July 2014 - mathfun@mailman.xmission.com

[math-fun] qAq*=A', qBq*=B' quaternion problem
by Henry Baker 20 Jul '14

20 Jul '14

I've been fascinated by this problem, because what should be a relatively simply problem is remarkably difficult to prove symbolically. Given 3D vectors A, A', B, B', q is a quaternion such that qAq*=A' and qBq*=B', where q* is the conjugate of q, and |A|=|A'|, |B|=|B'|. It is simpler to deal with non-unit quaternions, so the constraints on q are qA=A'q and qB=B'q. The answer is remarkably elegant: q = (B'-B)A + A'(B'-B) = dB A + A' dB -q = (A'-A)B + B'(A'-A) = dA B + B' dA (Juxtaposition here means quaternion multiplication.) *** Believe it or not, these two expressions represent the _same_ quaternion(!), because rigid rotations preserve dot products, and so A'.B'=A.B. *** Proof: Here are the same expressions in terms of dot and cross products: q = -(A'+A).dB + dAxdB = dA.(B'+B) + dAxdB QED Given q = dB A + A' dB, it is now easy to prove that qA=A'q: qA = (dB A + A' dB)A = dB A^2 + A' dB A = A^2 dB + A' dB A (A^2 is scalar) = A'^2 dB + A' dB A (|A'|=|A|) = A'(A' dB + dB A) = A'(dB A + A' dB) = A'q Similarly, since q = - dA B - B' dA, qB = (- dA B - B' dA)B = - dA B^2 - B' dA B = - B'^2 dA - B' dA B = B'(- B' dA - dA B) = B'(- dA B - B' dA) = B'q Now the next problem is trying to come up with a general method for solving the simultaneous equations qA=A'q, qB=B'q, which would lead to these types of solutions: q = (B'-B)A + A'(B'-B) = dB A + A' dB -q = (A'-A)B + B'(A'-A) = dA B + B' dA I wrote a quaternion and vector expression simplifier in Maxima to be able to manipulate these various expressions, but it is still missing an automatic solution procedure for simultaneous quaternion equations of this sort.

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Re: [math-fun] return of vacuum tube?
by Warren D Smith 18 Jul '14

18 Jul '14

>> It also occurs to me that a vacuum-tube analogue of the semiconductor >> "solar cell" could be built. >> >> /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | >> /\ | /\ | /\ | /\ | >> ======================================== >> >> where the === denotes an insulating substrate, /\ denotes >> triangular-prism strips of metal A, (going into the page) and | >> denotes "walls" of metal B. > > I don't completely understand the physics of this device, but why do > you need lots and lots of tiny pieces of the two metals, rather than > just 2 or 3 large ones? --if the strips are too wide then an ejected electron will be likely to merely re-land on the same one it started from, which would be a huge waste, so they cannot be too wide. Other remarks: if the metal strips were actually extremely narrow, e.g. 50 nm, then the same idea the vacuum-tube paper had should also hold here: we would not actually need vacuum, noble gas atmosphere would work. Indeed, it seems plausible to me that just the right gas pressure might actually yield MORE efficiency than vacuum. > The fact that the 50% numbers for both ejection and flying to the > right destination appear to have been derived by rectal extraction, > rather than some calculation that includes the size of the metal > pieces and distances between them, does not provide much confidence > that the engineering to build a device like this is feasible. --indeed, I just made up the 50% numbers. A larger error I made in my previous post, was claiming that the vacuum solar cell would be best suited to UV light. Actually, here is a table of wavelength=h*c/E in nanometers where h=planck constant, c=light speed, E = photon energy in electron volts: E(eV) wavelength(nm) 1 1240 (infra red) 2 620 (red) 3 413 (violet) 4 310 (ultraviolet) 5 248 UV 6 207 UV indicating that 2-4 eV work functions for the anodes will work starting from the near UV into the red range... lower work function is better since then more of the range will be covered. If you went really low, it'd be counterproductive... but I'd actually guess that about 2eV would be near optimal for our sun. In other words, this is a more promising idea than I'd initially thought. > I were to say "I would expect that only .5% of the electrons would fly > to the right destination, giving the cell an overall efficiency of > .2%", how would we determine which of our expectations is more > reasonable? > Andy --well, it is largely just a guess at present, but my suspicion is, with good design, my 50%*50%=25% numbers should be achievable while yours are very pessimistic. Evidence: Commercial "photomultiplier tubes" used to detect single photons indeed do achieve detection efficiencies in the 25% range, illustrating that this concept probably has some legs. For example here is a plot of photon detection efficiency for one random commercial such tube: http://www.thorlabs.com/images/TabImages/PMTSS_Quantum_Efficiency_Graph_780… indicating efficiency>30% between about 210 to about 420 nm, and above 20% from about 150 to 570 nm. My solar cell idea should probably be able to obtain slightly greater photon-->electron efficiency than a photomultiplier tube because the latter have geometric constraints caused by their desires for multi-stage "multiplication" while the solar cells do not have to worry about that. And also the phototube guys are counting only photons which generate electrons which successfully start a multiplication "avalanche" whereas in the solar cell application, we would count all electrons which reach the cathode, a wider class. On the other hand: these solar cells if commercialized would want to use low priced materials in contrast to expensive photomultiplier tubes where they feel free to use expensive materials. And the phototubes are artificially charged to high voltages in an effort to guide the electrons where they want, and give them more oomph. So of course, the real way to answer this is experimentally. I'm going to send the authors of the vacuum tube paper, my thoughts, after some further editing, since they actually do do experiments. I just armchair theorize. Important question: is it possible to fabricate simple repetitive patterns like parallel metal strips in some very economical manner like "rolling them" as opposed to photolithography? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] return of vacuum tube?
by Warren D Smith 18 Jul '14

