In the context of Green's functions for the Free Klein-Gordon field, the following integral occurs: $$\int_m^{\infty}{\rho e^{-\rho r}\over\sqrt{\rho^2-m^2}}\; d\rho.$$ Here $m$, is a positive constant in the set of real numbers. One may write: $$\begin{align}\int{\rho e^{-\rho r}\over\sqrt{\rho^...

Suppose $W:[0,1]\times \Omega \to \mathbb{R}$ is a standard Brownian motion on the unit interval. With $L^2[0,1]$ denoting the space of real-valued square-integrable functions with standard norm $\|\cdot\|_2$, consider the probability space $(L^2[0,1], \mathcal{B}(L^2[0,1]), P)$ defined by $P(B)=...

Let ($X_n$)$_{n \in \mathbb N}$ be a sequence of random variables defined on a probability space ($\Omega,F,P$). Suppose that $X_n \geq 0$ and $E(X_n ^4) < \frac{1}{n^\frac{3}{2}}$ , for all $n \in \mathbb N$. Show that $X_n$ converges to $0$ almost surely as $n \rightarrow \infty $ I know I need...

Assume a king sits on a1 on an 8x8 chessboard. The king is restricted so he can only move up, right, or diagonally towards top-right. How many paths are there to h8. I know this is a duplicate question that has been asked once before, but no one gave a final numerical answer. I am currently getti...

I'm a relativist. Hence, I have a working knowledge of Riemannian geometry, but I never really studied differential geometry of curves and surfaces. I know the traditional path is to start with curves and surfaces and then go to smooth manifolds, Riemannian geometry and etc, but I would like to k...

I posted this question here Do 2 Normal Distributions always produce another Normal Distribution? where it was suggested to me that Characteristic Functions can be used to show that the distribution for the sums of independent Normal Distributions are still Normal. I had originally tried to prove...

It's a very well known result by Gauss that $(\mathbb{Z}/2^n \mathbb{Z})^\times = \langle -1 \rangle \times \langle 3 \rangle \cong C_2 \times C_{2^{n-2}}$. Consider a faithful action $\mathrm{mul}: C_{2^{n-2}} \to \mathrm{Sym}(\mathbb{F}_2^n)$ that is obtained by representing each element of $\m...