June 2007 - mathfun@mailman.xmission.com

[math-fun] Passive walking & hopping
by Henry Baker 27 Jun '07

27 Jun '07

Cool "passive" walking & hopping toys: http://www.hevanet.com/psmfg/web%20site/passive.html

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[math-fun] Rolling & hopping ovals?
by Henry Baker 27 Jun '07

27 Jun '07

In Fig. 4 of the following paper, http://ruina.tam.cornell.edu/research/topics/locomotion_and_robotics/papers… , an "oval" is show to roll and hop along a horizontal line without losing any energy. The center-of-gravity (cog) of the oval is a periodic sequence of upward-facing circular arcs and downward facing parabolic arcs. The speed of travel along the x-coordinate is synchronized with the rotation of the oval, so that the oval touches down in a way entirely symmetrical with the way it lifts off. Furthermore, only one side of the oval ever touches the ground, as the other side is up in the air when the oval revolves before touching down again (I assume that this situation could be generalized to do more than 1/2 rotation in the air). The question: what is the mathematical shape of the oval to achieve this overall lossless rolling & hopping behavior? (I don't know the answer, and the paper didn't give any references that I could download from the Internet.) Additional questions might be: does the oval have to be symmetrical about its short axis? (Maybe not.) Does the cog have to traverse a circular arc during the actual rolling motion? (I think not, so long as the velocity & angular velocity on point of contact and velocity & angular velocity are the "same" (modulo mirror images).) Has anyone else run across this type of oval before? Apparently, a football rolling end-over-end can approximate some of this behavior. I'm surprised that no one has built a children's toy based on this principle -- a hopping hoop would be much more fun than a simple rolling hoop.

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RE : [math-fun] Fungi and scientific points of view
by Eric Angelini 24 Jun '07

24 Jun '07

Neil: >>It's scary in the way that The Hot Zone was: rusts and molds could easily destroy civilisation. >James Brown: > Not only does The Triumph focus on eye-catching but unrepresentative diseases, but much of it is written in a flippant manner which does not help to explain some unfamiliar and occasionally obscure science. (...) http://www.the-ba.net/the-ba/CurrentIssues/ReportsandPublications/ScienceAn… <http://www.the-ba.net/the-ba/CurrentIssues/ReportsandPublications/ScienceAn…> I'll order this bookasap to make my opinion! Best, (thanks for having pointed that book, Neil) E.

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[math-fun] Fungi
by N. J. A. Sloane 23 Jun '07

23 Jun '07

Funsters, I just read a truly great - funny, scholarly and deeply scary - book called "The Triumph of the Fungi - A Rotten History" by Nicholas P. Money. It's the kind of book you come across every ten years that you want to buy copies of for all your friends. It got a great review in Nature. It's scary in the way that The Hot Zone was: rusts and molds could easily destroy civilisation. Neil

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[math-fun] moderator outage for Math-Fun
by Richard Schroeppel 22 Jun '07

22 Jun '07

I'm off to see Dan Asimov's wedding. I'll be out of town for three weeks. Michael Kleber (michael.kleber at gmail.com) is the temporary Math-Fun moderator. Rich

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[math-fun] Still an open problem?
by Christian Boyer 20 Jun '07

20 Jun '07

Euler had published the complete parametric solution in rationals of x^3 + y^3 = z^3 + w^3 (1) but as said by Hardy and Wright in their book: "The problem of finding all integral solutions is more difficult". In the Rocky Mountain Journal of Mathematics, 1998, Ajai Choudhry published a "complete" parametric solution in integers of (1). But it seems that some number theorists don't agree with this solution or any other claimed solution. Do you know if the complete parametric solution in integers of (1) is still an open problem? Christian.

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Re: [math-fun] Continuous derivations on the reals
by Eugene Salamin 19 Jun '07

19 Jun '07

There has been a data type error here. A derivation is a linear operator on the space of functions, not a function. If 1 denotes the constant function, then d(1) = d(1.1) = d(1).1 + 1.d(1) = 2.d(1), so d(1) = 0, and thus constant functions are in the null space of all derivations. On the other hand, I seen no reason why d(x) must be 1. If d(x) = q (i.e. the function q(x)), then for any function f, it looks like d(f) = q.D(f), where D is the standard differentiation operator. Gene ----- Original Message ---- From: "mcintosh(a)servidor.unam.mx" <mcintosh(a)servidor.unam.mx> To: math-fun(a)mailman.xmission.com Cc: mcintosh(a)servidor.unam.mx Sent: Tuesday, June 19, 2007 10:57:40 AM Subject: Re: [math-fun] Continuous derivations on the reals Quoting David Wilson <davidwwilson(a)comcast.net>: > Let [1] d(uv) = d(u) v + u d(v) > > Then [2] d(u^2) = 2u d(u) not when the quantities are noncommutative, as is typically the case when the object of discussion is a Lie Algebra. As for this implying a vanishing Maclaurin series, what does this say about ordinary functions, say polynomials, which do indeed have (non-zero) derivatives? - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ____________________________________________________________________________________ No need to miss a message. Get email on-the-go with Yahoo! Mail for Mobile. Get started. http://mobile.yahoo.com/mail

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[math-fun] Continuous derivations on the reals
by Dan Asimov 19 Jun '07

19 Jun '07

Marc wrote: << Is there term for when f(a b) = a f(b) + b f(a)? And the answer, supplied by Neil, is "derivation". Okay, sure, if say C^oo(R) is the set of infinitely differentiable functions from R to R, then D: C^oo(R) -> C^oo(R) defined by D(f)(x) = f'(x) is perhaps the epitome of a derivation. -------------------------------------------------------------------- But what are the continuous derivations d: R -> R ? I.e., for all real u,v, we must have d(uv) = d(u)*v + u*d(v) The 0 function works. What is the general solution? --Dan

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Re: [math-fun] Continuous derivations on the reals
by Dan Asimov 19 Jun '07

19 Jun '07

Ki Song wrote: << I thought 'derivation' was restricted to linear operators. Perhaps these operators should be called something like quasi-derivation, or pseudo-derivation? >> Derivations are defined for rings, fields, and algebras. Cf. http://en.wikipedia.org/wiki/Derivation_%28abstract_algebra%29 . --Dan

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Re: [math-fun] term for f(a b) = a f(b) + b f(a)?
by N. J. A. Sloane 17 Jun '07

17 Jun '07

Marc asked: Is there term for when f(a b) = a f(b) + b f(a)? If not, what would you suggest? This is called a "derivation" and is better written f(a b) = a f(b) + f(a) b See e.g. Bourbaki, Lie Groups, Vol 1 , Chap 1, section 1, para 1, page 2 ! Neil

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