18 Jul '14

It also occurs to me that a vacuum-tube analogue of the semiconductor "solar cell" could be built. /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | /\ | ======================================== where the === denotes an insulating substrate, /\ denotes triangular-prism strips of metal A, (going into the page) and | denotes "walls" of metal B. Metal A has low photoelectric "work function" and B has higher work function. Everything is in vacuum. Sunlight shining on it from top ejects photoelectrons from metal A, and some of them fly to come to rest on metal B, result is solar-powered current source. Example work-function numbers in electron volts: A: Na=2.36, K=2.29, Sr=2.59. B: Ti=4.33, Ni=5.2, Fe=4.7, Mn=4.1. I think lower work functions are available than listed in the "A" row by use of special surface treatments and alloys. I would expect fairly good efficiency for incoming photons in the right frequency range, e.g. if 50% of photons eject an electron and 50% of electrons fly to the right destination, then about 25% efficiency, say 20%. Comparison versus semiconductor solar cells: (a) Unfortunately it appears the vacuum solar cell would be powered by UV light, not visible or IR. (b) Vacuum cell presumably more radiation-tolerant and more immune to temperature change, than semiconductor cells. That plus availability of vacuum makes it seem well suited for space uses. (c) no high purity low-defect crystals of semiconductor needed. (d) vacuum cell could be degraded by surface poisoning due to, e.g, oxidation, so important it really be in vacuum.

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[math-fun] The first congruent number
by Bill Gosper 18 Jul '14

