I had tried to help someone on math.StackExchange to prove the identity: $$ (1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4$$ I guess you could argue the left hand side is …

I came across the following differential equation when considering some direct scattering problems: $$ N'_x(x,z)=G(x,z)N(x,z) $$ where $N(x,z)$ is a $2\times2$ complex matrix wit …

Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$ $$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and th …

Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can …

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$? More generally,we can obtain a power series from decim …

Let $s \in R^{n}$ (meaning $s$ is $n \times 1$ vector), where $n$ is the dimension of the vector. The ideal sliding term, $\nu$ is taken to be: \begin{equation} \nu = \frac …

I am just asking it so that somebody can close it as "not a real question".

I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve. I want to predict the nature of use …

I am wondering whether there is a Lie algebraic version of principal bundle for Lie group over a given manifold $M$. The first thing I try to think of is group cocycle picture of p …

QUESTION I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to …

I am considering the twiced tangent bundle $T(TM)$ of manifolds $M$. Locally, if $M=R^d$ then $T(TM)=R^{4d}=\oplus^3 TM$. My attempt is to see whether $T(TM)\cong \oplus^3 TM$ natu …

I'm searching for a proof that a finitely generated projective module over a Von Neumann Regular ring is free. I know that this result is true, because a friend of mine have proved …

The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but …

Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …

Hi. Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I …