March 2015 - mathfun@mailman.xmission.com

[math-fun] Elliptical problems
by Bill Gosper 26 Mar '15

26 Mar '15

Subject [math-fun] (More) area on a sphere. From Bill Gosper <billgosper(a)gmail.com>[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_sear…> Sender math-fun <math-fun-bounces(a)mailman.xmission.com>[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_sear…> To math-fun(a)mailman.xmission.com[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_sear…> Reply-To math-fun <math-fun(a)mailman.xmission.com>[image: Add contact] <https://ma.sdf.org/rc/?_task=mail&_action=show&_uid=17446&_mbox=INBOX&_sear…> Date 2015-03-10 13:08 I love playing with NeilB's (& Gene's) solid angle integral. In terms of latitude and longitude, the area enclosed by a series of arcs, (lat(t),long(t)), 0<t<1 is just the sum (over the arcs) of spharea[lat_, long_] := long[1] - long[0] - Integrate[D[long[t], t]*Sin[lat[t]], {t, 0, 1}] (times r²). E.g., for the four sides of a "rectangle", (lat(t), long(t) = {{south*(1 - #) + north*# &, east &}, {north &, east*(1 - #) + west*# &}, {north*(1 - #) + south*# &, west &}, {south &, west*(1 - #) + east*# &}}; In[4]:= Factor[Total[spharea @@ # & /@ %]] Out[4]= (east - west) (Sin[north] - Sin[south]) consistent with the "hatbox" (cylindrical projection) equal area theorem. The projection of an a×b (semiaxes) by (height) z ellipse onto the unit sphere has area 2 (π- (2 b z EllipticPi[1 - b^2/a^2, (a^2 - b^2)/(a^2 + z^2)])/(a Sqrt[a^2 + z^2])) (done in two halves), presumably a crypto-symmetric function of a and b. (Does the area formula for the general a,b,c ellipsoid have six-way crypto-symmetry?) ------ Indeed, Macsyma and Mma get six different but numerically equivalent expressions. Neither can prove the equality of any two, nor even expand them in comparable series. ---------------- Checking, replace z by 1 and a,b by h a, h b, small h, In[37]:= Series[% /. {a -> a*h, b -> b*h}, {h, 0, 2}] Out[37]= SeriesData[h, 0, { 2 Pi - 2 a b^(-1) (a^(-2) b^2)^Rational[1, 2] Pi, 0, a b Pi}, 0, 3, 1] In[38]:= Simplify[Normal[%], a > b > 0] Out[38]= a b h² π I.e., the area of an h a × h b ellipse. -------- Similar were my series solutions (recounted here) to finding the semiaxes of an ellipsoid given its principal girths. ------------- I think I once mentioned here NeilB's proud ownership of a plastic-laminated cheat-sheet -------- gosper.org/cheetsheat.pdf (scanned with Neil's IPhone!) --------- of handy-dandy geometry formulas, purporting to give the area of an elliptical cone in elementary functions. Probably planted by the KGB or Red Chinese during the Cold War. ------------- But lurking in Canada. I misrecalled: it was an elliptical cylinder, not an elliptical cone. Note also the a b c sin theta volume for the general parallelepiped, given only one of the angles between adjacent edges. This is known as the a b c conjecture. Finally, the reverse of the card helpfully protects you from looking like an idiot by reminding you how to spell "frustrum". --rwg _______________________________________________

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Re: [math-fun] Polynesian (Pacific Ocean) navigation pre-clocks, pre-GPS
by Warren D Smith 26 Mar '15

26 Mar '15

Bees have an internal clock, and they also keep track of the sun. The combination gives them direction information adequate for their (short distance only) needs. I think some other animals track stars, but do not recall the details. (It would probably be long distance migrating birds or insects.) Some bacteria also have magnetic compasses, which they use to keep track of "up" versus "down." Question: How do plant seeds, when germinating deep, know which way is "up"?

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[math-fun] Migratory birds
by Warren D Smith 25 Mar '15

25 Mar '15

Many birds have an additional magnetic sense, sort of like having a built-in compass.

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Re: [math-fun] Polynesian (Pacific Ocean) navigation pre-clocks, pre-GPS
by Henry Baker 25 Mar '15

