44 AM, Bill Gosper <billgosper(a)gmail.com> >>> wrote: >>> >>>> Thanks Ed! I had rejected that because it wasn't an actual image, >>>> lacked >>>> Z labeling, but mostly >>>> because I hadn't thought to click on Decay Mode to override the >>>> defaulted >>>> Half Life, which is so >>>> colored as to make it almost impossible to notice the instability of >>>> technetium. I subsequently >>>> found http://elementdata.net/chart_of_the_nuclides.jpg , but Mike >>>> Stay's >>>> https://upload.wikimedia.org/wikipedia/commons/9/92/NuclideMap.PNG >>>> is exactly the ticket. Thanks, Mike! >>>> --Bill >>>> And here I thought SF stood for San Francisco. >>>> >>>> >>>> On Tue, Jul 28, 2015 at 8:46 AM, Edward Fredkin <ed(a)fredkin.com> wrote: >>>> >>>>> Yes! >>>>> >>>>> Look at http://www.nndc.bnl.gov/chart/help/index.jsp >>>>> >>>>> Ed >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> From: Bill Gosper <billgosper(a)gmail.com> >>>>> Reply-To: "billgosper(a)gmail.com" <billgosper(a)gmail.com> >>>>> Date: Tuesday, July 28, 2015 at 11:01 AM >>>>> To: "math-fun(a)mailman.xmission.com" <math-fun(a)mailman.xmission.com> >>>>> Subject: nuclide map >>>>> >>>>> >>>>> Does anybody know where to find a full sized version of >>>>> >>>>> https://upload.wikimedia.org/wikipedia/commons/7/79/NuclideMap_stitched_sma… >>>>> ? Even though 4000 pixels wide, it's too small to read. (Will >>>>> settle >>>>> for unstitched.) --rwg > > >

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Re: [math-fun] nuclide map
by Bill Gosper 29 Jul '15

29 Jul '15

Thanks Ed! I had rejected that because it wasn't an actual image, lacked Z labeling, but mostly because I hadn't thought to click on Decay Mode to override the defaulted Half Life, which is so colored as to make it almost impossible to notice the instability of technetium. I subsequently found http://elementdata.net/chart_of_the_nuclides.jpg , but Mike Stay's https://upload.wikimedia.org/wikipedia/commons/9/92/NuclideMap.PNG is exactly the ticket. Thanks, Mike! --Bill And here I thought SF stood for San Francisco. On Tue, Jul 28, 2015 at 8:46 AM, Edward Fredkin <ed(a)fredkin.com> wrote: > Yes! > > Look at http://www.nndc.bnl.gov/chart/help/index.jsp > > Ed > > > > > > From: Bill Gosper <billgosper(a)gmail.com> > Reply-To: "billgosper(a)gmail.com" <billgosper(a)gmail.com> > Date: Tuesday, July 28, 2015 at 11:01 AM > To: "math-fun(a)mailman.xmission.com" <math-fun(a)mailman.xmission.com> > Subject: nuclide map > > Does anybody know where to find a full sized version of > > https://upload.wikimedia.org/wikipedia/commons/7/79/NuclideMap_stitched_sma… > ? Even though 4000 pixels wide, it's too small to read. (Will settle > for unstitched.) --rwg > > -- > You received this message because you are subscribed to the Google Groups > "mathfuneavesdroppers" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to mathfuneavesdroppers+unsubscribe(a)googlegroups.com. > For more options, visit https://groups.google.com/d/optout. >

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[math-fun] Somsky gears page
by Tom Rokicki 29 Jul '15

29 Jul '15

This page lets you play with meshing Somsky gears. http://tomas.rokicki.com/somsky.html I've tested it on Chrome and Safari; if someone lets me know if it works on firefox or IE, I'd appreciate it. -- -- http://cube20.org/ -- [Golly link suppressed; ask me why] --

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Re: [math-fun] Flakey and/or compromised computer arithmetic
by Henry Baker 28 Jul '15

28 Jul '15

"Blinding", thanks! Is there an algebraic theory of blinding that is sufficiently general to cover both crypto blinding and things like "pre-conditioning" in numerical matrix theory? At 02:16 PM 7/28/2015, Mike Stay wrote: >It's called "blinding" in multiparty secure computation protocols. > >On Tue, Jul 28, 2015 at 1:57 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> There have been proposals down through the years to incorporate checking logic on ALU's to make sure that the calculations were correct. I believe that IBM made some progress along these lines back in the 1960's. >> >> More recently, timing/power/radio/sound emission attacks on encryption have led to computations in which random numbers are inserted early in the calculation, only to later drop out, in order to mask the side-channel information of the actual calculation being performed. >> >> I don't know the precise name for these "masking numbers & calculations", but they fall short of fully homomorphic encryption, which would allow an *arbitrary* but unknown calculation to be performed. >> >> There are also worries of compromised hardware which looks for *specific constants* being used as operands, and upon encountering such operands, the hardware squirrels away some of the other -- presumably private -- data for illicit use. >> >> Some forms of "masked arithmetic" could also reduce the capabilities of this type of compromised hardware. >> >> Are there any papers which develop a theory of "masked arithmetic" which would be substantially simpler (& more efficient) than fully homomorphic encryption?

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[math-fun] Flakey and/or compromised computer arithmetic
by Henry Baker 28 Jul '15

28 Jul '15

There have been proposals down through the years to incorporate checking logic on ALU's to make sure that the calculations were correct. I believe that IBM made some progress along these lines back in the 1960's. More recently, timing/power/radio/sound emission attacks on encryption have led to computations in which random numbers are inserted early in the calculation, only to later drop out, in order to mask the side-channel information of the actual calculation being performed. I don't know the precise name for these "masking numbers & calculations", but they fall short of fully homomorphic encryption, which would allow an *arbitrary* but unknown calculation to be performed. There are also worries of compromised hardware which looks for *specific constants* being used as operands, and upon encountering such operands, the hardware squirrels away some of the other -- presumably private -- data for illicit use. Some forms of "masked arithmetic" could also reduce the capabilities of this type of compromised hardware. Are there any papers which develop a theory of "masked arithmetic" which would be substantially simpler (& more efficient) than fully homomorphic encryption?

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[math-fun] Purported proof that there are no odd perfect numbers
by Warren D Smith 28 Jul '15

28 Jul '15

>link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis with >the intriguing title, *A Proof of the Odd Perfect Number Conjecture*. --Simon Davis' only other paper on arXiv was http://arxiv.org/abs/hep-th/0312101 "Effectively Closed Infinite-Genus Surfaces and the String Coupling" 2003. If he proved the odd perfect number conjecture in 2004 it is somewhat, er, odd, he got no attention. Although given that he published it on hep-th (high energy physics) topic rather than "number theory" it would be rather easy for number theorists to miss it. This is yet another excellent example of why the arrogant irresponsible damned idiots in charge of arxiv.org have been massively remiss by refusing to include a commenting facility.

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