We give a survey on the contribution of our research group to the cell theory of weighted Coxeter groups. We present some detailed account for the description of cells of the affine Weyl group $ \widetilde{C}_n $ in the quasi-split case and a brief account for that of the affine Weyl group $ \widetilde{B}_n $ in the quasi-split case and of the weighted universal Coxeter group in general case.
In recent years, the relativity between domains with specific metrics and complex Euclidean spaces has been a topic of interest in the study of complex variables. Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with their induced metrics. A Cartan-Egg domain is a type of bounded non-homogeneous domain. Its Bergman kernel function can be constructed as an explicit expression using the expansion principle. In this paper, the relativity between a Cartan-Egg domain with Bergman metrics and a complex Euclidean space with canonical metrics is explored. In relation research of complex Euclidean spaces, the working premise is that a Bergman kernel function is a Nash function. However, the Bergman kernel function of Cartan-Egg domains are not necessarily Nash functions. Therefore, existing methods cannot be used directly. By analyzing the algebraic properties of a Bergman kernel function’s partial derivative function of a Cartan-Egg domain, we show that a Cartan-Egg domain with Bergman metrics is not related to a complex Euclidean space with canonical metrics.
Linear traveling-wave transformations are usually applied when constructing exact traveling-wave solutions for nonlinear evolution equations. Herein, for the first time, specific nonlinear traveling-wave transformations are introduced to extend the $N$ -soliton decomposition algorithm and an inheritance-solving strategy to a variable-coefficient nonlinear evolution equation. Two higher-dimensional nonlinear evolution equations with time-dependent coefficients, the Boiti-Leon-Manna-Pempinelli (BLMP) equation and the cylindrical Kadomtsev-Petviashvili (cKP) equation, are solved. The direct algebraic method and inheritance-solving strategy are used to construct several different types of multiwave-interaction solutions for the BLMP equation, specifically, the horseshoe-like solitons and their interaction with lump as well as different periodic waves. Using the $N$ -soliton decomposition algorithm, the higher-order interaction solutions between the horseshoe-like solitons, breathers, and lump waves of the cKP equation are established. These new multiwave-interaction solutions contribute to the existing solutions of nonlinear evolution equations with variable coefficients, enriching the repository of solutions to a certain extent.
LaSalle’s invariance principle is an important tool for studying the stability of stochastic systems. Considering the influence of time delay and pure-jump path on the stability of the system and using the convergence theorem for special semi-martingale, the LaSalle’s invariance principle for a class of stochastic delay differential equations driven by $\alpha$ -stable processes is established in this study. The sufficient conditions for the asymptotic stability of a class of delay equations are given by LaSalle’s invariance principle.
In this paper, blow-up of solutions to a class of weakly coupled semilinear double-wave systems with nonlinear terms of derivative type is considered. By choosing suitable functionals and using an iteration technique, the weakly coupled phenomena are studied in-depth for the case when $ p\ne q $ . For the case when $ p=q $ , the solution is degenerated to a single semilinear double-wave equation with a nonlinear term of derivative type. Furthermore, the nonexistence of global solutions to the Cauchy problem in the subcritical case is proven. Meanwhile, the upper bound estimate of the lifespan of solutions is also derived.
This paper presents an investigation of the weighted Drazin inverse $A^{d, W}$ of matrices based on the weighted core-EP decomposition of the pair $\{A, W\}$ . Some characterizations and representations of the weighted Drazin inverse are presented using the weighted core-EP decomposition of the pair $\{A, W\}$ . Further, the limit representations and the integral representations of the weighted Drazin inverse are discussed. Furthermore, an example is presented.
A graph is called a two-degree graph if its vertices have only two distinct degrees. A two-degree tree of order at least three have two degrees, $ 1 $ and $ d $ for some $ d\geqslant 2; $ such a tree is called a $ (1,d) $ -tree. Given a positive integer $ n, $ we determine: (1) the possible values of $ d $ such that there exists a $ (1,d) $ -tree of order $ n; $ (2) the values of $ d $ such that there exists a unique $ (1,d) $ -tree of order $ n $ , and (3) the maximum diameter of two-degree trees of order $ n. $ The results provide a new example showing that the behavior of graphs may sometimes be determined by number theoretic properties.
In this paper, we consider the inverse problem of quadratic eigenvalue for a Hermitian R-antisymmetric matrix. By using the matrix block method, singular value decomposition, vector straightening, and the Moore-Penrose inverse, we prove the existence of a Hermitian R-antisymmetric solution. In addition, we provide the general expression for a Hermitian R-antisymmetric solution, and discuss the best approximation thereof. Finally, an example is offered to validate the theory.
