May 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Hexagonal analogue of Langton's ant
by Steve Witham 24 May '15

24 May '15

I had a link to an paper proving some basic-sounding theorems about the bee. At the moment, I can't find it among my links about Langton's Ant. The alternative to the path being unbounded is that the bug cycles; there are "initial" states producing very short cycles for the bee. (Scare quotes because such a state is just one of the states in the loop.) The bee I'm talking about has a hexagonal cell with one bit of state: turn left or right and toggle the state, as with Langton's Ant. I believe Patterson's worm has a lattice where each edge has a bit of state, and once eaten, an edge can't be traveled again (and so it's sometimes possible for the worm to paint itself into a corner and halt). --Steve > Date: Thu, 21 May 2015 18:41:36 -0400 > From: James Propp <jamespropp(a)gmail.com> > > I studied a very simple combination of initial-state and rule-string that > demonstrably gives states of the universe with bilateral symmetry > infinitely often, and such that the path of the bee appears to be > unbounded, but I was never able to prove the latter assertion; I looked in > vain for a combinatorially-defined "arrow of time" for this system. > > Since a couple of decades have passed, maybe this problem has been solved > while I was thinking about others things! > > Jim Propp > > ------------------------------ > Date: Fri, 22 May 2015 04:24:01 +0200 > From: "Adam P. Goucher" <apgoucher(a)gmx.com> > > The most well-known hexagonal analogues of Langton's ant are Paterson's Worms.

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[math-fun] algebraic fcn simplification
by Bill Gosper 24 May '15

24 May '15

I just spent hours simplifying (64 t^2 - 16 Sqrt[-32 t^2 + t^2/Sqrt[1 - t] + 31 Sqrt[1 - t] t^2 + 32 t^3 - 17 Sqrt[1 - t] t^3 + 10 t^4 + Sqrt[1 - t] t^4])/(128 t^2) to 1/2 + (4 - 5 t)/(8 (1 - t)^(1/4) t) - ((1 - t)^(1/4) (4 + t))/(8 t) using Corey's denester on numeric special cases. How should I have done this? Plotting the difference shows they only agree when arg t lies within π/4 of the real axis. Elsewhere it seems to be -1. Mma gives 0 for Limit t->0, but Plot3D rather consistently spikes at the origin. I don't know how to elicit that spike value except by plotting. Bug? --rwg

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[math-fun] "Antifragility" and mathematical logic
by Henry Baker 23 May '15

23 May '15

I am a huge fan of Nassim Nicholas Taleb, who invented the concept of "antifragility": https://en.wikipedia.org/wiki/Antifragility "Antifragility is a concept developed by Professor Nassim Nicholas Taleb, and a term he coined in his book, Antifragile. Antifragility refers to systems that increase in capability, resilience, or robustness as a result of mistakes, faults, attacks, or failures. As Taleb explains in his book, antifragility is fundamentally different from the concepts of resiliency (i.e. the ability to recover from failure) and robustness (that is, the ability to resist failure)." "Simply, antifragility is defined as a convex response to a stressor or source of harm (for some range of variation), leading to a positive sensitivity to increase in volatility (or variability, stress, dispersion of outcomes, or uncertainty, what is grouped under the designation "disorder cluster"). Likewise fragility is defined as a concave sensitivity to stressors, leading a negative sensitivity to increase in volatility. The relation between fragility, convexity, and sensitivity to disorder is mathematical, obtained by theorem, not derived from empirical data mining or some historical narrative. It is a priori". ---- Taleb gives some examples of fragility: a porcelin teacup or a human bone; you can walk around all day without damaging your bones, but if you jump down 10 meters onto a hard surface you will certainly break some bones, and possibly die. Taleb gives some examples of robustness/resilience: something with a more linear response: small insults produce small damage; large insults produce large, but not spectacular damage. Taleb gives some examples of antifragility: the larger the deviation, the larger the gain -- e.g., the human exercise training effect, where the stressor stimulates a positive outcome. Also, venture capital, which uses optionality & limited liability to create huge gains from a large number of small random bets. Also Maxwell's Demon: http://en.wikipedia.org/wiki/Maxwell%27s_demon ---- Mathematical logic/computer programs would seem to be the ultimate "fragile" system -- a single contradiction destroys the entire system: you can then prove anything as well as its negation. Every bit error in a computer program makes the program worse; only a few such errors can guarantee the program's failure with essentially 100% certainty. Indeed, a number of AI researchers in the 1960's, including Marvin Minsky, searched for a type of logic that wouldn't be so fragile. However, such a logic wasn't found, and eventually AI gave up on logic entirely, and "modern" AI doesn't care for it at all. ---- But fragile classical logic lives on in the form of computer programs, which can fail spectacularly from a single bug. This problem has become increasingly important for computer security, where a single flaw can open up the entire computer system, and indeed an entire enterprise, to catastrophic compromise. Dan Geer has been worried about these issues, but not from a mathematical logic perspective: https://en.wikipedia.org/wiki/Dan_Geer http://geer.tinho.net/pubs ---- Q: What would an "antifragile" computer look like? Such a computer would not only _tolerate_ HW&SW failure, it would _thrive_ on it and run even better because of it. In this sense, such a computer would not only embrace randomness, but feed off it. Most cryptographic protocols require a source of randomness -- e.g., Linux's /dev/random -- in order to operate, and the security of these protocols requires randomness of the highest quality in order to achieve security. So perhaps cryptographic protocols can provide us with some inspiration about how randomness & failure can actually become a blessing instead of a curse. I obviously don't know the answer, but throw this question out for discussion.

