February 2013 - mathfun@mailman.xmission.com

Re: [math-fun] Meteor strike!
by Dan Asimov 15 Feb '13

15 Feb '13

From < http://www.nytimes.com/2013/02/16/world/europe/meteorite-fragments-are-said… >: Debris and a Boom, Likely From a Meteor, Hit Siberia Fire in the Russian Sky: The Times’s Ellen Barry and Richard P. Binzel, a Massachusetts Institute of Technology professor, discuss the explosive event over western Siberia on Friday. By ELLEN BARRY and ANDREW E. KRAMER MOSCOW — Bright objects, apparently debris from a meteor, streaked through the sky in western Siberia early Friday, accompanied by a thunderous shock wave that damaged buildings across a vast territory. Russia’s Interior Ministry said more than 1,000 people were injured, 200 of them children, mostly from shards of glass that shattered when the meteor entered the atmosphere. . . . . . .

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Re: [math-fun] Attesting to Atoms
by Henry Baker 15 Feb '13

15 Feb '13

After reflecting more upon this conversation, I now think that the _physics_ community should press forward with quantum factoring experiments as a very high priority -- perhaps higher priority than the LHC. Quantum factoring may very well be the _first_ place to demonstrate discreteness in quantum _phase angle_. Quantum factoring is a win-win experiment: either way, you get a Nobel prize. If quantum factoring succeeds, you get a Nobel prize for proving the existence of a new & useful type of computational principle. If quantum factoring fails due to showing imperfections in our understanding of basic quantum theory, then you get a Nobel prize for that. At 08:23 PM 2/9/2013, Rowan Hamilton wrote: >Last time I checked the most accurate physical theory in human history >was Quantum Electro-Dynamics. The value of the anomalous magnetic >moment of the electron has been measured and compared to the QED >theory to one part in 10^12.

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[math-fun] Reverse is harder?
by Warren Smith 14 Feb '13

14 Feb '13

Much or all of cryptography depends on the claim that there is a bijective map F such that computing y=F(x), is a lot easier than computing x=Finverse(y). For example, if F maps (a,b,c) to (b^a mod c, b, c) where gcd(b,c)=1, then many people think/hope that Finverse is a lot harder than F ("discrete logarithm problem"). But my QUESTION is: what is the best result one can actually PROVE of the form "Finverse is a lot harder than F"? ============================================== Dhananjay S. Phatak pointed out to me (then I improved by consulting Joerg Arndt's book...) that the "Gray coding" problem is a provable (pitifully weak, but provable) example. In C-language notation g = x ^ (x>>1) will convert bitstring x to Gray code g. This is implementable by a parallel circuit with depth=1, and #gates=wordlength-1. Reverse operation accomplished by: x = g; x ^= x>>1; x ^= x>>2; x ^= x>>4; x ^= x>>8; x ^= x>>16; ... keep going until word length. This can be done by a circuit with O(N) gates, but the depth is now log2(wordlength) not 1, and this is best possible in the sense that no circuit made of (<=2-input, <=2-output) binary logic gates can do it with less depth. --- More generally, multiplying y=Mx where M is an NxN sparse matrix, and x,y each are N-vectors (in some field) is easy to do forward (assume at most a constant number of nonzeros per row of M): order N field operations and time; parallelizes to depth=O(1) if at most. But computing x=Minverse y is provably harder, requiring depth of order log(N) at least. The proof is to observe that in general each entry of x will depend on all N entries of y. [I actually do not know how to do it in less than order N^2 operations, but I do not have a proof that's best possible.] Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Q re ellliptic functions
by Henry Baker 13 Feb '13

13 Feb '13

Are elliptical regions (those conic sections with 2 foci) in the complex plane particularly interesting in the study of "elliptic functions" ?? I don't know enough about elliptic functions to have any insights here.

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[math-fun] Intermediate value theorem for complex numbers
by Warren Smith 13 Feb '13

13 Feb '13

I invented such a theorem in the late 1980s. Patrick Morton told me I should publish it, but I thought it was too trivial. Let F(z) be an analytic function. My idea was that there is some curve A in the complex plane, on which Re(F)=0. There is another curve B on which Im(F)=0. If you know that for some domain D in the complex plane, topologically a disk (F analytic everywhere inside), that A and B cross the boundary of the disk in the cyclic order along that boundary ABAB, then we are topologically forced to have the curve A crossing the curve B somewhere within D. And at that crossing point, F=0. The curves A and B generically cross each other orthogonally, and neither can be self-tangent due to conformal properties of analytic functions. In theorem above, you need to assume that those 4 points of the boundary of D must be unique and that the curves actually cross the boundary there, e.g. the right one of Re(F)' and Im(F)' where here ' represents derivative along boundary, each is nonzero at the 4 points (without these, theorem inapplicable). If you do not want to assume F is analytic, well, mere continuity suffices for some but not all parts of the argument... --- One can also obtain complex versions of Rolle's theorem...

