April 2011 - mathfun@mailman.xmission.com

[math-fun] Gould's sequence and 'mean spectra'
by Adam P. Goucher 03 Apr '11

03 Apr '11

Gould's sequence: http://oeis.org/A001316 This sequence is very erratic, but it is straightforward to apply some calculations to determine various means of the first n terms: Arithmetic mean: O(n^(log(3)/log(2)-1)) Quadratic mean: O(n^(log(5)/log(4)-1/2)) Cubic mean: O(n^(log(9)/log(8)-1/3)) Quartic mean: O(n^(log(17)/log(16)-1/4)) ... Clearly, these converge to O(n). The first formula is derived from the Sierpinski triangle; subsequent formulae are based on higher-dimensional generalisations. The harmonic mean can be calculated in a very different way, by analysing A000120. Let the sum of the first 2^a terms be denoted omega(a). We have the recurrence relation: omega(a+1) = 2*omega(a) + 2^a Expanding this yields: omega(a+1) = 2^a + 2*2^(a-1) + 4*2^(a-2) + ... + 2^a = (a+1)*2^a The arithmetic mean, omega(a+1)/2^(a+1) = (a+1)/2. So, the arithmetic mean of the first 2^a terms is a/2. The arithmetic mean of the first n terms is asymptotically (log(n)/log(2))/2 = log(n)/log(4). The geometric mean of Gould's sequence is thus: O(2^(log(n)/log(4))) = O(sqrt(n)). The harmonic mean of Gould's sequence is the reciprocal of the arithmetic mean of the reciprocal of Gould's sequence. The sum of the first 2^a terms is (3/2)^a; the mean is (3/2)^a/2^a = (3/4)^a. The reciprocal of this is (4/3)^a. Taking logs yields: (4/3)^a = exp(log((4/3)^a)) = exp(a*log(4/3)) = O(exp(log(n)*log(4/3)/log(2))) = O(n^(log(4/3)/log(2))) = O(n^(2-log(3)/log(2))) Specifically, the exponents of the means (henceforth referred to as the 'mean spectrum') are as follows: -infM: 0 ... -xM: 1-log((2^x+1)/2)/log(2^x) ... -4M: log(32/17)/log(16) = 0.228134... -3M: log(16/9)/log(8) = 0.276692... -2M: log(8/5)/log(4) = 0.339036... HM: log(4/3)/log(2) = 0.415037... GM: 1/2 = 0.500000... AM: log(3/2)/log(2) = 0.584963... QM: log(5/2)/log(4) = 0.660964... CM: log(9/2)/log(8) = 0.723308... 4M: log(17/2)/log(16) = 0.771866... ... xM: log((2^x+1)/2)/log(2^x) ... infM: 1 There is an elegant reflection formula here, which is that the opposite exponents sum to unity. The spectrum is sigmoid, or S-shaped, similar to the logistic function, hyperbolic tangent function, and cumulative normal distribution function. The spectrum has the formula: f(x) = (log_2(2^x + 1) - 1)/x. A natural analogue of the function is g(x) = (ln(e^x + 1) - 1)/x, which is also a sigmoid function. Differentiating with respect to x produces: g'(x) = (x*e^x/(e^x+1) - ln(e^x + 1) + 1)/x^2. It can be verified that this is indeed an even function. Which other discrete sequences produce interesting mean spectra? For the sequence of natural numbers, we have f(x) = 1 for x >= 0, and f(x) = -1/x for x <= -1. I'm not sure what the spectrum is like between -1 and 0. Sincerely, Adam P. Goucher

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[math-fun] Reflection formula for QPochhammer
by Bill Gosper 02 Apr '11

02 Apr '11

(q shifted factorial, for penciling in to Appendix I of your BHS): %i log(z) %i sqrt(z) theta (---------, sqrt(q)) 1 2 (z; q) = ------------------------------------- = inf q 1/8 (q, -; q) q z inf 2 2 log (z) 2 %pi - -------- ------ 2 %pi z 2 log(q) %pi log(z) log(q) sqrt(- -------) %e theta (----------, %e ) log(q) 1 log(q) --------------------------------------------------------- . q 1/8 (q, -; q) q z inf (q/z;q) might sometimes be easier to compute than (z;q), e.g. for |q| near 1. Theta[1] and (q;q) (which is also a Theta[1]) are relatively easier. --rwg

