June 2016 - mathfun@mailman.xmission.com

[math-fun] A value of the Rogers-Ramanujan continued fraction
by Bill Gosper 25 Jun '16

25 Jun '16

Jim's latest essay, https://mathenchant.wordpress.com/2016/06/16/sri-ramanujan-and-the-secrets-… cites https://en.wikipedia.org/wiki/Rogers–Ramanujan_continued_fraction, which says Rogers was first to "discover" the continued fraction with unit denominators and geometrically progressing numerators. Presumably, that means he had the infinite product, but how come nobody noticed the publication of such a startling result? Even less clear from the Wikipedium is whether Rogers knew " It can be evaluated explicitly for a broad class of values of its argument." I.e., (as NeilB had to point out to me), the infinite product can be expressed as an algebraic function of Dedekind etas. The article gives two cases: q = e^(-2π) (the one that startled Hardy) and e^(-2√5π). I wonder what Hardy would have thought of this one: 1 -------------------------------------------------- 4/5 (i - 1) i 1 + ---------------------------------------------- 8/5 (i - 1) i 1 + ------------------------------------------ 12/5 (i - 1) i 1 + -------------------------------------- 16/5 (i - 1) i 1 + ---------------------------------- 4 (i - 1) i 1 + ------------------------------ 24/5 (i - 1) i 1 + -------------------------- . . . 1/4 1 6 5 1/5 (- (-9 + Sqrt[5] + -----------------)) 2 Sqrt[GoldenRatio] = ----------------------------------------- 4/25 (i - 1) i ~ 0.8460733945195074 + 0.2172347253807965 i --rwg

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[math-fun] Sin(angle)? No, angle(sign)
by Bill Gosper 25 Jun '16

25 Jun '16

This sounds silly. Driving home just now I slowed down to observe signs that were indeed angled towards me in such a way as to briefly reflect my headlights. But even then, the return was swamped by the Scotchlite beads, which are just as bright if angled away from you! The only way I can imagine getting dazzled by a shiny sign is by the Sun rising or setting somewhere behind me. But that geometry changes day-to-day, so the only logical tilt angle is slightly face down. (OCD extra credit: Affinely distort the content to compensate the angle.) Then we still have Scotchlite dazzle if the Sun is exactly behind us. (Which is dwarfed by the problem of the Sun being exactly in front of us.) --rwg On 2016-06-22 15:27, Tom Rokicki wrote: > Yep! I originally wrote "would anyone like to speculate ... " but decided > not to give it away so easily. > > On Wed, Jun 22, 2016 at 2:55 PM, Mike Speciner <ms(a)alum.mit.edu> wrote: > >> So you don't get direct reflection from your headlights? >> >> >> On 22-Jun-16 17:19, Tom Rokicki wrote: >> >>> In the US, road signs are mounted slightly *counterclockwise* >>> (when viewed from above). >>> >>> Without looking it up, does anyone want to guess why this might be? >>> >>> -tom >>> >>> On Wed, Jun 22, 2016 at 7:31 AM, Bill Gosper <billgosper(a)gmail.com> >>> wrote: >>> >>> I threw an Aerobie down the street, trying to reach a certain road sign. >>>> Standing at the Aerobie, I was disappointed to see the face of the sign, >>>> across the street. But wait a minute, it's crooked! Eyeballing a >>>> perpendicular to the street, I'm a couple feet past the sign, dammit. >>>> >>>> Whoa, maybe roadsigns are twisted slightly clockwise on purpose? >>>> To exactly face traffic from where they first become visible? --rwg

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[math-fun] GrayLife?
by Leo Broukhis 25 Jun '16

