July 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Donald Saari on Dark Matter
by Henry Baker 18 Jul '15

18 Jul '15

Saari's analysis could have been performed by Maxwell; indeed, it essentially *was* performed by Maxwell, as Saari relies heavily on Maxwell's analysis of ring systems. Saari's work is classical Newtonian; no GR need apply. On the other hand, the equations that the "dark matter" folk have been relying on are also classical Newtonian. I wonder how much of Saari's analysis would fail in a GR setting? At 10:30 AM 7/17/2015, Paul R. Pudaite wrote: >Any thoughts on the following? > >http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.407 >https://www.siam.org/pdf/news/2094.pdf > >...Surely large scale computational models of discrete galactic dynamics >have been developed, but I haven't found reports.

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[math-fun] LOOP: it's pool, but backwards.
by Alex Bellos 18 Jul '15

18 Jul '15

I first read about elliptical pool tables in Martin Gardner. So I decided to go make one. http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/… <http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/…>

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[math-fun] Some pretty Saari slides
by Henry Baker 17 Jul '15

17 Jul '15

FYI -- (250 slides; 25 MBytes) http://www.mathnet.ru:8080/PresentFiles/6920/russia_dark_matter.pdf

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Re: [math-fun] Donald Saari on Dark Matter
by Henry Baker 17 Jul '15

17 Jul '15

At 12:30 PM 7/17/2015, Warren D Smith wrote: >If anybody can provide a pdf of Saari's paper, please email it >to me warren.wds AT gmail.com . Ditto to me, thx. You wont be sarri...

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[math-fun] Donald Saari on Dark Matter
by Warren D Smith 17 Jul '15

17 Jul '15

Saari's objection seems to be based on the existence of atypical special star configurations which violate "star soup" approximation. But many computer simulations of galaxies have been done with randomized initial conditions. They duplicate reality best when they include dark matter. Saari's objection has no force about such simulations. If anybody can provide a pdf of Saari's paper, please email it to me warren.wds AT gmail.com . I personally think there are valid grounds to worry about and be skeptical about dark matter, but Saari's grounds almost certainly have no relevance to reality. Nevertheless, he is correct that the rigorous foundations for this and many other continuum approximations in physics, are lacking.

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[math-fun] Donald Saari on Dark Matter
by Paul R. Pudaite 17 Jul '15

17 Jul '15

Any thoughts on the following? http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.407 https://www.siam.org/pdf/news/2094.pdf ...Surely large scale computational models of discrete galactic dynamics have been developed, but I haven't found reports. Paul

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[math-fun] Multiplicativity of arccos identities
by Warren D Smith 17 Jul '15

17 Jul '15

Two identities of the form A*arccos(B/C) + D*arccos(E/F) = G*(180 degrees=pi radians) (A,B,C,D,E,F,G integer) having the SAME (A,D), can be "multiplied" to produce a third with that same A,D also. That is, using the complex number formulation, see https://dl.dropboxusercontent.com/u/3507527/AcosMiracles128.out you multiply the corresponding complex numbers M3=M1*M2, N3=N1*N2. This multiplication does not always "work" in the sense that it often yields, say, non-integer B or E. However, "squaring" always works. This partially explains the "powers of 2,3,5" observation. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Dark Matter models
by Henry Baker 17 Jul '15

