September 2014 - mathfun@mailman.xmission.com

[math-fun] chess ratings
by Warren D Smith 05 Sep '14

05 Sep '14

Amazingly enough, my Bachelor's thesis was on this very subject -- rating systems. The internet chess club rating system is, or at least was, based on my work. See http://rangevoting.org/WarrenSmithPages/homepage/works.html paper #20 and paper #88. Paper 20 discusses among other things, a "fluid dynamics" model of learning as reflected in ratings, and discusses the rating distribution, and how to optimize rating system parameters. The rating distribution has features of a "extreme value distribution." Paper 88 discusses a new approach. Glickman kind of annoyed me by a combination of ignoring and appropriating my work, which I told him all about before he got going on the topic. But I do not know what Glickman's latest stuff is, haven't looked at this in years.

1 0

[math-fun] Herstory of email?!?
by Henry Baker 04 Sep '14

04 Sep '14

There's a real howler going on on Huffingtonpost.com about the history of email. An MIT professor (!) named Deborah Nightingale claims that email was invented in 1978. Perhaps she hasn't talked with anyone at MIT/Stanford/CMU/BBN/Xerox/IBM/DEC/BellLabs/... about this? The Boy Who Invented Email -- History of Email http://www.huffingtonpost.com/larry-weber/the-history-of-email-boy-who-inve…

6 7

[math-fun] Herstory of email?!?
by Bill Gosper 04 Sep '14

04 Sep '14

Henry> There's a real howler going on on Huffingtonpost.com about the history of email. An MIT professor (!) named Deborah Nightingale claims that email was invented in 1978. Perhaps she hasn't talked with anyone at MIT/Stanford/CMU/BBN/Xerox/IBM/DEC/BellLabs/... about this? The Boy Who Invented Email -- History of Email http://www.huffingtonpost.com/larry-weber/the-history-of-email-boy-who-inve… <HGB Just adding my fuzzy memories to the confusion. The MIT AI Lab Incompatible Timesharing System had an instant messaging command. E.g. to talk to Tom Knight, :SEND TK mumble mumble <ctrl something, probably z> . But if he wasn't logged in, as soon as you typed TK, ITS would insert "(MAIL)". ("Oh cr@p, he just logged out on me.") I.e., mail was mostly considered just a poor substitute for chatting. I came West in '74. I'm pretty sure this was in place well before. The Stanford AI Lab independently developed a mail system, and many of its old timers think they were the pioneers. "Electronic Mail" started gaining attention in the 80s. The Postal Service, wanting to get in on the action, held a big conference someplace. But they thought "electronic mail" meant faxing. --rwg

4 3

[math-fun] mixed orthogonal groups
by Fred Lunnon 04 Sep '14

04 Sep '14

I've never been very comfortable with this notation for matrix groups; but I cannot see anything wrong with extending it to degenerate forms, apart from the fact that for some reason it's just not "standard". So I will just define O(p, q, r) == real (or whatever) n x n matrices preserving (x_1)^2 + ... + (x_p)^2 - (x_(p+1))^2 - (x_(p+q))^2 , where n = p+q+r . Now the usual representation of Euclidean 3-space isometries by homogeneous projective matrices becomes E(3) = PO(3, 0, 1) --- not as earlier botched --- and the Poincaré / Laguerre 3-space group would be PO(3, 1, 1) . Presumably this representation is well-known, though I have not seen it mentioned. Speaking of botches --- I managed to gain a extra space dimension earlier, my muddle arising from Mob(2) acting on the unit 2-sphere in 3-space. Should have read (groan) Lorentz == 2-space Möbius = Mob(2) = PO(3, 1) , dimension 6 ; Poincaré == 3-space Laguerre = Lag(3) = PO(3, 1, 1) , dimension 10 . WFL