18 Jul '14

In what was probably Europe's first public math contest, the great Fibonacci was asked for three rational squares with common difference 5. (This is equivalent to finding a rational Pythagorean triangle with area 5.) Somehow, Fibonacci found (31/12)^2, (41/12)^2, (49/12)^2. This is the first of an infinite sequence of such solutions, found these days by repeated "addition" of the point {5*5/4, 6*5^2/4^2} along the "elliptic" curve y^2 = (x-5) x (x+5), producing the sequence 5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, ..., whose elements *of even index* are Fibonacci's desire. Elliptic curve addition doesn't quite provide a nice recurrence formula. Empirically, there's a mysterious, period 8 sign pattern: pisano[1] = 5/2; pisano[2] = 41/12; pisano[n_Integer] := (5 \[Sqrt](-100 + 9 pisano[-1 + n]^2 + 4 pisano[-1 + n]^4 - 12 (-1)^(1/8 (-2 + n) (1 + n) (3 + n) (4 + n)) Sqrt[-25 pisano[-1 + n]^2 + pisano[-1 + n]^6]))/ (Abs[-5 + 2 pisano[-1 + n]] (5 + 2 pisano[-1 + n])) Try a few: Table[pisano[n], {n, 8}] {5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, 312098738002194296165/128615821825334210638, 249850594047271558364480641/5354229862821602092291248} Test the "even" ones: Sqrt[#^2 + {-5, 5}] & /@ %[[Range[2, 8, 2]]] {{31/12, 49/12}, {113279/1494696, 4728001/1494696}, {518493692732129/178761481355556, 767067390499249/178761481355556}, {249563579992463717493803519/5354229862821602092291248, 250137278774864229623059201/5354229862821602092291248}} Rich suggested the solutions might obey a Somos recurrence. So far, I've only found one (Somos5) for the numerators: a[n] -> (124558 a[-3 + n] a[-2 + n] - 781 a[-4 + n] a[-1 + n])/a[-5 + n] The 124558 and 781 are very probably minimal. "Fitting" a 𝝑 fcn to these numerators held a surprise: a[0], a[1], a[2], ... , a[n] = 1,5,41,11285,... = (I (-1)^n 5^(1/2 Mod[n, 2]) E^(- I ArcTan[3/5]/2) (π/2)^(3/4) u^n^2* EllipticTheta[2, n z, (-1)^(-(13/34) + I/34)])/(2 17^(1/4) Gamma[5/4]) (Note the crazy (constructible!) 34th root of unity. Unfortunately, u and z are simply what they need to be to make a[1] and a[2] work. ISC doesn't recognize them, and I can't find a way to symbolically separate them. z -> 4.908105290682877137963508159098016296309287655031188041917052851155165374957836660235707132115102`69. - 0.197404925101320072627868756915313864010305639948302119404764160895568125915396715987289924743617`69. I, u -> -0.910971694653082823579468669395203737157839231764166792826687755862129892782777001356185727331806`65.8+ 1.430029427800740885258427207242669226859970344275381501850156729643487583694210398313066894949425`66. I} Mathematica's Root notation lets you exactly specify an implicitly defined root, say, the solution to a transcendental equation. Newton's method is getting to be like another button on your calculator. But how do we handle the solution vectors of simultaneous transcendental (say) equations, like for u and z here? --rwg (with much help from NeilB)

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[math-fun] Left/right parity in websites
by hbaker1 16 Jul '14

16 Jul '14

I went to an Israeli web site, but I couldn't read it in Hebrew (which reads right-to-left). I clicked on the "English" button, and the web site then appeared in English, _but all the pictures (jpg & png) were also L/R mirrored_ ! I couldn't tell if this L/R mirroring of the images was a bug or a clever tongue-in-cheek joke. (I suppose that the mirroring could also be a sign of mild laziness; the same page layout might work better with everything mirrored, rather than simply mirroring the text.) Perhaps I need to inquire at this website about how CPT is working there; perhaps I've contacted a web site in an alternate universe!

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[math-fun] return of vacuum tube?
by Warren D Smith 16 Jul '14

