April 2020 - mathfun@mailman.xmission.com

[math-fun] Existence from parity
by James Propp 20 Apr '20

20 Apr '20

What are people’s favorite examples of existence proofs that show that a set is not empty by showing that its cardinality is odd? Jim Propp

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[math-fun] Social Distancing and A005260
by Brad Klee 19 Apr '20

19 Apr '20

https://oeis.org/A005260, see comment from April first 2019. Now that it is dangerous for world-lines to intersect or to nearly intersect, should we also count tandem walks according to how well they will adhere to social distancing guidelines? for walks with minimum social distance 0,1,2, I get that: a[n_, SocialDistance_] := 2 Total[ Times @@ #[[{1, -1}]] & /@ Partition[ Binomial[n, #]^2 & /@ Range[0, n], SocialDistance + 1, 1]] Minimum social distance zero (A005260): a[#, 0]/2 & /@ Range[10] Out[] = {2, 18, 164, 1810, 21252, 263844, 3395016, 44916498, 607041380, 8345319268} Minimum social distance one (nAn): a[#, 1] & /@ Range[10] Out[] = {2, 16, 198, 2368, 30100, 392544, 5248782, 71501056, 989177508, 13859716000} Minimum social distance one (nAn): a[#, 2] & /@ Range[10] Out[] = {0, 2, 36, 656, 10400, 159750, 2402764, 35841344, 532676736, 7910428500} See also: triadisches ballett, ha ha ha ha. https://youtu.be/mHQmnumnNgo?t=258 Obviously this was filmed before COVID... --Brad

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[math-fun] Numberphile Tribute to John Horton Conway
by Tom Karzes 19 Apr '20

19 Apr '20

Here's a Numberphile tribute to John Horton Conway, including some inverview clips: https://www.youtube.com/watch?v=WsecAiJDI8s Tom

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[math-fun] Testing N people for viruses with log(N) tests
by Keith F. Lynch 19 Apr '20

19 Apr '20

For some reason there's still an extreme shortage of covid-19 tests. If only everyone could be tested, preferably at least once a week, most of social isolation could end. Negative people could associate with negatives. Positive people could associate with positives. People who were formerly positive but are now negative could associate with everyone, especially if they test positive for antibodies. Fortunately, it's not necessary to have N tests to test N people unless the expected infection rate is close to 50%. If the expected infection rate is, say, one in a hundred, samples can be taken from 128 people, and each sample thoroughly stirred then divided into 7 equal parts. The first part from all 128 would be mixed together, thoroughly stirred, then tested with one test. If the test is negative, then all 128 people are negative, and we're done. If that test is positive, the second part from 64 of the 128 would be mixed together and tested, and the second part from the other 64 of the 128 would be mixed together and tested, requiring two more tests. Probably one of the tests would be negative, in which case those 64 would be negative. The process would then continue with the third, fourth, etc., parts, subdividing it further and further to find the probably just one or two people in the set who test positive. The identical approach could be used with the test for antibodies. The sets of 128 should be chosen so that no two family members or coworkers are in the same set, to avoid sets with a high likelihood of correlations. If some sets have half the people in them infected, that will result in a lot more tests being needed, more than would be compensated for by a similar number other sets having nobody in them infected. I'll leave it to others to calculate what the expected number of tests needed would be, but it's obviously O(log2(N)) where the expected infection rate is 1/N. We're accustomed to thinking in terms of terabytes and petabytes. But it's worth remembering that fractional bits of information are meaningful too. For instance if just one person in the world is expected to test positive for something (e.g. for being the one who left DNA at a specific crime scene), learning which person that is is only 33 bits of information. That means that testing any one person provides just 4 nanobits of information, hence that relatively few tests which provide one bit of information need to be used. Of course the above assumes that the tests are reliable. Testing in this way amplifies any unreliability in either direction. However, this can be compensated for by multiple tests in ways that are much more efficient than simply testing everyone multiple times. Just as unreliable data transmissions can be made more reliable in ways more efficient than simply repeating the transmission multiple times. Parity bits and CRC checks can reduce the error rate of a data transmission to as low as is desired (albeit never quite to zero). I wonder if the reason why doctors aren't doing testing in the way I suggest is because of specialization. Doctors need to talk to experts on data transmission and encoding, such as the inventors of the JT65 mode which made it possible for hams with battery-operated radios and hand-held antennas to communicate by bouncing signals of the moon. With larger antennas, this has been done with a transmitter of just 3 milliwatts.

