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Dynamic Analysis of a Predator-Prey Model with State-Dependent Impulsive Effects LI Yong-feng, ZHU Cheng-zhi, LIU Yan-wei Chinese Quarterly Journal of Mathematics    2023, 38 (1): 1-19.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.001 Abstract （121）      PDF（pc） （2367KB）（255）       Knowledge map    Save  In this paper, a pest-dependent model and integrated pest management strategy is proposed, that is, when pest populations reach levels that impair economic development, we will use a combination of strategies, such as biological, cultural and chemical control strategies reduce pests to a reasonable level. First, we investigated the system without control measures, and discussed the existence and stability of equilibria, we also proved the system has no limit cycle. Then, a state feedback impulsive model is constructed, the existence and uniqueness of the order-one periodic solution are proved by means of the successor function method to confirm the feasibility of the biological and chemical control strategy of pest management. Secondly, the stability of system is proved by the analogue of the Poincar ´ e criterion. Finally, we give an example and numerical simulations to explain the mathematical conclusions. Related Articles | Metrics

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Complexity on Proximal Quasi-Newton Methods for a Class of Nonconvex Composite Optimization JIN Ling-Zi Chinese Quarterly Journal of Mathematics    2023, 38 (1): 62-84.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.005 Abstract （105）      PDF（pc） （404KB）（221）       Knowledge map    Save This paper studies a class of nonconvex composite optimization, whose objective is a summation of an average of nonconvex (weakly) smooth functions and a convex nonsmooth function, where the gradient of the former function has the Hölder continuity. By exploring the structure of such kind of problems, we first propose a proximal (quasi-)Newton algorithm wPQN (Proximal quasi-Newton algorithm for weakly smooth optimization) and investigate its theoretical complexities to find an approximate solution. Then we propose a stochastic variant algorithm wPSQN (Proximal stochastic quasi-Newton algorithm for weakly smooth optimization), which allows a random subset of component functions to be used at each iteration. Moreover, motivated by recent success of variance reduction techniques, we propose two variance reduced algorithms, wPSQN-SVRG and wPSQN-SARAH, and investigate their computational complexity separately. Related Articles | Metrics

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Space-Time Legendre Spectral Collocation Methods for Korteweg-de Vries Equation WANG Chuan, QIAO Yan Chinese Quarterly Journal of Mathematics    Accepted: 12 April 2023

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Stability and Hopf Bifurcation of an Eco-Epidemiological Model with Delay BAI Hong-fang Chinese Quarterly Journal of Mathematics    2023, 38 (2): 157-183.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.005 Abstract （126）      PDF（pc） （1289KB）（173）       Knowledge map    Save In this paper, an eco-epidemiological model with time delay is studied. The local stability of the four equilibria, the existence of stability switches about the predation- free equilibrium and the coexistence equilibrium are discussed. It is found that Hopf bifurcations occur when the delay passes through some critical values. Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Some numerical simulations are carried out to illustrate the theoretical results. Related Articles | Metrics

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Differential Identities in Prime Rings with Involution HUANG Shu-liang Chinese Quarterly Journal of Mathematics    2023, 38 (2): 134-144.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.003 Abstract （87）      PDF（pc） （319KB）（163）       Knowledge map    Save Let R be a prime ring of characteristic different from two with the sec- ond involution ∗ and α an automorphism of R . An additive mapping F of R is called a generalized ( α,α )-derivation on R if there exists an ( α,α )-derivation d of R such that F ( xy )= F ( x ) α ( y )+ α ( x ) d ( y ) holds for all x,y∈R. The paper deals with the s- tudy of some commutativity criteria for prime rings with involution. Precisely, we describe the structure of R admitting a generalized ( α,α )-derivation F satisfying any one of the following properties: ( i ) F ( xx∗) −α ( xx∗) ∈Z ( R ). ( ii ) F ( xx∗ )+ α ( xx∗ ) ∈Z ( R ). ( iii ) F ( x ) F ( xx∗) −α ( xx∗) ∈Z ( R ). ( iv ) F ( x ) F (x∗)+ α ( xx∗) ∈Z ( R ). ( v ) F ( xx∗) −F ( x ) F (x∗ ) ∈Z ( R ). ( vi ) F ( xx∗) −F (x∗) F ( x )=0 for all x∈R . Also, some examples are given to demonstrate that the restriction of the various results is not superfluous. In fact, our results unify and extend several well known theorems in literature. Related Articles | Metrics

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On the Eigenvalues and Eigenfunctions of the Sturm-Liouville Operator with the Barrier Potential SARWAR Qanitah, HUANG Zhen-you, ZAHID Abdul Hannan, XU Xin-Jian Chinese Quarterly Journal of Mathematics    2023, 38 (1): 50-61.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.004 Abstract （153）      PDF（pc） （279KB）（161）       Knowledge map    Save We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval [0 ,π ] when λ> 0. Generally, the eigenvalue problem for the Sturm-Liouville operator is often solved by using integral equations, which are sometimes complex to solve, and difficulties may arise in computing the boundary values. Considering the said complexity, we have successfully developed a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for Sturm-Liouville operator with barrier potential. The results are of significant interest in the field of quantum mechanics and atomic systems to observe discrete energy levels. Related Articles | Metrics

