January 2016 - mathfun@mailman.xmission.com

[math-fun] In Argentina, Largest Titanosaur
by Henry Baker 19 Jan '16

19 Jan '16

FYI -- Time to bring back my theory that pre-K-T oxygen levels were far in excess of today's oxygen levels. Animals this large wouldn't be able to survive in today's <21% O2 levels; they would feel like we do today at 20,000' altitudes (~50% O2 saturation levels). These excessively high levels of O2 pre-K-T were created by a billion+ years of plant-life CO2 reduction, leading to *huge* amounts of surface carbon in the form of coal/oil/etc. The K-T event 66 million years ago set of a planet-wide fire that burned for years and years, and finally brought atmospheric O2 levels down to nearly today's levels. (Tiny models of such fires burn today in West Virginia, some of which have been burning for >100 years.) https://en.wikipedia.org/wiki/Atmosphere_of_Earth https://en.wikipedia.org/wiki/CretaceousÂPaleogene_extinction_event ----- In Argentina, RancherÂs Discovery Leads to Largest Titanosaur http://www.nytimes.com/2016/01/19/science/titanosaur-argentina-american-mus… "These herbivores lived about 100 million years ago, in the Late Cretaceous Period, on all continents, including Antarctica. They seemed especially plentiful in southern lands." "The Patagonian skeleton was not an easy fit in its New York home. At 122 feet in length, it was a bit too long for the gallery. Part of its 39-foot-long neck extends through an opening in a wall toward the elevator banks, as if to welcome visitors to the fossil floors." "This titanosaur was a young adult, gender undetermined. Its appetite for all kinds of vegetation must have been prodigious. Based on bone sizes, researchers estimated that this individual weighed 70 tons Â as much as 10 African elephants, the heaviest land animals today. Think of its possible heft if it were fully grown." "The size and distinctive shape of an *eight-foot femur* of one specimen astonished scientists."

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[math-fun] KIC 8462852 -- "Alien megastructure" star, something serious happening
by Warren D Smith 19 Jan '16

19 Jan '16

> From: Brent Meeker <meekerdb(a)verizon.net> > The luminosity of KIC 8462852 is about 4.5 to 5 times that of the sun > and its mass is 1.45 solar masses. This means the star has a lifetime > of 2 to 2.5 billion years, which is short for the evolution of > intelligent life. SETI pointed their radio telescopes at it, but saw no > signal. I'd guess the most likely explanation is some chaotic behavior > of the nuclear reactions in the star. It's massive for an F-class star, > close to being a B-class star and low in metals; so it's plausible that > some kind of instability may occur in its radiative zone. > Brent --intelligent life if any may have evolved faster than on earth's 4 Gyr. Or it might be, e.g. Von Neumann machine from a different star. --Even if all nuclear reactions turned off instantly, the outside of the star would not notice for a long time? Cooldown time scales should be 1000-1000000 years? Not 1 day for 20% dimming. --Only certain kinds of stars brighten and dim, this kind is not one of them. It seems to be pretty normal F3-type star, and those just do not do this. Compare with other F3 low-metal stars. No others have ever been seen that do this. Cepheid variables, enormous stars that are extremely bright (100K sun) extremely huge fusion rates, extremely high radiative losses, and extremely short lived (10^6 years) pulsate with periods 2-110 days. The prototype "Delta Cephei" has period=5.4 days, surface temperature oscillates from 5500 to 6600 K, diameter changes about 15%, and its brightest is about 1.9 times its dimmest. --If it really were the star, then its spectrum should change quite noticeably during a dimming event. This if so (or if not) is going to become clear soon because they're pointing spectroscopes at it. https://en.wikipedia.org/wiki/Variable_star discusses. There are many kinds of variable stars in that article, mainly A- and B-type. The only F-types among them are "delta-scuti variables", which this isn't, periodic with rapid period of 15 minutes to 4 hours, whereas KIC 8462852 is very unperiodic and irregular. Further, no star has ever been noticed to have timescales of 100 years for some 20% brightness change, which Schaeffer found for KIC 8462852 by examining old photographic plates at Harvard library. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] KIC 8462852 -- "Alien megastructure" star, something serious happening
by Warren D Smith 19 Jan '16