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[math-fun] Predicting earthquakes
by Henry Baker 01 Apr '11

01 Apr '11

There has been a lot of discussion in the media about earthquakes as a result of the recent Haiti, New Zealand, Japanese and Chilean earthquakes. "Freakonomics" radio (by the same folks who wrote the book) discussed predicting earthquakes just this week. My simple mathematical model of earthquake faults is this: techtonic plates move relative to one another at about the rate that our fingernails grow--i.e., inches per year. If the plates slide smoothly, then there are no apparent earthquakes. However, if something on one plate hangs up on the other plate, stresses build up linearly over time until the stress becomes greater than the strength of the material(s), and there is a sudden slippage which produces an earthquake which releases much/most of the energy built up while the plate movement was stymied. So, if one looks at the earthquakes along a particular fault line, one could presumably estimate the time & energy along that fault line, and "predict" the size of the next earthquake along that fault line, based on how long it has been since the previous earthquake along that line. So far, so good. But we've only been gathering more-or-less accurate earthquake data for perhaps 150 years, and only anecdotal evidence previously. Yes, one can find ruptures of earthquakes hundreds/thousands of years ago & try to estimate how long ago these happened by weathering & other processes around the rupture, but what about earthquakes that didn't produce surface ruptures, or whose ruptures are hidden beneath the ocean? If we continue with our simple earthquake model, what can we say about areas that _have not_ had earthquakes in the last 150 years? If there have not been even small earthquakes in the area, then either there is _no_ stress buildup or _the hangup between plates is so large that an earthquake that eventually does happen in the area will be incredibly massive_. For example, while California is generally considered to be earthquake country, Missouri is not. Yet Missouri apparently had some massive earthquakes in the 1811-1812 time period: http://earthquake.usgs.gov/earthquakes/states/events/1811-1812.php There has been recent discussion about the vulnerability of the Oregon/Washington portions of the U.S. West Coast to very large earthquakes. The non-existence of medium sized earthquakes may lead many to believe that California is more earthquake-prone than Oregon/Washington, but that may merely be an illusion--the result of California having a slightly-better-lubricated set of plate boundaries. The bottom line seems to be: the existence of quakes of order N in a particular area seems highly correlated with quakes in the area of order <=N, but the _non-existence_ of quakes of order N in a particular area says _nothing_ about the likelihood of quakes of orders >N in the area. The simple model might conceivably be used to predict damage over a long period of time. If the damage is proportional to the energy released, then the damage -- integrated over a long enough period of time -- will be proportional to the amount of energy of the underlying movements & temporarily stored & released in earthquakes of every size. Unless we have any strong reason to suspect anything different, the integrated damage _may not be localized_ to "earthquake-prone" regions at all, because the rare earthquakes in the "non-earthquake" regions will be so powerful as to "catch up" in overall damage very quickly. Indeed, the economic damage in "earthquake-prone" regions may actually be less than that in the "non-earthquake" regions, because the more common the earthquakes, the less the peak stresses being relieved, and the more likely it is for the buildings & other structures to be "earthquake-resistant". The relevance of this email has to do with the prediction of the size & likelihood of earthquakes near nuclear power plants in the U.S. Based on what I've heard, all those multi-million-dollar "studies" of earthquake likelihoods near existing nuclear power plants aren't worth the paper they've been printed on. These earthquake "studies" seem to have even less validity than the "Value At Risk" (VAR) models used by the financial community prior to the financial meltdown of 2008; these models gave probabilities of events occurring only once every few centuries to events that occurred nearly every decade. ---- Does anyone in the Bay Area of California recall all the "studies" that were about how much traffic congestion would be produced if the Embarcadero Freeway were to have been demolished? Well, the Loma Prieta earthquake finished off the Embarcadero Freeway, and none of this massive congestion materialized. I don't recall any of the firms who did those traffic studies apologizing or giving the money back for their faulty predictions. http://en.wikipedia.org/wiki/1989_Loma_Prieta_earthquake If we can't even predict our own above-ground traffic, what hope is there of predicting movements of the earth miles beneath the surface?

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