November 2011 - mathfun@mailman.xmission.com

Re: [math-fun] bussard ramjet
by Warren Smith 15 Nov '11

15 Nov '11

>>WDS: THIS WHOLE "BUSSARD RAMJET" IDEA IS JUST IRREDEEMABLY STUPID. >Schroeppel: > Your argument assumes that we stop the hydrogen, react it, > and use the reaction energy to eject helium. > However, if you can capture the energy of the impinging > hydrogen, and apply it to the departing helium, you have a > net gain in the ship's momentum. --Suppose the hydrogen is stopped and its energy is not lost, instead only a fraction F of its energy is lost. We then fuse the H to get He, which we eject out the back using both the fusion energy and the non-lost (1-F) fraction. Ship velocity = Vship. Outgoing velocity (if F=1) is Vfusion. Outgoing velocity (general F with 0<F<=1) is Vout. Assume relativistic effects small enough so we can use Newton laws. Vout^2 = Vfusion^2 + (1-F)*Vship^2 . The ship will keep accelerating while Vout>Vsip, but will stop accelerating when Vship=Vout. Solving, we find this happens when Vship[max possible] = Vfusion / squareroot(F). My old claim was implicitly assuming F=1. However, F=0 is not possible if perfect efficiency not possible. With any fixed F with 0<F<=1 there is an uppermost speed. I think 90% efficiency would be rather optimistic (the best heat engines humanity has only about 40% efficient, steam turbines, right?)... and would give ship at most sqrt(10)*Vfusion. Well actually, I think even 1% efficiency would be amazing and have no idea how to reach it. But anyhow: there is an upper limit. Meanwhile rockets, unlike jets, have no upper limit (if make the fuel tank big enough).

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Re: [math-fun] Re .... structure seen from space...?
by Henry Baker 15 Nov '11

15 Nov '11

Potemkin Village? http://en.wikipedia.org/wiki/Potemkin_village http://en.wikipedia.org/wiki/Operation_Fortitude http://en.wikipedia.org/wiki/Ghost_Army The visual deception arm of the Ghost Army was the 603rd Camouflage Engineers. It was equipped with inflatable tanks, cannons, jeeps, trucks, and airplanes that the men would inflate with air compressors, and then camouflage imperfectly so that enemy air reconnaissance could see them. At 12:54 PM 11/15/2011, Stuart Anderson wrote: >"Perhaps it's some kind of targeting or calibrating grid for Chinese >spy satellites? .... Nobody really knows," technology website Gizmodo >speculated. > >http://maps.google.com/maps?q=40.458148,93.393145&hl=en&ll=40.458018,93.392… > >... others said the structures, some of which are near Jiuquan, >China's space city, could just be training sites for cosmonauts >practising their landings and walks. > >They may also be practice ranges for the People's Liberation Army. > >"The picture of the circle looks very like a missile test range, with >target and instrumentation set out to record weapon effects. The >Americans have lots of these in Nevada - Area 51," Tim Ripley of >military magazine Jane's Defence Weekly told the Telegraph. > >Paul Marks, of the New Scientist magazine, said that, by zooming in on >the images, viewers could possibly make out some military planes. > >"One of the other formation ... looking tantalisingly like Stonehenge >from a great height, zooming in reveals three aircraft sitting at its >heart. Clearly, it is some kind of military target for airstrike or >gunnery practice," he wrote. > >"Another 4 x 4 piece grid some 200 metres across has some pieces >clearly blown to smithereens, again supporting the target practice >theory, and a dummy runway in garish bluish-white is probably not for >style-conscious aliens but air-to-ground strafing practice." > >Of course, Google Maps may have been hacked and the images may be part >of an elaborate hoax. They could be the "21st century version of crop >circles", Marks wrote. " > >Read more: http://www.smh.com.au/technology/technology-news/chinas-area-51-mystery-des… > >Stuart Anderson

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[math-fun] Re .... structure seen from space...?
by Stuart Anderson 15 Nov '11

