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[math-fun] and the famous formula of A. Einstein (manuscript)
by Simon Plouffe 19 May '03

19 May '03

In one of the manuscripts, we can read, E= mc^2 on the top of the page. http://www.alberteinstein.info/gallery/images/01-148p01z_high.jpg Simon Plouffe

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[math-fun] collected papers of Albert Einstein
by Simon Plouffe 19 May '03

19 May '03

hello, This site holds pretty much all of what he wrote I believe, it goes back to 1896. http://www.alberteinstein.info/ ...interesting! Simon Plouffe

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[math-fun] "Canonical" 1-1 correspondences between sets
by John McCarthy 19 May '03

19 May '03

When two sets are proved to have the same number of elements, e.g. the number of partitions of n into odd parts and the number of partitions of n into distinct parts, it seems natural to ask whether the proof should be given by establishing a 1-1 correspondence between the sets. In the above example it was done but long after the equality was established. [Have I got the facts right?] Of course, whenever there is a 1-1 correspondence between two sets, there are k! of them. We can ask whether there is a "canonical" 1-1 correspondence. Here's a case where there are 1-1 correspondences but where it seems that none are canonical. Consider a vector space X of dimension n over a finite field. Also consider its dual space X* and its double dual X**. X has a canonical isomorphism F with X**, namely F(x)(f) = f(x), but it doesn't seem to have a canonical isomorphism with X*. Therefore, a proof that X and X* have the same number of elements wouldn't be as natural as a proof that X and X** have the same number of elements. Does this phenomenon account for the fact that it is so often easier to establish that two sets are equipollent than to establish an elegant 1-1 correspondence? Is the question discussed in the literature on cominatorics?

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[math-fun] Re: your mail
by Richard Guy 19 May '03

19 May '03

It may seem clumsy, but I write and pronounce Mertens's, Jones's, Gauss's, Menelaus's, Pythagoras's, etc. Fowler, writing around WW1, said `It was FORMERLY [my stress] customary, when a word ended in -s, to write its possessive with an apostrophe but no additional s. In verse, & in reverential contexts, this custom is retained, & the number of syllables is the same as in the subjective case. But elsewhere we now add the s & the syllable. Mertens' might be appropiate if two people named Merten had collaborated. R. On Mon, 19 May 2003, Joshua Zelinsky wrote: > Hi Richard, > > Carl Pomerance and Ivic have pointed out a few minor mistakes in my > paper. First, I neglected to include the reference to where Ivic proved his > improvement of the Erdos bounds: "The distribution of primitive abundant > numbers." Studia Scien. Math. Hungarica 20(1985), 183-187. Second, where I > wrote "Merten's Theorem", that should be "Mertens' Theorem." Third, > Pomerance has alerted me that he had already produced a similar heuristic > argument for the non-existence of odd perfect numbers which he presented at > a lecture at Berkeley in 2000. Thus, the claim that the paper presents the > first such heuristic is inaccurate. > > > Regards, > Josh Z > > _________________________________________________________________ > STOP MORE SPAM with the new MSN 8 and get 2 months FREE* > http://join.msn.com/?page=features/junkmail >

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[math-fun] Partition(s)
by asimovd＠aol.com 19 May '03

19 May '03

Just saw a play, "Partition", performed in Berkeley by the Aurora Theatre Company, which is all about the interaction between Hardy & Ramanujan. Thoroughly enjoyed it. I don't know if it will be performed in other cities, but its run in the Bay Area has just ended. It's nice, the number of math-related plays recently. Alas, I haven't seen either Arcadia or Proof. Has anyone out there seen either of these, and if so, how did you like it? ALSO: Was moved to check out the extensive MathWorld entry on partitions, at <http://mathworld.wolfram.com/PartitionFunctionP.html>, and was reminded of the mystical asymptotic formula P(n) ~ exp(pi sqrt(2n/3)) / (4 sqrt(3) n) discovered by Hardy & Ramanujan. Is anyone familiar enough with a proof of this -- or a heuristic argument -- to post a sketch of it to math-fun? --Dan

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[math-fun] distinct and square-free factorizations
by Marc LeBrun 18 May '03

18 May '03

Call a factorization distinct if there are no repeated factors, and call a factorization square-free if it contains no squares. For all positive integers > 1 I claim the number of each match. For example of the sixteen factorizations of 72: A 72 B 8.9 C 6.12 D 4.18 E 3.24 F 3.4.6 G 3.3.8 H 2.36 I 2.6.6 J 2.4.9 K 2.3.12 L 2.3.3.4 M 2.2.18 N 2.2.3.6 O 2.2.2.9 P 2.2.2.3.3 seven (GILMNOP) have repeated factors, and seven (BDFHJLO) contain 4, 9 or 36. Do you know any citation for this? (Assuming it's in fact true!<;-) Also, I'd like to see your proofs, to see how they differ from mine. Particularly interesting would be an explicit construction mapping the factorizations 1-1. (At first I thought this was easy--just split q^2 into q.q--but it's unclear to me what the images of J or P, say, ought to be). Thanks!

