In this paper, we study the spectral radius of bipartite graphs. Let $G$ be a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph. Suppose that $G$ is not a complete bipartite graph and $(e-1,e+1)$ is not a pair of twin primes. We describe the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1,e+1)$ of integers to be a pair of twin primes.

We follow the ideas of Khare and Ramakrishna-Khare and prove the modularity lifting theorem in higher weights. This approach somehow differs from that using Taylor-Wiles systems.

In this paper, we prove that if the Fourier coefficients of a vector-valued modular form satisfy the Hecke bound, then it is cuspidal. Furthermore, we obtain an analogous result with regard to Jacobi forms by applying an isomorphism between vector-valued modular forms and Jacobi forms. As an application, we prove a result on the growth of the number of representations of $m$ by a positive definite quadratic form $Q$.

Let $R$ be a ring, $\alpha$ and $\beta$ two automorphisms of $R$. An additive mapping $d \colon R \to R$ is called an $(\alpha,\beta)$-derivation if $d(xy) = d(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$. An additive mapping $G \colon R \to R$ is called a generalized $(\alpha,\beta)$-derivation if $G(xy) = G(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$, where $d$ is an $(\alpha,\beta)$-derivation of $R$. In this paper we introduce the definitions of $b$-generalized $(\alpha,\beta)$-derivation and $b$-generalized $(\alpha,\beta)$-biderivation. More precisely, let $d \colon R \to R$ and $G \colon R \to R$ be two additive mappings on $R$, $\alpha$ and $\beta$ automorphisms of $R$ and $b \in R$. $G$ is called a $b$-generalized $(\alpha,\beta)$-derivation of $R$, if $G(xy) = G(x) \alpha(y) + b\beta(x) d(y)$ for any $x,y \in R$.

Let now $D \colon R \times R \to R$ be a biadditive mapping. The biadditive mapping $\Delta \colon R \times R \to R$ is said to be a $b$-generalized $(\alpha,\beta)$-biderivation of $R$ if, for every $x,y,z \in R$, $\Delta(x,yz) = \Delta(x,y) \alpha(z) + b\beta(y) D(x,z)$ and $\Delta(xy,z) = \Delta(x,z) \alpha(y) + b\beta(x) D(y,z)$.

Here we describe the form of any $b$-generalized $(\alpha,\beta)$-biderivation of a prime ring.

In this paper we study proper holomorphic maps between bounded symmetric domains when the source domain is not irreducible. More precisely, we provide sufficient conditions for semi-product proper holomorphic maps to be product proper. As an application we characterize proper holomorphic maps between equidimensional bounded symmetric domains.

We study the notion of sensitivity on $G$-spaces and through examples observe that $G$-sensitivity neither implies nor is implied by sensitivity. Further, we obtain necessary and sufficient conditions for a map to be $G$-sensitive. Next, we define the notion of Devaney's chaos on $G$-space and show that $G$-sensitivity is a redundant condition in the definition.

This paper deals with blow-up phenomena for an initial boundary value problem of a porous medium equation with time-dependent coefficients and inner absorption term in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficient and nonlinear functions to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, the upper and lower bounds for $t^{\ast}$ are derived in the higher dimensional space. Finally, some examples are presented to illustrate applications of our results.

In this paper, we analyze the exact controllability problem for wave equations endowed with switching controls. The goal is to control the dynamics of the system by switching among different actuators such that, in each instant of time, there are as few active actuators as possible. We prove that the system is exactly controllable under suitable geometric control conditions.

This paper is mainly concerned with the Toeplitz operator $T_{\phi}$ over the Dirichlet space $\mathcal{D}$ with the symbol $\phi$ in the Sobolev multiplier algebra $M(W^{1,2}(\mathbb{D}))$, thereby extending several known ones in a very different manner.

By using variational approaches, we study a class of quasilinear Schrödinger equations involving critical Sobolev exponents \[ -\Delta u + V(x)u + \frac{1}{2} \kappa [\Delta(u^2)]u = |u|^{p-2}u + |u|^{2^*-2}u, \quad x \in \mathbb{R}^N, \] where $V(x)$ is the potential function, $\kappa \gt 0$, $\max \{ (N+3)/(N-2),2 \} \lt p \lt 2^* := 2N/(N-2)$, $N \geq 4$. If $\kappa \in [0,\overline{\kappa})$ for some $\overline{\kappa} \gt 0$, we prove the existence of a positive solution $u(x)$ satisfying $\max_{x \in \mathbb{R}^N} |u(x)| \leq \sqrt{1/(2\kappa)}$.

In this paper, we propose fixed point results through the notion of measure of noncompactness (MNC) in partially ordered Banach spaces. We also prove some new coupled fixed point results via MNC for more general class of function. To achieve this result, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. Further, we apply the obtained fixed point theorems to prove the existence of mild solutions for fractional integro-differential evolution equations with nonlocal conditions. At the end, an example is given to illustrate the rationality of the abstract results for fractional parabolic equations.

In this paper, we consider sublinear second order differential equations with impulsive effects. Basing on the Poincaré-Bohl fixed point theorem, we first will prove the existence of harmonic solutions. The existence of subharmonic solutions is also obtained by a new twist fixed point theorem recently established by Qian etc in 2015 [18].

In this paper, we consider the fractional complex Ginzburg-Landau equation in two spatial dimensions with the dissipative effect given by a fractional Laplacian. The periodic initial value problem of the fractional complex Ginzburg-Landau equation is discretized fully by Galerkin-Fourier spectral method, and the dynamical behaviors of the discrete system are studied. The existence and convergence of global attractors of the discrete system are obtained by a priori estimates and error estimates of the discrete solution. The numerical stability and convergence of the discrete scheme are proved.

This paper deals with the Fréchet and Mordukhovich coderivatives of the normal cone mapping related to the parametric extended trust region subproblems (eTRS), in which the trust region intersects a ball with a single linear inequality constraint. We use the obtained results to investigate the Lipschitzian stability of parametric eTRS. We also propose a necessary condition for the local (or global) solution of the eTRS by using the coderivative tool.