2 4

Re: [math-fun] dragon gif
by Bill Gosper 04 Sep '14

04 Sep '14

Jörg>* Bill Gosper <billgosper(a)gmail.com> [Sep 02. 2014 08:16]: Jörg>* Bill Gosper <billgosper(a)gmail.com> [Sep 01. 2014 08:00]: [...] Staggering. So somewhere in all that are these three "sevens"gosper.org/IMG_0245.JPGgosper.org/IMG_0246.JPGgosper.org/IMG_0247.JPG Yes, see http://jjj.de/tmp-rwg/ Indeed there are 3 shapes of order 7: http://jjj.de/tmp-rwg/search-r07-curves.txt here a "same = #" means that the shape already appeared at the earlier curve (named "R07-#"). So the images cited above where plotted in the 1970s? <Jörg Yep, 180dpi Xerographic printer (UV crt discharges selenium cylinder). Same machine on which DEK developed TeX and Metafont. Zoom in. See the jaggies? But no developer beads. I was a good day. Those pages were cut from a roll *after* printing. (I wonder if the Computer History Museum has one.) Jörg>As an extra here is order 13 (there are 15 shapes), see http://jjj.de/tmp-rwg/search-r13-curves.txt Eyeballing http://jjj.de/tmp-rwg/all-r13-curve-decompositions.pdf should be enjoyable. <Jörg Hey, you could serialize these things as the L-System Technical Journal. (hardcopied in the 70s, pre-laser.) Also, did you notice this way of mis-teardropping the France fractal?gosper.org/jelly7.bmp Jörg>Never heard of the term "France fractal".<Jörg So what am I going to call it, the Smarandache Fractal? Dropping the left red part in http://gosper.org/jelly7.bmp gives "Gosper's island", right? If you insist.-) But note that's the gnarlier of two ways to do it. Trying to find the less gnarly, instead I found http://www.tweedledum.com/rwg/frac5Image4.gif and http://www.tweedledum.com/rwg/frac5Image2.gif . Ah, here it is: http://www.tweedledum.com/rwg/tear7Image4.gif --rwg Lastly, I had forgotten about these two teardroppings of Mandelbrot's seven Snowflake recursion:gosper.org/trozeImage8.gifgosper.org/trozeImage13.gif Jörg>Neat! Maybe someone can do his thirteen, but I can't do it in my head. I just found this: http://www.meden.demon.co.uk/Fractals/frozenteardrop.html . Yes, I know this one. Sadly, the recipes are (as almost always) left in the dark. Btw. many of the images given are the generalization of the unit square with complex numeration systems (such as the Heighway dragon with radix -1+i and digits 0 and 1). Best, jj Jörg> [...]

2 1

[math-fun] Stupid question about geometrical transformations
by Fred Lunnon 03 Sep '14

03 Sep '14

What is generated by the union of projective and conformal (Moebius) groups? Since these two intersect in similarities, the super-group in n-space has dimension at least (n^2-1) + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + n - 1 ; just how big is it? How should such transformations be represented for computational purposes? Why don't I know the answers to these apparently obvious questions? [Uh, maybe don't answer that one right now ...] Physicists have previously devoted some thought to this matter: in particular, a promising paper by Wolfgang Bertram (2001) at http://www.emis.de/journals/AG/2-4/2_329.pdf launches into discussing "Jordan functors", which will however surely cost this innocent much gruesome effort to decode. [Pascual Jordan certainly seems put himself about, despite which I don't recall ever having encountered him before this week.] Fred Lunnon

3 21

[math-fun] "champion" probability distributions
by Henry Baker 03 Sep '14

03 Sep '14

[Is there another name for these distributions?] If I take a Gaussian distribution with mean m and standard deviation s, then it would appear that integrate(pdf,x,m+6s,infinity) ~ 10^-9, so this upper tail integrates to 1 in a billion. Now suppose we want to form a "team" of k champions taken from this 6sigma tail. I should be able to convolve this distribution k times to get the pdf for the team. Is there a closed form for this pdf? How about simply the pdf for a pair of champions? I should be able to represent the 1-champion pdf as pdf_gaussian(x,m,s)*unit_step(x-(m+6*s))/tailintegral I'd like to laplace-transform this, exponentiate it k times, and then inverse transform it. Even if I can't find a closed form, I'd be interested in the mean & std of the k-champion distribution. Maxima seems to fail me. Does Mathematica do any better?

1 0

[math-fun] Riemann hypothesis equivalent to bounding the count of divisors?
by Warren D Smith 03 Sep '14

