The Lusztig correspondence is a bijective mapping between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^{*})^{0}$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^{*}$. In this article we discuss the unicity and ambiguity of such a bijective correspondence. In particular, we show that the Lusztig correspondence for a classical group can be made to be unique if we require it to be compatible with the parabolic induction and the finite theta correspondence.

Kohnen proved a non-vanishing result for $L$-functions associated to Hecke eigenforms of integral weights on the full group. In this paper, we show a non-vanishing result for the averages of $L$-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms of weight $k \in \frac{1}{2} \mathbb{Z}$ on the full group. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize the result of Kohnen to $\Gamma_{0}(N)$ and prove the analogous result for Jacobi forms.

Let $(A,\mathbb{N}^{2},\alpha)$ be a dynamical system consisting of a $C^{*}$-algebra $A$ and an action $\alpha$ of $\mathbb{N}^{2}$ on $A$ by automorphisms. Let $A \times_{\alpha}^{\mathrm{piso}} \mathbb{N}^{2}$ be the partial-isometric crossed product of the system. We apply the fact that it is a full corner of a crossed product by the group $\mathbb{Z}^{2}$ in order to give a complete description of its primitive ideal space.

This paper deals with a pseudo-parabolic equation involving variable exponents under homogeneous Dirichlet boundary value condition. The authors first develop the potential well method to prove a threshold result on the existence or nonexistence of global solutions to the equation when the initial energy is less than the mountain pass level $d$. In the case of high energy initial data, a new characterization for the nonexistence of global solution is also given. These results extend and improve some recent results obtained by Di et al. [9].

This paper is to derive the precise asymptotic spreading behavior for an epidemic model with nonlocal dispersal. The proof is based on a Liouville type theorem on the positive bounded entire solutions. This Liouville theorem holds for a general class of reaction-diffusion systems with nonlocal dispersal which can be useful for reaction-diffusion systems arising in ecology and epidemiology.

We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system with perturbations and two parameters $\lambda_{1}$ and $\lambda_{2}$ on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-$(p,q)$ conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-$(p,q)$ conditions which is weaker than the well-known Ambrosetti–Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters $\lambda_{1}$ and $\lambda_{2}$.

In this paper, we explore the multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearities \[ \begin{cases} -\Delta u + V(\epsilon x)u = \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, dx = c, \end{cases} \] where $\mu \gt 0$, $c \gt 0$, $2 \lt q \lt 2+4/N \lt p \lt 2N/(N-2)$, $N \geq 3$, $\epsilon \gt 0$ is a parameter and $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. By employing the minimization techniques and the truncated argument, we obtain that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\epsilon$ is sufficiently small.

The aim of this paper is to study the stability of perturbed set optimization problems via general ordering sets. Firstly, for a set optimization problem (SOP) via general ordering sets, four kinds of concepts about the minimal solutions of (SOP) are given. Then, some properties of the four kinds of solution sets and the level set of objective mappings are investigated. Finally, by employing the recession cone technique, sufficient conditions of upper Painlevé–Kuratowski convergence of minimal approximate solution sets, Painlevé–Kuratowski convergence of weak minimal approximate solution sets of (SOP) are obtained, where the feasible set is perturbed. Some examples are given to illustrate the mainly results in the paper.