April 2020 - mathfun@mailman.xmission.com

[math-fun] J.H. Conway
by Michael Kleber 14 Apr '20

14 Apr '20

Reports on Twitter that Conway has died, from COVID-19. https://twitter.com/grahamfarmelo/status/1249069889534472192 --Michael -- Forewarned is worth an octopus in the bush.

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[math-fun] John Conway
by Cris Moore 13 Apr '20

13 Apr '20

This is the first time a cellular automaton has made me cry: https://xkcd.com/2293/ <https://xkcd.com/2293/> Cris Moore moore(a)santafe.edu

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[math-fun] Whopper math problems
by Henry Baker 13 Apr '20

13 Apr '20

Burger King offers a free Whopper (hamburger) to students who solve this math problem (and also download the Burger King surveillance app^H^H^Hmalware): -sum((-1)^j*((5!*5)/(4!))^(1/2),j,1,461) (all of you can solve this w/o a computer or even a pencil) Perhaps one of you can convince Burger King's advertising agency (Paris-based "Buzzman") to sneak some real math problems into this promotion? https://www.adweek.com/creativity/burger-king-lets-students-unlock-free-who… BTW, what high-school-level math problems would be fun for this sort of promotion?

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[math-fun] A footnote about John Conway
by Neil Sloane 13 Apr '20

13 Apr '20

This is where John was living when he got ill: Parker at Landing Lane, Nursing Care Residence, 501 Easton Avenue, New Brunswick NJ, 08901

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Re: [math-fun] Covid19 R_e considered misguided
by Keith F. Lynch 12 Apr '20

12 Apr '20

Henry Baker <hbaker1(a)pipeline.com> wrote: > Reading between the lines, I'm coming to the conclusion that > the concept of "R_e" ("effective" R), may be fatally flawed, > as it tries to capture some sort of "mean" or "average" R in > an inherently exponential setting. A geometric mean is meaningful in an exponential setting. > 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. Are you sure none of the power means will do it? The higher power means give larger numbers more weight than smaller numbers. > For example, in Massachusetts in the early going, almost 100% of the > confirmed cases stemmed from a single meeting of a single company in > a single downtown Boston hotel. No smooth model will work for tiny numbers. Fortunately for modelers, and unfortunately for everyone else, the numbers are no longer tiny. > So if it is true that there are "super spreaders", both in terms of > individual people and/or individual events, there must be a better > way to quickly identify and isolate these people and events other > than putting the entire world on lockdown. Unfortunately, there's no way to know who those people are until far too late. Ideally we would have plentiful, inexpensive, reliable tests for both the virus and antibodies to it, so that everyone could be tested once a week. Those who are negative for the virus would be free to associate with each other, but not with those who are positive for the virus. Those who are positive for the virus would be free to associate with each other, but not with those who are negative for the virus. Those who are positive for antibodies but negative for the virus would be free to associate with everyone. Unfortunately, current antibody tests are not only scarce, but have such a high error rate that, if given to everyone, the majority of positives would be false positives. > Perhaps fractures in glass or metal may have relevance, as it may > only take a single flaw in a crystal structure to destroy the entire > structure. It's quick, easy, inexpensive, and reliable to detect fractures, e.g. with ultrasound. Unfortunately, it's not yet easy, inexpensive, or reliable to detect either viruses or antibodies. Another approach would be to find a way to mass-produce inexpensive, reliable, and self-contained biohazard suits, which would allow everyone to freely associate, though people would need to eat alone. The suits would need power, but for stationary jobs and social events that could be done from an ordinary power outlet.

