March 2017 - mathfun@mailman.xmission.com

[math-fun] Big Splat theory
by James Propp 04 Sep '17

04 Sep '17

Has anyone ever proposed in the Annals of Improbable Research or a similar venue that, in addition to undergoing periodic reversals of its magnetic field, Earth is also subject to occasional short-lasting reversals of its gravitational field? Such a phenomenon would handily explain selective extinction of megafauna: a mouse could survive an upward fall for five seconds followed by a downward fall for five seconds followed by an extreme deceleration, but a mastodon would not fare as well. :-) Jim Propp

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[math-fun] Grandi valuations
by James Propp 05 May '17

05 May '17

If S is an eventually-periodic subset of {0,1,2,...}, then |S|_q, defined as the sum of q^n as n ranges over S, is a rational function. The map from S to |S|_q is a finitely-additive measure ("valuation" is probably a better word to use) taking its values in the ring of rational functions in q. If we order that ring "at 1", we get a way to compare sizes of sets of natural numbers. In fact, it is a total ordering (though not a well-ordering) of the set of eventually-periodic subsets of {0,1,2,...}. This may remind some of you of the ordered fields that you get by adjoining to R a formal infinitesimal or a formal infinity. In both cases, you've got the field R(x), and all that differs is the ordering you put on it. My proposal is similar, except that I'm ordering the field at 1 rather than 0 or infinity. Writing |S|_q as A / (1-q) + B + lower order terms, we can focus on just A and B. A measures the density of S, and B captures finer information. The first thing to notice about B is that it's sensitive to perturbations that don't affect the density; if you add an element to S, you increase B by 1, and if you remove an element from S, you decrease B by 1 (even though A doesn't change). But B conveys subtler information too: For instance, compare {0,2,4,...} with {1,3,5,...}; the former has B = 1/4 while the latter has B = -1/4. Informally, we say that sliding the set of even nonnegative integers over the right by 1 reduces the "size" of the set by 1/2. And that makes sense, because if you slide it over to the right again, "losing" half an element again, you get {2,4,...}, which is missing precisely 1/2 + 1/2 = 1 of the elements of {0,2,4,...}. (Yeah, this way of measuring sets is not translation-invariant. But you gain something nice in return: a proper subset of a set is always strictly smaller than the set.) I recently learned that Ilya Chernykh has been exploring the same circle of ideas (see http://mathoverflow.net/questions/215762/non-standard-numbers-and-exponenti…) but I don't know of anyone else who's fished in these waters. (You could argue that analytic number theorists know all this already and don't think it's interesting.) Anyway, the map that sends S to (A,B) should I believe be called a "Grandi valuation", since it gives a rigorous framework for making sense of Grandi's formula 1 - 1 + 1 - 1 + ... = 1/2. (Indeed, Bernoulli's formula 1 + 0 - 1 + 1 + 0 - 1 + ... = 2/3 can be understood in the same way: {0,3,6,...} exceeds {2,5,8,...} by 2/3 of an element, if we measure size using this valuation.) More precisely one might call the map the "Grandi-Abel valuation", since it uses Abel regularization of the summation of the indicator function of S. Anyway, other ways of taming divergent series exist, so there will be other Grandi valuations. And maybe the term should also be applied to the non-truncated valuation (the one that takes values in the ordered field R(q)), which gives the sets {0,3,4,7,8,11,...} and {1,2,5,6,9,10,..} different "sizes" (the former is infinitesimally larger). In the non-truncated valuation, we get a translation-covariance property that might console us for the loss of translation-invariance: shifting a set corresponds to multiplication by q. One can try to extend the Grandi valuation to sets that aren't eventually periodic, as I do in my MathOverflow post, and to some extent this works. However, one can construct sets that are incomparable, so if you're a fan of number systems that satisfy trichotomy, it's important not to overreach. I haven't given a definition of what a Grandi valuation is in general, but a hallmark is that a Grandi valuation should assign to any eventually periodic set S a value in some module from which the quantity I called "B" can be derived. Anyway, I have a bit of a crush on this notion, and I'd like to put it to some sort of use. My current target is a general uniqueness theorem in the context of one-dimensional packing. Jim P.S. It has occurred to me that when we measure infinite sets in this way, a cute name for this kind of generalized cardinality would be "grandiosity". But I think it's prudent to avoid introducing this term until I have a better idea for how it should be defined!