15 Nov '11

"Perhaps it's some kind of targeting or calibrating grid for Chinese spy satellites? .... Nobody really knows," technology website Gizmodo speculated. http://maps.google.com/maps?q=40.458148,93.393145&hl=en&ll=40.458018,93.392… ... others said the structures, some of which are near Jiuquan, China's space city, could just be training sites for cosmonauts practising their landings and walks. They may also be practice ranges for the People's Liberation Army. "The picture of the circle looks very like a missile test range, with target and instrumentation set out to record weapon effects. The Americans have lots of these in Nevada - Area 51," Tim Ripley of military magazine Jane's Defence Weekly told the Telegraph. Paul Marks, of the New Scientist magazine, said that, by zooming in on the images, viewers could possibly make out some military planes. "One of the other formation ... looking tantalisingly like Stonehenge from a great height, zooming in reveals three aircraft sitting at its heart. Clearly, it is some kind of military target for airstrike or gunnery practice," he wrote. "Another 4 x 4 piece grid some 200 metres across has some pieces clearly blown to smithereens, again supporting the target practice theory, and a dummy runway in garish bluish-white is probably not for style-conscious aliens but air-to-ground strafing practice." Of course, Google Maps may have been hacked and the images may be part of an elaborate hoax. They could be the "21st century version of crop circles", Marks wrote. " Read more: http://www.smh.com.au/technology/technology-news/chinas-area-51-mystery-des… Stuart Anderson

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Re: [math-fun] How to generate an infinite number of true-random bits in hardware
by Warren Smith 15 Nov '11

15 Nov '11

>The new Intel random number generator, code name "Bull Mountain", starts from thermal noise at 3 GHz, conditions the bits, and then uses these numbers as seeds to generate 511 pseusorandom numbers per true random number. It is this last step that I dislike and don't trust. --Aha. However, this last step probably will be good enough to assure Intel bits pass all the usual test packages like TestU01 (unless they've been total idiots... which they might have been). It likely will destroy cryptographic security, but you could just only take 1 number per 512 to get that back. If Intel is doing this I don't see why they needed 3 GHz, why couldn't (3/512) GHz have sufficed? >Thermal noise has power 4kT per unit bandwidth. If you're building your own circuit, 10 MHz is a reasonable bandwidth, and at temperature 300 K, the noise is 1.6e-13 W. With a 1 Mohm input impedance, and using P = V*V/R, the RMS noise voltage is 400 microVolt. --You already blew it because the input capacitance of typical op-amp pin is 4 pF, and R*C for R=1megaOhm and C=4pF is 4 microsec, so you just cut off your "10 MHz" bandwidth right there. > An amplifier gain of 2500 brings this up to a convenient 1 V, and this can be done in two or three stages using op-amps, with AC coupling between stages to avoid amplifying the op-amp bias voltages. --Hi-gain amps are vulnerable to oscillation from tiny parasitic capacitances coupling output and input. You can try to avoid using fancy shielding, but, why even go there if you don't have to? This is especially worrying when your amp is high frequency (10 MHz is pretty high and with several cascaded amps you maybe need 3X more speed). Most commercial op-amps are designed for lower frequencies, or at least this used to be true. By the way, if you really want to take the "big amp" approach I disparage, a better source of noise is a neon light. Lots more than thermal noise in a resistor. --Intel and your schemes both seem vulnerable to crosstalk from nonrandom noise, i.e. from other signals within your computer (if phsyically located within your computer, e.g. on the processor chip) Your "random" bits would then be highly "predictable" to the right predictor. Again you can try to shield, but why go there if you do not have to?

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[math-fun] How to generate an infinite number of true-random bits in hardware
by Warren Smith 15 Nov '11

