I am having trouble proving a result from a paper, which of course includes no proof. I wonder if the author had a simple - but flawed - argument in mind, or if I'm just being a dunce. It is Theorem 3 here: https://www.jstor.org/stable/2372339 It involves the following property: If $X$ is a m...

This is a very general question, but hopefully some people find it interesting. I'm working in the setting of compact metric spaces, so most of the basic topological properties will be satisfied. When the space $X$ is also connected, then I know that being locally connected is sufficient to be ...

I was trying to come up with a counterexample for something and kept failing and failing, so I was hoping someone could help me out. The setting is compact, connected, metric spaces (continua) so everything is nice. Here are the definitions I'm using: $X$ is locally connected at $x \in X$ if i...

I aim to prove $P ∩\bigcup_α Q_α =\bigcup_α (P ∩Q_α)$. For $P,Q$ to be open subschemes of an affine scheme $X$. I have been told that every open scheme is a union of complements of hypersurfaces, an I have attempt write all the $Q_\alpha$'s to be a union of complement $S$ of some hypersurface, ...

$X_2(s)=-\sum_{i=0}^{h-1}N^is^i(N+U(s))$, where $N$ is nilpotent matrix of order $h$, Could anyone help me to understand inverse laplace transformation of $X_2(s)$? $\delta$ is dirac delta function. Thanks for helping. I am attaching the actual expression. But I guess I will understand the origi...

Let $W$ and $B$ be two Brownian motions with $\text{d}\langle W, B\rangle_t = \rho \text{d}t$ under some probability measure $\mathbb{P}$, where $\rho$ is a constant. Let $\mathbb{Q}$ be an equivalent measure to $\mathbb{P}$. Does $\text{d}\langle W, B\rangle_t = \rho \text{d}t$ under $\mathbb...