October 2015 - mathfun@mailman.xmission.com

[math-fun] Someone at Wolfram.com is messing around.
by Bill Gosper 17 Oct '15

17 Oct '15

In[679]:= Sum[(-1)^n/n^2*Cos[Sqrt[\[Pi]^2*n^2 - 9]], {n, \[Infinity]}] Out[679]= -(\[Pi]^2/(12 E^3)) How the hockeysticks did it know that? It's one of the weirder formulas in www.tweedledum.com/rwg/idents.htm, except with the summand written more sanely, exactly as in (1.2) of https://projecteuclid.org/download/pdf_1/euclid.ijm/1255987146 But (1.2) is the b = 3i case of (1.1): In[680]:= FunctionExpand@Sum[(-1)^n/n^2*Cos[Sqrt[\[Pi]^2*n^2 + b^2]], {n, \[Infinity]}] == \[Pi]^2/4*(Sin[b]/b - Cos[b]/3) Out[680]= n 2 2 2 (-1) Cos[Sqrt[b + n Pi ]] 1 2 Cos[b] Sin[b] Sum[----------------------------, {n, Infinity}] == - Pi (-(------) + ------) 2 4 3 b n which, as you can see, stumps FunctionExpand. Why would anyone code up this very special case of an obscure identity? To snow a casual observer? To get a rise out of yours truly?? WRI works in mysterious ways. But they stopped short. If only they had (1.1), they could've expanded at b=3i into an infinitude of obscurities: {Sum[((-1)^n*Sin[Sqrt[-9 + n^2*Pi^2]])/(n^2*Sqrt[-9 + n^2*Pi^2]), {n, Infinity}] == -(((-7 + E^6)*Pi^2)/(216*E^3)), Sum[(9*(-1)^n*Cos[Sqrt[-9 + n^2*Pi^2]])/(2* n^2*(-9 + n^2*Pi^2)), {n, Infinity}] + Sum[-(((-1)^n*Pi^2* Sin[Sqrt[-9 + n^2*Pi^2]])/(2*(-9 + n^2*Pi^2)^(3/2))), {n, Infinity}] == (55*Pi^4*Sinh[6])/31104, Sum[((-1)^n*Pi^2*Cos[Sqrt[-9 + n^2*Pi^2]])/(-9 + n^2*Pi^2)^2, {n, Infinity}] + Sum[((-1)^n*(-27 + 2*n^2*Pi^2)* Sin[Sqrt[-9 + n^2*Pi^2]])/(n^2*(-9 + n^2*Pi^2)^(5/2)), {n, Infinity}] == (5*(-7 + E^6)*Pi^2)/(1944*E^3)} --rwg

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Re: [math-fun] Dead trojan horse, and complaints about clouds generally
by Warren D Smith 17 Oct '15

17 Oct '15

"libration."

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[math-fun] Dead trojan horse, and complaints about clouds generally
by Warren D Smith 16 Oct '15

16 Oct '15

Hi, I thought of a simple argument which pretty much kills my "trojan horse" idea. Consider any cloud of anything whose opacity is at least 20%. Any particle in this cloud will therefore collide, with probability at least about 20%, after performing 1/pi libations. Hence 1 to a few libations times later, the cloud will have greatly changed, i.e. its existence is clearly temporary. The libation time for the jupiter trojans in our solar system has been estimated to be 150 years. It doesn't seem reasonable to suppose any cloud responsible for KIC 8462852 could have libation times over 100K years. Therefore, (1) my "trojan horse" idea loses its whole advantage of having gigayear lifetimes, rendering it now unattractive. (2) other ideas of clouds arising from "aftermaths" intuitively seemed temporary, and this argument now verifies that intuition. Further, the whole aftermath temporary cloud idea seems to me to have a hell of a lot of difficulty explaining how there are lots of different durations and magnitudes of the brightness dips. Why the hell should there be lots of different temporary clouds, all different sizes and/or in way-different orbits? I mean, asking for just one already was implausible due to its short life compared to that of the star, but several at the same time seems absurd. So... alien megastructures seem called for. If that's what it is, those aliens seem pretty damn awe inspiring. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] possible explanation for KIC 8462852
by Warren D Smith 16 Oct '15

16 Oct '15

To T.Boyajian & J. Wright 15 Oct 2015 I have a possibly stupid and possibly correct explanation for KIC 8462852. (If the former, you likely can rapidly recognize that.) See, it would be handy to explain it, if there were big dense clouds of asteroids, sometimes occluding the star. But it seems a priori nutty for a suitable cloud to be able to exist, and stably enough that it still is here. But actually: two such clouds, called the TROJAN ASTEROIDS, do exist in our solar system, 60 degrees ahead and 60 degrees behind, Jupiter. And they are stable enough to still be here over 4 billion years later. And our trojan "clouds" are physically very large in diameter, I think easily larger than the sun. No problem about that. But our trojans produce only a tiny occlusion effect on our sun as viewed by some suitable extrasolar observer. KIC's would have to be a much more severe occulder. So postulate KIC 8462852 contains 1 (or perhaps more) jupiter-analogues orbiting it -- which might be massive planets, perhaps more compact than jupiter and more massive. Whatever parameters are needed to make it work. Or perhaps they might even be a cold neutron star or white dwarf, not a "planet." The question of whether those jupiter-analogues exist could be addressed by more observation including doppler wobble observations, incidentally. Have any been done? The jupiter(s) (my hypothesis would guess) are herding clouds of trojans, with much more surface area than our solar system's trojan clouds, hence capable of much more occluding. But having enough random motions and/or low enough mass so these clouds stay clouds, not condensing into new planets. (There are stability demands, known since Routh, about systems of this type which force the clouds of trojans each to have much less mass than the star and the jupiter, anything below 1% should suffice.) So, that's about it. Oh, of course, if this turns out to be a good candidate, it has to be named the "trojan horse." -- Warren D. Smith (PhD) http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Characteristic polynomial question
by Mike Stay 15 Oct '15

