In this paper, we introduce the definitions of simsun succession statistics and simsun patterns. In addition to its original definition by Brenti, we give two more combinatorial interpretations of the $q$-Eulerian polynomials using simsun successions. We also present a bijection between permutations avoiding the simsun pattern $132$ and set partitions.

Unidirectional hypercubes and unidirectional folded hypercubes are generalizations of hypercubes and folded hypercubes to digraphs. The super-$\lambda$ property of a digraph is a index for network reliability, which can be measured by the restricted arc-connectivity quantitatively. In this paper, we first show that the restricted arc-connectivity of the $n$-dimensional unidirectional hypercube is $n-1$ when $n$ is even and is $n-2$ when $n$ is odd, and then we show that the restricted arc-connectivity of the $n$-dimensional unidirectional folded hypercube is $n-1$ when $n$ is even and is $n$ when $n$ is odd. As a consequence, we prove that both unidirectional hypercube and unidirectional folded hypercube are super-$\lambda$.

This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum \[ \sum_{\substack{a \in \mathbb{Z}_n^{\ast} \\ b_1, \ldots, b_k \in \mathbb{Z}_n}} \gcd(a-1, b_1, \ldots, b_k, n) \chi(a) \lambda_1(b_1) \cdots \lambda_k(b_k), \] where for a positive integer $n$, $\mathbb{Z}_n^{\ast}$ is the group of units of the ring $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, $\gcd$ represents the greatest common divisor, $\chi$ is a Dirichlet character modulo $n$, and for a nonnegative integer $k$, $\lambda_1, \ldots, \lambda_k$ are additive characters of $\mathbb{Z}_n$. Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].

We present a simple proof and a generalization of a Menon-type identity by Li, Hu and Kim, involving Dirichlet characters and additive characters.

In this paper, we obtain an expectation formula with respect to the $q$-probability distribution $W(x,y;q)$ based on a Maclaurin expansion. The formula has many applications in mathematics. Some of the applications are also given, which include a probability version of the Al-Salam and Verma $q$-integral.

In this paper, we investigate the distribution of branch points of some algebroid functions with infinitely many branch points. A sufficient condition that algebroid function has infinitely many branch points is given. By the Ahlfors' theory of covering surfaces, we also obtain the relationship between the Borel cluster directions of branch points and the Borel directions.

In this paper, a fourth order singular elliptic problem involving $p$-biharmonic operator with Dirichlet boundary condition is considered. The existence of at least one weak solution is proved in two different cases of the nonlinear term at the origin. The results are obtained by applying the critical points principle of Ricceri, variational methods and Rellich's inequality. Also an example is presented to verify the results.

For a normal Hilbert $B(H)$-module $X$ gramian Schauder basic $X$-valued measures are considered. Some equivalence conditions are given for an $X$-valued measure to be gramian Schauder basic. As an application gramian uniformly bounded linearly stationary $X$-valued processes are characterized.

In this paper, we prove the existence of decay integral solutions to a class of fractional differential inclusions with finite delays and estimate their decay rate. For these purposes, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of fractional PDE with almost sectorial operator is also given.

A space of boundary values is constructed for a minimal symmetric operator, generated by a discrete singular Hamiltonian system, acting in the Hilbert space $\ell_{\mathbf{A}}^{2}(\mathbb{N}_{0}; E \oplus E)$ ($\mathbb{N}_{0} = \{ 0,1,2,\ldots \}$, $\dim E = m \lt \infty$) with maximal deficiency indices $(m,m)$ (in limit-circle case). A description of all maximal dissipative, maximal accumulative, self-adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a self-adjoint dilation of a maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We establish a functional model of the dissipative operator and construct its characteristic function in terms of the scattering matrix of the dilation. Finally, we prove the theorem on completeness of the system of eigenvectors and associated vectors (or root vectors) of the dissipative operator.

This paper is concerned with the breathers of discrete one-dimensional nonlinear Schrödinger equations in inhomogeneous media. By using a constrained minimization approach known as the Nehari variational principle or the Nehari manifold approach, we obtain the existence of nontrivial breathers.

Our purpose is to characterize the eigenvalue interval for a system of boundary value problems with a one-dimensional perturbed $q$-Laplace operator. We consider both sublinear and superlinear nonlinearities with a possible singularity at zero. The tools applied here are based on variational methods and properties of the Fenchel transform.

This paper is concerned with arbitrary high-order energy-preserving numerical methods for stochastic canonical Hamiltonian systems. Energy and quadratic invariants-preserving (EQUIP) methods for deterministic Hamiltonian systems are applied to stochastic canonical Hamiltonian systems and analyzed accordingly. A class of stochastic parametric Runge-Kutta methods with a truncation technique of random variables are obtained. Increments of Wiener processes are replaced by some truncated random variables. We prove the replacement doesn't change the convergence order under some conditions. The methods turn out to be symplectic for any given parameter. It is shown that there exists a parameter $\alpha_{n}^{*}$ at each step such that the energy-preserving property holds, and the energy-preserving methods retain the order of the underlying stochastic Gauss Runge-Kutta methods. Numerical results illustrate the effectiveness of EQUIP methods when applied to stochastic canonical Hamiltonian systems.

In this paper, we present an efficient fully-discrete local discontinuous Galerkin (LDG) method for nonlinear reaction-diffusion systems, which are often used as mathematical models for many physical and biological applications. We can derive numerical approximations not only for solutions but also for their gradients at the same time, while most of methods derive numerical solutions only. And due to the strict time-step restriction ($\Delta t = O(h^2_{\min})$) of explicit schemes for stability, we introduce the implicit integration factor (IIF) method based on Krylov subspace approximation, in which the time step can be taken as $\Delta t = O(h_{\min})$. Moreover, the method allows us to compute element by element and avoid solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical experiments about the reaction-diffusion equations with exact solutions and the well-studied Schnakenberg system are conducted to illustrate the accuracy, capability and advantages of the method.

In this paper, we first explore some properties of $M$-tensors by showing that the Fan product of two $M$-tensors is an $M$-tensor, then establish lower bound estimations and upper bound estimations on the minimal eigenvalues of for Fan product of two $M$-tensors. Some inclusion relations among them are also obtained.