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Re: [math-fun] Re: Parallel Universes and random numbers from [0,1)
by asimovd＠aol.com 18 Apr '03

18 Apr '03

> From: Helger Lipmaa <helger(a)saturn.tcs.hut.fi> > Subject: [math-fun] Parallel Universes > > http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC5880… > > "In infinite space, even the most unlikely events must take place > somewhere." In serious probability theory this question is typically not addressed, although I think it should be. The general situation is where there is an event whose probability is 0, but which is not impossible -- like picking a number at random from the unit interval and getting 1/3. When the experiment of picking the random number is repeated enough times, we, or at least I, would expect the probability to be 1 that 1/3 has come up at least once. One thing is for sure -- if the experiment is repeated a countably infinite (aleph_0) number of times, the the probability is 1 that every number in the unit interval will have arbitrarily nearby numbers having been picked. When attempting to apply this to the real world, one could fairly say that in an infinite universe (assuming appropriate randomness and a continuous state space), then with probability 1, everything that might happen in a given finite portion of spacetime is approximated with arbitrary precision -- .somewhere. (This ignores the possibility of black holes, alternative universes, and just about all the weirdness that is likely to be true.) Back to the unit interval [0,1). Suppose we were to pick a number at random from [0,1) an infinite number of times -- let's call the infinite cardinal "beta" (which of course satisfies beta >= aleph_0). Then what is the chance that we never picked 1/3 ? (Where 1/3 could clearly be replaced by any other x in [0,1).) Heuristically, this probability should be (1 - 1/c)^beta, where c = continuum = # of points in [0,1). This expression is meaningless in ordinary cardinal arithmetic but can be made to make sense in, say, the surreal (Conway) numbers. In surreal numbers, the nearest real number to (1 - 1/c)^beta is 0 if beta < c, 1 if beta > c. I'm not sure whether (1 - 1/c)^c = 1/e (to the nearest real), but wouldn't that be nice? Let's call this probability P(beta). However, picking a random number from [0,1) once for each t in [0,1) -- which heuristically yields 1/e for P(c) -- leads to a paradox. Since the index set {t} = [0,1) is bijectively equivalent to [0,2), this means that doing the experiment once for each t in [0,1) should yield the identical probability as doing it once for each t in [0,2). This means that P(c)^2 = P(c), so P(c) = 0 or 1, and NOT 1/e. This is frustrating to me, since morally it "should" be true that P(c) = 1/e. --Dan

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Re: [math-fun] Re: Parallel Universes
by Henry Baker 18 Apr '03

18 Apr '03

It is possible that multiple parallel universes can co-exist, but they must be linked in a very tight fashion, because the existing laws constrain things pretty tightly. In particular, the basic laws of the universe are _reversible_ -- i.e., information-conserving -- so any interaction with a parallel universe must interchange the same number of bits in both universes. E.g., when a quantum state is "measured" in one universe, other universes measure the same state, but the result of the measurement will be different. Quantum states that are entangled in one universe will necessarily be entangled in the other universe, and so on. This is the essence of the Everett-Wheeler model. At 06:01 AM 4/18/03 -0400, Ken Roberts wrote: >> From: Helger Lipmaa <helger(a)saturn.tcs.hut.fi> >> Subject: [math-fun] Parallel Universes >> >> http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC5880… >> >> "In infinite space, even the most unlikely events must take place >> somewhere." > >A nice article but that "infinite" word jars a bit. >I wish the Cantor diagonalization process had been >discussed. I suppose it is considered in with-math >version of the article, but that article is so long >that it could not be fit within the 2^N size of an >issue of SciAm or the 2^M size of a reader's mind.

