April 2009 - mathfun@mailman.xmission.com

[math-fun] math, existence, and God
by James Propp 29 Apr '09

29 Apr '09

Bask on March 25, Andy Latto wrote: >My view is that philosphers often worry too much about ontology (which >things "exist" and which don't), when it really isn't a very >interesting or important question, and it's just the nature of our >language that misleads us into thinking it is. [followed by a whole bunch of stuff, all of which I agree with and think is expressed extremely well, including a very trenchant thought-experiment about mental experience] Have any theologians tried to develop a theology without ontology, along lines suggested by 20th century mathematics? For instance, if one believes that the differing moral verdicts that two people reach are in principle reconcileable by way of a sufficient amount of dialogue and/or shared experience, one might define God as a kind of projective limit in the category of people-who-could-exist, in which the existence of the limit is not as important as what it entails about mappings between finite beings. This actually brings us back to qualia: Presumably you and I could reach agreement about a whole lot of things if I could experience your life from birth to the present day from your own perspective, and vice versa; but what does it mean for one person to have the experiences of another? I suspect that some science fiction writer must have explored this avenue for the future evolution of religion, even if the theologians haven't! I also suspect that some category theorists have thought along similar lines. Jim Propp

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Re: [math-fun] orrery problem
by Dan Asimov 28 Apr '09

28 Apr '09

It might be relevant that, according to rumor, the years of the various planets have ratios that are approximately rational with small denominators. Seems that a random solar system is likely to approach one like this for stability reasons. (But I don't have documentation.) --Dan << My brother is interested in building an orrery, and posed the following (somewhat ill-defined) problem: given a bag of gears and a set of periods, how do you partition them to get the best set of gear chains? Continued fractions, of course, give the best rational approximations to a given period, but that assumes you can choose the number of teeth. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] orrery problem
by Mike Stay 28 Apr '09

28 Apr '09

My brother is interested in building an orrery, and posed the following (somewhat ill-defined) problem: given a bag of gears and a set of periods, how do you partition them to get the best set of gear chains? Continued fractions, of course, give the best rational approximations to a given period, but that assumes you can choose the number of teeth. -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

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[math-fun] bicycle gear ratios & continued fractions
by Henry Baker 28 Apr '09

28 Apr '09

Perhaps someone has already done this. The sensor on my bike outputs both cadence (crank speed in revs/sec) and wheel speed (revs/sec). I'm trying to infer which gear the bike is in by looking at the ratio of these two numbers. My plan was to use the equivalent of Scheme's "rationalize" function (which uses a continued fraction algorithm internally), which produces an exact fraction which is within some epsilon of a given number -- in this case the wheel/crank speed. This should give me an approximation to the ratio (#front teeth)/(#rear teeth). http://www.r6rs.org/final/html/r6rs/r6rs-Z-H-14.html Unfortunately, there is enough error in the measurements of the crank speed and the wheel speed so that the output isn't stable. So far, we haven't utilized any information about which chainrings (front teeth) and cogs (rear teeth) are mounted on the bike, except for the fact that each must have an integer # of teeth. Of course, I can always make up a table of all of the gears and simply find the closest one & output it. But if the number of cogs grows large and the number of chainrings grows large (we're mathematicians, after all!), we'd rather not search a large table. Suppose now that we have upper & lower limits on the teeth on the chainrings (28 to 52) and the cogs (12 to 26), but otherwise don't know how many chainrings or cogs are mounted on the bike. How do we modify the continued fraction algorithm used within rationalize to produce fractions with numerators and denominators in the appropriate range ?

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Re: [math-fun] 1 to 46 at random
by Eric Angelini 27 Apr '09

27 Apr '09

I've received the full set, thanks to Paul P. Best, É.

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[math-fun] 1 to 46 at random
by Eric Angelini 27 Apr '09

27 Apr '09

Could someone please send me the 46 integers 1 to 46 at random ? Thanks, É.

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[math-fun] Future of Financial Mathematics?
by Ray Tayek 25 Apr '09

