November 2014 - mathfun@mailman.xmission.com

[math-fun] Microwave ovens
by James Propp 02 Feb '15

02 Feb '15

Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)? Jim Propp

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Re: [math-fun] analytic function for new Airbnb logo?
by Bill Gosper 14 Dec '14

14 Dec '14

Henry>I wanted an approximation that was entire over the complex plane, so the straight lines and discontinuities would have to be fudged. If I take the image, convert it to black-and-white, and erode the boundaries to a single pixel width, I will get a "continuous" curve that goes around the origin twice. If I consider this curve in (r,theta) coordinates, I can now consider the Fourier transform. I'd like to reset the origin in such a way that the "fundamental" is zero -- i.e., the lowest frequency (which takes 2 revolutions) has a coefficient of zero. (There are other possible locations for the origin: the balancing point, the center of gyration, etc.) That having been done, I could alternatively use (log(r),theta) coordinates, which have some nicer properties, because it is the actual complex log function. Fourier analysis can approximate the discontinuous changes in curvature; hopefully modifying these coefficients a little will remove these discontinuities with a minimum of ringing. The sides of your rounded edge squares are impressively flat. <HGB At 03:48 PM 7/30/2014, Bill Gosper wrote: HGB> My approach would be to take the Fourier transform of the polar plot, rwg>Defined as piecewise linear|circular? and then simplify to a small number of coefficients. Of course, it will take 2 circuits around the origin to complete the figure. The sides are perhaps a little too flat, which will blow up the number of coefficients required. I guess if you transform [0,4pi) to [0,1) you could use some sort of Cheby polynomial approximation. It would be nice to be able to transform it back to p(x,y)=0, where p is a polynomial in x,y. At 02:56 PM 7/29/2014, Whitfield Diffie wrote: HGB>Anyone game for an analytic function to approximate Airbnb's new logo? They should manufacture a paper clip and give it out at trade shows. Whithttp://www.underconsideration.com/brandnew/archives/airbnb_logo_detail.… rwg> You can get some impressively straight segments with theta fcns: gosper.org/halph.pdf (10MB?!) or see http://gosper.org/thetpak.html (scrolled 69%). (.3281... is âˆš(one ninth constant), which I claim should be named Halphen's. It's the largest q for which the curve is convex.)) As I mentioned here before, as 0<q<1, the curve continuously deforms from a tiny "circle" to a big "+" sign, sweeping out a fairly convincing spaceplane: http://gosper.org/sst.png --rwg But a problem with this logo is that it switches curvatures discontinuously. <rwg Another possibility: modulate the arcspeed (slow down for curves, speed up for straightaways) before taking the Fourier.

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[math-fun] Fred Helenius email?
by Allan Wechsler 30 Nov '14

30 Nov '14

I'm getting bounces on fredh(a)ix.netcom.com. What is the correct way to reach Fred?

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[math-fun] Aerobie aerodynamics
by Bill Gosper 30 Nov '14

30 Nov '14

Many years ago, Alan Adler produced an excellent 38 min. video on the invention of the Aerobie flying ring. He sent the master it to an L.A. cassette duplication outfit who then threw it in a dumpster and flew-by-night. I just uploaded a low quality recording at gosper.org/aerobieaero.mp4 . If you find it boring, you're paying insufficient attention. --Bill

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Re: [math-fun] Aerobie aerodynamics
by Bill Gosper 30 Nov '14

30 Nov '14

On 2014-11-29 09:09, Fred Lunnon wrote: > Using Firefox on recent iMac, I get sound but no vision; > length indicator at bottom right constantly varies between > 1 and 6 hours(!) > > WFL > > On 11/29/14, Bill Gosper <billgosper(a)gmail.com> wrote: >> Many years ago, Alan Adler produced an excellent 38 min. video on the >> invention of the Aerobie flying ring. He sent the master to an L.A. cassette >> duplication outfit who threw it in a dumpster and flew-by-night. I >> just uploaded a low quality recording at >> gosper.org/aerobieaero.mp4 . If you find it boring, you're paying >> insufficient attention. >> --Bill Jörg's Worte der Weisheit: * rwg <rwg(a)sdf.org> [Mar 06. 2014 07:44]: Try http://gosper.org/brinicle.mp4 . My FireFox won't play it, but you can probably Save Page As and play it with Quicktime, Miro, Flip Player, RealPlayer, ... --Bill Fascinating. Plays fine with VLC, see http://en.wikipedia.org/wiki/VLC_media_player Rule of thumb: if it doesn't play with VLC, forgetaboutit. Best, jj

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[math-fun] Chernick numbers
by Bill Gosper 29 Nov '14

29 Nov '14

"J. Chernick[6] <http://en.wikipedia.org/wiki/Carmichael_number#cite_note-Chernick1939-6> proved a theorem in 1939 which can be used to construct a subset <http://en.wikipedia.org/wiki/Subset> of Carmichael numbers. The number [image: (6k + 1)(12k + 1)(18k + 1)] is a Carmichael number if its three factors are all prime." (I used to think these were the only Carmichael numbers.) Isn't it fairly immediate that this can be simplified to p*(2p-1)*(3p-2), p>3, all three factors prime, since 2p-1 can't be prime when mod(p,6)=5? --rwg

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[math-fun] Prime[1.] [Was Chernick numbers]
by Bill Gosper 29 Nov '14

