Wikipedia:Reference desk/Archives/Mathematics/2013 September 5

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September 5

Elliptic functions

I'm interested in the solutions to the differential equation

$y''={\frac {1}{2}}y(y-1)(y+1)$

I have identified them as

$y={\sqrt {\frac {2k}{1+k}}}\,\,\mathrm {sn} \left({\frac {x-x_{0}}{\sqrt {2(1+k)}}},k\right)$

where $x_{0}$ is a constant of integration, $k$ is the elliptic modulus and $\mathrm {sn} (\cdot ,k)$ is the Jacobi elliptic function. I understand that $k$ is usually defined on the interval [0,1], however the function appears to be defined outside this range.

In have even been able to plot this for arbitrarily large $k$, though the behaviour of $\mathrm {sn} (\cdot ,k)$ seems to change, with the oscillation amplitude (which is always 1 for $|k|<1$) becoming smaller for larger k.

I however cannot find information on the properties of the function outside of the range $|k|<1$, I would be particularly interested in knowing the form of the pre factor which causes the decay.

Thanks. — Preceding unsigned comment added by 109.144.130.145 (talk) 00:10, 5 September 2013 (UTC)

The Jacobian elliptic functions are still well-defined for any real value of the modulus $m=k^{2}$, but these can all be expressed in terms of elliptic functions with modulus between 0 and 1. Sławomir Biały (talk) 13:06, 5 September 2013 (UTC)

Is it possible to provide a link for a relationship between those defined inside and outside the region $|k|<1$? — Preceding unsigned comment added by 144.82.173.230 (talk) 15:51, 5 September 2013 (UTC)

Markov Processes, Feller SemiGroups and Evolution Equations

I work in a laboratory studying cognitive science and methodology, and I've been asked to "familiarize" myself with the contents of the book, "Markov Processes, Feller SemiGroups and Evolution Equations," by Jan A van Casteren, for a series of upcoming experiments we will be working on sometime next year. I don't have a particularly strong mathematical background; I went through a traditional education up to and including what is usually called Calculus I. In doing an overview of the book I've been asked to read, it seems quite daunting and very much beyond my skill level to understand without having some background first. I have a good amount of time in which to get up to speed, but I'm not sure where to start. What sort of mathematics courses, textbooks, or other materials would be most helpful in working up to a point where I can start to understand the topics included in the book? Lord Arador (talk) 20:16, 5 September 2013 (UTC)

Judging from the table of contents, this book takes a fairly abstract, pure-math, approach to Markov processes and stochastic differential equations. If you are doing experimental cognitive science, it is a strange choice--an applied math perspective would be more useful for cognitive modeling. At any rate, I recommend learning the basics of Markov processes and stochastic differential equations first. Once you have developed some experience with those, concepts like Feller processes will possibly make more sense. Start with sources like Wikipedia and Encyclopedia of Mathematics for brief overviews of the topics you are interested in. Then branch out with tutorials on the very basics, such as that for Markov chains and processes and applied stochastic differential equations. --Mark viking (talk) 21:31, 5 September 2013 (UTC)

Thanks so much for the advice. I think this will help a lot. Lord Arador (talk) 01:42, 6 September 2013 (UTC)