25 Mar '15

Re: Migratory birds 1. Birds tend to fly a heck of a lot faster (& cheaper) than humans in boats, so the cost of any error is at least an order of magnitude lower. 2. Birds have been at this for tens of millions of years. Modern humans (i.e., Homo Sapiens) have been doing this for only 30k-40k years, max, and may have been doing this only since the last ice age ~20k years ago. So it is a lot more likely that birds have evolved additional capabilities not available to humans. 3. European navigators, for all of their "sophistication", never developed the navigational capabilities supposedly available to Polynesians, and hence did _mostly_ "festoon"-type sailing within a day or so of land. Of course, the ancient Romans lived mostlu within the Mediterranean Sea & the Black Sea, and didn't have to develop the capabilities to master the Pacific Ocean. At 09:44 AM 3/25/2015, Fred Lunnon wrote: ><< even partial skills won't be transferred to the next generation >> > >An elegant argument which embarrassingly fails to conform to observation. >For instance, how did migratory birds manage to acquire the same capability? > >WFL > >On 3/25/15, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I heard a recent program on the ability of ancient Polynesians to navigate >> the Pacific Ocean w/o clocks/longitude or GPS. >> >> Supposedly, the Polynesian navigators use information about >> waves/swells/etc. to detect the presence & direction of islands over the >> horizon. >> >> I'm a bit skeptical about this, because unless these skills are relatively >> easily acquired, anyone with less than a certain critical mass of skills >> will die, and even partial skills won't be transferred to the next >> generation. (This is semi-analogous to the spread of a communicable >> disease, only in this case we _want_ the navigational information/techniques >> to propagate.) >> >> On the other hand, if this information is available in the wave/swell >> patterns, then presumably an "AI" computer program could learn these >> techniques by sending out lots of little buoys with GPS, sky-facing camera >> (for sun & star info) & wave-sensing capabilities, and eventually learn how >> to correlate these wave/swell patterns with positions relative to known >> islands. >> >> http://web.archive.org/web/20090917235953/http://pvs.kcc.hawaii.edu/navigat…

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[math-fun] car talk puzzler
by Scott Huddleston 25 Mar '15

25 Mar '15

Last week's car talk puzzler http://www.cartalk.com/content/puzzlers has an interesting pseudo-probabilty twist. The puzzle (3/14/15, President's Day): My friend, Max says to me, "I just read that three of the first five presidents of the United States died, ready for this? On the Fourth of July. What do you think the odds of that happening are?" I reply, "No idea. But, I'll give you 10-to-one odds that I can name at least one of the three." Now I don't know anything about the first five presidents, he writes, except that their names are: Washington, Adams, Jefferson, Madison and Monroe. In that order. Others might enjoy looking at this, so I won't give my analysis now. Enjoy, Scott

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[math-fun] Theorem-dependancies
by Guy Haworth 24 Mar '15

24 Mar '15

Thanks to previous responses and to Dan, Mark, Mike and Tom ... a good set of contributions. I was asking the question because I was thinking about a database of theorems, showing - individual theorems as a set of initial 'constraints' leading to a 'result', but also - showing how other theorems are used to produce the new theorem. The 'graph' of these theorems is not a hierarchy or even a lattice ... but theorem-equivalences show that it is a more general graph. Has anyone produced this kind of database of theorems on a computer - with access-software, GUI etc?! I guess 'Principia Mathematica' was a first demonstration, pre computer. Somewhere well downstream, we get to '1 + 1 = 2' with the footnote "The above proposition is occasionally useful." Thank again - Guy

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Re: [math-fun] Computational effort
by Bill Gosper 24 Mar '15

24 Mar '15

From: Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: Eugene Salamin <gene_salamin(a)yahoo.com>, math-fun < math-fun(a)mailman.xmission.com> All methods (that they've told us about) for enriching uranium are slow and require many stages. In this situation, given the U-235 fractions x[F] in the feed, x[P] in the enriched product, and x[W] in the depleted waste (or tails), the SWU is a measure of the amount of effort required to achieve that enrichment. The customer would be charged so many $/SWU. There's lots of Google hits on Separative Work Unit. I found this paper to be quite interesting. arxiv.org/pdf/0906.2505 The usually cited reference book, and one which explains the formula for calculating SWU, is Karl Cohen "The Theory of Isotope Separation", out of print and hard to find, but is in the collection at the King Library at San Jose State University. The gas diffusion and centrifuge methods are so slow because they depend on the mass difference. Fortuitously, natural fluorine consists of the single mass-19 isotope, so UF6 molecules differ only in the uranium. Laser isotope separation offers greater specificity by relying on isotopic differences in electronic spectra of uranium atoms or vibrational spectra in UF6 molecules. So far, laser isotope separation hasn't displaced centrifuges as the preferred technology (at least as far as they're telling us). In principle, i.e. without violating any laws of physics, there could exist a membrane or barrier permeable to only one isotope. This would allow instant single-stage separation. Diffusion of hydrogen through palladium foil is very effective since the D/H mass ratio is 2. The obstacle to its use is the energy required to electrolyse huge amounts of water for the 1/5000 deuterium fraction. If hydrogen sees use as a fuel, isotope separation can piggy-back on it. -- Gene I recall a Manhattan Project documentary where they were debating the feasibility of isotope separation, and someone suggested trying deuterium first. I can't find the quote, but Oppenheimer said something like "Hell, we should be able to do that with our bare hands." --rwg From: rwg <rwg(a)sdf.org> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Sunday, March 22, 2015 8:02 PM Subject: Re: [math-fun] Computational effort On 2015-03-22 12:25, Eugene Salamin via math-fun wrote: > There already is a unit for isotope enrichment effort. It is the > Separative Work Unit, SWU. > -- Gene Gene, are you saying there's a theoretical proof of no shortcut to enrichment? That sounds like a dangerous assumption. > > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Sent: Sunday, March 22, 2015 12:18 PM > Subject: Re: [math-fun] Computational effort > > Separating U235 from U238 is a pretty decent "proof of work" system. > > "Uraniumcoin" ? > > "Isotopecoin" ? > > "Hex[a]fluorocoin" ? > > "Centricoin" ? > > These "coins" put the "rich" into "enriched"... > > https://en.wikipedia.org/wiki/Enriched_uranium > > Of course, you may only want to use Oralloy certificates > in commerce, exchangeable* for the real thing. Who wants > to carry around a lead wallet, with even heavier contents? I've heard of deep pockets, but I can't picture one holding a critical mass of U235coins. If that's a problem, enrich just short of finite critical mass. Short of criticality, the lead would be superfluous. Bcc: Tehran mint --rwg > > * (Except by certain not-so-friendly countries.) > > At 07:16 AM 3/20/2015, James Propp wrote: >> (b) the Manhattan Project, > Henry> Having an upper limit on the # of U235 coins you can have cuts down on inequality; Dr. Piketty would be pleased: https://en.wikipedia.org/wiki/Thomas_Piketty Re lead wallets: I thought that U235 was reasonably radioactive, and the radioactivity increases non-linearly with the amount of material gathered together. :-) Also, lead wallets keep well-heeled friends from exploding with delight from all that hugging. Deep pockets, maybe; heirs, maybe not! ------ Would you disinherit your grandchild merely for having two heads?

