March 2020 - mathfun@mailman.xmission.com

[math-fun] Occam's Razor and COVID analysis.
by Brad Klee 31 Mar '20

31 Mar '20

Part of the problem using the COVID-19 calculator is that it has too many parameters relative to available data, especially with the outbreak only 99% ended in China (for now). If we take only one data set, for example Chinese deaths/day, then we are better off fitting to a one or two parameter model. Quite amazingly, COVID data looks enough like the Popillia japonica seasonality data from iNaturalist. It is also well-fit by a logistic map (thanks Cris!). For reference: https://mathworld.wolfram.com/LogisticMapR=2.html With China data, we need to bin data 6 days into one bin, and then find 98.9% accuracy by overlap dot-product. From peak outbreak to 99.98% quell is six intervals, or 36 days. Depending on peak-flattening the extracted parameter 6 (days / bin) should be expected to change by region. This is a good way to compare distinct outbreaks, only when stated next to a distribution accuracy statistic. Another interesting data set is the iNaturalist seasonality for the Chaser and King Skimmer dragonflies of genus Libellula. Good news, there should be an outbreak soon! --Brad

1 0

[math-fun] PDF link Re: Hyperbolic pairing
by Steve Witham 30 Mar '20

30 Mar '20

I've put up the "Hyperbolic Pairing" draft so one can see the math. http://www.tiac.net/switham/2020/03/HyperbolicPairing/index.html V0.4, slightly improved. There is a link to the PDF in the middle of the (short) page. Your browser should complain that it is not secure. I need to get and install a certificate. --Steve

2 1

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?

3 2

[math-fun] socks
by Cris Moore 30 Mar '20

30 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

2 3

[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

1 0

[math-fun] Other exponential trends.
by Brad Klee 29 Mar '20

29 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

1 0

[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

2 1

[math-fun] Inverse "flowsnake" "curve".
by Bill Gosper 25 Mar '20

25 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 >

1 1

[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

5 6

[math-fun] Fwd: Sure looks like 1/Log@4
by Bill Gosper 23 Mar '20

23 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

1 0