July 2013 - mathfun@mailman.xmission.com

Re: [math-fun] Really stupid physics question
by Henry Baker 28 Jul '13

28 Jul '13

Perhaps I'm too dense to understand your answer. In the simplest example of N=2, what does the crystal look like & what are its modes? My thought was that the DFT's come from circulant matrices, so if you had a kind of necklace of masses bouncing against one another, this might work. But the other major problem is if you think of Fourier Transforms in terms of frequency & time, you now have discrete frequencies, which is ok, but discrete times (in a physical system) ? So perhaps we model the system with a set of finite difference equations ? At 01:31 PM 7/28/2013, Eugene Salamin wrote: >The vibrations of a finite crystal is what you want. The DFT is the expansion into normal modes. > > -- Gene >>________________________________ >> From: Henry Baker <hbaker1(a)pipeline.com> >>To: math-fun(a)mailman.xmission.com >>Sent: Sunday, July 28, 2013 12:13 PM >>Subject: [math-fun] Really stupid physics question >> >>We've all seen the classical analysis of a vibrating string, which leads to classical Fourier theory. >> >>In the 20th century, we all became converts to DFT's (Discrete Fourier Transforms). >> >>So.... >> >>I'd like to see the exact mechanical counterpart to the DFT. >> >>Given N, the _length_ of the DFT, how do I arrange for N masses & (?) Hookean springs so that the dynamics give me exactly the DFT ? >> >>In particular, for very small N (N<10, nowhere near the continuum case), what do such systems look like?

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[math-fun] quantum theory foundational issues, my theory of how they should be resolved
by Warren D Smith 28 Jul '13

28 Jul '13

Message: 2 Date: Sun, 28 Jul 2013 11:53:37 -0700 From: meekerdb <meekerdb(a)verizon.net> To: math-fun <math-fun(a)mailman.xmission.com> Subject: Re: [math-fun] quantum theory foundational issues, my theory of how they should be resolved Message-ID: <51F568B1.3030004(a)verizon.net> Content-Type: text/plain; charset=ISO-8859-1; format=flowed On 7/28/2013 7:12 AM, Warren D Smith wrote: >> From: Gareth McCaughan <gareth.mccaughan(a)pobox.com> >> On 27/07/2013 11:18, Warren D Smith wrote: >> So for example, why do you not experience being in a superposition of >> living in Paris and Tokyo? Because [...] > Why would you expect > > a superposition of experiencing being in Paris and experiencing > being in Tokyo > > to resemble, in any way, > > experiencing being in a superposition of Paris and Tokyo? > > In default of some reason to expect that, I don't see why the > question arises at all. And I know of no such reason, though > I'm not a QM expert and maybe there is one that I've never > encountered. > > --this is an excellent point, and there's plenty of arguments of that > vague ilk which may > sound good..., > but the problem is, these questions don't seem addressable by > mathematics or precise reasoning. So when various philoso-physicists > (at least those in the Everett camp) are going around saying "the > foundational problems of QM are solved" it's all hogwash. It's > basically religion, and hopes+dreams, not science. > That debate probably would never end and it'd never be clear if it > really solves the problem. > > However, with my approach to the foundations of QM, the gordian knot is cut. > It could really be science. > > And not only that, I can kick their asses with more paradoxical > issues. For example, why is the "position basis" favored over others? It isn't in all cases. Atoms, for example, are usually found in energy eigenstates. --the "energy of atom" operator in this case commutes with "position of atom" operator so this is not a valid objection. However, I would agree electrons in metals, say, can be regarded as delocalized. > Decoherence tends to localize things. > Give them precise positions. Why? With pure QM unitary symmetry, no > basis favored over any other. Well, my view explains that. And if > (say) in the early universe, everything were in a momentum eigenstate, > with pure QM, no localization of anything would ever be possible, > translation symmetry could never be broken. It's because the interaction terms in the Lagrangian are position dependent. --no they are not. A better try is: quantum field theory is "local" i.e. is partial differential equations in position space. In other space it would not be local. So that is something special about the position basis. That is nice, but still does not imply localization will occur via decoherence, see below. > So even if everything were in a momentum eigenstate (and everything is in a superposition of energy-momentum eigenstates) there would be finite probabilities of localized interaction. --In quantum field theory... the thing is, a momentum eigenstate has got translation invariance -- any translate is still a same-momentum eigenstate. If interaction occurs, this will still yield translation invariant states only. If you dispute this, then: I will take whatever you say is the output state and I will translate it. My translated state is as valid as your untranslated state. Therefore, you had to be wrong in your assertion. QED. So no localization could ever occur, in quantum field theory, if initially momentum eigenstates. > This (in flat spacetime) is a theorem of mathematics. So the > physicists who somehow convince themselves that positions being the > "pointer basis" is somehow caused by QM, thru mysterious > measurement-simulating environmental effects, are full of crap. The > true explanation is quantum gravity, I claim via my reasoning. > > What about the "direction of time"? QM with measurement features a > time direction. QM without features CPT symmetry. The direction of physical time is determined by expansion of the universe (which allows entropy to increase) - which in a sense is a gravitational effect. Brent --Bunk. If the universe began contracting, entropy still would be increasing.

