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I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these. If I understand correctly, mathematicians tend to be quite happy working with …

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to f …

I have heard that differential equations on $\mathbb{S}^1$ of the form \begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) …

Consider the differential equation on $\mathbb{R}^2$ whose polar-coordinate representation is given by $$ \begin{array}{r c l} \dot{r} & = & r(\alpha - r^2) \\ \dot{\theta} & = & -sin(\theta).\end{arr …

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support. Definition. The …

The claim is true. (As proved here, it is quite unique to sinusoidal vector fields. The same reference also mentions, in its introduction, existing applications of equation (1) with $g$ taking the par …

Let $V$ be a normed vector space, let $l \in V$, and let $(a_n)$ be a sequence in $V$. We say that $a_n$ is Cesàro-convergent to $l$ if $\frac{1}{n}\sum_{i=1}^n a_i \to l$ as $n\to\infty$. Now I will …

Given any $g \in L^\infty(\mathbb{R})$, we define the associated multiplier operator $T_g \colon L^2(\mathbb{R}) \to L^2(\mathbb{R})$ by $$ \mathcal{F}(T_g f) \ = \ g.\mathcal{F}f $$ where $\mathcal{F …

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order: The answer to Q2 is yes; the structure of the proo …

Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the Haus …

Apologies if this question is too basic for MO. I think it should be the case that for any decreasing $f \colon [A,\infty) \to [0,\infty)$ and $k \geq 0$, if $\int_A^\infty f(x) e^{kx} \, dx < \infty …

Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields $$ \mathcal{S} = \{b,\sigma_1,-\sigma_1, …

I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before. So I'm wondering if there are any papers/text …

Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$. Fix $L \geq 1$. I will say that $\mu_n$ converges in op …

Dear all, I would like to know whether the following concept is one that is commonly studied, or has a name, or if there are any textbooks that make reference to it: We say that a function $f:[a,b] …