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[math-fun] Another pic of Hilbert's spacefill
by Bill Gosper 19 Apr '20

19 Apr '20

to accompany should've been "sampHilbert" <http://www.tweedledum.com/rwg/sampeano.htm> Hilbert's [spacefilling] function is dense with quadruple points, i.e. points in [0,*i*] with four distinct preimage points in [0,1]: gosper.org/hilbquad.png . E.g., the preimage of 1/2 + *i*/4 is {5/48, 7/48, 41/48, 43/48}. This picture connects, in the order they were swept, all the points in [0,*i*] with preimages having denominators ≤ 3⨉4096. (All quadruple point preimages have denominators 3⨉2ⁿ.) E.g., the lower-leftmost vertex is 1/32 + *i*/64, the image of 5/12288, 7/12288, 41/12288, and 43/12288. The white segments are retraced boundary, retroflexed in the middle, (making quadruple points). (Spacefilling functions map closed intervals to closed sets.) The Heighway Dragon is dense with mere triple points. Here's one: Hi res Heighway <http://gosper.org/dragtrip!.png> . —rwg

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Re: [math-fun] Testing N people for viruses with log(N) tests
by Keith F. Lynch 18 Apr '20

18 Apr '20

Henry Baker <hbaker1(a)pipeline.com> wrote: > Better to sample and test *sewage* at strategic branching locations > in the sewer system, where the "mixing" has already taken place. > (Yes, fecal matter does contain the virus.) You would need an army (navy?) of small autonomous robots, each equipped with the ability to quickly do large numbers of virus tests. They'd also need access to a complete and accurate map of the city's sewer system, to the level of every individual household. (Do such maps even exist?) And they'd have to have some way to keep track of where they were, despite being in conductive pipes that neither GPS nor cell phone signals can penetrate. And they'd need a way to report their results. And once they did find and report that the virus is in a household, someone would still have to test every person in that household. I did come up with the idea a few years ago of sequencing all human DNA found in sewers, and cross-referencing it with a database of the DNA of fugitives. It's more feasible because there are far fewer fugitives than there are people infected with covid-19. "Hilarie Orman" <ho(a)alum.mit.edu> wrote: > It is being done. https://www.nature.com/articles/d41586-020-00973-x Not the same thing. That article is about testing at the downstream end. That's very useful for determining if there's any of the virus in the city, and mildly useful for determining how much (if you assume the ratio of baths to flushes is roughly constant), but useless at learning which people are infected. By now, the answer to "is it in the city?" is almost certainly yes, whether the city is London or Timbuktu. But, assuming a technology that can quickly sequence and analyze *all* DNA, yottabytes of data, it would be useful for detecting new or known dangerous viruses, bacteria, parasites, etc. Probably a project for the last half of this century, if not the next one. The latter would be close to the level of technology necessary to make a complete backup of a living human brain. That may be the ultimate answer to viruses and the thousand natural shocks that flesh is heir to: Upload ourselves into computers. Then we need only fear *computer* viruses.

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Re: [math-fun] Covid19 R_e considered misguided
by Keith F. Lynch 18 Apr '20

18 Apr '20

Henry Baker <hbaker1(a)pipeline.com> wrote: > Thus, if one person has an R_e of 0.9 and another person has an R_e > of 40.0, there isn't a good way to average the two R_e's to compute > a composite R_e. On second thought, why isn't the arithmetic average appropriate? Wouldn't those two people cause just as many cases as two people with an R_e of 20.45 would? Of course I'm assuming that what's different is the behavior of the two people, not the virus. The 0.9 person always stays home, and the 40 person works in a crowded grocery store without a mask. Since the virus is only a few months old, I doubt different strains of it yet vary in non-trivial ways. If they did, that would of course quickly select for high values of R_e, so R_e would be going up everywhere. I haven't heard of anything like that being observed.