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[math-fun] Lunar Eclipse tonight
by Richard Schroeppel 15 May '03

15 May '03

Remember to look for the Lunar Eclispse tonight. Rich

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Re: [math-fun] Re: (not math, but maybe fun)
by Mike Stay 13 May '03

13 May '03

>Not math, but maybe fun: > > >=Eugene Salamin > > Every chemical is toxic. For sucrose the LD50 (the dosage which kills > > 50% of the subjects) for a 170 pound rat is 5 kg. > >And I thought the one I killed in my garage this weekend was big! (Maybe >they should lay off the sugar?<;-) > > <Insert obligatory "Princess Bride" reference here. ;-> -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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[math-fun] Ian Stewart takes a crack at explaning Poincare
by Thane Plambeck 13 May '03

13 May '03

This is from Nature, 8 May 2003: Deftly avoiding those manifolds and balls in R^n.... * * * * * The Clay Mathematics Institute is offering a million dollars for a solution to the PoincarÃ© conjecture, and Grisha Perelman may have found one. What is the conjecture, and why does it matter? In 1904, Henri PoincarÃ© was making fundamental advances in topology - the multi-dimensional study of mathematical properties such as 'knotted' or 'connected', which are unchanged by continuous deformations. Buried in his work was an unjustified assumption about three-dimensional spaces. When he noticed it, and failed to find a proof, the assumption became first a question, and then a conjecture. Unsolved almost a century later, the PoincarÃ© conjecture joined other problems (such as the Riemann hypothesis and Yang-Mills theory) on a list of 'Millennium Prize Problems' for whose solutions a substantial cash prize is offered1. Grisha Perelman of the Steklov Institute of Mathematics in St Petersburg, Russia, may now have found the answer. The PoincarÃ© conjecture can be understood by analogy with the case in two dimensions. A two-dimensional space, or surface, is like a bubble made from an infinitely thin film of soap. If the bubble is round, the surface is a sphere, but many other shapes are possible. A torus, for instance, is shaped like an American doughnut; a two-holed torus is like two doughnuts joined together, and so on. The early topologists proved that any surface is topologically equivalent to a torus with a finite number of holes. A lumpy potato is the same as a sphere, a coffee-cup is the same as a torus, and a teapot is the same as a two-holed torus4. PoincarÃ© sought a comparable understanding of three-dimensional spaces. The simplest is the three-sphere, analogous to a sphere but having three dimensions instead of two. This is almost the same as a solid ball, but only if you pretend that the entire outer surface of the ball is a single point. The key problem is to decide whether a given surface - a lumpy potato, say - is equivalent to a sphere, or to something more exotic. PoincarÃ©'s predecessors had solved that problem in two dimensions, by working out how closed loops in the surface behave when they are moved. Any loop on a sphere can be continuously deformed, or shrunk, to a point. But on a torus, or a more complicated surface, many loops cannot be shrunk (Fig. 1a). Better still, if every loop can be shrunk then the surface must be a sphere. So the 'shrinkability' of loops provides a definitive test for spherical topology. What about three dimensions? It is easy to prove that every loop in a three-sphere can be shrunk, but PoincarÃ© tacitly assumed the converse: if every loop can be shrunk, then the space is topologically a three-sphere. Only later did it dawn on him that this statement - now called the PoincarÃ© conjecture - was not obvious, and perhaps might not even be true. Throughout the twentieth century, the conjecture remained as mysterious as ever. Analogous statements in all numbers of dimensions from four upwards had been proved true, but PoincarÃ©'s original three-dimensional problem still held out. The PoincarÃ© conjecture had become a serious obstacle to any understanding of three-dimensional topology. Because topology is fundamental to many branches of mathematics, and to several areas of mathematical physics, this missing piece of the mathematical tool kit had become a serious embarrassment. How can we hope to understand the shape of our own Universe, for instance, when we don't even know the range of possibilities? And what about the even harder topological problems arising in quantum field theory (for example, in relation to string theory, where space-time has extra dimensions that curl up in some topological manner)? Around 1983, William Thurston had devised an entirely new approach to the PoincarÃ© conjecture, by relating it to classical geometry. Starting, again, in two dimensions, Thurston wondered why there are lots of shapes for a