April 2015 - mathfun@mailman.xmission.com

Re: [math-fun] California water crisis
by Henry Baker 23 Apr '15

23 Apr '15

Good idea, because it allows for a very low-tech solution(!). The very good news: once you have a 900' reservoir, your work(!) is done: just turn on the tap. Problems: seawater is usually at sea level, so this requires pumping seawater to a high reservoir. Next problem: reservoirs at 900' altitude are typically fresh water reservoirs, so this seawater reservoir would have to be lined to keep the seawater from leaching into the soil & causing other environmental damage. Final problem: you then have to pump the fresh water back up to where it is consumed (this is already a problem in San Diego, where the pumping cost from their desal plant is not insignificant). At 05:49 PM 4/23/2015, rcs(a)xmission.com wrote: >During the 197x Energy Crisis, there was an article in Science pointing >out the osmotic pressure difference between fresh water & sea water was >equivalent to a 900 foot dam. The authors had a proposal to extract >some of that as usable energy, building some dam-like gadget at the >mouth of a river. Points in favor: sizable new energy source, tolerable >cost, known physical chemistry. Against: serious eco-damage, possible >membrane issues.

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[math-fun] pronunciation of base 16 numbers
by Thane Plambeck 23 Apr '15

23 Apr '15

Scroll down to the bottom for the proposal itself, although the video is good, too. But surely someone's done something like this before? http://www.bzarg.com/p/how-to-pronounce-hexadecimal/ -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] Water desalination
by Warren D Smith 23 Apr '15

23 Apr '15

If desalination cost 5 kWhour/meter^3, then to desalinate 11 trillion gallons, which is of the order of what California currently wants above natural supply, https://www.nasa.gov/press/2014/december/nasa-analysis-11-trillion-gallons-… would take 8*10^17 joules. The total human world energy consumption per year has been estimated to be 5*10^20 joules http://en.wikipedia.org/wiki/World_energy_consumption of which CA's proportionate share (by population vs world) would be 3*10^18 joules which means California could afford this energy expenditure. But it would be very large compared to the energy that used to be spent to supply water. If humankind had throughout its history spent anywhere near that amount of energy to get water, there would not have been a human race. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] big prime gaps
by Warren D Smith 23 Apr '15

23 Apr '15

The widest prime gap among primes<X (assuming X>100), is claimed in this 40-page paper to exceed C * lnX * lnlnX * lnlnlnlnX / lnlnlnX for some computable positive constant C. This is a major breakthrough but still well short of proving Cramer's conjecture. arxiv.org/abs/1412.5029 Long gaps between primes by Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, Terence Tao April 2015 -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Droodle: What is polygon 10/2?
by Bill Gosper 23 Apr '15

23 Apr '15

On Wed, Apr 22, 2015 at 9:02 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > A: The Israeli DoD building. > Rich sez he bumped the xmission byte limit to 65K, so I tried inserting > the 53k inline > image gosper.org/x1y882d377o987e1.png , the (circularized) Minsky "circle" > x=1, y= 882 or 883 with multipliers d=377/987 (and no other) and e=1. > However, > as NeilB explained to me, GMail ballooned the image to 78K of base64, so here I send again. For 144/377 < d < 377/987, you get a visually similar figure with period 35801. For 377/987 < d < 233/610 you get a period 59640 ring of ten of those fractal pentagrams I keep mentioning. --rwg

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[math-fun] "new" kind of fourier series
by Warren D Smith 22 Apr '15