25 Jun '16

Am I reinventing the wheel again? Has this not been done before? Let's consider an extension of cellular automata to an orthogonal grid containing arbitrary real values, such that a grid populated by exactly two different values behaves exactly as the corresponding binary CA (say, the lesser = dead, the greater = alive) for any two values. This necessarily means that the rule must not contain B0 and must contain S8, e.g. EightLife. Also it follows that the function new_center_value(neighborhood) must not contain any magic numbers, and the values of the cells in the neighborhood should be sufficient to determine which cells are considered alive or dead, and which new value, if any, to assign to the center cell. Multiple variants are possible. I've come up with the following mechanisms giving more or less interesting results: 0. If max(neighborhood) == min(neighborhood), do nothing, else 1a. Compute the mean. The values greater than the mean are alive, the rest are dead. 1b... I haven't considered yet any alternative ways to select the cut-off value. 2a. The new value of the center cell may be either kept as is, if the cell stays alive or stays dead, or 2b. It can be given a new value in all cases. 3a. Use max and min of the neighborhood as the new alive and dead values. This quickly decays into binary CA if the new value is always assigned. 3b. Use mean(alive cells) and mean(dead cells) as the new values for "alive" and "dead". 3c. Find the largest gap in the sorted list of cell values to classify the values, use the mean of the lesser class as the new "dead" value, and of the greater class as the new "alive" value. In case of multiple largest gaps populate the classes once per such gap, e.g. for 1 3 4 6 the lesser class contains 1 1 3 4, and the greater class contains 3 4 6 6. Unlike binary CA, in one step a cell can be alive for one neighborhood and dead for another [allusions to human culture omitted]. An example of 3b is http://ic.pics.livejournal.com/spamsink/7162160/49128/49128_original.gif An example of 3c is http://ic.pics.livejournal.com/spamsink/7162160/48828/48828_original.gif Both are 200 steps from a gradient-shaded circle (pgmrap -e 500 500|pnminvert). [the images may not be 100% correct, but they do convey the style of the result] Another aspect is that some rules resulting in trivial behavior of binary CA may exhibit a more complex behavior in grayscale CA. Regards, Leo

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Re: [math-fun] Speaking of arithmetic puzzles
by Leo Broukhis 24 Jun '16

24 Jun '16

On Fri, 17 Jun 2016 11:14:54 -0700, Leo Broukhis <leob(a)mailcom.com> wrote: > > X/1 + X/2 + X/3 + X/4 - X/12 = 1 > 1 / (X/2 + X/3) = X/7 > > The meaning (not "value") of X is the same in both equations. > What is the meaning of X? Or, more specifically, which Unicode character > does X stand for? > > From the dearth of responses, I conclude that mathematics is not yet ready for such problems. The answer is U+09F8 ৸ Bengali Currency Numerator One Less Than the Denominator Leo

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[math-fun] Function question
by David Wilson 23 Jun '16

23 Jun '16

For integer a >= 0, b >= 0, let f(a, b) = 0 if b = 0; f(a div b, a mod b) + 1 otherwise. Q1: Suppose f(a, b) = n. What is the smallest possible value of a in terms of n? Q2: What is lim(a -> inf; SUM(1 <= b <= a; f(a, b)) / a)?

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[math-fun] Super-QM: Super Spooky Action at a Distance ??
by Henry Baker 22 Jun '16