19 Jan '16

The article is here http://arxiv.org/abs/1601.03256 and they're going to have a hell of a time explaining that. (I used to know that guy, Bradley Schaeffer, in college.) Hoo boy. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] mega awesome space aliens? Now what?
by Warren D Smith 18 Jan '16

18 Jan '16

https://en.wikipedia.org/wiki/KIC_8462852 summarizes situation. I think there now is a decent chance (>10%) the explanation for this is, there are mega awesome space aliens hanging out near the star KIC 8462852. Basically because nobody can think of any alternative explanations that seem plausible. At least I can't. If this really is little green men, they're using something like 20% of the entire power produced by their star. Which means, wow. In comparison, the Earth intercepts 5*10^(-10) of the sun's power. So, herein, let's do some elementary calculations related to: Do those aliens know about us? And what might they do about it? One can also ask, if so, then what *are* they going to do about it? Sure hope they don't dislike us. If they for some reason wanted to destroy humanity, they might be able to. So, first of all, if they were going to spot us by telescope, then one could imagine at best (?) they have a telescope with aperture the size of the solar system (radius 5*10^15 meter), looking at us in UV light (wavelength=200 nm=2*10^(-7) meter), and, e.g, to detect life on earth they detect the ozone layer indicating free oxygen. At worst one might imagine they have a telescope only 1000 meters in diameter checking us out. Assume diffraction limited optics. They are 1400 lightyears = 2.3*10^19 meters away. Earth-sun distance is 1.5*10^11 meter. Earth diameter is 1.3*10^7 meter. To see earth as distinct from sun, they need [ 5*10^15 * 1.5*10^11 ] / [ 2*10^(-7) * 2.3*10^19] big. To see features on Earth itself, they need [ 5*10^15 * 1.3*10^7 ] / [ 2*10^(-7) * 2.3*10^19] big. To directly see, say, a city like ancient Rome, now using 500 nm light, they'd need [ 5*10^15 * 10^3 ] / [ 5*10^(-7) * 2.3*10^19] big. Those are best case for the aliens using telescopes. Worst case, they'd need: [ 10^3 * 1.5*10^11 ] / [ 2*10^(-7) * 2.3*10^19] big and [ 10^3 * 1.3*10^7 ] / [ 2*10^(-7) * 2.3*10^19] big. for the first two tasks. So clearly the aliens should already know Earth exists. They might not yet have noticed it contains life -- that depends how hard they tried to find out. If they tried hard to look at us, then they'd know. If they then tried even harder to look at us, they might even be able to already know Earth has intelligent life. They also could have sent out probes which would somehow have reported that information back to them -- depending how old they are. Now in about 1300 years the aliens will first be able to receive radio transmissions from Earth. Furthermore, we could starting soon make an intentional effort to beam signals at them, in which case they might receive them in 1400 years. Suppose we beamed 100 megawatts of signal power at them using earthwide network of linked radio telescopes (effective aperture 2*10^6 meter) using 1mm wavelength and kept doing it for years. That seems about the most noticeable thing we could do within the next 10-20 years. If so, then I believe we would be able to make ourselves both (a) distinguishable from the sun, and (b) detectable even by current earth based radio telescopes (if aimed at us from there). The detection problem then would be comparable to us receiving signals from current spaceprobes in the outer solar system. I do not know if that'd be enough to make them notice us, but it'd have a decent chance of working. We could then send them whatever at bitrates comparable to outer solar system spaceprobes sending us data. And they, 1400 years later, could send us return mail, if so reaching us an additional 1400 years later. It helped in these estimates to be able to assume those aliens are tremendously advanced. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] mini-me black hole in center of Milky Way
by Warren D Smith 17 Jan '16

17 Jan '16

As is nowadays well known, the Milky Way contains a big central black hole, mass = 4*10^6 Msun. (The mass of the whole Milky Way is of order 10^12 Msun.) http://iopscience.iop.org/article/10.3847/2041-8205/816/1/L7 SIGNATURE OF AN INTERMEDIATE-MASS BLACK HOLE IN THE CENTRAL MOLECULAR ZONE OF OUR GALAXY Tomoharu Oka, Reiko Mizuno, Kodai Miura, and Shunya Takekawa Ap.J. letters now claims evidence that orbiting it only about 200 light years away is a second hole with mass about 10^5 Msun. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] math-fun Digest, Vol 155, Issue 25
by Warren D Smith 17 Jan '16