29 Nov '14

DanA>Yes, it's fairly immediate. (Though the quoted formula is more explicit.) --Dan P.S. Can someone please tell me why my naïve Mma summation NSum[1/(Prime[k]^2 - 1), {k,1,Infinity}] gets the output: ----- Prime::intpp: Positive integer argument expected in Prime[16.]. Prime::intpp: Positive integer argument expected in Prime[17.]. Prime::intpp: Positive integer argument expected in Prime[18.]. General::stop: Further output of Prime::intpp will be suppressed during this calculation. Out[20]= 0.550863 ----- I mean, who asked it to put a float into Prime[] ? (Not me.) (The actual value is about .55[168] .) The problem is Mathematica perpetuating the *evil* myth that floats are fuzzy. They should be regarded as *exact* dyadic rationals liable to fuzzy operations. Thus in Macsyma, prime(1.) was 2. . In Mma 10, In[2]:= NSum[1/(Prime[k]^2 - 1), {k, 1, Infinity}, NSumTerms -> 9999] During evaluation of In[2]:= Prime::intpp: Positive integer argument expected in Prime[10000.]. >> During evaluation of In[2]:= Prime::intpp: Positive integer argument expected in Prime[10001.]. >> During evaluation of In[2]:= Prime::intpp: Positive integer argument expected in Prime[10002.]. >> During evaluation of In[2]:= General::stop: Further output of Prime::intpp will be suppressed during this calculation. >> During evaluation of In[2]:= SequenceLimit::seqlim: The general form of the sequence could not be determined, and the result may be incorrect. Out[2]= 0.551692946 --rwg On Nov 28, 2014, at 7:53 PM, Bill Gosper <billgosper(a)gmail.com> wrote: "J. Chernick[6] < http://en.wikipedia.org/wiki/Carmichael_number#cite_note-Chernick1939-6> proved a theorem in 1939 which can be used to construct a subset < http://en.wikipedia.org/wiki/Subset> of Carmichael numbers. The number [image: (6k + 1)(12k + 1)(18k + 1)] is a Carmichael number if its three factors are all prime." (I used to think these were the only Carmichael numbers.) Isn't it fairly immediate that this can be simplified to p*(2p-1)*(3p-2), p>3, all three factors prime, since 2p-1 can't be prime when mod(p,6)=5? --rwg

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[math-fun] primes constellations as proof of work
by Warren D Smith 28 Nov '14

28 Nov '14

That sounds like an excellent idea. If you ask, for, say N consecutive equispaced primes, for decently large N these become hellishly hard to find -- exponentially hard as a function of N as far as anybody knows -- but once you find them they are trivial to verify and concise to state. If instead of "prime" we asked for "Fermat-test-passing number," then so what? Still seems a perfectly legitimate problem to me.

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[math-fun] "Riecoin" = prime # proof-of-work digital $$
by Henry Baker 28 Nov '14

28 Nov '14

FYI -- http://riecoin.org/faq.html Finding a prime number p takes O( log(p) ^ 4 ) work, while verifying if it's prime requires O( log(p) ^ 3 ) using the Rabin-Miller test. That's not much difference, so finding prime numbers is not practical for a PoW. One possible solution for this is looking for prime number constellations (like twin prime numbers, or triplets, etc), that is n "consecutive" prime numbers. Consecutive in this case means that they are grouped as closely as possible minimizing the distance between the first and the last one. This takes O( log(p) ^ (n + 3) ), while verification still takes the cube of the log. This allows us to make the generation arbitrarily more difficult than the verification. Difficulty can be adjusted by changing the length of the prime numbers. Riecoin currently uses constellations of size 6 (also known as prime sextuplets) which have the form p, p+4, p+6, p+10, p+12, p+16. So each block represents six prime numbers. ... How is Riecoin different from Primecoin? Primecoin uses the Fermat primality test, which has some flaws. Carmichael numbers are not prime and still pass Fermat's test for all bases, however those are relatively rare. Secondly, in general, if Fermat's test says a number is prime, it has at least a 50% probability of being prime. Primecoin uses only one Fermat test with base 2. While base 2 may provide more confidence than the general bound of 50%, still many composites will pass as primes. What's worst, is that Euler-Lagrange-Lifchitz test used for the other primes in the chain assumes the previous number in the chain is prime. So if the chain starts with a number that is not prime, then the Euler-Lagrange-Lifchitz test is not guaranteed to work, and all numbers in the chain may be composite. Short version: Primecoin numbers are not guarranteed to be prime, they may be Fermat pseudoprimes to the base 2. There is an infinite list of Fermat pseudoprimes to the base 2 (oeis.org/A001567) Riecoin uses enough Rabin-Miller tests with random bases, so the probability of a number that is not prime being accepted by the majority of the Riecoin network is negligible.

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[math-fun] Junk "DNA" found in malware
by Henry Baker 27 Nov '14

27 Nov '14

>From "Kaspersky_Lab_whitepaper_Regin_platform_eng.pdf", which describes the recently-analyzed so-called "Regin" malware: "We can assume the attackers take various low-level open-source projects or Windows DDK source codes and merge them together with their malicious loader. Hence, each stage 1 loader looks very different from others, as *** it contains random useless code from various other programs. This technique makes it more difficult to build reliable detection for the loaders. ***" Perhaps classical biological viruses use "junk DNA" for the same reasons?

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