22 Jun '16

FYI -- Although people have been working on this since the early 1990's, this is the first time I ever heard of such things. Is there much interest in these "super" quantum theories? "It's possible that nature could have been even more nonlocal [than standard QM] while still respecting relativity." "These super-quantum boxes have become known as Popescu-Rohrlich (PR) boxes." "Given their super-efficiency, PR boxes could do computation even faster than quantum computers." http://www.theatlantic.com/science/archive/2016/06/quantum-mechanics-weird/… How Quantum Mechanics Could Be Even Weirder And what it could mean for the future of computing Philip Ball 10:00 AM ET Science Why doesn't the world make sense? At the fundamental level of atoms and subatomic particles, the familiar "classical" physics that accounts for how objects move around gives way to quantum physics, with new rules that defy intuition. Traditionally these are expressed as paradoxes: particles that can be in two places at once, cats that are simultaneously alive and dead, apparently impossible faster-than-light signaling between distant particles. But quantum rules are perfectly logical and consistent--the "paradoxes" are the result of our trying to impose on them the everyday reasoning of classical physics. What's more, over the past several decades we've come to understand that the classical and quantum worlds don't exactly operate by "different" rules. Rather, the classical world emerges from the quantum in a comprehensible way: you might say that classical physics is simply what quantum physics looks like at the human scale. All the same, we're confronted with the question: why is the quantum world the way it is? Why do fundamental particles dictate this set of rules and not some other? Normally that question carries an implication that quantum particles are being a bit perverse by not behaving like billiard balls, reassuringly solid and definite and thing-like. But that might be the wrong way to think about it. Last December, I spoke with Romanian-British physicist Sandu Popescu of Bristol University in England, who told me that things could have been even stranger than quantum. In fact, Sandu said, we're not even completely sure that things aren't even stranger. Maybe we just haven't detected this extra strangeness yet. Scribbling on his whiteboard with infectious enthusiasm, Sandu explained that this hypothetical "super-quantum" world comes into view by thinking about what now seems to be the defining characteristic of quantum theory: nonlocality. Increasingly, it looks as though we have come at quantum mechanics from the wrong direction. At first it seemed to be about how energy is not continuous but is divided up into discrete chunks (quanta). Then it seemed to be about how quantum objects have to be described by smeared-out, wavelike mathematical entities called wave functions. Then, the question became how all the possible states of an object encapsulated by a wave function get crystallized into just one state when we measure it using classical apparatus. But in 1935, Einstein and two younger colleagues unwittingly stumbled upon what looks like the strangest quantum property of all, by showing that, according to quantum mechanics, two particles can be placed in a state in which making an observation on one of them immediately affects the state of the other--even if they're allowed to travel light years apart before measuring one of them. Two such particles are said to be entangled, and this apparent instantaneous " action at a distance" is an example of quantum nonlocality. Erwin SchrÃ¶dinger, who invented the quantum wave function, discerned at once that what later became known as nonlocality is the central feature of quantum mechanics, the thing that makes it so different from classical physics. Yet it didn't seem to make sense, which is why it vexed Einstein, who had shown conclusively in the theory of special relativity that no signal can travel faster than light. How, then, were entangled particles apparently able to do it? We now know that entanglement and nonlocality don't violate relativity. Although a measurement here does seem to instantly affect what happens there, you can't actually send any faster-than-light signal or information this way--because you can only verify the effect of the measurement elsewhere by exchanging information classically. Arguably it's better to forget this picture of cause-and-effect: you could say that two entangled particles aren't really two particles at all, but have actually become one single, nonlocalized quantum entity. Alternatively, nonlocality can be regarded as an indication that the properties of a quantum object needn't all be located on the object itself. There's no classical analogue: it doesn't really mean anything to say that the speed or color of a tennis ball aren't entirely situated on the ball itself. But that's what the quantum world is like. Quantum nonlocality seems almost ingeniously designed, then, to allow an event at one place to have instant consequences elsewhere without violating relativity. But in the late 1990s Sandu asked a curious question: are the rules