17 Jul '15

I attended a lecture about Dark Matter last night, and found the modelling to be quite troubling. "If it isn't dark, it doesn't matter." Basically, the mass of dark matter exceeds that of ordinary matter by such an amount that to a first approximation, *ordinary matter may be ignored*. Furthermore, not only does dark matter not "interact" with ordinary matter, it does not appear to "interact" with itself, either. https://en.wikipedia.org/wiki/Dark_Matter So far, so good. So what does dark matter consist of? Model 1. Dark matter consists of "atomic" particles all of the same size. In order to avoid weird bosonic behaviors like superfluidity, I would guess that dark matter would consist of fermions. Since these dark matter particles "interact" only through gravity, this model -- except for using general relativity instead of Newtonian gravity -- would be essentially an ideal Maxwellian gas, and would be subject to the barometric formula for pressure. https://en.wikipedia.org/wiki/Barometric_formula Of course, since black holes form due to the extreme curvature of space-time, and since the type of mass that creates this curvature doesn't matter, the center of clusters of dark matter would have to include black holes. Furthermore, once such a black hole formed, any additional mass -- whether dark or not -- would fall into the black hole and add its momentum -- thus adding to the angular momentum of the black hole. (We assume that any charge added to the black hole due to normal matter falling in would be very small relative to the total size & mass of the black hole, so these charge effects can be ignored.) So the center of these dark matter clusters would still be interesting places, having black holes which are at least mildly rotating, but some of which are wildly rotating. With black holes revolving about black holes, there must be some amount of gravitational wave generation going on. Thus, a dark matter cluster would be a more-or-less spherically-symmetrical object with a relatively small overall rotation and a black hole center. The surface of this object would be ill-defined, as the pressure would fall off exponentially according to the barometric formula. The ONLY way for such an object to lose energy would be from gravitational waves. Furthermore, since the particles are small relative to the size of the cluster, there is no mechanism for many of these particles to escape. As a result, this object would not evolve much over the present lifetime of the universe. I would imagine that one could place a limit on the physical size of these dark matter particles (measured in units of mass) by carefully measuring the various statistics (i.e., the classical "thermodynamics") of these dark matter clusters. For example, if the particles were too large, a measurable number of them could statistically "evaporate" and leave the system. Model 2. Same as Model 1, except that the dark matter consists of a multiplicity of different particle types, still "interacting" only by means of gravity. Once again, this is reminiscent of a Maxwellian gas composed of different "partial pressures", and more-or-less classical "thermodynamics" of these ideal Maxwell particles would hold. With enough disparity in the size distribution, it is conceivable that a measurable amount of the smallest size particles could "evaporate", so these types of dark matter clusters might evolve somewhat in the course of 14 billion years. ----- The professor giving the lecture indicated that many of the recent experiments have lowered the limits of the possible interactions between dark and normal matter to such absurd values that the simplest explanation is that there are NO interactions (other than gravity) whatsoever. He personally thinks that these particles may be some sort of "atomic" black holes left over from the formation of the universe. Of course, what keeps these (presumably small) black holes from coalescing (and generating gravitational waves in the process) isn't at all clear. Perhaps they are small enough to have quantum properties that keep them from coalescing -- e.g., fermionic behavior ? These small black holes would also be chargeless, else they could interact with normal matter.

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[math-fun] ArcCosine Miracles List, and explanation
by Warren D Smith 17 Jul '15

