To characterize the effects of stochastic environment and major mutation factors on populations, we consider a class facultative population system based on Markov chains and pure-jump stable processes. First of all, the existence and uniqueness of a global positive solution of the proposed model is discussed. Then, sufficient conditions for ergodicity are specified. Finally, conditions for positive recurrence of the model are presented.
An ${\rm{E}} $ -total coloring of a graph $G $ is an assignment of several colors to all vertices and edges of $G $ such that no two adjacent vertices receive the same color and no edge receive the same color as one of its endpoints. If $f $ is an ${\rm{E}} $ -total coloring of a graph $G $, the multiple color set of a vertex $x $ of $G $ under $f $ is the multiple set composed of colors of $x $ and the edges incident with $x $. If any two distinct vertices of $G $ have distinct multiple color sets under an ${\rm{E}} $ -total coloring $f $ of a graph $G $, then $f $ is called an ${\rm{E}} $ -total coloring of $G $ vertex-distinguished by multiple sets. An ${\rm{E}} $ -total chromatic number of $G $ vertex-distinguished by multiple sets is the minimum number of the colors required in an ${\rm{E}} $ -total coloring of $G $ vertex-distinguished by multiple sets. The ${\rm{E}} $ -total colorings of cycles and paths vertex-distinguished by multiple sets are discussed by use of the method of contradiction and the construction of concrete coloring. The optimal${\rm{E}} $ -total colorings of cycles and paths vertex-distinguished by multiple sets are given and the ${\rm{E}} $ -total chromatic numbers of cycles and paths vertex-distinguished by multiple sets are determined in this paper.
Tridiagonal sign pattern matrices and paw form sign pattern matrices were analyzed with respect to their potential for ensuring algebraic positivity. The necessary conditions allowing algebraic positivity of the two classes of sign pattern matrices were given using combinatorial matrix theory and graph theory. Finally, the equivalent conditions that would ensure algebraic positivity of tridiagonal sign pattern matrices and paw form sign pattern matrices of order $n $ were determined.
An involution ring is called a *r-clean ring if every element is the sum of a projection and a *-regular element. Some extensions of *r-clean rings are discussed, and a characterization of the element in a *-abelian *r-clean ring is given.
In this paper, some sufficient conditions for forced oscillation of impulsive multi-delay fractional partial differential equation solutions with damping term are established by using the method of differential inequalities under Robin and Dirichlet boundary conditions, an example is given to verify the validity of the main results.
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.