First jsMath Test

Lets try to do some mathematics with jsMath.

The weighted centroid problem

Consider a parameter $\kappa$ with $0\lt\kappa\lt 1$. Then the weighted centroid problem is the following equation
\[
(1-\kappa)\int_a^x e^{\theta(y)}\,\mathrm{d}y=\kappa\int_x^b e^{\theta(y)}\,\mathrm{d}y
\]
For $\theta(y)$ we impose just some decay condition. Say
\[ \lim_{y\to\infty}\theta(y)=-\infty. \]

There are several different ways take a look on this equations. First one could fix $a$ and $b$ and then seek for a solution $x=x(\kappa,a,b)$. Another point of view could be to fix $x$ as a parameter and seek for a solution $b=b(a)$. In this setting, there could an explosion occur. I.e. think of $\theta(y)=-y$, then of course the total mass on the positive real line of $e^{-y}$ is finite. Then for a certain parameter regimes $E(x)$ with $(a,\kappa)\in E(x)$ doesn’t exist $b(a)$.

In the following part we want to investigate the solution for $b(a)$ and the explosion regime $E(x)$ for a certain class of functions $\theta(y)$. The first class will be the linear, quadratic, linear-logarithmic functions. With the help of this class we want to obtain estimates for a class of more general functions leading to harder transcendental equations.