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[math-fun] Re: Parallel Universes
by Ken Roberts 18 Apr '03

18 Apr '03

> From: Helger Lipmaa <helger(a)saturn.tcs.hut.fi> > Subject: [math-fun] Parallel Universes > > http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC5880… > > "In infinite space, even the most unlikely events must take place > somewhere." A nice article but that "infinite" word jars a bit. I wish the Cantor diagonalization process had been discussed. I suppose it is considered in with-math version of the article, but that article is so long that it could not be fit within the 2^N size of an issue of SciAm or the 2^M size of a reader's mind. Given countable number of physical constants in binary choose new constant differing from each one previous, etc. Or any other measurement of context difference. Voila - a universe not matching any of the existing. Actually we have to imagine "voila" but the "whole" idea is unobservable imaginings so that's consistent. Some of the article's phraseologies are stimulating... "Imagine moving planets to random new locations, imagine having married someone else, and so on. At the quantum level, there are 10 to the 10118 universes with temperatures below 108 kelvins. That is a vast number, but a finite one." If the imaginations must be embedded in finite imaginers, then there are only a finite number of imaginings too. "All of physics is ultimately a mathematics problem: a mathematician with unlimited intelligence and resources could in principle compute..." Same difficulty. There exists "in the world" no such mathematician. My favourite version is a restating of Hamlet: "There are more philosophies dreamt in heaven than exist in our earth". "Mathematical structures have an eerily real feel to them. They satisfy a central criterion of objective existence: they are the same no matter who studies them." That I like. There might be an interesting analogue to the recent discussion of deriving a "fair coin" probability distribution by tossing a weighted coin. At least an opportunity to do some more math-stuff! Apropos of "the" set of natural numbers, author says... "Which is simpler, the whole set or just one number? Naively, you might think that a single number is simpler, but the entire set can be generated by quite a trivial computer program, whereas a single number can be hugely long. Therefore, the whole set is actually simpler." The whole set is simpler only if we believe that the algorithm is sufficient to create a representative of such a set in the physical universe. Otherwise the single number is simpler. Even that may be trouble, since "twoness" requires a recognizer capable of all the complexity of mathematics. (Though I do know a very young man who presently counts only "2, 2, 2..." in a somewhat monotonous fashion, he is ultimately capable of more nuanced mathematical complexity.) The best mathematical construct I have encountered to adapt to the operational role of "universal" thinking, is that of dual space of functions. If the physical universe exists, the mappings of physical onto some flatness (eg paper, via scribbling or photographs) is a universe of imaginings, with enhanced structure. It is also an optimistic outlook, since an I-universe need not be subject to 2nd law of thermodynamics in the same way as a P-universe. Only its implementation and its output representations in P-universe have that particular constraint. It is conceivable, if time is directional, that information increases in I-universe though P-universe becomes less thermodynamically rich. That counting algorithm describing the generation of the set of all natural numbers can exist in various P-universe representations (Peano, computer pgm etc), but can also be implemented in I-universe and has its fullest output there, which can then be selectively projected onto P-universe. The problem is not with the math but with ourselves. A pleasant entertainment for the start of this weekend which celebrates the spring season of rebirth. Best to all, Ken Roberts

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Re: [math-fun] Peter Sarnak on the Poincare work, with photo of Perelman at ...
by asimovd＠aol.com 18 Apr '03

18 Apr '03

In the article about Perelman's putative proof of Bill Thurston's geometrization conjecture that Thane mentioned was in The Princetonian, the writer got things almost exactly backwards when he wrote: << In fact, there are infinitely many distinct two-dimensional surfaces. The ball is different from the donut is different from a two-holed donut is different from a three-holed donut, ad infinitum. But this is not the case in higher dimensions. As was first suggested by former University mathematics professor William Thurston, when dealing with three-dimensional surfaces, there are only eight shapes that are topologically distinct. This statement, known as the Geometrization Conjecture in three dimensions, is currently unproven. >> The latter paragraph is exactly wrong. There are assuredly infinitely many topologically distinct 3-manifolds; in some sense each dimension is more complicated than lower dimensions, but 3 dimensions may be in another sense the most complicated of all. What Bill proved for a large class of (compact) 3-dimensional manifolds and conjectured for all of them is that each such manifold can be decomposed into pieces each of which can be given one of only 8 different types of geometries. For surfaces, the analogue is that each compact surface can be given a metric of constant curvature, either = 1, or = 0, or = -1. Furthermore, surfaces don't need to e decomposed into pieces first. Three of the eight geometries in the conjecture are the 3D analogues of the 3 constant curvature metrics in 2D; the other 5 might be called hybrids. --Dan