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[math-fun] Intermediate value theorem for complex numbers
by rcs＠xmission.com 13 Feb '13

13 Feb '13

The two-dimensional version of the intermediate value theorem. Suppose F(z) is a continuous function from C->C over the complex numbers, or at least over the unit square. We are silent about the differentiability of F. Suppose that the real part of F, on the vertical edges of the square, has ReF(0+iy)=0 and ReF(1+iy)=0, for 0<=y<=1. On the horizontal edges, the imaginary part of F is similarly constrained: ImF(x+0i)=0 and ImF(x+1i)=0, for 0<=x<=1. (This forces F(z)=0+0i at the four corners of the square.) I presume that this implies that there is some point z inside the square with F(z) = 0+0i. The puzzle is to come up with a proof, ideally adaptable to a higher number of dimensions. The one-dimensional version can be proved by (iterated) bisection of the domain, with no fancy topology needed. This doesn't seem to extend to higher dimensions though. Rich

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[math-fun] ACC and MCC lattices in chemistry?
by Warren Smith 13 Feb '13

13 Feb '13

The new lattices ACC and MCC found by Fields, Conway, Sloane, and me... do they occur in chemistry? My answers: The radioactive element Protactinium has a tetragonal crystal structure within 1% of ACC; it has a/c=1.212 instead of ACC's 1.2247. For MCC, I have been unable to find any chemical element with this structure although perhaps under unusual conditions (high pressure, temperature?) it happens. Widening my search to all inorganic compounds I found A.Leclaire, M.M.Borel, A.Grandin, B.Raveau: A novel form of molybdenum (V) phosphate: delta-K Mo2 P3 O13, Zeitschrift fur Kristallographie 188 (1989) 77-83 http://rruff.info/doclib/zk/vol188/ZK188_77.pdf If I understand their claims aright, the potassium atoms in this stuff form a lattice apparently within 5% or less (in all geometric respects) of being an MCC lattice.

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Re: [math-fun] Moebius-ish function question
by Keith F. Lynch 13 Feb '13

13 Feb '13

Dan Asimov <dasimov(a)earthlink.net> wrote: > Keith's February puzzle led me to wonder: > Let pf(n) denote the number of prime factors of n, > counting multiplicity. ... I don't know, but I do know that if you instead said that any number with a repeat prime factor isn't counted at all, your question would be equivalent to the Reimann Hypothesis. In other words, add 1 for any number that's the product of even number of unique primes, add 0 for any number with any duplicate prime factors (i.e. any number divisible by a square), and subtract 1 for any number that's the product of odd number of unique primes. (This is the Mertens function, Sloane's A002321). > Such questions are always philosophically tricky, since the outcomes > of independent random trials will have probability 0 of conforming > exactly to any number-theoretic function like Q. The question that's equivalent to the RH is whether the Mertens function exceeds Big-O(n^(1/2 + epsilon)).

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Re: [math-fun] Weird leap-year rule
by Keith F. Lynch 13 Feb '13

13 Feb '13

Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: > Surely that proves only that there are *some* N such that the > average over the first N years is log 2. I don't understand what you mean. Since every year's length is a whole number of days, the average length of years 1 through N will always be a rational number, hence will never be the natural log of 2. (Though it will get closer and closer as N grows larger.)

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[math-fun] Neutron stars are superfluid, superconductor?
by Warren Smith 11 Feb '13

11 Feb '13

Dany Page, Madappa Prakash, James M. Lattimer, and Andrew W. Steiner: Rapid Cooling of the Neutron Star in Cassiopeia A Triggered by Neutron Superfluidity in Dense Matter, Phys. Rev. Lett. 106,8 (2011) 081101 http://arxiv.org/abs/1011.6142 provides, allegedly, the first direct evidence of this. The critical temperature for superfluidity in this star's neutron-matter is estimated to be 5*10^8 Kelvin.

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