16 Jul '14

Jin-Woo Han & Meyya Meyyappan: Introducing the Vacuum Transistor: A Device Made of Nothing, IEEE Spectrum June 2014. J-W Han, Jae Sub Oh, M. Meyyappan: Vacuum nanoelectronics: Back to the future?--Gate insulated nanoscale vacuum channel transistor, Applied Physics Letters 100 (2012) 213505. What they call the "vacuum transistor" is actually a miniature vacuum tube. They point out that (a) if you make it very small (50 nanometers, say), you do not even need vacuum, since the path length before an electron hits a gas atom is smaller or comparable to the size of the tube. (b) you do not need a heated filament, since "field emission" from sharp corners of tiny pointed anodes is adequate. (c) you can fabricate it using micro-lithography techniques like those used to make silicon chips, constructing integrated circuits made of these vacuum tubes. (d) at low voltages using noble gas, there is not enough electron energy to ionize gas atoms, which supposedly prevents erosion of cathodes by ions. (e) might be more radiation-tolerant than semiconductor electronics. They've made these devices run at 460 GHz, which is way faster than solid state transistors can go. I also point out that (f) no more ultrapure single crystal low-defect silicon needed -- maybe can now use cheaper less-pure metal materials. I don't see why semiconductors are needed at all, although you might want hybrid circuit using both vacuum tubes and semiconductor devices, for example so you could have "flash RAM". This all sounded pretty promising to me. It seems possible to me this could speed up computer chip internals by factors 10-100 while also decreasing costs. Was the whole transistor approach the wrong approach? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] GHash > 50% of Bitcoin's computational power
by Henry Baker 15 Jul '14

15 Jul '14

FYI -- http://arstechnica.com/security/2014/06/bitcoin-security-guarantee-shattere… Bitcoin security guarantee shattered by anonymous miner with 51% network power In a first, one player got a monopoly of Bitcoin's total computational power. by Dan Goodin - Jun 15, 2014 7:15 pm UTC For the first time in Bitcoin's five-year history, a single entity has repeatedly provided more than half of the total computational power required to mine new digital coins, in some cases for sustained periods of time. It's an event that, if it persists, signals the end of the crypto currency's decentralized structure. Researchers from Cornell University say that on multiple occasions, a single mining pool repeatedly contributed more than 51 percent of Bitcoin's total cryptographic hashing output for spans as long as 12 hours. The contributor was GHash, which bills itself as the "#1 Crypto & Bitcoin Mining Pool." During these periods, the GHash operators had unprecedented powers that circumvented the decentralization that is often held up as a salient advantage Bitcoin has over traditional currencies. So-called 51 percenters, for instance, have the ability to spend the same coins twice, reject competing miners' transactions, or extort higher fees from people with large holdings. Even worse, a malicious player with a majority holding could wage a denial-of-service attack against the entire Bitcoin network. Like tremblers before a major earthquake, most of GHash's 51-percent spans were relatively short. Few people paid much attention, since shortly after a miner loses the majority position, it also loses its extraordinary control. Then, on June 12, GHash produced a majority of the power for 12 hours straight, a sustained status that enables precisely the type of doomsday scenario some researchers have warned was possible. Do not pass Go There's no evidence the anonymous operators of GHash exercised any of those abilities. Still, the mere possibility undermines a core Bitcoin tenet that it be decentralized so it can't be controlled by a single entity. "A 51 percenter can control which Bitcoin transactions happen," wrote Ittay Eyal, a post-doctorate researcher in Cornell's Department of Computer Science, in an e-mail to Ars. "It becomes a monopoly. It can set arbitrarily high transaction fees, for example, or even extort someone to allow them to perform transactions. It could block or delay all transactions but its own. One of Bitcoin's goals was to be a free system, independent of anyone's control. With small pools, no one has this kind of control. With a 51 percenter, there is." GHash's ascendency to a majority miner comes even as its operators pledged never to cross the 51-percent threshold. It also comes less than a year after GHash was accused of using its considerable hashing power to attack a gambling site. Emin GÃ¼n Sirer, a Cornell professor who works with Eyal, agreed there was no evidence GHash or its operators at CEX.io took advantage of the recent majority positions. In his own e-mail, he added: But having a single entity in GHash's position, of holding 51 percent of the mining power, of being in a monopoly position, of being able to launch any of these attacks at will, completely violates the spirit and intent of Bitcoin as a currency. Bitcoin's value proposition stems from its technological foundation, which in turn is based on building distributed trust. People flock to Bitcoin because they do not trust the fiat infrastructure, they hold Bitcoin because they are worried that the people in charge of USD can inflate it at will or usurp money from their accounts. But now, with a monopoly miner, they are suddenly in a position where they have to, once again, trust a single entity to remain benign. This completely collapses the Bitcoin narrative that the Bitcoin community has been using to draw in new users. If we are to trust GHash's good will and ongoing benign behaviors, we might as well do away with the entire Bitcoin protocol and replace the system with a simple database server kept on GHash's premises. Worse, no one knows who exactly is behind GHash/CEX.io. They have had an episode where they did a double-spend attack against a gambling site in the past. But even if GHash could be trusted right now, a single entity in command of the currency represents a single point of failure for the Bitcoin economy. Officials with GHash didn't respond to an e-mail seeking comment for this article. Eyal and Sirer's previous research into Bitcoin weaknesses has been criticized by some as being exaggerations that aren't possible in practice. Matt Green, a Johns Hopkins University professor specializing in cryptography and the security and anonymity of Bitcoin, said whatever mitigations there may be, GHash's 51-percent stake is a significant development. "If they tried to do [something malicious] people would notice and there would be blowback," Green told Ars. "So it's not the end of the world. Nonetheless, the security model of Bitcoin depends on no miner having (even close to) majority. Right now that assumption is being violated." Eyal and Sirer reported their observations of GHash in a blog post published Friday headlined It's Time For a Hard Bitcoin Fork. Such a redesign should recognize the existing Bitcoin blockchain to ensure backwards compatibility, but it should incorporate changes to fix three fundamental problems threating the viability of Bitcoin as a currency that's immune from manipulation. The changes include restructuring blocks in a way that disincentives mining pools, closes the threat of so-called "selfish mining" attacks that allow a small number of miners to control Bitcoin, and improves visibility into any attempts to manipulate the Bitcoin blockchain. The Cornell researchers rejected arguments some Bitcoin advocates have asserted that attacks on the Bitcoin blockchain are infeasible because they would require 51 percenters to act against their own investments in hardware and interest in a stable digital currency. From the start, the researchers said, the point of Bitcoin has been to incorporate strong cryptography as a self-policing system that ensured all players were on equal footing. "Overall, there is absolutely no reason to trust GHash or any other miner," they wrote in Friday's post. "People in positions of power are known to abuse it. A group with a history or double-expenditures just blithely went past the 51 percent psychological barrier: this is not good for Bitcoin."

