We establish a Dynkin formula and a Courrège-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\infty}(\mathds{R}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: \begin{equation*} Af(x) = \sup_{\alpha \in I} (-q_{\alpha}(x,D) f)(x). \end{equation*} As an immediate consequence, we obtain a representation formula for infinitesimal generators of sublinear Markov semigroups with a sufficiently rich domain. We give applications in the theory of non-linear Hamilton-Jacobi-Bellman equations and Lévy processes for sublinear expectations.

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation \begin{gather} \label{A} i\partial_{t}u+\frac{1}{2}\partial_{x}^{2}u-\frac{1}{4}\partial_{x}^{4} u=u^{3},\text{ }t>0,\text{ }x\in\mathbb{R}, u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in\mathbb{R.} \notag \end{gather} The aim of the present paper is prove the global existence of solutions to (0.1) if the initial data $u_{0}\in\mathbf{H}^{1}\cap\mathbf{H}^{0,1}$. Also we find the large time asymptotics of solutions.

In this paper we study the behavior of the total energy and the $L^2$-norm of solutions of two coupled hyperbolic equations by velocities in exterior domains. Only one of the two equations is directly damped by a localized damping term. We show that, when the damping set contains the coupling one and the coupling term is effective at infinity and on captive region, then the total energy decays uniformly and the $L^2$-norm of smooth solutions is bounded. In the case of two Klein-Gordon equations with equal speeds we deduce an exponential decay of the energy.

The Markov chain approximation of a one-dimensional symmetric diffusion is investigated in this paper. Given an irreducible reflecting diffusion on a closed interval with scale function $s$ and speed measure $m$, the approximating Markov chains are constructed explicitly through the trace of the Dirichlet form corresponding to the diffusion. One feature of our approach is that it does not require uniform ellipticity on diffusion coefficient of the limit object or uniform regularity on conductances of the approximative Markov chains, as imposed usually in the previous related works.

We classify isomorphism and similarity classes of pointed reflection spaces of residue $łeq 2$. This leads to the classification of reduced extended affine root systems whose involved pointed reflection spaces have residue $łeq 2$.

Aguiar and Ardila defined the Hopf monoid ${\rm GP}$ of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to submonoids. We define the Hopf monoid of directed graphs and show that it also embeds in ${\rm GP}$. The resulting basic invariant coincides with the strict chromatic polynomial of Awan and Bernardi.

Let a torus $T$ act smoothly on a compact smooth manifold $M$. If the rational equivariant cohomology $H^*_T(M)$ is a free $H^*_T(pt)$-module, then according to the Chang-Skjelbred Lemma, it can be determined by the $1$-skeleton consisting of the $T$-fixed points and $1$-dimensional $T$-orbits of $M$. When $M$ is an even-dimensional, orientable manifold with 2-dimensional 1-skeleton, Goresky, Kottwitz and MacPherson gave a graphic description of the equivariant cohomology. In this paper, first we revisit the even-dimensional GKM theory and introduce a notion of GKM covering, then we consider the case when $M$ is an odd-dimensional, possibly non-orientable manifold with $3$-dimensional $1$-skeleton, and give a graphic description of its equivariant cohomology.

The neighborhood complex $N(G)$ is a simplicial complex assigned to a graph $G$ whose connectivity gives a lower bound for the chromatic number of $G$. We show that if the Kronecker double coverings of graphs are isomorphic, then their neighborhood complexes are isomorphic. As an application, for integers $m$ and $n$ greater than 2, we construct connected graphs $G$ and $H$ such that $N(G) \cong N(H)$ but $\chi(G) = m$ and $\chi(H) = n$. We also construct a graph $KG_{n,k}'$ such that $KG_{n,k}'$ and the Kneser graph $KG_{n,k}$ are not isomorphic but their Kronecker double coverings are isomorphic.

We give a explicit description of gluing stability conditions on ruled surfaces by introducing gluing perversity. Moreover, we describe a destabilizing wall of skyscraper sheaves on ruled surfaces by deformation of stability conditions glued from $\widetilde{GL^{+}}(2,\mathbb{R})$-translates of the standard stability condition on the base curve.

We consider a viscoelastic wave equation with weak, strong damping and power nonlinearity. We have already obtained a global solution and its decay rate in [8]. In this paper, we apply the concavity method in order to show that the solution blows up in finite time under non-classic constraint on $\mu$.

We consider one-dimensional stochastic differential equations driven by Cauchy processes with drift. This driving process is also known as a strictly $1$-stable process. In this paper, we study the pathwise uniqueness of the solution to the stochastic differential equations under a non-Lipschitz condition on the diffusion coefficient.

A symmetric union in the 3-space $\mathbb{R}^3$ is a knot, obtained from a knot in $\mathbb{R}^3$ and its mirror image, which are symmetric with respect to a 2-plane in $\mathbb{R}^3$, by taking the connected sum of them and moreover by connecting them with some vertical twists along the plane, which is a generalized operation of the connected sum of a knot and its mirror image. In this paper, we show that a satellite symmetric union with minimal twisting number one such that the order of the pattern is an odd number $\ge 3$ has at least two disjoint non-parallel essential tori in the complement.

Let ${\mathbb F}={\mathbb R}$, ${\mathbb C}$ or the Hamilton's quaternions ${\mathbb H}$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional ${\mathbb F}$-hyperbolic space. Let ${\rm U}(n,1; {\mathbb F})$ be the linear group that acts by the isometries of ${\bf H}_{\mathbb F}^n$. A subgroup $G$ of ${\rm U}(n,1; {\mathbb F})$ is called \textit{Zariski dense} if it does not fix a point on ${\bf H}_{\mathbb F}^n \cup \partial {\bf H}_{\mathbb F}^n$ and neither it preserves a totally geodesic subspace of ${\bf H}_{\mathbb F}^n$. We prove that a Zariski dense subgroup $G$ of ${\rm U}(n,1; {\mathbb F})$ is discrete if for every loxodromic element $g \in G$, the two generator subgroup $\langle f, g \rangle$ is discrete, where $f \in {\rm U}(n,1; {\mathbb F})$ is a test map not necessarily from $G$.

New results pertaining to the invariant manifolds of stochastic partial differential equations are presented. We prove the existence of local and global invariant manifolds for a non-autonomous stochastic evolution equation. These manifolds are constituted by trajectories of the solutions belonging to particular function spaces and the theory of Ornstein-Uhlenbeck process.

We study criticalities of generalized Schrödinger operators in terms of the Schrödinger forms induced by generalized Feynman-Kac perturbations of symmetric Markov processes, which extend earlier work due to Takeda [26]. The related functional inequalities and analytic characterizations of criticalities of Schrödinger forms are given. As applications, we establish some maximum principles via the analytic characterization.