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[math-fun] Really stupid physics question
by Henry Baker 28 Jul '13

28 Jul '13

We've all seen the classical analysis of a vibrating string, which leads to classical Fourier theory. In the 20th century, we all became converts to DFT's (Discrete Fourier Transforms). So.... I'd like to see the exact mechanical counterpart to the DFT. Given N, the _length_ of the DFT, how do I arrange for N masses & (?) Hookean springs so that the dynamics give me exactly the DFT ? In particular, for very small N (N<10, nowhere near the continuum case), what do such systems look like?

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[math-fun] quantum theory foundational issues, my theory of how they should be resolved
by Warren D Smith 28 Jul '13

28 Jul '13

I should make it clear that gravitons, neutrinos, etc are very small "measuring" effects normally as compared with photons, air molecules, etc. The usual physicist's patter about how those latter things cause decoherence is fine, except that they sort of inherently depend on the "environment" (air, material near slit, etc) being aleady non-wierd, well localized, un-entangled with the state of whatever they "measure." Without that assumption those explanations get hard to be sure of. With my graviton based explanation, this regime also is taken care of. Point is, you need some "ground truth" at the bottom of the cascade of bull. If you provide it, ok. If you do not, things get dubious.

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[math-fun] Yet another weighing problem
by Victor Miller 28 Jul '13

28 Jul '13

In the Numberplay blog of the NY Times, they posed the following problem: You have 14 identical appearing coins, 7 of which are heavy (of the same weight), and 7 are light (of the same weight). You can't distinguish them by appearance or feel. You have a special kind of balance available to you: you can insert piles of coins on the left side and the right. The machine will either report that the two piles are equal in weight -- in which case it will return all the coins. If it reports that one side is heavier than the other, it will exact a tribute from the heavy side -- it will take one of the heavy side's coins at random, and return all the rest. The original question is: devise a strategy of weighing which will allow you to always exhibit one of the heavy coins. So the obvious generalization is to consider pairs (m,n) of positive integers with m >= n, which means that you have m coins in total of which n are heavy and m-n are light. For which (m,n) can you devise a strategy as above? Call such pairs (m,n) *feasible*. I would think (is there a simple proof?) that if (m,n) is feasible then so is (m,n+1) (provided n+1 <= m). If so, this would seem to give a sequence for OEIS (which is probably already there). One thing that the original problem didn't ask is -- devise a strategy (which might allow you to use random choice) for which the expected number of weighings is smallest. There are also two alternatives 1) You're allowed to label the individual coins, which are preserved through the weighings. 2) You're not allowed to label. Victor

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[math-fun] NYTimes: quantum theory w/o observers
by Henry Baker 28 Jul '13

28 Jul '13

Quite a discussion going on here: "to quote the physicist J.S. Bell: "It would seem that the theory is exclusively concerned about Âresults of measurement,Â and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of ÂmeasurerÂ? Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system Â *** with a Ph.D.? **** If the theory is to apply to anything but highly idealized laboratory operations, are we not obliged to admit that more or less Âmeasurement-likeÂ processes are going on more or less all the time, more or less everywhere. (ÂAgainst ÂMeasurementÂ Â)" http://opinionator.blogs.nytimes.com/2013/07/25/return-of-the-stingy-oddsma…

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[math-fun] quantum theory foundational issues, my theory of how they > should be resolved
by Warren D Smith 27 Jul '13

27 Jul '13

> But the weird effects of QM are observed in systems that are (temporarily) > isolated from > the environment. > > ... Is the double-slit experiment isolated from the environment? > http://en.wikipedia.org/wiki/Double-slit_experiment --yes, more precisely, quantum mechanics is free to be weird until it decoheres, and it depends on the environment and the phenomenon, how long that takes. However in most human situations the world seems non-quantum and non-weird, which is since the decoherence times are smaller than human perception time scales in nearly all situations humans get to see with naked eyes.

