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Re: [math-fun] COVID Calculator
by Keith F. Lynch 30 Mar '20

30 Mar '20

Brad Klee <bradklee(a)gmail.com> wrote: > A truly frightening scenario is one with an extremely adaptive > super-virus whose successful mutation time roughly matches time > of human immunization response. > Let?s hope this possibility remains science _fiction_! Isn't that exactly what HIV is? Fortunately, HIV is not as contagious as covid-19. I wonder if a virus as deadly as HIV or rabies, but as contagious as covid-19 or the common cold is possible. I hope not. Here's an easy (?) math problem to get our minds off viruses: What do 3076521984, 3718250496, and 6398410752, and no other numbers, have in common?

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[math-fun] socks
by Cris Moore 29 Mar '20

29 Mar '20

In an effort to think about something other than Covid-19, as I was taking socks out of the dryer the other day, the following problem crossed my mind. Suppose that we have n pairs of socks, and that individual socks are removed from the dryer according to a random permutation of the 2n socks. At any given time during this process, some number of socks will be unmatched (i.e. one of the pair is out of the dryer and the other is still in). In the unlikely event that no socks have been lost, what is the typical maximum number of unmatched socks when n is large? I solved this first with a system of differential equations, by thinking about each pair as having a state of 0, 1, or 2 socks removed so far. These three classes of pairs can be thought of as susceptible, infected, and recovered… oh wait. Then I found a much simpler solution. Best, Cris Moore moore(a)santafe.edu In the forgetting of what has scarcely transpired there resonates the fury of one who must first talk himself out of what everyone knows, before he can then talk others out of it as well. — Theodor Adorno

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[math-fun] Hyperbolic pairing Re: A simple naive question
by Steve Witham 29 Mar '20

29 Mar '20

> Subject: [math-fun] A simple naive question > From: Ã‰ric Angelini <bk263401(a)skynet.be> > Date: 3/10/20, 1:30 PM > To: math-fun <math-fun(a)mailman.xmission.com> > > Hello Math-Fun, > I would like to transmit two _composite_ numbers A and B > in the form of single number C. How can I do that? > > For instance, if Alice concatenates A et B, Bob will have > a problem in reconstructing those numbers out of C: > A = 865445 > B = 12377007 > C = 86544512377007 > > Is there a clever, economic way to say where C must > be split? Or do you know another technique (dealing > perhaps with prime numbers)? This has nothing to do > with cryptography (I guess) -- just a (silly) question. > Best, > Ã‰. Hi Ã‰ric-- Here's a technique that is economical (but only in the number of bits) and (as it happens) does have to do with prime factoring. This is just the intro, table of contents, and the basic definition. I hope the latex-flavored markdownisn't too hard on the eyes, I will work on putting a PDFsomewhere tomorrow. Charles R Greathouse IV and (indirectly) Richard Sladkey are referred to later in the body. I like Cantor's pairing because it's "mathy." But with Cantor and with the interleaving-digits way, the size of the final thing is twice the size of the *max* of the two inputs. Someone mentioned 2^x 3^y. Very compact (to describe). I have always admired Goedel's tuple-encoding function: 2^a 3^b 5^c 7^d 11^e... What's the word for that? Profligate? Hubristic? I like the BCD digits punctuated by a hex 'A' because it's roughly the size of the concatenation. But... what'sthe *best* way!? In particular, if (as it seems) I want theencoded number to be around the combined size of theinputs, what's the theoretical least overhead? Plus I wanted it to be mathy. Just a silly answer, --Steve ## Hyperbolic pairing function A pairing function is an $f(x, y)$ that takes two numbers (in our case any two positive integers) and somehow encodes them uniquely into a number $c$. Then an inverse function $f^{-1}(c)$ gives back the pair $(x, y)$. The granddad of pairing functions, Georg Cantor's, indexes pairs by scanning diagonals in an x, y grid. Here we define a pairing that scans hyperbolas that pass through integer points,for which the sequence $a(n)$ in https://oeis.org/A006218 is helpful. This definition (one of several there)... $$ a(n) = \sum_{k=1..n} d(k), \\ \text{where }d(k) = \text{number of divisors of k,}\\ $$ means that the half-open interval $[a(n - 1), a(n))$ has just enough room for those pairs whose product is $n$. If we but assign the pairs locations within the interval, a pairing function is defined. #### Contents Encoding $f(x, y)$ <br> Decoding <br> Cost in bits<br> Calculating $c = a(n)$ <br> - Exactly - Approximately - Bounds on $c$ Calculating the inverse $n = a^{-1}(c)$ - This means search - Approximating the inverse - Bounds on $n$ Digression about harmonic numbers ### Encoding $f(x, y)$ To encode a pair $(x, y)$, first let $$ n = xy. $$ Arrange the prime-power factors of $n$ in the usual way $$ n = \prod_j \large{p_j^{t_j}} \\ \text{where } p_k < p_j \text{ whenever } k < j.\\ $$ $x$ and $y$ are products of different powers of the same $p_j$'s: $$ x = \prod_j \large{p_j^{r_j}}\\ y = \prod_j \large{p_j^{s_j}}\\ $$ Encode $x$'s "$r$ digits in base $(t+1)$," *viz*, and bada-boom, $$c = f(x,y) = a(n-1) + \sum_j r_j \prod_{k \lt j} (t_k+1).$$ (I believe this is easier than, e.g., enumerating all the possible $x$'s and sorting them by $x$. And n.b., that would give a different ordering.) (end of extract) --Steve