potato, but only one round sphere. What makes that shape special? It has constant (positive) curvature: every bit of a round sphere is bent the same amount as any other. Similarly, there is a 'canonical' geometry for a torus, and in this case the curvature is zero. The usual torus does not look flat, but a rectangle with opposite edges identified is flat, and topologically that is a torus. Finally, tori with two or more holes can be realized as surfaces of constant negative curvature. Thurston suggested that something similar should happen in three dimensions. He proved that exactly eight different 'geometries' can occur. It turned out to be too much to expect every three-dimensional space to have a single canonical geometry, but Thurston's 'geometrization conjecture' asserted the next best thing: every three-dimensional space can be cut up, in a systematic way, so that each piece has precisely one of those eight geometries. This new conjecture was much more ambitious than PoincarÃ©'s: it aimed at understanding all three-dimensional spaces, not just the three-sphere. In particular, the PoincarÃ© conjecture was an easy consequence of the geometrization conjecture: the condition on loops meant that only one piece would be needed, and the associated geometry must be that of the three-sphere. Perelman's work2, 3 is a new approach to the geometrization conjecture. If it pans out, it will constitute a huge leap forward in three-dimensional topology and mathematical physics. It is based on the Ricci flow, an idea of Richard Hamilton's5: if a soap bubble is deformed away from its normal round shape, surface tension will pull it back to a perfect sphere. The Ricci flow is an analogous way to deform any surface so that its curvature 'tries' to become constant. Along the way, though, it may have to split into separate pieces; for example, a dumbbell-shaped bubble must break up into two round ones (Fig. 1b). Hamilton defined the Ricci flow in three dimensions, and developed a programme to prove that when a three-dimensional 'bubble' follows the flow, the pieces it breaks into are essentially those predicted by the geometrization conjecture. Perelman's papers do not carry out the entire Hamilton programme, but it looks as though they might establish enough of it to prove the geometrization conjecture. That in turn will prove the PoincarÃ© conjecture. If everything hangs together, a 99-year search is at an end. If not, Perelman's papers will still shed a huge amount of light on the Ricci flow, which is important in quantum field theory and pure mathematics, and may well pave the way to an eventual proof of the two conjectures. Either way, the spin-off is likely to make profound changes to how we think about topology, space-time and quantum field theory. References 1. http://www.claymath.org/Millennium_Prize_Problems 2. Perelman, G. Preprint math.DG/0211159 at <http://arXiv.org> (2002). 3. Perelman, G. Preprint math.DG/0303109 at <http://arXiv.org> (2003). 4. Stewart, I. Flatterland (Macmillan, London, 2001). 5. Hamilton, R. S. J. Diff. Geom. 17, 255-306 (1982). Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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RE: [math-fun] (Tides and) Public education (more a soliloquy than a rant)
by Henry Baker 13 May '03

13 May '03

Erik: This page looks very cool! I tried playing with a few of the animations, and it was fun. Just one thing to think about, though. In order to understand your explanations, you already have to know calculus, which won't help the high school student who doesn't yet know calculus. I think that the problem with asking about tides & other things in this thread was also a question about how simple/intuitive an explanation is possible. On another subject -- I'm wondering what would have happened to physics had gravity _not_ produced conic sections for orbits. Since mathematicians of these early times had only studied conic sections, they weren't ready to "see" anything else. As an analogy, the discovery of "smooth" calculus blinded mathematicians and physicists for the next several hundred years to the possibilities of non-smooth thingy's such as fractals. At 04:24 PM 5/10/03 -0700, Erik Neumann wrote: >>The computer & internet have the capability to go far beyond >>the library in its ability to engage & challenge children. I >>just wish that Feynmann had the internet at his disposal when >>he did his famous lectures. Perhaps someone of Feynmann's >>caliber is (or soon will be) working on a similar project. > >I'm not of Feynmann's caliber, but I'm trying to work in this area... if >you have an opportunity to take a look at my website, >http://www.myphysicslab.com, and give me some feedback on possible >improvements I'd be very grateful. > >Regards, > >Erik Neumann erikn(a)MyPhysicsLab.com >http://www.myphysicslab.com

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