July 2016 - mathfun@mailman.xmission.com

[math-fun] International Mathematics Olympiad 2016
by Dan Asimov 19 Jul '16

19 Jul '16

The U.S. took first place in the IMO a few days ago in Hong Kong. The U.S. was also first in 2015, and this is the first time it has ever been first for 2 years in a row (in the 42 years that it has been competing). In both years the leader has been Po-Shen Loh, a math prof at Carnegie Mellon. Each member of the team of 6 got a gold medal! (Last year's team won 5 golds and 1 silver.) (Of course, it is a totally non-level playing field for countries with vastly differing populations and training resources to compete against each other. If if were up to me, this would be a strictly individual competition.) In any case, the *problems* in this year's competition are downloadable (in almost any language you can think of) at this URL: https://www.imo-official.org/problems.aspx Some of the problems look pretty interesting. —Dan

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Re: [math-fun] Dinos died from 2nd hand smoke
by Henry Baker 17 Jul '16

17 Jul '16

'When an asteroid hit Earth 66 million years ago, only those feathered maniraptorans that had downsized to about 1 kilogram or soÂthe birdsÂwere able to survive, probably because their small size allowed them to adapt more easily to changing conditions, the team concludes online today in PLOS Biology.' 'This size reduction was essential for the evolution of flight, says Luis Chiappe, a paleontologist at the Natural History Museum of Los Angeles County in California, who was not involved in the study. "Flight is easier for smaller animals" because it is "a lot less energetically demanding," he says. And during all those millions of years when maniraptorans were changing body size more quickly than other dinos, Chiappe says, "they were experimenting with various degrees of birdness." ' There's only one problem with this "smaller is better to fly" theory: the Pteranodon. While not technically a dinosaur, it lived in the same period, and was presumably closely related to whatever animals birds consider as ancestors. However, I do agree with the "smaller is better to survive asteroids" theory. https://en.wikipedia.org/wiki/Pteranodon 'Pteranodon ... is a genus of pterosaurs which included some of the largest known *flying* reptiles, with wingspans over 6 metres (20 ft).' 'Pteranodon was not a dinosaur. By definition, all dinosaurs belong to eitherorder within Dinosauria, eitherSaurischia or Ornithischia. As such, this excludes pterosaurs. Nonetheless,Pteranodon is frequently featured in dinosaur media and is strongly associated with dinosaurs by the general public.' 'Pteranodon species are extremely well represented in the fossil record, allowing for detailed descriptions of their anatomy and analysis of their life history. Over 1,000 specimens have been identified, though less than half are complete enough to give researchers good information on the anatomy of the animal.' 'An even larger specimen is known ... with a wingspan of 7.25 metres (23.8 ft)' 'While most of a Pteranodon flight would have depended on soaring, like long-winged seabirds, it probably required an occasional active, rapid burst of flapping, and studies of Pteranodon wing loading (the strength of the wings vs. the weight of the body) indicate that they were capable of substantial flapping flight, contrary to some earlier suggestions that they were so big they could only glide' At 11:52 AM 7/17/2016, James Buddenhagen wrote: >no math here ... but to the questions of birds surviving > >http://www.sciencemag.org/news/2014/05/how-birds-survived-dinosaur-apocalyp… > >On Sun, Jul 17, 2016 at 8:18 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> "So how did birds survive?" >> >> Excellent question! >> >> Most scientists think that the primary victims were *large* animals, with the possible exception of large animals living in the oceans. >> >> There remained plenty of oxygen in the atmosphere, but also lots of possibly poisonous gasses. >> >> Smaller animals in burrows might have avoided some of the worst fire conditions in the aftermath of an asteroid hit. >> >> We now (post-GPS) know that some birds -- even some very small birds -- can migrate prodigious distances (thousands of miles), in which case birds may have survived on far-flung islands. >> >> I don't know how many burrowing birds there are.

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[math-fun] Dinos died from 2nd hand smoke
by Henry Baker 17 Jul '16

17 Jul '16

FYI -- I've been arguing for a massive fire after the asteroid for a number of years now; although my hypothesized fire would have been even bigger... ...because back then, *everything* was "an oil field", so the asteroid didn't have to have very good aim. http://www.latimes.com/science/sciencenow/la-sci-sn-dinosaurs-asteroid-soot… After asteroid hit, a giant cloud of smoke led to dinosaursÂ demise, study says Researchers from Japan argue that the 6-mile-wide chunk of rock slammed into an oil field in the present day YucatÃ¡n Peninsula and triggered a burning inferno that launched a massive cloud of smoke into the sky. The resulting layer of soot that enveloped the globe would have been just the right thing to kill the dinosaurs and most other land-dwelling creatures, said study lead author <http://db.tohoku.ac.jp/whois/e_detail/1559a90e9e8a12af48d792940027a6c6.html>Kunio Kaiho, a paleontologist at Tohoku University in Japan. ...