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[math-fun] Other exponential trends.
by Brad Klee 28 Mar '20

28 Mar '20

The data on iNaturalist has obvious anthropocentric biases, but it can still be fun to contribute and analyze, example: https://www.inaturalist.org/observations/40799158 For six species of Papilionidae that live in Arkansas, it appears that annual die-off follows an exponential trend, see: https://0x0.st/iMzO.png >From the log-linear fits, the time constants are approximately: 15, 18, 21, 18, 25, and 22 days for glaucus, polyxenes, philenor, troilus, cresphontes, and marcellus respectively. I did not find reliable data about adult swallowtail lifetimes, did see 1-2 weeks or less or more. It could be that 20 days is a viable upper bound. Yet this analysis is suggestive at best, possibly coincidental or biased. It would be interesting to look at a wider range of annual insects and see if exponential die-off is generally predictive of adult lifetime. Also, In many cases there's enough data to bin by day, and doing so can give a better idea of how many broods, etc. Glaucus, polyxenes, philenor, and marcellus all have prominent spikes during the last week of April. Is this species dependent or a consequence of more humans wanting to go outdoors? Does anyone else have ideas about how we can make good use of the data on iNaturalist (now up to 33M observations)? --Brad

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[math-fun] New Tetrads, New orchard solution (A006065)
by ed pegg 28 Mar '20

28 Mar '20

I've put in an update at http://www.mathpuzzle.com/ --Ed Pegg Jr

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[math-fun] Inverse "flowsnake" "curve".
by Bill Gosper 24 Mar '20

24 Mar '20

On Thu, Mar 19, 2020 at 5:40 PM Bill Gosper <billgosper(a)gmail.com> wrote: > I was chagrinned to realize that the popularity of this curve is probably > due to how carefully it > self-avoids when you sample it at the natural frequencies of 7^n. > Ironically, like all true spacefillers, > it's actually dense with triple points: > > In[299]:= unflow[3/7 + I/7/\[Sqrt]3] > > Out[299]= {5/42, 11/42, 17/42} > > In[296]:= FlowS /@ {5/42, 11/42, 17/42} > > Out[296]= {3/7 + I/(7 Sqrt[3]), 3/7 + I/(7 Sqrt[3]), 3/7 + I/(7 Sqrt[3])} > > In[300]:= unflow[223/686 - (19 I)/(686 Sqrt[3])] > > Out[300]= {257/2058, 263/2058, 281/2058} > > In[297]:= FlowS /@ {257/2058, 263/2058, 281/2058} > > Out[297]= {223/686 - (19 I)/(686 Sqrt[3]), 223/686 - (19 I)/(686 Sqrt[3]), > 223/686 - (19 I)/(686 Sqrt[3])} > (Without Julian bailing me out!) I finally found a bug that was driving me nuts. flowsnake triple points of the form n/6/7/7/7 <http://gosper.org/flotrips2.png> I believe all the triple points are of the form n/6/7^k. —rwg >