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[math-fun] Peter Sarnak on the Poincare work, with photo of Perelman at blackboard
by Thane Plambeck 17 Apr '03

17 Apr '03

http://www.dailyprincetonian.com/archives/2003/04/17/news/7979.shtml Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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[math-fun] steam locomotives- yes
by Chris Landauer 17 Apr '03

17 Apr '03

hihi, all - no. little wheels in front (even if zero!), then no. drivers (big wheels, main transfer of power to track), then no. little wheels in back (the little ones are guide wheels i think) so some were 2-4-2 and others 0-10-0, and there are many other configurations more later, cal

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Re: [math-fun] four-by-four
by asimovd＠aol.com 17 Apr '03

17 Apr '03

Please note that the original question was about the *etymology* of the term. --Dan

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Re: [math-fun] four-by-four
by asimovd＠aol.com 17 Apr '03

17 Apr '03

David Wilson asks: << Does anyone know the etymology of the term four-by-four or 4x4 in relation to an off-road vehicle? >> I have to agree with Bernie here: The 4-speed transmission, and the 4-wheel drive. --Dan

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[math-fun] Problem(N): Use all KxL rectangles of area <= N to tile some square.
by asimovd＠aol.com 16 Apr '03

16 Apr '03

An interesting problem -- and its solution -- appeared in the most recent American Math. Monthly: a) Given integer N > 0, let R(N) be the set of all KxL rectangles with integer sides whose area is <= N (with no two isometric). Find an N > 1 such that all the members of R(N), each used exactly once, can be used to tile a square? b) Show the analogous question in 3D has no solution: If C(N) is the set of rectangular solids with integer sides of volume <= N (with no two isometric), then for N > 1 there is no cube that can be tiled by all the members of C(N), each used exactly once. --Dan

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[math-fun] biased to unbiased coin flips in real life
by Thomas Colthurst 16 Apr '03

16 Apr '03

So I assume you know the old problem of "convert a sequence of biased coin flips (with unknown bias) into a (not necessarily same length) sequence of unbiased coin flips". I recently learned two interesting things about this problem: it was apparently first posed (& of course solved) by Johnny von Neumann and that it is used in certain Intel chips as part of a hardware random number generator process. (See http://www.cryptography.com/resources/whitepapers/IntelRNG.pdf for details.) While looking at this, I wondered if there was a more efficient scheme that processed more than two bits at a time. A little bit of thought revealed that there was, for example, this 4 bit code: 0000, 1111, 1100, 0011 -> output nothing 0001, 1110, 0101 -> 00 0010, 1101, 0110 -> 01 0100, 1011, 1001 -> 10 1000, 0111, 1010 -> 11 This is a win because the 2 bit code would only output 1 bit for sequences like 0001, while here we output 2 bits. In general, good codes can be obtained by mapping an n bit block with k 1's in it to an output with round_down( log_2( choose(n,k) ) ) bits, but these will only be more efficient than the two bit code for all biases when n is a power or two. The reason is that for very low or very high biases, the efficiency depends most on what you do when there is only one 0 or 1 in the input block. The two bit code will always output one bit in this circumstance, while most of the others (including those for all odd n) will sometimes output no bits. In fact, if we define the efficiency of a code as the ratio of the expected output length to the input length, then the efficiency of the 2^m block codes goes to 1 as m goes to infinity. The worst case is when p is very small (or very large); for these the efficiency is approximately p m. On the other hand, if the bias is known to be near 0.5, then odd length codes seem to be more efficient. (The length 7 code has efficiency 29/56, which is not beaten by any code of length less than 15.) I'm a little bit curious as to why Intel didn't implement a more efficient code. One guess is that they didn't know about them. Another is that the 2 bit code was good enough so as to make wasting silicon on more a complicated code not worth it. Still another possible reason is that the bias of the input coin flipper isn't exactly constant, but has (say) a small linear trend over time which is small enough to ignore over 2 bits but maybe not over longer lengths. -Thomas C

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