06-1350/Some Equations by Kurlin

2007-04-10T23:25:40Z

200.238.102.170:

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'''Claim''' {{ref|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

<center>
<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
</center>

Satisfies

<center>
<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>
</center>

The claim is, of course, very easily verifiable and if this class is successful, at some point in the future I will tell you why I care.

Typically long equations have no closed form solutions. Yet note that Kurlin first derived the equation and only then found its solution. Was he merely lucky that a solution existed, or was there some deeper principle guarding him from failure? I don't believe in luck and I prefer to believe in principles. Yet, at what level did the principle arise? Is it that all solvable equations of this kind have closed-form solutions, or was there an a priori reason why the math that Kurlin was doing lead him to an equation with a closed form solution?

Here are a few concrete questions to expose my ignorance of simple algebra. Note that <math>f</math> is is a rational function in its variables and their exponentials. Let's call such functions "tame". The equation <math>f</math> satisfies involves rational expressions in the variables and their exponentials and linear substitutions applied to <math>f</math>. Let us call such equations "tame". (The equation is also linear in <math>f</math>. You may or may not wish to include this in the definition of "tame"). So we see that this particular tame equation has a tame solution.

* Is that always the case?
** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
** If there are further ones, why haven't we heard about them? Or else, where do we read about them?
** If there aren't further ones, this is a lovely "closure" property enjoyed by exponentials. How come I don't know it yet?

* Assuming not all tame equations have tame solutions, what was special about our equation, that lead to the existence of a tame solution? Can you write a very general condition on tame equations that will ensure the existence of a tame solution? In other words, was Kurlin simply lucky that his equation had a tame solution (and clever that he found it!), or are there some general rules that guarded him against the wilderness?

{{note|Kurlin}} ''Compressed Drinfel'd Associators,'' V. Kurlin, {{arXiv|math.GT/0408398}}.

Talk:06-240/Homework Assignment 4

2007-04-10T16:48:25Z

200.238.102.170:

== Divisibility by Prime Number==

Pls correct me if I were wrong.
The operation of cut away the unit digit is a distraction. If we consider the unit digit, the operation basically is a deduction of a number, and that number is divisible by 7.
The whole operation is shown as follow:

{|
|<div align="right">8641<s>5</s>
|-
|<div align="right"> 10<s>5</s>
|

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06-1350/Some Equations by Kurlin

Talk:06-240/Homework Assignment 4