03 Sep '14

If d(N) is the number of divisors of N, I can prove -- except for a gap -- the following: The Riemann Hypothesis is equivalent to the claim that there exists a constant J such that d(N) < a certain known function of N and J for all N>0. Specifically: The Riemann hypothesis is equivalent to the existence of a constant J such that log2( d(N) ) <= Li(lnN) + K*Li((lnN)^K) - (lnN)^K / lnlnN + J*(lnN)^(1/2) / (lnlnN)^2 for all integers N>1. Here K=log2(3/2)=0.5849625... Actually I could probably compute a suitable J with more work, in which case the bound would become a fully specified function of N only. Most of the work behind "my" proof was done by Ramanujan 1915 and I am taking some of his claims on trust, I mostly have not checked his work: S.Ramanujan: Highly composite numbers, Proc. London Math. Soc. (2) 14 (1915) 347-409. http://ramanujan.sirinudi.org/Volumes/published/ram15.pdf THE MISSING LEMMA (which I need to complete the proof): Letting z be the non-real zeros of the Riemann zeta function, I need that S2(x) = SUM x^z / z^2 must obey S2(x) < -x^P for some fixed P>1/2, for an infinite number of integers x>0, if the Riemann hypothesis is false. This "missing lemma" is presumably true and presumably has nothing particularly to do with the Riemann hypothesis -- all you really (presumably) need is the fact that if the RH is false the set of z obey: 1. If z exists, then ComplexConjugate(z) also exists. [Which causes S2(x) to be real-valued.] 2. The z all obey 0<Re(z)<1 and with at least one z having Re(z)>1/2. 3. There are O(T*logT) values of z having |Im(z)|<T, for T>20. 4. Indeed, there are O(logT) values of z having T<Im(T)<T+7, for T>20. 5. There are a countably infinite number of z. 6. All the z are non-real, indeed |Im(z)|>14.. I.e. any set of z with these 6 properties would presumably work. The lemma is easy to see it is true if the set of z were finite (with, say, a million elements) because then the finite subset of z with maximum real part, alone, would suffice to prove it with P=max(Re(z))>1/2 because the non-maximal Re(z) have effects on S2(x) which become relatively negligible for large x. The difficulty for me is when the set of z is infinite. It is possible this lemma is already known or standard (to those who know it). STRONG EXPLICIT BOUNDS ON THE d(N) FUNCTION: Empirically, my computer says that log2( d(N) ) < Li(lnN) + K*Li((lnN)^K) - (lnN)^K / lnlnN + 1.19113*(lnN)^(1/2) / (lnlnN)^2 is true where K=log2(3/2)=0.5849625... for all integers with 1<N<exp(8140) thus "confirming" the Riemann hypothesis. Also, for all N with 1<N<exp(8140) that are "highly composite" [have greater d(N) than all lesser numbers] my computer says log2( d(N) ) > Li(lnN) + K*Li((lnN)^K) - (lnN)^K / lnlnN - 3.71217*(lnN)^(1/2) / (lnlnN)^2 Also, for all N with 1<N<exp(8140) that are "highly composite" [have greater d(N) than all lesser numbers] my computer says Li(lnN) + ((lnN)^K)/(K*(lnlnN)^2) - 0.96824 *(lnN)^(1/2) / (lnlnN)^2 < log2( d(N) ) < Li(lnN) + ((lnN)^K)/(K*(lnlnN)^2) - 0.94223 *(lnN)^(1/2) / (lnlnN)^2 which seems pretty damn impressively tight, although I do not think a bound window of this form can be valid for all N>1. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

Re: [math-fun] Really, really stupid question about Lorentz group
by Warren D Smith 03 Sep '14

03 Sep '14

The "Lorentz" (sub)group is normally not including translations and the "Poincare" group does include them. In informal speech where it does not terribly matter, one often speaks of "Lorentz invariance" or some such, really meaning Poincare, or really meaning the manifold (GR) version of same, figuring the listener understands what you meant. If you really wanted to be careful & precise, you should actually define it each time you use it. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

2 1

[math-fun] Really, really stupid question about Lorentz group
by Fred Lunnon 02 Sep '14

