Let $\mathbf X := (X, \mathcal S, \mu)$ be a probability space without atoms. We say two measure preserving transformations $T$ and $F$ on $\mathbf X$ are $\delta$-close if $ \int_X Tg - Fg \ d\mu < \delta ||g||_{L^\infty}$ for all $g \in L^\infty(X)$. For any ergodic transformation $T$, and rea...

Let $u: \mathbb R^n \to \mathbb R$ be a $BV$ function with no jump part, i.e., writing $Du = D^a u + D^s u + D^j u$ for the decomposition of $Du$ into absolutely continuous, Cantor, and jump part respectively, we have $D^j u = 0$. Does the weak derivative of such a function $u$ vanish on level se...

Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$ be the isotropy group of $x$. If $f$ is a continuous, complex valued, compactly supported function on $\mathcal G$, denote by $\rho (f)$ the restriction of $f$ to $\mathcal G(x)...

Recall that a structure $\mathcal{M} = \langle M, I^\sigma_M \rangle$ in a signature $\sigma$ is categorically axiomatized by a second-order theory $T$ when, for any $\sigma$-structure $\mathcal{N} = \langle N, I^\sigma_N \rangle$, $\langle N, \mathcal{P}(N), I^\sigma_N \rangle \vDash T$ just in ...

Let $q_1,q_2,\dots,q_m$ be a collection of prime powers such that $q_i = p_i^{k_i}$. I have the following questions. When the products $\prod_{i=1}^m(q_i-1)^{r_i}$ and $\prod_{j=1}^m(q_j-1)^{s_j}$ (where $r_i, s_i \in \{0,1\}$ for all $i$ and $r_j \ne s_j$ for some $j$) are equal? Is there any ...

Question: Is there a real valued function $f:\mathbb{R}\to\mathbb{R}$ such that its set of discontinuities is $\mathbb{Q}$ and its set of zeros $\{x\in \mathbb{R}\mid f(x)=0\}$ is also $\mathbb{Q}$? It's well known that the Thomae function has as discontinuities the rationals. However, its zero s...