17 Jul '15

ArcCosine Miracles List ===Warren D. Smith====July 2015========= Here's a fairly large list of "arccosine miracles" i.e. equations of the form A*arccos(B/C) + D*arccos(E/F) = G*(180 degrees) with A,B,C,D,E,F,G all integers, and min(A,D,G)>=0 and max(|A|,|B|,|C|,|D|,|E|,|F|)<101 and |B|<C<71. I have not proven the exact validity of these, but they are valid to at least 11 decimal places. Various rather trivial infinite families, such as when B=C or E=F or D=0 or B=0 or E=0 or |B/C|=|E/F|, or integer multiples of listed identities, have been omitted from this list to try to keep it short. It occurs to me that if M is a complex number with |M|=C and re(M)=B both integer, while N is a complex number with |N|=F and re(N)=E both integer, then im( (M/C)^A * (N/F)^D ) = 0 where A and D are ordinary nonnegative integers, holds if and only if our arccos identity does. That seems to be a way to rephrase the problem in a way that does not mention any arc-trig functions, and lives solely in the land of complex numbers whose real and imaginary parts are expressible using integers and their square roots only. This rephrasing should allow both speeding up the search, and making it yield rigorous proofs instead of perhaps-merely-approximately-valid identities. And any of our alleged identities may be rigorously proven (or disproven) in this way (I haven't, but you easily could). CLOSE BUT NO CIGAR: 74*acos(9/11)+57*acos(-51/89) =9719.99999999397 12*acos(-15/26)+68*acos(-89/98) =12060.00000001333 45*acos(10/31)+64*acos(-31/71) =10620.00000002814 28*acos(25/33)+67*acos(72/97) =3959.99999997569 76*acos(29/35)+47*acos(-44/65) =8820.00000001841 91*acos(18/43)+71*acos(19/79) =11340.00000000618 37*acos(20/47)+66*acos(1/65) =8279.99999999370 77*acos(47/53)+56*acos(-65/79) =10260.00000000485 7*acos(-19/56)+66*acos(-46/57) =10260.00000002915 8*acos(-29/57)+85*acos(-73/100) =12600.00000001594 66*acos(46/57)+7*acos(19/56) =2879.99999997085 94*acos(-1/58)+25*acos(-19/77) =11160.00000000605 23*acos(-2/59)+8*acos(-67/70) =3420.00000002234 5*acos(59/60)+72*acos(35/74) =4499.99999998777 71*acos(53/62)+42*acos(-89/100) =8639.99999999190 66*acos(-1/65)+37*acos(-20/47) =10260.00000000630 47*acos(44/65)+76*acos(-29/35) =13319.99999998160 8*acos(67/70)+23*acos(2/59) =2159.99999997766 APPARENTLY GENUINE IDENTITIES: 3*acos(-2/3)+1*acos(-22/27) = 540.0 2*acos(-1/3)+1*acos(-7/9) = 360.0 3*acos(1/3)+1*acos(-23/27) = 360.0 2*acos(2/3)+1*acos(1/9) = 180.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-1/4)+1*acos(-7/8) = 360.0 3*acos(1/4)+1*acos(-11/16) = 360.0 2*acos(3/4)+1*acos(-1/8) = 180.0 4*acos(-4/5)+2*acos(-24/25) = 900.0 2*acos(-3/5)+2*acos(-4/5) = 540.0 2*acos(-2/5)+1*acos(-17/25) = 360.0 2*acos(-1/5)+1*acos(-23/25) = 360.0 4*acos(3/5)+2*acos(-24/25) = 540.0 2*acos(4/5)+2*acos(3/5) = 180.0 2*acos(-1/6)+1*acos(-17/18) = 360.0 3*acos(1/6)+1*acos(-13/27) = 360.0 2*acos(5/6)+1*acos(-7/18) = 180.0 2*acos(-3/7)+1*acos(-31/49) = 360.0 2*acos(-2/7)+1*acos(-41/49) = 360.0 3*acos(-1/7)+3*acos(-11/14) = 720.0 3*acos(1/7)+3*acos(-13/14) = 720.0 2*acos(4/7)+1*acos(17/49) = 180.0 2*acos(5/7)+1*acos(-1/49) = 180.0 2*acos(6/7)+1*acos(-23/49) = 180.0 5*acos(-7/8)+2*acos(-61/64) =1080.