02 Sep '14

==================================================================== Just exactly what is the "Lorentz group"? ________________________________ I want to tell you about a clutch of closely related groups of geometrical transformations --- Lie groups --- some familiar, some less so. Each is essentially defined by preserving a quadratic form on m variables, which acts like a metric to specify the topology in some suitable coordinate system. Now it is well-known (trans: I think it's true, but can't find the reference --- anybody?) that the signature of this form --- the numbers (p, n, z) of respectively positive, negative, zero coefficients in the diagonal form --- is an invariant of the group (modulo p,n transposition anyway). Signature provides a concise method for distinguishing groups which are NOT isomorphic. A less technical means to achieve the same end is comparing their topological dimension: unfortunately, all these groups happen to boast the same dimension m(m-1)/2 . The coincidence has in the past contributed to much confusion, a scarcely credible example of which will later be unveiled. Such transformations are traditionally constructed as compositions of elementary "reflectors" leaving invariant prime (maximal nontrivial) subspaces of the underlying space. However the groups so generated are neither connected nor compact, and engineering or physical applications usually restrict attention to their "kinematic" subgroups: connected to the identity, and generated by "rotators" composed of two reflectors. Compactification for computational purposes incorporates non-invertible projections as limits, which may then be identified with members and subspaces of the space. *Exhibit* EucKin(m-1) , proper Euclidean group in real (m-1)-space. Eschewing improper reflections in hyperplanes, this is generated instead by rotations around coline axes, together with translations. Continuous rigid motions arise as infinitesimal limits of compositions of rotators. Each isometry decomposes uniquely in general into at most [m/2] orthogonal commuting rotators: eg. when m = 5 a 3-space helical screw with freedom 6 factorises into rotation composed with translation along the same axis line. Translations constitute an (m-1)-dimensional normal subgroup. *Signature* (m-1, 0, 1) . *Exhibit* MobKin(m-2) , kinematic Moebius group in (m-2)-space: conformal, conserving points and spheres. Again, the usual geometric inversions are disconnected reflectors, and instead we compose rotators, of three flavours: elliptic (including EucKin(m-2) rotations), parabolic (including EucKin(m-2) translations), hyperbolic (including Euclidean dilations). Hyperbolic rotators ("boosts") are easiest to visualise, with eigenspace comprising two real, distinct "limit points": a test sphere initially dilates from source point and finally shrinks into sink point, meantime remaining tangent to some arbitrary (m-2)-set of guiding (m-3)-spheres meeting both points (so fixed by the motion). *Signature* (m-1, 1, 0) . *Exhibit* LagKin(m-2) , kinematic Laguerre group in (m-2)-space: equilong (preserving distance between tangencies), conserving primes and "oriented" spheres (alternatively, half-spaces internal or external to spheres). Besides being disconnected, reflectors are dismally unintuitive; rotators include EucKin(m-2) rotations and translations, exclude similarities (this convention is regrettably not universal). A (hyperbolic) rotator has a pair of (real) "limit primes": a test sphere initially detaches from source prime, finally merges into sink prime, meantime remaining tangent to fixed m-2 spheres, themselves tangent to both primes. A noteworthy parabolic rotator is the "offset", under which the radius of every sphere increases by the same extent; offsets constitute a 1-dimensional normal subgroup. *Signature* (m-2, 1, 1) . *Exhibit* LieKin(m-3) , kinematic Lie-sphere group in (m-3)-space: conserving contact (oriented tangency), acting on & conserving oriented spheres (not separately lines or points). Includes MobKin(m-3) and LagKin(m-3) , which in turn both include EucKin(m-3) . Rotators are as before, except that their eigenspaces now comprise pairs of arbitrary spheres (conjugate complex, coincident, or real). Despite its failure to preserve angles, physicists are reputed to to describe this group as "conformal". *Signature* (m-2, 2, 0) . Note MobKin(m-2) is transitive on points, and separately on other spheres; in contrast LagKin(m-2) leaves fixed the single prime at inversive infinity, where lurk eigenspaces of translations, offsets etc. MobKin(m-2) has no small normal subgroups, essentially because every (say) elliptic rotator is conjugate to every other elliptic rotator equipped with the same (angular) extent; whereas LagKin(m-2) has its one-dimensional normal subgroup of offsets. So those two groups are distinct on these grounds alone. It may also be worth emphasising the distinction between continuous Moebius dilation: hyperbolic, with distance from some fixed centre point an exponential function of extent; and continuous Laguerre offset: parabolic, with all radii a linear function of extent, but no detectable finite axis. Summing up then: if you believe in their signatures (or even if you don't), despite their sharing the same dimension, for m > 3 *** all four groups above are non-isomorphic in pairs! *** Now take say m = 5 ... ==================================================================== Why have I been bending your ear and inflicting this obscure guff on you? https://en.wikipedia.org/wiki/Lorentz_group http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html refer to various luminaries who apparently accept unquestioningly that the (kinematic) Lorentz group is isomorphic to what I have called MobKin(3) above. Whereas an impressive list of authorities --- including Cartan, Poincare, Klein, Blaschke, Coolidge, Bateman, Pottmann --- are alleged at https://en.wikipedia.org/wiki/Spherical_wave_transformation to assert equally firmly that the same group is isomorphic to LagKin(3) instead. [ Implicit in these and similar surveys seems to be an assumption that the z component of the signature may be casually disregarded, the remaining p,n somehow guaranteeing isomorphism; although the quadratic forms are not positive definite, and do not entail classical metrics. ] At this stage in my life, I am resigned to the prospect of never personally acquiring a more than rudimentary understanding of relativistic mechanics. However it is most disconcerting to discover that the grasp of their subject commanded by a substantial proportion of cognoscenti, to whom I might have entrusted the task of allaying my ignorance, appears evidently feebler than my own. Come on, people --- 127 years after Michelson-Morley experimented and Fitzgerald-Lorentz contracted, surely we should be able to get our act together sufficiently to engineer agreement about whether their bloomin' group either (1) has degenerate signature, and/or some one-dimensional normal subgroup, and/or some fixed plane; or else (2) has not? It isn't rocket science, is it ... On second thoughts, it IS rocket science --- Heaven help us all ... Fred Lunnon, Maynooth 02/09/14 ====================================================================

2 4