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[math-fun] http://mathworld.wolfram.com/SphericalSpiral.html still disputes my old rhumb (= loxodrome) math-fun
by Bill Gosper 17 Apr '17

17 Apr '17

Despite the constant-heading definition, the graph (http://mathworld.wolfram.com/images/eps-gif/SphericalSpiral_700.gif) clearly shows a course approximating due west near the poles, but about 30º north of west near the equator. Here is the old(!) mail minus a long list of ~b (BCC), some deceased. math-fun(a)optima.cs.arizona.edu rhumb lines ~c [Eric Weisstein](a)wolfram.com Date: (unfortunately missing) Under "Spherical Spiral", Eric's Stupendium (at least the versions I can access) gives "The path taken by a ship which travels from the south pole to the north pole of a Sphere while keeping a fixed (but not Right) Angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles. It is given by the parametric equations [...]" which are equivalent to [x,y,z]= cos(t) sin(t) a t [---------------, ---------------, - ---------------]. 2 2 2 2 2 2 sqrt(a t + 1) sqrt(a t + 1) sqrt(a t + 1) He also gives [the above-mentioned] 3D plot for some unspecified value of a, which presumably depends on the ship's heading. Leaving aside the problems of navigability at the south pole, and of having any initial heading other than due north, this still can't be right. The coils of a rhumb line do not, in fact, appear to bunch up at the poles because the curve approaches a logarithmic spiral, and has finite total arc length. It indeed has infinite winding number, but due to their exponential shrinkage, only a few coils are visually discernable. E.g., for a northwest heading, each orbit of the pole gets you a factor of e^(2 pi) ~ 535.5 closer, so you don't even see a full turn. As a further check, I get as the unit due north (tangent to the unit sphere) vector from the above curve: a t cos(t) a t sin(t) 1 [---------------, ---------------, ---------------], 2 2 2 2 2 2 sqrt(a t + 1) sqrt(a t + 1) sqrt(a t + 1) which dotted with the unit tangent to the curve gave me a - --------------------, 2 2 2 sqrt(a t + a + 1) which should not depend on t. For a heading of a radians west of north, I propose instead the formula t t [sin(t) cos(tan(a) log(tan(-))), sin(t) sin(tan(a) log(tan(-))), - cos(t)], 2 2 which runs from south pole to north as 0 < t < pi. This paramaterization has the virtue that arc length s is merely t sec a. [Now that there's Mathematica, Out[398]= {Cos[Log[Tan[t/2]] Tan[a]] Sin[t], Sin[t] Sin[Log[Tan[t/2]] Tan[a]], -Cos[t]} In[399]:= ArcLength[%, {t, 0, t}] Out[399]= t Sqrt[Sec[a]^2] ] Thus sailing northwest [a = -π/4, gosper.org/NW rhumb.png] will get you to the pole only sqrt 2 times slower than sailing due north. Unfortunately, you are killed by centrifugal whiplash [assuming you personally have a positive radius] before being able to contemplate the miracle of having sailed northwest to a point not northwest of anywhere. [ --rwg]

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[math-fun] probability
by Cris Moore 04 Apr '17

04 Apr '17

Is there a good source for intro-level probability “paradoxes” that would give me an opportunity to pit cognitive biases against mathematical level-headedness? I have in mind things suitable for e.g. 6th-graders, things like: “I flip 8 coins. Which is more likely, that they come up HTTHHTTT or HHHHHHHH?” - Cris

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Re: [math-fun] More power series Q's
by Henry Baker 30 Mar '17

30 Mar '17

At 12:51 PM 3/29/2017, Schroeppel, Richard wrote: >Easy transform for your 3x equations: >Set g(x) = K f(x), with K tbd. >Then >G(3x) = K f(3x) = 3K f(x) + Kb f(x)^3 > = 3 G(x) + (Kb/27) G(x)^3. > >Adjust Kb/27 to be whatever you like; I suggest trying +-4. Indeed! Thanks! (%i1) bar; 3 (%o1) f(3 x) - b f (x) - 3 f(x) (%i2) f(x):=K*sinh(x); (%o2) f(x) := K sinh(x) (%i3) %o1,eval; 3 3 (%o3) - b sinh (x) K + sinh(3 x) K - 3 sinh(x) K (%i4) %/K,expand; 3 2 (%o4) - b sinh (x) K + sinh(3 x) - 3 sinh(x) (%i5) %,K^2=4/b,expand; 3 (%o5) sinh(3 x) - 4 sinh (x) - 3 sinh(x) (%i6) %,trigreduce,expand; (%o6) 0 So our b is simply a weird way to scale sinh().