17 Jan '16

>Might black hole distribution at the center be a cause or effect its overall form? One might ignorantly surmise that elliptical galaxies have a single central black hole, spiral galaxies two, some irregulars have more (or none). --I do not know. It looks like all(?) galaxies have a giant black hole. This suggests that they are important either in helping galaxy form, or in keeping it stable preventing it from self-destructing. The fact (if it is a fact) that Milky Way has a second big hole may or may not matter much. I'd guess "doesn't matter" since it is about 40 times lighter than the Big One and pretty near to it. The enormous energy releases occurring during the formation of the giant holes ("quasars") may have been important for purposes like keeping galaxies separated and ionizing gas. There has been some speculation that in the future, when Andromeda and the Milky Way probably will merge, their big holes may once again light up as quasars, outshining the whole rest of the galaxy. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Angle treasure hunt
by Bill Gosper 16 Jan '16

16 Jan '16

I may once have quizzed this group: Where would you go to find an angle of π/19? I just stumbled on 2π/11. Spoilers below. Hubcap voyeurs: Who has the biggest album of angle pictures? --rwg Where? The Bottoms. http://previews.123rf.com/images/guesswho/guesswho0911/guesswho091100009/63… eri ng-whe el-Stock-Photo-sh ip.jpg

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[math-fun] Strange truth from FunctionExpand
by Bill Gosper 15 Jan '16

15 Jan '16

At the Seattle Joint Meeting last week, NeilB delivered gosper.org/mizan.pdf, my brief tribute to the late q-hypergeometric wizard Mizan Rahman, co-author (with George Gasper) of Basic Hypergeometric Series. My note mentions Ramanujan's sum of dyadic rationals converging rapidly to 1/π, Sum[(42 k + 5)*Binomial[2 k, k]^3/2^(12 k + 4), {k, 0, oo}], unwittingly rediscovered in Acceleration of Series - DSpace@MIT <https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uac…> . Expecting nothing more than 5/16 HypergeometricPFQ[{47/42, 1/2, 1/2, 1/2}, {1, 1, 5/42}, 1/64] I randomly typed it into Mathematica: Sum[(42 k + 5)*Binomial[2 k, k]^3/2^(12 k + 4), {k, 0, \[Infinity]}] // FunctionExpand 1/\[Pi] + (5 EllipticK[1/16 (8 - 3 Sqrt[7])]^2)/(4 \[Pi]^2) - ( 5 Gamma[1/7]^2 Gamma[2/7]^2 Gamma[4/7]^2)/(64 Sqrt[7] \[Pi]^4) I.e. it got 1/π plus an exotic 0 containing one of the EllipticK s for the 7th singular value! Solving for it, EllipticK[1/16 (8 - 3 Sqrt[7])] == (Gamma[1/7] Gamma[2/7] Gamma[4/7])/(4 7^(1/4) \[Pi]) (an old result). Checking, In[548]:= ContinuedFraction[EllipticK[1/16 (8 - 3 Sqrt[7])]/ EllipticK[1 + 1/16 (-8 + 3 Sqrt[7])], 22] Out[548]= {0, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4} Well, I guess Wolfram isn't just regurgitating canned results! I probably mentioned earlier Mma startling me by evaluating the infinite product of factorials divided by (an improved) Stirling's formula, (d15) in http://www.tweedledum.com/rwg/idents.htm, a result Askey once considered "wild". Now if Mma could only denest radicals. --rwg I recently had occasion to measure the diagonal of an 8" square: 11+1/π inches. Oops, sorry teacher, you want mixed numbers: 11 1/π"

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[math-fun] highly symmetric integer lattices
by Warren D Smith 14 Jan '16

14 Jan '16

also, another issue is, you can have, say, a 12-dimensional sublattice (like K12) of Z^24. In other words it's integer, and it's N-dimensional, but the coordinates are only integer in some higher dimension D>N on which we restrict to some N-dimensional subspace. .

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[math-fun] Voronoi tilings from grids with periodic deleted points
by Thane Plambeck 14 Jan '16

14 Jan '16

Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle. This is the picture you get (use VoronoiMesh[] in Mathematica) https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/ Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm) https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/ Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap. (1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way? -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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