In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

We establish an integral test on the lower escape rate of symmetric jump-diffusion processes generated by regular Dirichlet forms. Using this test, we can find the speed of particles escaping to infinity. We apply this test to symmetric jump processes of variable order. We also derive the upper and lower escape rates of time-changed processes by using those of underlying processes.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ , and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is −1. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$ such that ${{p}^{2}}\,\parallel \,N$ , and $p$ is inert in $O$ . Although there are no Heegner points on ${{X}_{0}}(N)$ attached to $O$ , in this article we construct such points on Cartan non-split curves. In order to do that, we give a method to compute Fourier expansions for forms on Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R\,=\,k\left[ x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1} \right]$ . Assume that $G$ contains a maximal $R$ -torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_{m}^{2}$ . We show that the natural map of non-stable ${{K}_{1}}$ -functors, also called Whitehead groups, $K_{1}^{G}\left( R \right)\,\to \,K_{1}^{G}\left( k\left( \left( {{x}_{1}} \right) \right)\cdots \left( \left( {{x}_{n}} \right) \right) \right)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

Continua $X$ and $Y$ are monotone equivalent if there exist monotone onto maps $f\,:\,X\,\to \,Y$ and $g:\,Y\to \,X.\,\text{A}$ . A continuum $X$ is isolated with respect to monotone maps if every continuumthat is monotone equivalent to $X$ must also be homeomorphic to $X$ . In this paper we show that a dendrite $X$ is isolated with respect to monotone maps if and only if the set of ramification points of $X$ is finite. In this way we fully characterize the classes of dendrites that are monotone isolated.

Let $M\,=\,\text{G}{{\text{L}}_{{{r}_{1}}}}\,\times \,\cdots \,\times \,\text{G}{{\text{L}}_{{{r}_{k}}}}\,\subseteq \,\text{G}{{\text{L}}_{r}}$ be a Levi subgroup of $\text{G}{{\text{L}}_{r}}$, where $r\,=\,{{r}_{1}}+\cdots +{{r}_{k}}$, and $\widetilde{M}$ its metaplectic preimage in the $n$-fold metaplectic cover $\widetilde{\text{G}{{\text{L}}_{r}}}$ of $\text{G}{{\text{L}}_{r}}$. For automorphic representations ${{\pi }_{1}},\ldots ,{{\pi }_{k}}$ of ${{\widetilde{\text{GL}}}_{{{r}_{1}}}}\left( \mathbb{A} \right),\ldots ,{{\widetilde{\text{GL}}}_{{{r}_{k}}}}\left( \mathbb{A} \right)$, we construct (under a certain technical assumption that is always satisfied when $n\,=\,2$) an automorphic representation $\pi $ of $\widetilde{M}\left( \mathbb{A} \right)$ that can be considered as the “tensor product” of the representations ${{\pi }_{1}},\ldots ,{{\pi }_{k}}$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v,\,{{\pi }_{v}}$ is equivalent to the local metaplectic tensor product of ${{\text{ }\!\!\pi\!\!\text{ }}_{1,\,v}},\ldots ,{{\text{ }\!\!\pi\!\!\text{ }}_{k,\,v}}$ defined by Mezo. Then we show that if all of the ${{\text{ }\!\!\pi\!\!\text{ }}_{i}}$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element and show the compatibility with parabolic inductions.

We give a normal form of the cuspidal edge that uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges that determine them up to order three. We also clarify relations between these invariants.

The notion of families of quantum invertible maps (${{C}^{*}}$-algebra homomorphisms satisfying Podleś condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular, Wang's quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further, the construction of the Hopf image of Banica and Bichon is phrased in purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or, more generally, a family of quantum invertible maps.

In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the $b$–principle and in dimension 2 approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension 2 to a closed simply connected manifold is bordant to a submersion. We show that the Madsen–Weiss theorem (the standard Mumford Conjecture) fits a general setting of the $b$–principle, namely, a version of the $b$–principle for oriented colored broken submersions together with the Harer stability theorem and Miller–Morita theorem implies the Madsen–Weiss theorem.