22 Apr '15

For approximating a function of D variables, the typical fourier series is SUM_A B_A * exp(i*X*A) where X=(x1,x2,...,xD) is the coordinate-vector, and A=(a1,a2,...,aD) is a vector of integers, and B_A is a real coefficient. Another kind of fourier series would be SUM_{A,T} C_{A,T} * sin(i*x1*a1)*cos(i*x2*a2)*cos(i*x3*a3)*...*sin(i*xD*aD) where each function can be either sin or cos, as specified by a D-bit word T. Now you might say "the two are the same" which'd be true in some sense... but... Each term in the first series is a "plane wave." But each term in the 2nd series is a function that is not a plane wave, but rather something "pebbly looking." Also, the 2nd series has more coefficients, a factor 2^D or 2^(D-1) more, Which it seems to me makes a difference. I mean, if you view series #2 as arising from expanding series #1, the "extra" coefficients do not supply any extra information. But if you truncate both series at some given set of terms, then series #2 can provide a bitter fit than series #1 can, by taking advantage of the underdetermination of the coefficients to provide a better least-square fit. So when you view it that way, you see series #2 is actually a strictly wider class of series, capable of more feats, than series #1. For some reason, everybody(?) has focused on series #1, but I would think series #2 is the better choice, a goodly fraction of the time, in applications. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Water in US west
by Warren D Smith 22 Apr '15

22 Apr '15

The Soviets drained the Aral Sea, turning it into an ecological disaster. We might conjecture the USA will divert the great lakes to feed the thirst of the dry west, draining them and creating a similar disaster. The entire West was predicated on intentionally planned unsustainable levels of water use, see the book "Cadillac Desert" by Marc Reisner 1993. Desalination would consume energy... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] CA minimum growth
by Warren D Smith 22 Apr '15

22 Apr '15

> From: "Adam P. Goucher" <apgoucher(a)gmx.com> > Mainly to Bill Gosper: > Theorem: There exists a universal constant K such that for any unbounded > weakly-increasing computable function f : N --> N (such as the inverse > Ackermann function), there exists an infinite-growth pattern such that > the population at time t is bounded above by f(t) + K. > Proof: Build a universal counter machine using a modified sliding-block > memory (where the machine waits for a return signal bouncing back from > the block before proceeding). Program it to successively evaluate > f(n) - f(n - 1)... --I think Goucher's proof is correct, at least if one ignores his details. The key word is "sliding block counter machine" where only a bounded number of counters are needed for turing universality; at any single step such a machine can choose a counter, increment or decrement it, change state (bounded number of states), and perform some simple tests on the counters. Now EXTENDED THEOREM: Goucher's upper bound is best possible in the sense that if f(t) were any (non-strictly) increasing function N-->N whose inverse was UNcomputably-rapidly increasing, i.e. increasing faster than any computable function [so f(t) itself increases to infinity but more slowly than any increasing computable function of t which does so] then Goucher's upper bound would become false. PROOF SKETCH: using a spiral-type enumeration, number all the cells in the CA outward from center. Using a turing machine, simulate the CA while also keeping track of the number of CA steps simulated, and the CA population. Keep going until reach t steps. This proves the population at time t is a computable function. This contradiction with the hypothesis proves the attempted Goucher-theorem-improvement is false, i.e. no such improved theorem is possible, i.e. Goucher's theorem is "best possible" in the sense claimed. QED.

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[math-fun] CA minimum growth
by Bill Gosper 22 Apr '15

22 Apr '15

Everybody knows that the area and population growth of a bounded 2D cellular automaton pattern is limited to ~ time^2, but it never occurred to me that there is a theoretically minimal positive (diameter) growth rate ~ sqrt(log t), achieved in Life by Adam Goucher's [Golly Help] [Online Archives] [Very Large Patterns] [O(sqrt(log(t)))], which counts in binary with an infinite triangular array of "boats". Question: Is sqrt(log(t)) also the theoretically minimum growth of population? --rwg

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[math-fun] What's the proof?
by Bill Gosper 20 Apr '15

20 Apr '15

DanA>I would like to second Neil's recommendation of this book. It is indeed a great book, totally readable and totally fascinating. I seem to recall it arose from a series of lectures Kac gave at Haverford College. --Dan On Apr 17, 2015, at 8:11 AM, Neil Sloane <njasloane(a)gmail.com> wrote: That identity is at the start (page 1, line 1) of one of the true great math books, Mark Kac's Statistical Independence in Prob., Analysis and Number Theory. He attributes it to Vieta.<njas But the cover art http://img2.imagesbn.com/p/9780883850251_p0_v1_s260x420.jpg is really stupid. --rwg

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