In this paper, we introduce the notion of strongly Gorenstein weak flat modules, and we subsequently provide homological characterizations of strongly Gorenstein weak flat modules. It is shown that a Gorenstein weak flat module is a summand of a strongly Gorenstein weak flat module.
The complete convergence of sequences of random variables under sublinear expectation was studied. Using the properties of extended negatively dependent (ND) sequences, under the condition that the $ \lambda $ -order Choquet integrals of the random variable are finite, the complete convergence of the weighted sums for extended ND sequences under a sublinear expectation was proved. The results generalize and improve the results of independent sequences in the classical probability space.
The dynamical behavior of a class of second-order semilinear singularly perturbed Neumann boundary value problems with a turning point are studied. Firstly, we establish sufficient conditions for the exchange of stabilities near the turning point. By correcting the regularized equation of the degenerate problem, the accuracy of the asymptotic solution to the original problem is improved. Secondly, the Nagumo theorem is used to prove the existence of a smooth solution. Finally, a specific example is used to verify the validity of the results.
This paper considers a Neumann boundary value problem of a singularly perturbed delay reaction-diffusion equation with a nonlinear reactive term. By using the boundary layer function method, contrast structure theory, and contraction mapping principle, the asymptotic expansion of the solution is constructed, and the existence of a uniformly valid solution is proven. Finally, an example is presented to show the effectiveness of our result.
The precision (inverse covariance) matrix generated by the periodic vector autoregressive model is a sparse block tridiagonal matrix. Based on this precision matrix, a new block trace function is proposed for testing the equality of block traces of two precision matrices, the asymptotic behavior under the null hypothesis is investigated. Numerical experiments show that the proposed testing procedure has both appropriate size and good power.
In this paper, de Moivre’s theorem for a matrix representation of a class of hyperbolic split quaternions is studied. Firstly, the study of hyperbolic split quaternions is transformed into the study of a matrix representation of hyperbolic split quaternions. Secondly, by using the polar representation of a hyperbolic split quaternion, the three forms of de Moivre’s theorem for a matrix representation of the hyperbolic split quaternion are obtained, and Euler’s formula is extended. Thirdly, the root-finding formula of the hyperbolic split quaternion matrix representation equation is obtained. Finally, the validity of the conclusions is verified with some examples.
This paper establishes a coarse version of finite asymptotic property C-decomposition complexity in the context of coarse spaces. In particular, permanence properties of finite asymptotic property C-decomposition complexity are studied, and it is shown that finite coarse asymptotic property C-decomposition complexity implies coarse property A. In addition, the paper explores coarse property C and coarse decomposition complexity.
This paper explores the existence of anti-periodic solutions for a class of nonlinear discrete dynamical systems with summable dichotomy. Using the Banach fixed-point theorem, sufficient conditions for the existence and uniqueness of anti-periodic solutions for nonlinear discrete dynamical systems are established. Lastly, an example is presented to illustrate the main results.
In this paper, the blow-up problem of a parabolic equation with a nonlinear gradient term in finite time is studied. By constructing an auxiliary function, using the method of energy estimation and the differential inequality technique, the lower bound of blow-up time is obtained. After limiting the parameters of the equation, the existence of a global solution is proved.
In this paper, we explore the blow-up of solutions to a class of nonlocal porous medium systems with space-dependent coefficients and inner absorption terms under nonlinear boundary conditions in ${\mathbb{R}}^{n}\left(n \geqslant 3\right)$ . By constructing an energy expression and using the differential inequality technique, we obtain sufficient conditions for the global existence of solutions to the problem. Then, upper bound and lower bound estimates of the blow-up time are derived by means of the Sobolev inequalities and other differential methods when blow-up occurs.
There are rare $q $ -congruences on double series in the literature. In this paper, we present several $q $ -congruences involving double series. When $q $ tends to 1, the proposed approach provides the corresponding conclusions for congruences.
Alternating direction iteration (ADI) scheme is an effective method for solving real positive definite linear systems; in many cases, however, the method requires that all the direction matrices involved are multiplication exchangeable, which severely limits the scope of application. In this paper, new revised alternating direction iteration (RADI) schemes are proposed, that do not stipulate the multiplication exchangeable requirement, thereby expanding the application scope. In parallel, measures to improve the efficiency of RADI schemes are also discussed.