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[math-fun] Theorem-dependance: possibly trivial question
by Guy Haworth 23 Mar '15

23 Mar '15

Let us say that, if Theorem 1 (T1) can be used to prove Theorem 2 (T2), then T1 --> T2. Are there simple examples of 'equivalence', T1 --> T2 and T2 --> T1 (setting up loop T1 --> T2 --> T1) that could be explained to school children? Are there cases of longer loops T1 --> ... --> Tn --> T1 which cannot, or cannot naturally, be shortened ? Schoolchildren-compatable preferred again ! Thanks - Guy

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[math-fun] Happy birthday Emmy.
by meekerdb 23 Mar '15

23 Mar '15

http://time.com/3753669/emmy-noether-mathematician-honored-google-doodle/ http://www.google.com/doodles/emmy-noethers-133rd-birthday A post by Vic Stenger (from 2013): http://www.huffingtonpost.com/victor-stenger/happy-birthday-emmy-noeth_b_29… Brent

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[math-fun] Mechanical thought experiment
by Henry Baker 23 Mar '15

23 Mar '15

(This thought experiment was inspired by space shuttles landing at 200+ knots, and having electric motors in the wheels that brought the landing gear wheels up to landing speed prior to actual touch-down, so that the landing would not generate any flat spots on the landing gear tires as a result of the landing.) The following thought experiment could have been performed in Newton's time, but I doubt anyone would have considered it until after the steam locomotive was invented. Consider a spoked wheel in which the vast majority of the mass is located in the rim; i.e., the mass of the spokes are insignificant relative to the mass of the rim. Furthermore, although the rim is heavy, it is also elastic and can stretch. Assume the wheel is moving on a perfectly flat surface with a uniform gravitational field perpendicular to the surface. Assume the wheel is moving in a straight line; thus the 3rd dimension in the problem can be ignored. Assume that there is a load on the axle in the center of the wheel. If the wheel is stationary, this load is distributed by the spokes in such a way that the spokes on the bottom are in compression and the spokes at the top are in tension. Since the wheel rim is elastic, the rim itself will be deformed slightly, but we will ignore this deformation for a moment. Now turn the wheel and have it roll along the surface without slipping at a pretty decent velocity. At some speed, the mass of the wheel rim will provide enough centrifugal force to counteract the compression force on the lower spokes, so that all of the spokes will be in tension, although the spokes at the top of the wheel will be in greater tension than the spokes at the bottom. Now that we are going fast enough that all of the spokes are under tension, we now allow the rim to disintegrate into N equal pieces, where N is the number of spokes. Each of the rim pieces becomes a point mass having 1/N of the total rim mass and still attached to its spoke. Let us further assume that there are small wires holding the pieces in a roughly circular shape, but these small wires carry a negligible load, and all of the load in the axle is now being carried by the impulses of the pieces of the rim hitting the "road". Another way to model this would be to have an N-spoked wheel without any rim at all, but the mass of each spoke is concentrated at the periphery; we also assume that each such spoke is constrained to stay within its "ray" from the center of the wheel--i.e., the weights of each spoke slide along its spoke ray, but are constrained to not go beyond a certain radius R from the center of the "wheel". Of course, there are all sorts of 2nd & 3rd order effects, but is there a scenario under which such a "dis-integrated wheel" would work ? I.e., actually carry a load so long as the speed was higher than some critical speed.

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