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[math-fun] Ben Laurie's criticism of bitcoin and suggested replacement
by Mike Stay 15 Jul '14

15 Jul '14

http://www.links.org/files/decentralised-currencies.pdf -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Digital money
by Warren D Smith 15 Jul '14

15 Jul '14

I also perceive the following extremely stupid design flaw in bitcoin. According to the bitcoin paper, you "prove work" by finding x so that hash(x) begins with n zeros in binary, an specifically, you keep incrementing x, starting from a known value x0, until such a new x is found. Flaw: a parallel search using 1000 computers, will find that x 1000 times faster. Better design: if instead of "incrementing" x, i.e. using x0, x1=1+x0, x2=1+x1, x3=1+x2, etc we tried x's in the order x0, x1=F(x0), x2=F(x1), x3=F(x2), etc where F is a strong encryption function, then this search would be inherently serial and not parallelizable. Advantage: The goal is, it is supposed to be "hard" to find the new x. "Hard" should mean "requiring a certain amount of time, no matter how much money you have to buy more parallelism." I repeat: this was an INCREDIBLY stupid design flaw in bitcoin, even assuming we do not debate, and simply accept, their whole design goals. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The Brazil-nut effect
by Henry Baker 15 Jul '14

15 Jul '14

...and its application to asteroids (I wonder if this effect is also evident here in seismic California, where I see lots of big boulders.) http://arxiv.org/abs/1407.2748 Out of the handful of asteroids that have been imaged, some have distributions of blocks that are not easily explained. In this paper, we investigate the possibility that seismic shaking leads to the *** size sorting *** of particles in asteroids. In particular, we focus on the so-called Brazil Nut Effect (BNE) that separates large particles from small ones under vibrations.

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