of quantum the only way to do that? When he and physicist Daniel Rohrlich looked closely at that question, they found that it isn't. It's possible that nature could have been even more nonlocal while still respecting relativity. If so, understanding why the strength of quantum nonlocality is limited could give some clues about where quantum mechanics comes from in the first place. What do we mean by "more nonlocal"? Picture two observers, called Alice and Bob, each of whom has a black box that dispenses a cuddly toy dog or cat when fed with a coin. The boxes will accept only dimes or quarters, and let's say Alice's box is designed so that a dime always elicits a cat. (Techie readers will see that these are just binary inputs and outputs: you could write them more prosaically as 1's and 0's.) So here's a challenge. Is it possible for these black boxes to achieve the following: if Alice and Bob both put in quarters, then the boxes will yield one cat and one dog, but any other combination of coins will yield either two cats or two dogs? If you work through the options, given also the constraint that a dime in Alice's box must produce a cat, you find it can't be done. Three times out of four, the boxes will fail to satisfy that requirement. The best you can get, then, is a success rate of 75%. What, though, if Alice's and Bob's boxes can communicate using quantum nonlocality? Now Bob's box can instantaneously use some information about what Alice's box has done to switch its output. It's possible to show that these quantum rules raise the possible success rate to about 85%--not perfect, but better. Can we get 100% success? Yes, Popescu and Rohrlich showed, we can--if we allow the boxes even more nonlocal exchange of information than the rules of quantum mechanics permit. That's possible, they say, still without violating relativity. These super-quantum boxes have become known as Popescu-Rohrlich (PR) boxes. This improved performance, they say, comes down to the efficiency of sharing information between PR boxes. In general, communications are very inefficient because they involve exchanging lots of information that doesn't actually feature in the final answer. Suppose, Sandu said, that he and I want to arrange a meeting. We're both very busy, but we compare diaries by phone. We might hit on a suitable date by fairly randomly asking "Are you free on 6th June?" and so on. But it could take a while if our diaries are very full. So let's say we look for an answer to what sounds like a simpler question: whether the number of possible days they're both free to meet is even or odd. (OK, that doesn't exactly help Â but just suppose we'd like to know anyway.) The answer is just one bit of information: say, 0 for "even", 1 for "odd." But the only way we can deduce this bit is for me to get a list of every day in the year that Sandu is free, to compare against my own calendar. Quantum nonlocality can reduce some of this redundancy of information, but not all. But if we have PR boxes, they can remove it all. The question can then be answered by feeding each of our PR boxes with the details in our respective diaries and having them exchange just one bit of information. For certain types of information processing like this, there is a sharp boundary between what can be done in quantum mechanics and what can be done with super-quantum PR boxes. So PR boxes tell us that quantum nonlocality is actually a measure of the efficiency with which different systems can communicate and share information. In other words, quantum mechanics is a set of rules with which some outcomes of information sharing and processing are possible, while some are not (like Alice and Bob achieving 100% success). Given their super-efficiency, PR boxes could do computation even faster than quantum computers. Could they ever exist, though? Sure, the world looks quantum-mechanical, not super-quantum; but it also looked classical for a long time, until we figured out how to spot quantum nonlocality. Could it be that we've just failed so far to uncover a stronger PR-box nonlocality that exists in the real world? But even if quantum nonlocality is the best we can hope for, PR boxes may offer clues about why that is. The question becomes not so much why nature isn't completely classical, but why it's not "more" quantum. We should then seek answers not by wondering why, say, objects are described by wave functions (or what a wave function is anyway), but by looking at a more fundamental matter of how information can be shunted about--of how efficient communication in nature can possibly be. What is it that apparently limits quantum nonlocality's ability to make information exchange more efficient? All this fits with a growing conviction among many physicists that quantum mechanics is at root a theory not of tiny particles, but of information. It's about how much we can deduce about the world by looking at it, and how that depends on intimate, invisible connections between here and there. Philip Ball is a writer based in London. His work has appeared in Nature, The New York Times, and Chemistry World. He is the author of Critical Mass: How One Thing Leads to Another.