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[math-fun] Newman = -1/Oldman
by Bill Gosper 30 Mar '17

30 Mar '17

This must have been found by Newman himself but I only just noticed. Enumerate the nonnegative rationals by iterating "Newman's function": In[96]:= NestList[1/(2*Floor[#] + 1 - #) &, 0, 9] Out[96]= {0, 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3} "Oldman's function" runs the sequence back to 0, counting the rationals: In[97]:= NestList[2*Ceiling[1/#] - 1 - 1/# &, 4/3, 9] Out[97]= {4/3, 1/4, 3, 2/3, 3/2, 1/3, 2, 1/2, 1, 0} E.g., 0 is the zeroth rational, 4/3 is the ninth. Oldman is completely unnecessary! Just negate and reciprocate the last value, and Newman runs backwards: In[100]:= NestList[1/(2*Floor[#] + 1 - #) &, -3/4, 9] Out[100]= {-3/4, -4, -1/3, -3/2, -2/3, -3, -1/2, -2, -1, ComplexInfinity} (with a much louder completion announcement). Or if you prefer, Oldman runs the negative rationals forwards: In[99]:= NestList[2*Ceiling[1/#] - 1 - 1/# &, -1, 9] Out[99]= {-1, -2, -1/2, -3, -2/3, -3/2, -1/3, -4, -3/4, -5/3} obviating Newman. --rwg

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[math-fun] ries strikes again. Mathematica answers!
by Bill Gosper 30 Mar '17

30 Mar '17

Date: 2017-03-27 18:21 From: "David Wilson" <davidwwilson(a)comcast.net> rwg, apologies for filling your head with all these √2's. > -----Original Message----- > From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On > Behalf Of Bill Gosper > Sent: Monday, March 27, 2017 8:25 PM > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] ries strikes again. Mathematica answers! > > In[69]:= ((1/2 ((-I)^(1 + √2) + I^(1 + √2)) - > Sqrt[-1 + 1/4 ((-I)^(1 + √2) + I^(1 + √2))^2])^(-1 + √2) + > (1/2 ((-I)^(1 + √2) + I^(1 + √2)) + > Sqrt[-1 + 1/4 ((-I)^(1 + √2) + I^(1 + √2))^2])^(-1 + √2)) // > FullSimplify > > Out[69]= 2 Sin[π 2√2] > > (How??) --rwg CNS Crash Report: Brain capacity exceeded. OS version: √2 Reboot. OS version √3 In[518]:= %367 /. m -> 2 - Sqrt[3] /. n -> Sqrt[3] + 2 // Simplify // tim During evaluation of In[518]:= 20.750373,2 Out[518]= 1/2 ((1/2 ((x - Sqrt[-1 + x^2])^(2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^( 2 - Sqrt[3])) - Sqrt[-1 + 1/4 ((x - Sqrt[-1 + x^2])^( 2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^(2 - Sqrt[3]))^2])^( 2 + Sqrt[3]) + (1/ 2 ((x - Sqrt[-1 + x^2])^(2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^( 2 - Sqrt[3])) + Sqrt[-1 + 1/4 ((x - Sqrt[-1 + x^2])^( 2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^(2 - Sqrt[3]))^2])^( 2 + Sqrt[3])) == x ∀ complex x ! (according to Plot3D, which can miss isolated point discontinuities, which I think can only occur along ridges of infinite curvature. --rwg)

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[math-fun] Biologists turn cells into biocomputers
by Henry Baker 29 Mar '17