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[math-fun] Interesting game, "Hanabi"
by Dave Dyer 27 Jul '13

27 Jul '13

...Interesting from the viewpoint of information theory, similar to optimum bidding strategies in Bridge. Synopsis: There are 5 suits, with values 1-1-1-2-2-3-3-4-4-5 in each. Each player has a hand of 4 cards, which are held backwards so the player can see everyone else's cards but not his own. The goal of the game is to cooperatively lay down a perfect solitaire style meld of 1-5 in each suite. On each turn, a player can either play a card, discard a card, or give jotto-style information to one other player. http://boardgamegeek.com/file/download/9gg381aeug/hanabi_rules_-_card_1_cor…

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[math-fun] dolphin names
by Warren D Smith 27 Jul '13

27 Jul '13

> From: Tom Duff <td(a)pixar.com> > http://languagelog.ldc.upenn.edu/nll/?p=5453 > > On Tue, 23 Jul 2013, Warren D Smith wrote: > >> http://www.guardian.co.uk/environment/2013/jul/22/bottlenose-dolphins-names… --I think that language log guy (G.K.Pullum) is complaining way too much, devoting great effort to attack a claim that was not made. The dolphin finding seems valid to me, no matter how much he whines about it, and it does not say (nor far as I could see was it trying to) that dolphins have human-like language. It indicates that they have a useful functionality of indicating to each other who is located where. Pullum completely fails to comprehend that that would be very useful functionality for a herd of dolphins, instead wondering why if anybody shouted out "G.K. Pullum" in the street he would respond "G.K.Pullum" -- he sees no reason, therefore the whole thing is "obviously spurious." No, it is "obviously correct" (unless simply fraudulent) and interesting, and a simple and reproducible experiment (and Pullum seems to me to be a pompous idiot), and furthermore it might cast light on the beginnings of actual human language since, e.g, such a functionality could have started the whole ball rolling on that. (Indeed, we might ask: when, in human history, did "names" first appear? Probably unanswerable.) Incidentally, other animals also have communication abilities in some ways resembling language, for example honeybees use a dance-based sign language which has been decoded using video and artificial bee-robots, and is extremely useful for them. The Australian stingless bees however do not do that and may communicate in a quite different manner (olfactory?).

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Re: [math-fun] 'Rationalize' a vector of real numbers ?
by Henry Baker 24 Jul '13

24 Jul '13

Wouldn't it be possible to pose this problem (for 2 real numbers) as an _integer linear programming_ problem, and aren't there pretty decent algorithms for such a relatively small integer programming problem? Of course, you would have to approximate the real numbers to some large number of digits -- perhaps 10 decimal digits (?) -- if you wanted to find the best rational approximations with max denominator < 10000. At 06:25 AM 7/24/2013, Simon Plouffe wrote: >Hello, > > I made a lot of research in that area of approximation of 1 real by a vector of reals. > > The thing is : There has been a lot of papers written on the generalisation of cont. frac. >T - fractions , U -fractions, any_letter- fractions. one problem being that there are no >natural way of doing that, well at least not any which is simple enough that will >give the 'best possible ' approximation. > >Let's say you have the simplest case where there is more than 1 number to approximate >by real numbers, take pi and e or exp(1), > >if we ask what is the best rational approximation to both of them at the same time , >well there is no clear answer, > >One simple algorithm is this one : > >for example take 2 numbers a and b > >in pseudo - code : > >do >{ | a - b | } = c >a <-- b >b <-- c >od > >So this will find X, Y, Z such that > >aX + bY + Z = 0 > >or said differently, you find 2 integers mutiplied by (in this example) by Pi and E >so that the error is as small as possible > >If you do a couple of iterations then one can find : 9257454 * E - 5824723 * pi = 6865462 > >BUT this example is not very good , with LLL or PSLQ it is possible to find smaller >integers that will give a better approximation and this is the whole point. >There are many ways to do that, > >When you have a vector of reals then there are many- many possible ways. > >if you look at the Maple doc. on ?kronecker or ?numtheory[minkowski] > > the algorithms shown are interesting in a sense that it adresses the problem >without necessarly finding the best answer. ... > >Best regards > >Simon plouffe

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