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[math-fun] Pandemics: Good and Bad News
by Brad Klee 17 Apr '20

17 Apr '20

Thanks again Ed Pegg, more documentation is now available explaining our idea on the simplest possible logistic fit model: https://demonstrations.wolfram.com/SummerInsectPandemicsInTheUnitedStates/ You can see the fit to COVID-19 China data. Relative to the discrete fit function, the data has a natural bin width 5, which determines by 5*11=55 the 99.7% peak width. I recalculated the U.S. fit with most recent data, and it looks like systematic error due to Easter was a factor in the second, over-optimistic estimate. The US natural width is now back to 11 at 99.9% overlap on four points to maximum. This is more than twice China's number. If the US tail is similar to the China tail, we can extrapolate 11*7 = 77 days to 99.7% quell, or with an estimated +/- 2 error, as 63-91days to 99.7% quell. Meanwhile... The business people and politicians who are running the country have given the first, seemingly arbitrary, guideline for reopening the country, the rule of two weeks, that reopening should begin after two weeks of declining numbers, with total reopen after six weeks. How they will measure and analyze this, I do not know, but in this analysis, two weeks is about 1.3 11 day intervals from time of peak. Here is the max=1 logistic peak: {0.0126, 0.212, 0.7241, 1.00, 0.844, 0.551, 0.315, 0.168, 0.0870, 0.0443, 0.0223, 0.0112, 0.00561, 0.00281, 0.00141} One interval off maximum reaches .844, two 0.551, so the rule of two weeks could get us to less than 50% reduction; whereas, in China, two weeks is 2.8 intervals, which is closer to 60-70% reduction. Third phase reopening could land on 17% reduction. This all sounds completely horrible, especially the part about millions of people going unemployed. Yet "there is as silver lining to every cloud", and as ever, the Viceroy butterflies of Limenitis archippus are returning and should last much longer than COVID-19. If they are governing on behalf of some monarch, it is no mortal man, so I am happier to follow them! --Brad

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[math-fun] More Ignorance about Monster Rep. Theory
by Brad Klee 16 Apr '20

16 Apr '20

Here are a few more questions, possibly interesting, but currently beyond my depth: Are all irreducible representations rotations? If no, are any of the monster irreps rotational? Does "Monstrous moonshine" imply that the J-invariant is a Hilbert Series for some rep. of monster group? Or more specifically, can J(q) be computed by some sort of Molien equation or correlation table? Do these last two questions make any sort of sense, or do they sound like drunken ravings? In memoriam John Conway, right now could be a good time to throw out some speculation, even if it is over the moon, so to speak. Cheers, Brad

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[math-fun] CDC COVID19 risk score, by profession
by Henry Baker 16 Apr '20

16 Apr '20

I'm absolutely fascinated by the 3-digit precision in these risk factors: Bigger # means higher risk of getting COVID19; don't know what the risk scale measures. Dental Hygienists 99.7/100 (highest; presumably higher than even "sex workers", but that's not a listed category) Urologists 88.3/100 Podiatrists 70.2/100 Embalmers 49/100 Baristas 40/100 Telemarketers 37.3/100 Microbiologists 36.2/100 Models 30.3/100 Nuclear Power Reactor Operators 29.8/100 Medical Scientists, Except Epidemiologists 25.1/100 Graduate Teaching Assistants 21.7/100 Mathematical Science Teachers, Postsecondary 20/100 Physicists 14.7/100 Epidemiologists 13.3/100 (!) Biostatisticians 8/100 Statisticians 6.7/100 Actuaries 5.2/100 Poets, Lyricists and Creative Writers 4.9/100 Mathematicians 4.3/100 Astronomers 2/100 (infection only via meteorites ?) Economists 1.5/100 (???)

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