15 Oct '15

What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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Re: [math-fun] Characteristic polynomial question
by Henry Baker 15 Oct '15

15 Oct '15

Perhaps it would be helpful to first solve this problem for n=3. Consider a cubic equation. The roots of the derivative quadratic are the foci of the inellipse of the triangle formed by the roots of the cubic (in the complex plane). This is the Siebeck-Marden theorem: https://en.wikipedia.org/wiki/Marden%27s_theorem The line through the foci of the inellipse is the least-squares approximation line through the roots of the cubic. So you're doing a flavor of "principle component analysis" https://en.wikipedia.org/wiki/Principal_component_analysis At 09:36 AM 10/15/2015, Mike Stay wrote: >What operation on a matrix M gives a new matrix M' whose >characteristic polynomial is the derivative of the char. poly. of M? >-- >Mike Stay - metaweta(a)gmail.com >http://www.cs.auckland.ac.nz/~mike >http://reperiendi.wordpress.com

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Re: [math-fun] Characteristic polynomial question
by Warren D Smith 15 Oct '15

15 Oct '15

> What operation on a matrix M gives a new matrix M' whose > characteristic polynomial is the derivative of the char. poly. of M? > ... oh sorry, "monicized derivative"? --no operation, if M' and M both are the same size, say NxN [since degree(P')+1=degree(P)=N]. So your only hope is some sort of operation that removes a row & column (among other things). If we restricted attention to matrices of "companion" form https://en.wikipedia.org/wiki/Companion_matrix then the answer would be "remove the 1st row and column, then multiply the entries of the final column by 1/N,2/N,3/N,4/N,...,(N-1)/N." Which of course fails for transposed companion form etc. But anyhow, that tells us that the following admittedly somewhat contrived operation works for any NxN matrix M: 1. Express M = A C A^(-1) where C is of companion form. [I believe it is possible to arrange for this factorization to be unique if certain restrictions are placed on A...] 2. Zero the 1st row and column of C, then multiply the entries of the final column of the not-zeroed (N-1)x(N-1) block by 1/N,2/N,3/N,4/N,...,(N-1)/N. Result is C'. 3. M' = A C' A^(-1) has charpoly which is the monicized derivative of charpoly(M)... except times x so there is an extra zero eigenvalue.

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[math-fun] Immortality and mortality
by Warren D Smith 15 Oct '15

15 Oct '15

As previously discussed, humans exhibit an exponential(time) death-rate curve, Gompertz law, which is really bad news for those who'd like to live. Furthermore, it looks like there are no remarkably long-lived mammals. The longest lived besides humans are (?) elephants, the oldest known elephant lived to age 86 in Taipei zoo. If we ask for animals that lived longer than the human record of 122 years, the only known advanced animals that clearly do that are several varieties of tortoises, which can live at least 250 years in some cases. And there are a few other birds and reptiles which might be able to exceed 122 years although it has never been observed to happen. Humans might be able to live longer via genetic engineering, more precisely genetic "theft," from a few of those long-lived animals. DNA sequencing could shed light. E.g. elephants were just sequenced and found to have about 50 copies of the p53 tumor-suppressor gene (humans have only 1 copy) which perhaps explains why elephants get cancer much less than humans do. Some primitive lifeforms can, however, entirely avoid Gompertz law -- which could be regarded as immortality, at least by comparison to us -- at least as far as anybody has been able to tell, their DeathRate(age) curves are asymptotically constant. These include "hydra." So perhaps Gompertz law only happens to "complex" animals, not to "simple" ones. Biochemically speaking, of course even hydra is a extremely complex object, but, e.g. it has few or no "organs" so perhaps the problem is the cell differentiation and macroscopic-development processes that happen in advanced animals, which perhaps are entirely absent in hydra. The mathematical question is: what really is the underlying cause of Gompertz law? Which as far as I know, was never really understood. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Terminology: "Characteristic Generating Function"?
by Warren D Smith 15 Oct '15

15 Oct '15

Interesting, I had not seen that kind of generating function discussed before, so quite likely nobody ever invented a terminology for it. "Lacunary series" is a term for "sparse power series" of that general ilk, though. I doubt this kind of series is very useful for the sort of combinatorial purposes GFs usually are used for. You mentioned a few cute examples, sure, but they seem likely to me to be about the best you'll ever get, in which case, combinatoricists should not be very interested. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] super-powerful alien race found?
by Warren D Smith 15 Oct '15

15 Oct '15

Kepler observed that the star KIC 8462852 underwent irregularly shaped, aperiodic dips in flux down to below the 20% level. The dipping activity can last for between 5 and 80 days. Some astronomers seem to think one of the most likely explanations is orbiting megastructures built by a super-powerful alien race near that star. http://www.theatlantic.com/science/archive/2015/10/the-most-interesting-sta… http://arxiv.org/pdf/1509.03622v1.pdf http://sites.psu.edu/astrowright/ http://www2.astro.psu.edu/~jtwright/Dyson/GHAT4.pdf -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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