In this paper we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz–Sobolev space. Here, we use the variational methods developed by Szulkin [34] combined with some properties of the weak* topology.

For a complex Lie group $G$ with a real form $G_0 \subset G$, we prove that any Hamiltonian automorphism $\phi$ of a coadjoint orbit $\mathcal{O}_0$ of $G_0$ whose connected components are simply connected, may be approximated by holomorphic $\mathcal{O}_0$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal{O}$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.

Let $d \in {\lbrace 1, 2, 3, \dotsc \rbrace}$ and $s \in (0, 1)$ be such that $d \gt 2s$. We consider a generalized non-local Schrodinger operator of the form\[L=L_K + \nu \; \textrm{,}\]where $L_K$ is a non-local operator with kernel $K$ that includes the fractional Laplacian $(-\Delta)^s$ for $s \in (0, 1)$ as a special case. The potential $\nu$ is a doubling measure subjected to a certain constraint. We show that the fundamental solution of $L$ exists, is positive and possesses extra decaying properties.

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero ideals of polynomial rings with at least two indeterminates over noetherian domains. Among others, we show that they are not transfer Krull but they share several arithmetic phenomena with Krull monoids having infinite class group and prime divisors in all classes.

We adapt ideas from Ekedahl [Eke84] to prove a Serre-type duality for Witt-divisorial sheaves of $\mathbb{Q}$‑Cartier divisors on a smooth projective variety over a perfect field of finite characteristic. We also explain its relationship to Tanaka’s vanishing theorems [Tan20].

We extend the $2$-representation theory of finitary $2$-categories to certain $2$-categories with infinitely many objects, called locally finitary $2$-categories, and extend the classical classification results of simple transitive $2$-representations of weakly fiat $2$-categories to this environment. We also consider locally finitary $2$-categories and $2$-representations with a grading, and prove that the associated coalgebra 1-morphisms have a homogeneous structure. We use these results to classify (graded) simple transitive $2$-representations of certain classes of cyclotomic $2$-Kac–Moody algebras.

In [15, Remark 4.1], Ohsawa asked whether it is possible to prove Theorem 4.1 and Theorem 0.1 in [15] using the Berndtsson–Lempert method. We shall answer Ohsawa’s question affirmatively in this paper. Our approach also suggests to introduce the Legendre–Fenchel theory and weak psh‑geodesics into the Berndtsson–Lempert method.

We show that for a uniformly irreducible random walk on a graph, with bounded range, there is a Floyd function for which the random walk converges to its corresponding Floyd boundary. Moreover if we add the assumptions, $p^{(n)} (v,w) \leq C \rho^n$, where $\rho \lt 1$ is the spectral radius, then for any Floyd function $f$ that satisfies $\sum^{\infty}_{n=1} nf(n) \lt \infty$, the Dirichlet problem with respect to the Floyd boundary is solvable.

A conjecture of T. Richards is proven which yields a quantitative version of the classical Gauss–Lucas theorem: if $K$ is a convex set, then for every $\varepsilon \gt 0$ there is an $\alpha_{\varepsilon} \lt 1$ such that if a polynomial $P_n$ of degree at most $n$ has $k \geq \alpha_{\varepsilon} n$ zeros in $K$, then $P^{\prime}_n$ has at least $k-1$ zeros in the $\varepsilon$-neighborhood of $K$. Estimates are given for the dependence of $\alpha_{\varepsilon}$ on $\varepsilon$.