In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas are studied. Combinatorial explanations are given for some equidistributions.

Mixed graphs are graphs whose edges may be directed or undirected. The first eigenvalue of a mixed graph is the least nonzero eigenvalue of its Laplacian matrix. We determine the unique mixed graphs with minimum first eigenvalue over all nonsingular mixed unicyclic graphs with fixed number of branch vertices, and the unique graph with minimum least signless Laplacian eigenvalue over all nonbipartite unicyclic graphs with fixed number of branch vertices.

We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a sequence subject to certain three-term recurrence.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an arbitrary $R$-module. Let $\mathcal{S}$ be a Serre subcategory of the category of $R$-modules. It is shown that the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $i \geq 0$, if and only if the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $0 \leq i \leq \operatorname{ara}(I)$. As an immediate consequence, we prove that if $R$ is a Noetherian (resp. $(R, \mathfrak{m})$ is a Noetherian local) ring of dimension $d$, then the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $i \geq 0$ if and only if the $R$-module $\operatorname{Ext}^i_{R}(R/I, M)$ belongs to $\mathcal{S}$, for all $0 \leq i \leq d+1$ (resp. for all $0 \leq i \leq d$). Also it is shown that if $I$ is a principal ideal up to radical, then the category of $I$-cominimax (resp. $I$-weakly cofinite) modules is an Abelian full subcategory of the category of $R$-modules.

It is shown that Rothstein's theorem holds for $(F, W)$-meromorphic functions with $F$ is a sequentially complete locally convex space. We also prove that a meromorphic function on a Riemann domain $D$ over a separable Banach $E$ with values in a sequentially complete locally convex space can be extended meromorphically to the envelope of holomorphy $\widehat{D}$ of $D$. Using these results, in the remaining parts, we give a version of Kazarian's theorem for the class of separately $(\cdot, W)$-meromorphic functions with values in a sequentially complete locally convex space and generalize cross theorem with pluripolar singularities of Jarnicki and Pflug for separately $(\cdot, W)$-meromorphic functions with values in a Fréchet space.

Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.

Using the $II$-metric and the $II$-Gauss map on a surface derived from the non-degenerate second fundamental form of a surface in the sphere, we establish some characterizations of compact surfaces including the spheres and the tori in the $3$-dimensional unit sphere.

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.

We investigate the existence of at least two nontrivial solutions to a Dirichlet problem on the Sierpinski gasket. We develop some general abstract multiplicity theorem which we apply to problem under consideration. Our approach relies on the fact that the action functional is a difference of two continuously differentiable convex functionals and therefore we can apply the ideas related to the Fenchel-Young conjugacy together to get one critical point together with the mountain pass geometry to get the other one.

In this paper, we prove a generalized critical point theorem under the condition (C), which is weaker than the (PS) condition. As its applications, we obtain the existence of the solutions for the Hamiltonian systems with a new super-quadratic conditions generalizing one in papers [2] and [12].

In the paper we consider points focusing entropy and such that this fact is influenced exclusively by the behaviour of the function around these points (i.e., it is independent from the form of the function at any distance from these points). Thus the notion of an $\mathcal{F}$-focal entropy point has been introduced. We prove that each edge periodic tree function and each continuous function mapping the unit interval into itself have such points. Moreover, we discuss the possibility of improving functions defined on some topological manifolds so that any fixed point of the function becomes its focal entropy point.

Associated with two Banach algebras $\mathcal{A}$ and $\mathcal{B}$ and a norm decreasing homomorphism $T: \mathcal{B} \to \mathcal{A}$, there is a certain Banach algebra product $\mathcal{M}_T := \mathcal{A} \times_T \mathcal{B}$, which is a splitting extension of $\mathcal{B}$ by $\mathcal{A}$. In this paper, we investigate approximate cyclic amenability of $\mathcal{M}_T$ which has been introduced and studied by Esslamzadeh and Shojaee in [5]. In particular, apart from the characterization of all cyclic derivations on the Banach algebra $\mathcal{M}_T$, we improve the results of [1, 2] for cyclic amenability of $\mathcal{M}_T$. These results paves the way for obtaining new results for (approximate) cyclic amenability of the Banach algebra $\mathcal{A} \times \mathcal{B}$ equipped with the coordinatewise product algebra and $\ell^1$-norm.

The aim of this paper is to present new results on the convergence of relative Pareto efficient sets and the lower semicontinuity of relative Pareto efficient point multifunctions under perturbations. Our results extend some results of Luc et al. [16, Theorem2.1], Bednarczuk [4, Theorem 4] and [5, Proposition 3.1], Lucchetti and Miglierina [17, Proposition 3.1]. Some remarks and examples are provided for analysing the results obtained and for comparing them with the preceding results.

In this paper, two open questions on strong $b$-metric spaces posed by Kirk and Shahzad [11, Chapter 12] are investigated. A counterexample is constructed to give a negative answer to the first question, and a theorem on the completion of a strong $b$-metric space is proved to give a positive answer to the second question.

In this study, we develop a high-order accurate discontinuous Galerkin scheme using curvilinear quadrilateral elements for solving elliptic interface problems. To maintain accuracy for curvilinear quadrilateral elements with $Q^k$-polynomial basis functions, we select Legendre-Gauss-Lobatto quadrature rules with $k+2$ integration points, including end points, on each edge. Numerical experiments show quadrature rules are appropriate and the numerical solution converges with the order $k+1$. Moreover, we implement the Uzawa method to rewrite the original linear system into smaller linear systems. The resulting method is three times faster than the original system. We also find that high-order methods are more efficient than low-order methods. Based on this result, we conclude that under the condition of limited computational resources, the best approach for achieving optimal accuracy is to solve a problem using the coarsest mesh and local spaces with the highest degree of polynomials.