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[math-fun] 24 sides 25 holes
by Leo Broukhis 24 Mar '20

24 Mar '20

I've just heard on a Russian YouTube stream (streamed live yesterday) that a polyhedron with 24 sides and 25 holes has been found, instead of the previously known 4095-sided polyhedron with 4096 holes. No picture was shown, unfortunately. Who posed the problem and when? Is a picture of the new solution available? Thanks, Leo

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[math-fun] Fwd: Sure looks like 1/Log@4
by Bill Gosper 22 Mar '20

22 Mar '20

---------- Forwarded message --------- To: Wolfram Technical Support *Out*[1013]= Sum[1/(n*Log[n]) - 1/(2*n*Log[2*n]) - 1/((-1 + 2*n)*Log[-1 + 2*n]), {n, 2,∞}] In[1015]:= 1/Log@4. Out[1015]= 0.721347520444482 In[1016]:= N[%1013, 22] Out[1016]= 0.7213475204444817036800 —Bill

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[math-fun] COVID Calculator
by Henry Baker 21 Mar '20

21 Mar '20

http://gabgoh.github.io/COVID/index.html Complete epidemic calculator, with sliders for various parameters. Model your favorite epidemic! --- If an asteroid were hurtling towards Earth, I guess I'd be playing with the orbital mechanics calculator. At 12:17 PM 3/1/2020, Henry Baker wrote: >Given the spread of the Covid-19, here's how to explain the >mathematics of epidemics. > >"The 'basic reproduction number' R0 is defined as the expected >number of secondary cases produced by a single infection in a >completely susceptible population. It is important to note that >R0 is a dimensionless number and not a rate." > >R0 is the largest eigenvalue of the 'next generation matrix'. > >For example, R0=phi=(1+sqrt(5))/2 for Fibonacci's rabbit problem; >R0='k' (neutron multiplication factor) in nuclear fission. > >Thus, R0 is the *base* of the exponentials & logarithms which >occur in the various solutions instead of Euler's 'e'. Clearly, >an epidemic won't go 'viral' unless |R0|>1. > >"Notes on R0" > >James Holland Jones, Stanford University, 5/1/2007 > >https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf

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Re: [math-fun] World's shortest Fibonacci formula
by Bill Gosper 21 Mar '20

21 Mar '20

Even shorter: Fib(v+1) = |Uᵥ(*i*/2)|. In[301]:= Table[Abs@ChebyshevU[n, I/2], {n, -1, 9}] Out[301]= {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55} —rwg On Mon, Jan 7, 2019 at 6:19 AM Bill Gosper <billgosper(a)gmail.com> wrote: > U_n(i/2)/iⁿ, where U is the special > function known as Chebyshev polynomial of the second kind: > > In[964]:= Table[ChebyshevU[n, I/2]/I^n, {n, 0, 9}] > > Out[964]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55} > --Bill > U_n(x):= > In[173]:= Sin[(n + 1) ArcCos@x]/Sin@ArcCos@x > > Out[173]= Sin[(1 + n) ArcCos[x]]/Sqrt[1 - x^2] > > In[174]:= FullSimplify@Table[%, {n, 0, 5}] > > Out[174]= {1, 2 x, -1 + 4 x^2, -4 x + 8 x^3, 1 - 12 x^2 + 16 x^4, 6 x > - 32 x^3 + 32 x^5} >

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