25 Apr '09

http://news.slashdot.org/article.pl?sid=09/04/25/1858223 An anonymous reader writes "<http://en.wikipedia.org/wiki/Nassim_Nicholas_Taleb>Nassim Nicholas Taleb, a famous '<http://en.wikipedia.org/wiki/Quantitative_analyst>Quant,' has long been a strong critic of the use of mathematics and statistics in the financial markets. He has been very vocal in his books <http://en.wikipedia.org/wiki/The_Black_Swan_%28Taleb_book%29>The Black Swan and <http://en.wikipedia.org/wiki/Fooled_by_randomness>Fooled by Randomness. In <http://www.edge.org/3rd_culture/taleb08/taleb08_index.html>his article on edge.org, he says 'My outrage is aimed at the scientist-charlatan putting society at risk using statistical methods. This is similar to iatrogenics, the study of the doctor putting the patient at risk.' After the recent financial crisis, wired.com ran an article titled 'Recipe for Disaster: The Formula That Killed Wall Street' in which the quant David Li and his <http://en.wikipedia.org/wiki/Gaussian_copula#Gaussian_copula>Gaussian Copula were crucified we <http://news.slashdot.org/article.pl?sid=09/03/03/036223&tid=98>discussed it at the time. Now, I've recently been admitted to a graduate program of good repute in Computational & Applied Mathematics. There is a wide range of subjects in which you can pursue your PhD, one of them being Financial Mathematics. I had a passing interest in it for quite some time. In the current scenario, how advisable it is to pursue a PhD in this topic? What would my options be five years down the line? Will the so-called 'quants' still be wanted by the banks and other financial institutions, or will they turn to more 'non-math' approaches? Would I be better off specializing in less volatile areas of Applied Mathematics? In short, what is the future of Financial Mathematics in light of the current financial crisis?" ... --- vice-chair http://ocjug.org/

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Re: [math-fun] Future of Financial Mathematics?
by Dan Asimov 25 Apr '09

25 Apr '09

Rumor I heard a month or two ago is that jobs are opening up in financial firms for "quants", since I guess financial firms want some new ideas. Don't really know for sure, though. As far as who to blame for the financial crisis goes, I mainly blame the regulators for not doing enough regulation. (Or whoever may have caused that to happen.) A company has enough trouble trying to figure out how to stay afloat without running afoul of regulators or laws, so I don't blame the companies. --Dan << Now, I've recently been admitted to a graduate program of good repute in Computational & Applied Mathematics. There is a wide range of subjects in which you can pursue your PhD, one of them being Financial Mathematics. I had a passing interest in it for quite some time. In the current scenario, how advisable it is to pursue a PhD in this topic? What would my options be five years down the line? Will the so-called 'quants' still be wanted by the banks and other financial institutions, or will they turn to more 'non-math' approaches? Would I be better off specializing in less volatile areas of Applied Mathematics? In short, what is the future of Financial Mathematics in light of the current financial crisis?" ... >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] completeness-of-the-reals for beginners
by Dan Asimov 25 Apr '09

25 Apr '09

I don't think there's always a "cut point" for Conway numbers S (surreals). For example, if the upper set is U = {x in S | x > all positive infinitesimals}, the the lower set (S - U) is L = (x in S | x <= some positive infinitesimal}. Then U has no minimum and L has no maximum. --Dan ------------------------------------------------------------------------ Rich writes: >> . . . what happens if you apply the Dedekind Cut construction to . . . Conway's Numbers. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] completeness-of-the-reals for beginners
by James Propp 24 Apr '09

24 Apr '09

Do any of you know of any authors of honors calculus texts who take as their completeness axiom the assertion that, if one partitions the reals into two non-empty sets A,B with a<b for all a in A and b in B, then there exists a cutpoint c such that either A=(-\infty,c] and B=(c,\infty) or A=(-\infty,c) and B=[c,\infty)? These partitions are close in spirit to Dedekind cuts, but here we are partitioning R, not Q, and we are axiomatizing the reals, not constructing them. Other popular completeness axioms for the reals are the least upper bound property and the bounded monotone sequence convergence property, but I think these are harder for undergraduates to grasp. It's fun to instigate a class discussion of "How can you partition the real line into two sets, one of which is completely to the left of the other?" at the start of the term; once the students have wrestled with it for a while, they're glad to accept the axiom. Then, a week or two later, when continuity is introduced, one can use the cut axiom to prove the Intermediate Value Theorem, and for a challenge problem on the homework one can ask the students to prove (with a few hints) that the IVT in turn implies the cut axiom. (If you could cut the reals into two non-empty sets A,B with A<B where A has no greatest element and B has no least element, then the function that's -1 on A and +1 on B is continuous.) Another approach some people take is to defer the proof of the IVT until the second semester, when sequences are traditionally introduced, so that the bounded monotone sequence convergence property won't seem so strange, and when the students have seen more proofs, but I like the idea of showing the students a non-trivial proof fairly early, and having them help turn an incomplete proof into a rigorous one. So, would one call this a small but genuine innovation in honors calculus pedagogy (perhaps worth publishing in a short note somewhere), or too minor a tweak to merit discussion, or a well-trodden pedagogical path that lots of teachers already know about? Jim

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