In this article we study two problems about the existence of a distance $d$ on a given fractal having certain properties. In the first problem, we require that the maps ${\psi }_i$ defining the fractal be Lipschitz of prescribed constants less than $1$ with respect to the distance $d$, and in the second one, we require that arbitrary compositions of the maps ${\psi }_i$ be uniformly bi-Lipschitz of related constants. Both problems have been investigated previously by other authors. In this article, on a large class of finitely ramified fractals, we prove that these two problems are equivalent and give a necessary and sufficient condition for the existence of such a distance. Such a condition is expressed in terms of asymptotic behavior of the product of certain matrices associated to the fractal.

Let $A={\mathbb{F}}_q\left[T\right]$, where ${\mathbb{F}}_q$ is a finite field, let $Q={\mathbb{F}}_q\left(T\right)$, and let $F$ be a finite extension of $Q$. Consider $\varphi$ a Drinfeld $A$-module over $F$ of rank $r$. We write $r=hed$, where $E$ is the center of $D:={End}_{\overline{F}}\left(\varphi \right)\otimes Q$, $e=[E:Q]$, and $d=[D:E{]}^{\frac{1}{2}}$. If $\wp$ is a prime of $F$, we denote by ${\mathbb{F}}_{\wp }$ the residue field at $\wp$. If $\varphi$ has good reduction at $\wp$, let $\overline{\varphi }$ denote the reduction of $\varphi$ at $\wp$. In this article, in particular, when $r\ne d$, we obtain an asymptotic formula for the number of primes $\wp$ of $F$ of degree $x$ for which $\overline{\varphi }\left({\mathbb{F}}_{\wp }\right)$ has at most $(r-1)$ cyclic components. This result answers an old question of Serre on the cyclicity of general Drinfeld $A$-modules. We also prove an analogue of the Titchmarsh divisor problem for Drinfeld modules.

We use a tensor $C^{\ast }$-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a ${}^{\ast }$-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal $C^{\ast }$-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the $C^{\ast }$-algebra in question. The quantum groups involved are $A_u\left(Q\right)$ and $B_u\left(Q\right)$.

Let $G$ denote $Sp(n,\mathbb{R})$, $\mathrm{U}(p,q)$, or ${\mathrm{O}}^{\ast }\left(2n\right)$. The main aim of this article is to show that the canonical quantization of the moment map on a symplectic $G$-vector space $(W,\omega )$ naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of $\mathfrak{g}:=Lie\left(G\right)\otimes \mathbb{C}$. More precisely, after taking a complex Lagrangian subspace $V$ of the complexification of $W$, we assign an element of the Weyl algebra for $V$ to $\langle \mu ,X\rangle$ for each $X\in \mathfrak{g}$, which we denote by $\langle \hat{\mu },X\rangle$. Then we show that the map $X\mapsto \mathrm{i}\langle \hat{\mu },X\rangle$ gives a representation of $\mathfrak{g}$. With a suitable choice of $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak{g}$.

Let $H$ be a numerical semigroup. We give effective bounds for the multiplicity $e\left(H\right)$ when the associated graded ring ${gr}_{\mathfrak{m}}K\left[H\right]$ is defined by quadrics. We classify Koszul complete intersection semigroups in terms of gluings. Furthermore, for several classes of numerical semigroups considered in the literature (arithmetic, compound, special almost-complete intersections, $3$-semigroups, symmetric or pseudosymmetric $4$-semigroups) we classify those which are Koszul.

Partially motivated by the study of topological Hamiltonian dynamics, we prove the following $C^0$-continuity of the Lagrangian capacity function ${\gamma }^{\mathrm{lag}}$:

$${\gamma }^{\mathrm{lag}}\left({\varphi }_H^1\right(o_N\left)\right):={\rho }^{\mathrm{lag}}(H;1)-{\rho }^{\mathrm{lag}}(H;[pt{]}^{\#})\to 0,$$ as ${\varphi }_H^1\to id$, provided the $H$’s satisfy $supp{X}_H\subset D^R\left(T^{\ast }N\right)\setminus o_B$ for some $R>0$ and a closed subset $B\subset N$ with nonempty interior. We also provide an estimate of the capacity in terms of the $C^0$-distance of $d_{C^0}({\varphi }_H^1,id)$ and the subset $B\subset N$ relative to $T^{\ast }N$.

In this article, we show the stability of the doubly Feller property of the resolvent for Markov processes, generalizing a work by Kai-Lai Chung on the stability of the doubly Feller property of a semigroup with multiplicative functionals. The stability of the doubly Feller property of the resolvent under time change is also presented.

Let $A$ be a Cohen–Macaulay local ring with $dimA=d\ge 3$, possessing the canonical module ${\mathrm{K}}_A$. Let $a_1,a_2,\dots ,a_r$ $(3\le r\le d)$ be a subsystem of parameters of $A$, and set $Q=(a_1,a_2,\dots ,a_r)$. We show that if the Rees algebra $\mathcal{R}\left(Q\right)$ of $Q$ is an almost Gorenstein graded ring, then $A$ is a regular local ring and $a_1,a_2,\dots ,a_r$ is a part of a regular system of parameters of $A$.

Actions of locally compact groups and quantum groups on W${}^{\ast }$-ternary rings of operators are discussed, and related crossed products are introduced. The results generalize those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed-point spaces for convolution operators generated by contractive, not necessarily positive measures, both in the classical and in the quantum context.