Enumeration of derangements in the symmetric group $\mathfrak{S}_n$ is classical. Extensions of the enumerative results to the hyperoctahedral group $B_n$ are combinatorially sound. That in the even-signed permutation group $D_n$ remains largely unexplored. Let $d_n^D(q) = \sum_{\sigma \in \mathcal{D}_n^D} q^{\operatorname{maj}(\sigma)}$ be the generating function of derangements in $D_n$ by their major indices. We study in this work properties of $d_n^D(q)$, including recurrence relations and factorial generating function. By proving the ratio monotonicity of $d_n^D(q)$, the unimodality, log-concavity and spiral property of $d_n^D(q)$ are also established.

Given a graph $G$ with $S \subseteq V_G$, we call $S$ a maximum dissociation set if the induced subgraph $G[S]$ contains no path of order $3$, and subject to this condition, the subset $S$ has the maximum cardinality. The dissociation number of $G$ is the cardinality of a maximum dissociation set. Inspired by the results of [26, 27] on the maximal number of maximum dissociation sets, in this contribution we investigate the maximal number of maximum dissociation sets in forests with fixed order and dissociation number. Firstly, a lower bound on the dissociation number of a forest with fixed order is established, and all extremal graphs are determined. Secondly, all trees (resp. forests) having the largest and the second largest number of maximum dissociation sets among trees (resp. forests) with given order and dissociation number are completely characterized. Finally, we show that the results in [26, 27] can be deduced by our results.

This work deals with the asymptotic behaviour of the electric field in the transverse magnetic (TM) mode, propagating in a bidimensional heterogeneous medium, composed by a homogeneous linear dielectric isotropic material surrounded by a thin layer of thickness $\varepsilon$ (destined to tend to $0$) and embedded in an ambient medium. Using the tools of multiscale analysis, an asymptotic expansion of the solution $u^{\varepsilon}$ to the Helmholtz problem with respect to the thickness $\varepsilon$ is derived. As a consequence, Ventcel-type transmission conditions on the limit interface $\Gamma$ are obtained modelling the effect of the thin layer with accuracy up to $O(\varepsilon^{3})$. A particular choice of the interface $\Gamma$ leads to a well-posed Ventcel's problem.

In this paper, the nonconforming virtual element method is used to solve the Stokes problem where the velocity and pressure are allowed to have the low regularity. With the help of an enriching operator, the consistency error is estimated under the low regularity condition. Then the optimal error estimates are obtained for the velocity and pressure approximations, which implies that the nonconforming virtual element method has the good convergence even for the Stokes problem with the low regularity.

In this paper, we investigate blow-up phenomena of the solution to a reaction-diffusion equation with nonlocal gradient absorption terms under Robin boundary condition on a bounded star-shaped region. Based on the method of auxiliary function and the technique of modified differential inequality, we establish some conditions on the nonlinearities for which the solution exists globally or blows up at finite time, when the sign of the constant $\sigma$ is either positive or negative. Moreover, upper and lower bounds for a blow-up time are derived under appropriate measure in higher dimensional spaces.

In this paper, we consider the asymptotic behavior of random reaction-diffusion delay equations driven by colored noise defined on unbounded domains. We firstly establish the existence and uniqueness of pullback random attractors for the continuous cocycle associated with the equation, and then consider the convergence of random attractors when colored noise approximates white noise. The methods of uniform tail-estimate and operator decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embeddings on unbounded domains.

We study in this paper the stability of a system of type Rao–Nakra sandwich beam in the whole line $\mathbb{R}$ under linear frictional dampings acting on the two wave equations. We prove that the $L^2(\mathbb{R})$-norm of solutions and their higher order derivatives with respect to the space variable converge polynomially to zero as time goes to infinity. These decay estimates are obtained independently of the parameters and in spite of the fact that the Euler–Bernoulli equation is not damped. Moreover, the decay rate is specified in term of the regularity of the initial data.

Our aim in this paper is to deal with the continuity of generalized Riesz potentials $I_{\rho,\tau} f$ of functions in Morrey spaces $L^{\Phi,\nu(\cdot),\kappa}(X)$ of double phase functionals with variable exponents over bounded non-doubling metric measure spaces. What is new in this paper is that $\rho$ depends on $x \in X$.