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[math-fun] Antifragile logic
by Warren D Smith 23 May '15

23 May '15

Error correction techniques originally explored by J.Von Neumann, and further worked on by many, e.g. A.Toom & Peter Gacs (cellular automata), Bruce Maggs (VLSI) make it possible for a computer to function exponentially long despite a small-enough random constant fraction of bit errors either due to mis-manufacturing or noise. In practice so far, these techniques have been practically unused; instead they've just made everything very reliable. (By the way, there already was the word "robust" hence no need to invent a new word "antifragile"?)

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[math-fun] EM propulsion drive
by Warren D Smith 22 May '15

22 May '15

A photon rocket would produce momentum p by consuming energy E=p*c. That's a very poor E/p ratio (since c is large). Meanwhile, old fashioned chemical rockets and ion drives emitting mass m at speed v, supply momentum p=m*v, at energy cost E=m*v^2/2. Here the E/p ratio is v/2. That is far superior to the photon drive. Like 30000 times better. But the photon rocket would have the advantage that no mass would be consumed (if the energy supply was free, e.g. came from solar power). Even that advantage might be small because you could save mass by making v large (e.g. using small particle accelerator as ion rocket) still tremendously outperforming the photon rocket on E/p ratio, while only consuming small mass. And also since your solar panel would provide more "thrust" than your photon rocket that it powered. So... I conclude the photon rocket idea is bullshit.

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[math-fun] Wireworld for light pulses
by Mike Stay 22 May '15

22 May '15

These guys used photolithography to make a specialized grating that works as a polarized beam splitter for 1550 nm light. It's built out of pixels 120nm on a side. It feels very much like wireworld, with pulses entering on one side, interfering with each other in the circuit in the middle, and exiting out one of two ports on the other side. http://www.nature.com/nphoton/journal/vaop/ncurrent/full/nphoton.2015.80.ht… -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Improving on counting backward from 100 by sevens
by Thane Plambeck 22 May '15

22 May '15

I'd like to know the origin of this commonly-deployed cognitive test. It is a nice one I think—the first three stepping stones, 93, 86, and 79—have obscure factorizations, and it's only at 72 where I'm first able to be entirely confident that I haven't screwed it up, since 100-72 is "obviously" 28, which is 7*4. I'm weak at mental arithmetic generally speaking though. Are there any other similarly-mathy tests like this that people use? -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] Origin of life via cooperating catalyst set: Probability of abiogenesis.
by Warren D Smith 22 May '15