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[math-fun] A063539 on OEIS
by Allan Wechsler 12 Apr '20

12 Apr '20

The referenced sequence records numbers whose largest prime factor falls short of the square root of the number. The graph of the sequence makes it clear that these numbers have a well-defined asymptotic density. Following a citation to HAKMEM 29 (Schroeppel), we find a claim equivalent to saying that this density is 1 - ln 2 ~ 0.30685 But empirically, the density looks considerably smaller, more like 0.27. The problem can't be whether or not to include exact squares of primes, because those have asymptotic density 0. We have one more reported reference, to Exercise 28 on p. 26 of Tenenbaum and Wu, Théorie analytique et probabiliste des nombres, solution on p. 34. But I don't have a copy, and my French isn't up to the task of understanding their solution. Can somebody check it and compare it with Schroeppel's result (which, typically for HAKMEM, is stated without proof)? Rich, do you remember your approach?

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[math-fun] Second Differences.
by Brad Klee 12 Apr '20

12 Apr '20

Using data up to today, total_cases.csv from: https://ourworldindata.org/coronavirus-source-data U.S. Data: ... 6427, 9415, 14250, 19624, 26747, 35206, 46442, 55231, 69194, 85991, 104686, 124665, 143025, 164620, 189618, 216721, 245540, 277965, 312237, 337635, 368196, 398809, 432132, 466033, 501560, 529951 Second Differences: {1847, 539, 1749, 1336, 2777, -2447, 5174, 2834, 1898, 1284, -1619, 3235, 3403, 2105, 1716, 3606, 1847, -8874, 5163, 52, 2710, 578, 1626, -7136} There is a systematic pattern of negatives corresponding to week ends, which suggests a bin-and-average: Mean[{ 5174, 2834, 1898, 1284, -1619}] = 1914 Mean[{3235, 3403, 2105, 1716, 3606, 1847, -8874}] = 1005 Mean[{5163, 52, 2710, 578, 1626, -7136}] = 499 Or Mean[{-2447, 5174, 2834, 1898, 1284}] = 1749 Mean[{-1619, 3235, 3403, 2105, 1716, 3606, 1847}] = 2042 Mean[{-8874,5163, 52, 2710, 578, 1626}] = 209 Either way it looks like US is not yet on peak of daily new cases, contrary to some reports, ex:

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[math-fun] Low-tech measurement of the wavelength of light
by James Propp 12 Apr '20

12 Apr '20

I once read about an experiment where you can measure the wavelength of light using a drop of oil and a pond. The idea is that the oil will disperse into a uniform film whose thickness (calculable as the volume of the drop divided by the surface area of the pond) is close to the frequency of visible light, and then you can observe chromatic effects. Can anyone provide a reference? I did a Google search but couldn't find the thing I recall reading. Thanks, Jim Propp

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[math-fun] peculiar log π series
by Bill Gosper 11 Apr '20

11 Apr '20

For positive integer k, Out[77]= Log[Pi] == EulerGamma + 2 Sum[Beta[1/(2*n), 1 + k, 0], {n, Infinity}] + Sum[(2^(1 - j)*Zeta[j])/j, {j, 2, k}] In[78]:= Table[%, {k, 4}] Out[78]= {True, True, Log[Pi] == EulerGamma + Pi^2/24 + Sum[2*Beta[1/(2*n), 4, 0], {n, Infinity}] + Zeta[3]/12, Log[Pi] == EulerGamma + Pi^2/24 + Pi^4/2880 + Sum[2*Beta[1/(2*n), 5, 0], {n, Infinity}] + Zeta[3]/12} where Beta[1/2/n,s,0] is In[83]:= Integrate[t^(s - 1)/(1 - t), {t, 0, 1/2/n}] Out[83]= ConditionalExpression[Beta[1/(2*n), s, 0], (2*Re[n] >= 1 || Re[n] <= 0 || NotElement[n, Reals]) && Re[s] > 0] || := Or. —rwg

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[math-fun] mental factoring true-or-false
by Bill Gosper 11 Apr '20

11 Apr '20

The three-(decimal)-digit number AB1 is obviously not divisible by the two-digit D5. So neither is the five digit number AB1D5? —rwg Grim news: Formidable mental factorer John Horton Conway has just died.

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