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[math-fun] Sin(angle)? No, angle(sign)
by Bill Gosper 22 Jun '16

22 Jun '16

I threw an Aerobie down the street, trying to reach a certain road sign. Standing at the Aerobie, I was disappointed to see the face of the sign, across the street. But wait a minute, it's crooked! Eyeballing a perpendicular to the street, I'm a couple feet past the sign, dammit. Whoa, maybe roadsigns are twisted slightly clockwise on purpose? To exactly face traffic from where they first become visible? --rwg

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[math-fun] Data storage in buckminsterfullerene
by Bill Gosper 22 Jun '16

22 Jun '16

Wow, a whole new meaning for buckybits. So Polya should count for us the number of distinguishable buckyballs with n 0s and 60-n 1s. But can it give us a UID from a 60 bit vector? --rwg On 2016-06-20 18:37, Eugene Salamin via math-fun wrote: > I have no comment on the practicality of this as a storage scheme. As > to the mathematical problem of counting the number of distinct > arrangements, this is made to order for the Polya Counting Theorem. > > -- Gene > > > > From: Keith F. Lynch <kfl(a)KeithLynch.net> > To: math-fun(a)mailman.xmission.com > Sent: Monday, June 20, 2016 5:44 PM > Subject: [math-fun] Data storage in buckminsterfullerene > > The recent discussion about data storage in diamonds made me wonder > about data storage in another stable form of carbon: C60, buckyballs. > > How many bits can be stored in one buckyball, i.e. fullerene molecule? > It's a truncated icosahedron with 32 faces (of which 12 are regular > pentagons and 20 are regular hexagons), 90 edges, and 60 vertices. > The 60 vertices are carbon atoms. Data can be stored by making some > of them C12 and some of them C13. But how much data? The base-2 log > of the number of distinct arrangements of C12 and C13 atoms. But > how many arrangements are there? If the molecules were not free to > rotate, there would be 2^60 arrangements, hence 60 bits. But of > course the molecules are free to rotate. The rotation group is > order 60. So there are at least 2^60/60 arrangements, hence at > least 54 bits. > > Can anyone think of a practical way to count the arrangements? The > obvious way would be to try all 2^60 arrangements and eliminate the > ones that are rotations of arrangements that have already been found. > But 2^60 is more than 10^18, so that's just barely computationally > feasible. Is there a better way? Has someone counted them already? > Thanks. > > This number might seem to be uselessly small, but it isn't, since data > could be stored in a large set of buckyballs. Each molecule could > contain a 50-bit sequence number and several bits of data, giving > several hundred terabytes of total storage. There would be many > duplicates to ensure that at least one molecule with each sequence > number is found. Several hundred terabytes may not seem like much by > today's standards, but it can store the whole of Wikipedia, plus more > books than anyone can read in a lifetime, plus more movies than anyone > is likely to view in a lifetime. It's a lot larger than the genome > of any organism, so it's a lot more practical than living DNA data > storage. And 2^50 buckyballs times 600 duplicates of each of them > would mass less than one milligram. > > _______________________________________________

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[math-fun] esoterica -- meeting on Lambert function
by Bill Gosper 22 Jun '16

22 Jun '16

On 2016-06-03 07:03, Veit Elser wrote: > Two fun W (Lambert function) facts: > > 1) The Taylor series of W records the number of spanning trees of the > complete graphs. > > 2) If 1/2+i y_n is the nth zero of the Riemann zeta function on the > critical line, then asymptotically (large n) > > y_n ~ 2 pi (n-11/8)/ W((n-11/8)/e) > > -Veit Less amazing, but useful: They appear in an inelegant but efficient series solution to Kepler's equation: http://www.tweedledum.com/rwg/pizza.html It's a bit disappointing that W doesn't solve Kepler's more elegantly. But suppose we posit that E(ε,M) := Kepler(ε,M) solves M = E - ε sin E . Does this dyadic function Kepler subsume Lambert W as a special case? If so, maybe we should embrace it. Failing that, I think "Dilbert Lambda"(y), which solves y = Λ exp Λ², is usually nicer than W. > >> On Jun 2, 2016, at 10:28 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> >> Thanks. I vastly prefer the Lambert W function when it is not defined with a branch cut but is allowed to extend to it full Lambertness on a Riemann surface. >> >> Whether considering the Lambert function or the more simply defined function of which it is the inverse function: >> >> f(z) = z exp(z), >> >> the aforesaid Riemann surface is just the subset of C^2 defined as >> >> {(z,w) | z = w exp(w)}. >> >> This is, in my opinion, the appropriate object of study. >> >> —Dan All we need is for you to tell us how it would look in Mathematica. --rwg

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[math-fun] Somos developments
by rcs＠xmission.com 21 Jun '16

21 Jun '16

Michael Somos has been developing polynomial-valued versions of the theta and sigma functions. These are somewhat related to Somos sequences, but more general. His work so far is at http://somos.crg4.com/wxyz.gp It's more than I can grasp in one sitting, but it looks promising. Rich

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