22 May '15

I had earlier remarked the usual sort of remark, that the estimated probability that some DNA sequence would represent "viable life" would be ultra-extremely small if you think about the length of the DNA in shortest possible free living life form today, etc. Like, 2^(-30000). Allan Wechsler than responded he thought there might have been (what I might call) "semi life" and for this sort of vague reason the chance of abiogenesis could be a lot higher than in any such naive calculation. It now occurs to me we can construct a cheap model of such a thing and solve it, thus "proving" the chance of abiogenesis is really quite large, completely the opposite of the previous kind of naive calculation. (Probably this new calculation also is naive, but anyhow, here it comes.) ---------------- Suppose there are N different chemicals in a puddle. Among them, if two are chosen randomly (call them A & B), assume C is the (small) chance that A catalyses the formation of B. QUESTION: In this model, what is the chance F(N) that our set of chemicals contains an "autocatalytic system" i.e. (sub)set of chemicals which in net catalyse the synthesis of every chemical in that subset? (I.e, the puddle contains "life"?) SOLUTION: Equivalent problem: In a random N-vertex directed graph with arc-existence probability C, determine F(N), the chance that there exists a subgraph S each of whose vertices has invalence>=1. There are binomial(N,K) such subgraphs with K vertices (0<=K<=N). The chance that any particular one of them works (i.e. has all invalencies>=1) is (1 - (1-C)^(K-1))^K. The expected number of them that work therefore is (and probability dependencies do *not* affect this): ExpectedNumber = binomial(N,K) * (1-(1-C)^(K-1))^K. Hence log(ExpectedNumber) = log(N)*K*log(1-(1-C)^(K-1)) / log(K) approximately when N>>K>>1. Hence if N>>K>>1/C>>1 we have ln(ExpectedNumber) = ln(N)*K*(1-C)^(K-1) / ln(K) Now choose C small, K=10/C, and N=10*K=100/C, which ought to be good enough for decent approximate validity of the above. Result: ln(ExpectedNumber) = 10*exp(-10)*ln(100/C)/ln(10/C) > 10*exp(-10) > 0.000454 > 1/2203. Finally use the fact ln(x)<x to see ExpectedNumber > 1/2203. CONCLUSION ("WARREN SMITH'S ABIOGENESIS THEOREM"): Given those assumptions, among 2203 puddles we expect to find at least one that is alive! (And among, say, 10000 puddles, it is virtually certain one will be alive.) Note, this theorem does not care how tiny C>0 is, provided the puddles each contain at least N=100/C chemicals. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Hexagonal analogue of Langton's ant
by James Propp 22 May '15

22 May '15

A few decades ago I played with a variant of Langton's ant in which the sites that the bug visits are hexagons rather than squares. I called it the "bee" (as opposed to "ant") model, and I could have sworn I wrote up something about it, but my efforts to dig up anything have been so far unavailing. Even if I didn't in fact ever publish anything about it, I'm sure others had the same idea, and perhaps they published something. If so, can any of you provide any links? Just as in the ant case (described in http://arxiv.org/pdf/math/9501233v1.pdf) one can show that for certain combinations of the initial state of the universe and the rule-string of the bee, the state of the universe will exhibit bilateral symmetry infinitely often. However, the proof of the "fundamental theorem of myrmecology" (the path of an ant is always unbounded) does not transfer over to melittology. (Maybe the boundedness claim is even false; I can't remember if I ever found a counterexample.) I studied a very simple combination of initial-state and rule-string that demonstrably gives states of the universe with bilateral symmetry infinitely often, and such that the path of the bee appears to be unbounded, but I was never able to prove the latter assertion; I looked in vain for a combinatorially-defined "arrow of time" for this system. Since a couple of decades have passed, maybe this problem has been solved while I was thinking about others things! Jim Propp

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[math-fun] David H. Bailey founds "science meets religion" web site
by Warren D Smith 21 May '15

21 May '15

DH Bailey was a math/computer guy working at L.Berkeley labs. He has a website about "science meets religion" here: http://www.sciencemeetsreligion.org/ I somehow subjectively find it/him annoying, but it probably has something good on it. Bailey did a lot of stuff related to high precision arithmetic, "experimental mathematics," and computations related to them, including playing a role in the BBP algorithm for computing digits of pi, and strong integer linear relations finding algorithms. But apparently he just retired and set up this website since this area (it is now revealed) is his true love? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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