29 Mar '17

FYI -- The cycle time is a little slow, but what the heck? Perhaps these glow-in-the-dark cells could become a bit-mapped display? Then someone could really say "read my lips"... http://www.sciencemag.org/news/2017/03/scientists-turn-mammalian-cells-comp… Scientists turn mammalian cells into complex biocomputers By Robert F. Service Mar. 27, 2017 , 11:00 AM Computer hardware is getting a softer side. A research team has come up with a way of genetically engineering the DNA of mammalian cells to carry out complex computations, in effect turning the cells into biocomputers. The group hasn't put those modified cells to work in useful ways yet, but down the road researchers hope the new programming techniques will help improve everything from cancer therapy to on-demand tissues that can replace worn-out body parts. Engineering cells to function like minicomputers isn't new. As part of the growing field of synthetic biology, research teams around the globe have been manipulating DNA for years to make cells perform simple actions like lighting up when oxygen levels drop. To date, most such experiments have been done in Escherichia coli and other bacteria, because their genes are relatively easy to manipulate. Researchers have also managed to link multiple genetic circuits together within a single cell to carry out more complex calculations in bacteria. Scientists have tried to extend this to mammalian cells to create genetic circuitry that can help detect and treat human diseases. But efforts to construct large-scale genetic circuits in mammalian cells have largely failed: For complex circuits to work, the individual componentsÂthe turning on and off of different genesÂmust happen consistently. The most common way to turn a gene on or off is by using proteins called transcription factors that bind to and regulate the expression of a specific gene. The problem is these transcription factors "all behave slightly differently," says Wilson Wong, a synthetic biologist at Boston University. To upgrade their DNA "switches," Wong and his colleagues steered clear of transcription factors and instead switched human kidney cell genes on and off using scissorlike enzymes that selectively cut out snippets of DNA. These enzymes, known as DNA recombinases, recognize two target stretches of DNA, each between 30 to 50 or more base pairs long. When a recombinase finds its target DNA stretches, it cuts out any DNA in between, and stitches the severed ends of the double helix back together. To design genetic circuits, Wong and his colleagues use the conventional cellular machinery that reads out a cell' s DNA, transcribes its genes into RNA, and then translates the RNA into proteins. This normal gene-to-protein operation is initiated by another DNA snippet, a promoter, that sits just upstream of a gene. When a promoter is activated, a molecule called RNA polymerase gets to work, marching down the DNA strand and producing an RNA until it reaches another DNA snippetÂa termination sequenceÂthat tells it to stop. To make one of their simplest circuits, Wong's team inserted four extra snippets of DNA after a promoter. The main one produced green fluorescent protein (GFP), which lights up cells when it is produced. But in front of it was a termination sequence, flanked by two snippets that signaled the DNA recombinase. Wong and his team then inserted another gene in the same cell that made a modified recombinase, activated only when bound to a specific drug; without it, the recombinase wouldn't cut the DNA. When the promoter upstream of the GFP gene was activated, the RNA polymerase ran headfirst into the termination sequence, stopped reading the DNA, and didn't produce the fluorescent protein. But when the drug was added, the recombinase switched on and spliced out the termination sequence that was preventing the RNA polymerase from initiating production of GFP. Voila, the cell lit up. As if that Rube Goldbergian feat weren't enough, Wong and his colleagues also showed that by adding additional recombinases together with different target strands, they could build a wide variety of circuits, each designed to carry out a different logical operation. The approach worked so well that the team built 113 different circuits, with a 96.5% success rate, they report today in Nature Biotechnology. As a further demonstration, they engineered human cells to produce a biological version of something called a Boolean logic lookup table. The circuit in this case has six different inputs, which can combine in different ways to execute one of 16 different logical operations. "It's exciting in that it represents another scale at which we can design mammalian genetic circuits," says Timothy Lu, a synthetic biologist at the Massachusetts Institute of Technology in Cambridge. Although the current circuits are a proof of concept, both Lu and Wong say synthetic biologists want to use them to create new medical therapies. For example, scientists could engineer T cells, sentinels of the immune system, with genetic circuits that initiate a response to wipe out tumors when they detect the presence of two or three "biomarkers" produced by cancer cells, Lu says. Another example being explored by Wong and others is to engineer stem cells so they develop into specific cell types when prompted by different signals. This could let synthetic biologists generate tissues on demand, such as insulin-producing cells, or cartilage-producing chondrocytes. DOI: 10.1126/science.aal0963

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Re: [math-fun] about a certain François Mendzina Essomba and surprising pi formulas
by Simon Plouffe 29 Mar '17

29 Mar '17

Hello, I programmed the first one, is is not like the AGM, the number of valid digits is proportional to n, something like approx. (5/2)^n, here is the maple routine : # F.M. Essomba formula es:=proc(n) option remember; if n = 0 then 1/2 elif n = 1 then sqrt(3.0)/2 else sqrt(2.0*es(n-1) + 2)/2.0 end if: end; a:=proc(n) 3*2.0^n/315*sqrt(1.0-es(n)^2)*(256 + 597*es(n) - 1024*es(n)^2 + 542*es(n)^3-56*es(n)^5);end; a(10000) differs from Pi at the 3979'th position. a (4000) differs from Pi at the 1591'th position, confirming that it converges rather slowly. It is not a record calculation, but certainly original (IMHO). I have not tried the other ones yet. Have a nice day and bonne journée.

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Re: [math-fun] posits, IEEE floats, asinh numbers
by Henry Baker 29 Mar '17

29 Mar '17

At 07:16 PM 3/27/2017, W. Edwin Clark wrote: >On Mon, Mar 27, 2017 at 8:41 PM, Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: >The book is apparently a "recent" ("revolutionary" ><https://www.amazon.com/gp/product/1482239868?keywords=the%20end%20of%20erro…>) > book by the author of the slides. http://johngustafson.net/ Has anyone been able to find a valid email for John Gustafson? The only one I found (john.gustafson(a)nus.sg.edu) doesn't work.

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