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[math-fun] Sandia storing information in encrypted DNA
by Henry Baker 17 Jul '16

17 Jul '16

FYI -- What happens when we're all "end2end" encrypted? Also, a whole new meaning for "junk" (DNA, that is.) https://share.sandia.gov/news/resources/news_releases/dna_storage/ Sandia storing information securely in DNA Sandia researchers explore a biologically inspired information storage system George Bachand, a Sandia National Laboratories bioengineer at the Center for Integrated Nanotechnologies, is exploring a better, more permanent method for ***encrypting*** and storing sensitive data: DNA. Compared to digital and analog information storage, DNA is more compact and durable and never becomes obsolete. Readable DNA was extracted from the 600,000-year-old remains of a horse found in the Yukon. ...

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[math-fun] The Limit of my credulity
by Bill Gosper 16 Jul '16

16 Jul '16

In[152]:= Limit[ArcSinh[x Log[x] - x] - Log[Log@x]^1, x -> \[Infinity]] Out[152]= -\[Infinity] In[167]:= Limit[ArcSinh[x Log[x] - x] - Log[Log@x]^(1 - 1/E^E^E^E^E^x), x -> \[Infinity]] Out[167]= \[Infinity] There was a brief shining moment when developmental Macsyma could actually do these, using Bill Dubuque's nhayat asymptotic expansion system, inspired by a certain moderator. --rwg

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[math-fun] Average position of Earth
by James Propp 15 Jul '16

15 Jul '16

Ignoring the effects of bodies other than the Earth and the Sun, and treating the Earth as a point mass, what is the average position of the Earth over the course of a year as it travels an elliptical orbit with the Sun at one focus? If you ask me "Are you interested in the average position of Earth relative to the the Sun, or relative to the position of the center of mass of the two-body system?", my answer is "I'm interested in both". I'm hoping that there's an embarrassingly easy way to see what the answer is using principles from physics that I must've been taught but haven't fully absorbed. I'm also interested in the limiting case where the ratio of the two masses in a two-body system goes to 0. Even if the answer to my original questions is "It's messy in either coordinate system", this limiting case (in which the two original questions coincide) might behave nicely. Jim Propp

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Re: [math-fun] Everting a torus
by Dan Asimov 15 Jul '16

15 Jul '16

Sure it's honest. IF by that you mean: ----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? ----- You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type. (OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.) —Dan ——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H: H: (T^2 - D)x[0,1] —> R^3 ** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way). -----Original Message----- >From: James Propp <jamespropp(a)gmail.com> >Sent: Jul 14, 2016 1:27 PM >To: math-fun <math-fun(a)mailman.xmission.com> >Subject: [math-fun] Everting a torus > >Compare the easy-to-follow choreography at >https://www.youtube.com/watch?v=S4ddRPvwcZI >(due to Greg McShane, I'm guessing) with the much subtler choreography at >https://www.youtube.com/watch?v=jA86M6fdm_Q >(due to Arnaud Cheritat, I'm guessing). > >Is the former eversion honest? Note that a circular patch has been cut away >from the torus to make the action easier to follow; this permits unintended >cheating. After all, you could make a video of a sphere-with-a-hole-in-it >being turned inside out, and it would look convincing, unless you think >hard about what happens to the boundary of the patch that bounds the hole. > >The same goes for a recent gif of Simon Pampena's (which I can view on >Twitter but can't find on the web). > >Jim Propp >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Wolfram Alpha stumper
by David Wilson 14 Jul '16

14 Jul '16

solve x^y = x + y = 2

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[math-fun] Everting a torus
by James Propp 14 Jul '16

14 Jul '16

Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing). Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole. The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web). Jim Propp

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Re: [math-fun] 9 * 99 * 999 * ...
by Dan Asimov 14 Jul '16

14 Jul '16

I would not phrase it this way. Which reminds me of newspapers reporting that Fermat's Last Theorem is essentially the Pythagorean theorem (solely because the same expression arises. Here is the Wikipedia article, in which Dedekind eta is defined: https://en.wikipedia.org/wiki/Dedekind_eta_function namely: eta(t) = exp(2pi*i*t/24) Product_{1<=k<oo} (1 - exp(2pi*i*t)^k) which is of course vaguely related to Product_{1<=k<oo} (1 - (1/10)^k) but with major essential differences. Dan ----- Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion. -----

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