15 Nov '11

How to generate true random bits in hardware Warren D. Smith Nov 2011 METHOD [I invented this at least 10 years ago]: 0. Let x be an analog electrical signal in -1<x<1 volt stored in "sample & hold" circuit #1. 1. Compute the analog signal y = 2*x*x-1 and feed it into sample & hold #2. 2. Compute the analog signal x = 2*y*y-1 and feed it into sample & hold #1. 3. return to step 1 (continue forever). Instead of the function f(x)=2*x*x-1 it also is possible to employ the function g(x)=2|x|-1, however the corner then would need to be sharp, which probably is not realistic for a fast electrical circuit. g(x) is computable exactly by certain circuits made of ideal op-amps and diodes. f(x) can be computed from essentially ANY nonlinear electrical element with the aid of op-amps. The random bits are: sign(x) and sign(y) after each update of x and y. With perfect computation of f(x) you will get true-random uncorrelated 50-50 coin toss bits, 1 bit per f- or g-iteration. With somewhat imperfect computation of f(x) we will lose some perfection in our randomness, so then it is better to output only the sign(y) as our random bits and discard the x's. WHY THIS WORKS: With g(x), the iteration is a self-map of [-1,1] with |slope|=2 everywhere, hence tiny electrical noise will get multiplied by 2^N after N iterations of g. This is, essentially, the left-shift map on the binary representation of a real number, hence we get 1 new bit of "randomness" shifted up from the low-significance bits per iteration. With f(x), the variable change x=cos(t) means the iteration f(x) is just doubling t (mod 2pi). This is the shift map on t/pi, exactly this time (and not merely "essentially" the shift map). With approximate f(x) the doubling factor might be only 1.9 (say) sometimes, and then we could get slightly less than 1 bit of true randomness per iteration but you are safely getting more than 1 bit per 2 iterations. I would prefer the bits be post-processed by digital algorithms to eliminate statistical problems caused by such imperfect manufacturing.

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[math-fun] structure seen from space...?
by Simon Plouffe 15 Nov '11

15 Nov '11

Hello, can anybody tell us what is this ? http://www.wired.com/dangerroom/2011/11/china-gigantic/all/1 You can see it from space, it is in mainland china, apparently nobody knows what it is. Me : I could not tell too, any clue someone ? Best regards and bonne soirÃ©e, Simon Plouffe

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[math-fun] Speech recognition, cochlear transform (and my approximation problem solved)
by Warren Smith 14 Nov '11

14 Nov '11

I just wrote a paper about speech processing. It invents a new signal transform, the "cochlear transform," which is sort of like unifying Fourier and Laplace, plus Fast ala the FFT. It includes in the final section a proof of a "Weierstrass-like approximation theorem" which settles (basically) an open problem posed by Aho, Stieglitz, and Ullman 1975 and which I had posed to this forum as an open problem, but nobody solved it for me :( Perhaps you experts on FFTs & ilk will have useful comments re the speech part and the stuff about how many arithmetic operations are used... send comments on the paper to warren.wds AT gmail.com. Paper draft here: http://dl.dropbox.com/u/3507527/SpeechProc.html -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Giant planet was ejected from the solar system ?
by Steve Witham 14 Nov '11

14 Nov '11

>From: Henry Baker <hbaker1(a)pipeline.com> > >http://swri.org/9what/releases/2011/giant-planet.htm > Wikipedia says escape velocity from the Sun's gravity starting at Jupiter's (current) orbit is 18.5 km/s. That's about one light year every sixteen thousand years. Wikipedia also says the Solar System is 4.6 billion years old. Starting when the Solar System was 600 million years old, and traveling at 18.5 km/s, an object would travel about 244 thousand light years (if the Milky Way weren't effecting its path). That's about twice the diameter of the Milky Way, or ten times the distance from the Sun to the center of the galaxy. In that same period, the Sun has orbited the center of the Galaxy 16 times. So I guess this means the ejected planet won't be found loitering between the nearby stars. It's truckin'. --Steve

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Re: [math-fun] Request for example elementary non-constructive proofs with "witnesses"
by Dan Asimov 13 Nov '11

13 Nov '11

Alon's nice example -- of a game (Chomp) where only the existence of a strategy for the first player is known, but not any actual example of such a strategy -- is also the case for Hex. --Dan ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] Request for example elementary non-constructive proofs with "witnesses"
by Dan Asimov 13 Nov '11

13 Nov '11

Not sure if this is exactly on point, but: It's an easy theorem in topology that any continuous function f: [0,1] -> [0,1] has a fixed point. But this says nothing about how to find it. If, however, it's known that there is a number c with 0 < c < 1 such that for all x,y in [0,1] we have |f(x) - f(y)| <= c |x - y|, then it's easy to show that for any x in [0,1] the sequence x, f(x), f(f(x)), . . . must converge to a fixed point. --Dan Marc wrote: << Could anyone supply me with elementary examples that illustrate the idea of a non-constructive proof, for those with a "Martin Gardner reader" level of mathematical sophistication that also has a not-too-trivial but reasonably easily-verified case? >> Even though kleptomaniacs can't help themselves, they do.

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