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 3*acos(-1/8)+2*acos(-9/16) = 540.0 1*acos(1/8)+2*acos(-3/4) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 1*acos(7/8)+2*acos(1/4) = 180.0 3*acos(-7/9)+2*acos(-23/27) = 720.0 2*acos(-4/9)+1*acos(-49/81) = 360.0 2*acos(-2/9)+1*acos(-73/81) = 360.0 1*acos(-1/9)+2*acos(-2/3) = 360.0 3*acos(1/9)+2*acos(-22/27) = 540.0 2*acos(5/9)+1*acos(31/81) = 180.0 1*acos(7/9)+2*acos(1/3) = 180.0 2*acos(8/9)+1*acos(-47/81) = 180.0 2*acos(-3/10)+1*acos(-41/50) = 360.0 2*acos(-1/10)+1*acos(-49/50) = 360.0 2*acos(7/10)+1*acos(1/50) = 180.0 2*acos(9/10)+1*acos(-31/50) = 180.0 2*acos(-5/12)+1*acos(-47/72) = 360.0 2*acos(-1/12)+1*acos(-71/72) = 360.0 2*acos(7/12)+1*acos(23/72) = 180.0 2*acos(11/12)+1*acos(-49/72) = 180.0 3*acos(-11/13)+3*acos(-23/26) = 900.0 2*acos(-5/13)+2*acos(-12/13) = 540.0 3*acos(11/13)+3*acos(1/26) = 360.0 2*acos(12/13)+2*acos(5/13) = 180.0 6*acos(-13/14)+3*acos(-47/49) =1440.0 3*acos(-11/14)+3*acos(-13/14) = 900.0 2*acos(-5/14)+1*acos(-73/98) = 360.0 2*acos(-3/14)+1*acos(-89/98) = 360.0 2*acos(-1/14)+1*acos(-97/98) = 360.0 2*acos(9/14)+1*acos(17/98) = 180.0 3*acos(11/14)+3*acos(1/7) = 360.0 3*acos(13/14)+3*acos(-1/7) = 360.0 2*acos(-11/16)+3*acos(-7/8) = 720.0 1*acos(-9/16)+3*acos(-3/4) = 540.0 2*acos(9/16)+3*acos(1/8) = 360.0 1*acos(11/16)+3*acos(-1/4) = 360.0 2*acos(-8/17)+2*acos(-15/17) = 540.0 2*acos(15/17)+2*acos(8/17) = 180.0 1*acos(7/18)+2*acos(-5/6) = 360.0 1*acos(17/18)+2*acos(1/6) = 180.0 3*acos(-13/19)+3*acos(-37/38) = 900.0 3*acos(13/19)+3*acos(11/38) = 360.0 1*acos(-7/25)+2*acos(-3/5) = 360.0 1*acos(7/25)+2*acos(-4/5) = 360.0 1*acos(17/25)+2*acos(2/5) = 180.0 1*acos(23/25)+2*acos(1/5) = 180.0 2*acos(24/25)+4*acos(-3/5) = 540.0 3*acos(-1/26)+3*acos(-11/13) = 720.0 3*acos(1/26)+3*acos(-23/26) = 720.0 3*acos(23/26)+3*acos(11/13) = 180.0 4*acos(-22/27)+3*acos(-79/81) =1080.0 2*acos(-13/27)+3*acos(-17/18) = 720.0 1*acos(-5/27)+3*acos(-5/6) = 540.0 1*acos(13/27)+3*acos(-1/6) = 360.0 1*acos(22/27)+3*acos(2/3) = 180.0 1*acos(23/27)+3*acos(-1/3) = 360.0 2*acos(-20/29)+2*acos(-21/29) = 540.0 2*acos(21/29)+2*acos(20/29) = 180.0 3*acos(-23/31)+3*acos(-59/62) = 900.0 3*acos(23/31)+3*acos(13/62) = 360.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 1*acos(-7/32)+2*acos(-5/8) = 360.0 1*acos(17/32)+4*acos(1/4) = 360.0 1*acos(23/32)+2*acos(3/8) = 180.0 1*acos(31/32)+4*acos(3/4) = 180.0 3*acos(-13/37)+3*acos(-47/74) = 720.0 2*acos(-12/37)+2*acos(-35/37) = 540.0 3*acos(13/37)+3*acos(-73/74) = 720.0 2*acos(35/37)+2*acos(12/37) = 180.0 3*acos(-11/38)+3*acos(-13/19) = 720.0 3*acos(11/38)+3*acos(-37/38) = 720.0 3*acos(37/38)+3*acos(13/19) = 180.0 2*acos(-9/41)+2*acos(-40/41) = 540.0 2*acos(40/41)+2*acos(9/41) = 180.0 3*acos(-11/43)+3*acos(-61/86) = 720.0 3*acos(11/43)+3*acos(-83/86) = 720.0 1*acos(-17/49)+2*acos(-4/7) = 360.0 1*acos(1/49)+2*acos(-5/7) = 360.0 1*acos(23/49)+2*acos(-6/7) = 360.0 1*acos(31/49)+2*acos(3/7) = 180.0 1*acos(41/49)+2*acos(2/7) = 180.0 1*acos(47/49)+2*acos(1/7) = 180.0 1*acos(-1/50)+2*acos(-7/10) = 360.0 1*acos(31/50)+2*acos(-9/10) = 360.0 1*acos(41/50)+2*acos(3/10) = 180.0 1*acos(49/50)+2*acos(1/10) = 180.0 2*acos(-28/53)+2*acos(-45/53) = 540.0 2*acos(45/53)+2*acos(28/53) = 180.0 2*acos(-11/61)+2*acos(-60/61) = 540.0 2*acos(60/61)+2*acos(11/61) = 180.0 3*acos(-13/62)+3*acos(-23/31) = 720.0 3*acos(13/62)+3*acos(-59/62) = 720.0 3*acos(59/62)+3*acos(23/31) = 180.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 2*acos(-33/65)+2*acos(-56/65) = 540.0 2*acos(-16/65)+2*acos(-63/65) = 540.0 2*acos(56/65)+2*acos(33/65) = 180.0 2*acos(63/65)+2*acos(16/65) = 180.0 end.

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[math-fun] NYMAG: Where That Malaysia Airlines Plane May Be
by Henry Baker 17 Jul '15

17 Jul '15

FYI -- http://nymag.com/daily/intelligencer/2015/02/jeff-wise-mh370-theory.html A little bit about the math used to estimate where MH370 ended up. Also, here's an audio podcast of this guy talking: http://traffic.libsyn.com/newamericanyc/NANYC-MH370_A_Year_Later.mp3 I won't spoil the story, but here's a hint: these planes are more complicated than most people realize.

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