The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed.

The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied. The existence of global solutions for this problem is proved by using potential well method, and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function. Meanwhile, blow-up of solutions in the unstable set is also obtained.

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an application, the results are applied to solving the zero problem and the equilibrium problem.

In this paper, a class of Kirchhoff type equations in {\mathbb{R}}^N(N⩾3) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

A Cartesian decomposition of a coherent configuration ✗ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of ✗ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration ✗ is thick, then there is a unique maximal Cartesian decomposition of ✗; i.e., there is exactly one internal tensor decomposition of ✗ into indecomposable components. In particular, this implies an analog of the Krull–Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.

We provide a constructive proof of H^1\left({\text{ℝ}}^d\right) (the classical Hardy space) factorization in terms of fractional commutators in Lorentz spaces. As a direct application, we obtain a characterization of functions in BMO space. Furthermore, we also obtain a Lorentz compactness characterization of fractional commutators.

The lump solutions and interaction solutions are mainly investigated for the (2+1)-dimensional KPI equation. According to relations of the undetermined parameters of the test functions, the N-soliton solutions are showed by computations of the Maple using the Hirota bilinear form for(2+1)-dimensional KPI equation. One type of the lump solutions for (2+1)-dimensional KPI equation has been deduced by the limit method of the N-soliton solutions. In addition, the interaction solutions between the lump and N-soliton solutions of it are studied by the undetermined interaction functions. The sufficient conditions for the existence of the interaction solutions are obtained. Furthermore, the new breather solutions for the (2+1)-dimensional KPI equation are considered by the homoclinic test method via new test functions including more parameters than common test functions.

We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials, and Narayana polynomials.

We use the concept of the inside-(a, η, h) domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space. As an application, we prove that the Hölder exponent \max \{1/a,1/(n+1)\} for the problem is optimal for any a\in [2,+\infty ].

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin–Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin–Vilkovisky algebra structures for them are described completely.

The (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation is a significant physical model. By using the long wave limit method and confining the conjugation conditions on the interrelated solitons, the general M-lump, high-order breather, and localized interaction hybrid solutions are investigated, respectively. Then we implement the numerical simulations to research their dynamical behaviors, which indicate that different parameters have very different dynamic properties and propagation modes of the waves. The method involved can be validly employed to get high-order waves and study their propagation phenomena of many nonlinear equations.

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states.

Let {\text{ℤ}}_m be the additive group of residue classes modulo m. Let s(m, n) denote the number of subgroups of the group {\text{ℤ}}_m\times {\text{ℤ}}_n, where m and n are arbitrary positive integers. For any x\ge 1, we consider the asymptotic behavior of D_s\left(x\right):=\sum m^2+n^2\le xS(M,n) and obtain an asymptotic formula by using the elementary method.