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Virtual Element Method of the Allen-Cahn Equation WANG Pei-zhen, TIAN Xu Chinese Quarterly Journal of Mathematics    2023, 38 (1): 20-29.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.002 Abstract （113）      PDF（pc） （307KB）（153）       Knowledge map    Save  In this article, the virtual element method of the Allen-Cahn equation on a polygon grid is discussed in the fully discrete formulation. With the help of the energy projection operator, we give the corresponding error estimates in the L2 norm and H1 norm. Related Articles | Metrics

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Initial Boundary Value Problem for Pseudo-Parabolic p-Laplacian Type Equation with Logarithmic Nonlinearity PAN Jia-hui, FANG Shao-mei Chinese Quarterly Journal of Mathematics    2023, 38 (4): 360-369.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.003 Abstract （56）      PDF（pc） （358KB）（137）       Knowledge map    Save  In this paper, we study the initial boundary value problem of pseudo-parabolic p-Laplacian type equation, which be use to model some important physical and biological phenomena. By using the potential well method, we obtain the global existence, asymptotic behavior and blow up results of weak solution with subcritical initial energy. Then we also extend these results to the critical initial energy. Related Articles | Metrics

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Weighted Analytic Torsion for Weighted Digraphs REN Shi-quan, WANG Chong Chinese Quarterly Journal of Mathematics    2023, 38 (1): 30-49.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.003 Abstract （142）      PDF（pc） （389KB）（131）       Knowledge map    Save  In 2020, Alexander Grigor’yan, Yong Lin and Shing-Tung Yau [6] introduced the Reidemeister torsion and the analytic torsion for digraphs by means of the path complex and the path homology theory. Based on the analytic torsion for digraphs introduced in [6], we consider the notion of weighted analytic torsion for vertex-weighted digraphs. For any non-vanishing real functions f and g on the vertex set, we consider the vertex-weighted digraphs with the weights ( f,g ). We calculate the ( f,g )-weighted analytic torsion by examples and prove that the ( f,g )-weighted analytic torsion only depend on the ratio f/g . In particular, if the weight is of the diagonal form ( f,f ), then the weighted analytic torsion equals to the usual (un-weighted) torsion. Related Articles | Metrics

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Independent Roman {2}-Domination in Trees LI Bei-bei, SHANG Wei-ping Chinese Quarterly Journal of Mathematics    2022, 37 (4): 386-393.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.006 Abstract （144）      PDF（pc） （336KB）（125）       Knowledge map    Save For a graph G = (V,E), a Roman {2}-dominating function f :V → {0,1,2} has the property that for every vertex v ∈V with f(v) = 0, either v is adjacent to at least one vertex u for which f(u) = 2, or at least two vertices u1 and u2 for which f(u1) =f(u2) = 1. A Roman {2}-dominating function f = (V0,V1,V2) is called independent if V1∪V2 is an independent set. The weight of an independent Roman {2}-dominating function f is the value ω(f) =\sumv∈V f(v), and the independent Roman {2}-domination number i{R2}(G) is the minimum weight of an independent Roman {2}-dominating function on G. In this paper, we characterize all trees with i{R2}(T) =γ(T)+ 1, and give a linear time algorithm to compute the value of i{R2}(T) for any tree T.  Related Articles | Metrics

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Induced Matching-Extendability of Halin Graphs ZHANG Qing-nan, HUI Zhi-hao, YANG Yu, WANG An, Chinese Quarterly Journal of Mathematics    2022, 37 (4): 380-385.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.005 Abstract （156）      PDF（pc） （344KB）（112）       Knowledge map    Save  Let G be a connected graph having a perfect matching. The graph G is said to be induced matching (IM) extendable if every induced matching M of G is contained in a perfect matching of G. In this paper, we show that Halin graph G =T ∪C is IMextendable if and only if its characteristic tree T is isomorphic to K1,3, K1,5, K1,7 or S2,2. Related Articles | Metrics

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Ergodicity of Bandwidth and Cutwidth on Families of Graphs and Trees LIN Yi-shu, CHANG Cai-bing, LIU Yan Chinese Quarterly Journal of Mathematics    2022, 37 (4): 355-365.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.003 Abstract （84）      PDF（pc） （346KB）（110）       Knowledge map    Save Bandwidth, cutwidth, cyclic bandwidth, bandwidth sum and cyclic bandwidth sum are well-known indices about optimal labeling of graphs applied in VLSI design, network communications, and other areas involving the graph layout. To design the graphs with the given indices, we need to study the ergodicity. Let F be a set of graphs under consideration and ϕ an integer-valued function defined on F, namely, ϕ is an index, such as bandwidth and cutwidth. If there exists a graph G ∈ F such that ϕ(G) =x for any integer x in the interval [a,b], where a and b are the minimum and maximum of ϕ on F, respectively, then ϕ is said to have ergodicity on F. Let Gn be the set of simple connected graphs with order n and Tn the set of trees with order n. In this paper, we investigate the ergodicity of bandwidth, cutwidth, cyclic bandwidth, the bandwidth sum and cyclic bandwidth sum on Tn and Gn. Related Articles | Metrics

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Some Applications of Surface Curvatures in Theoretical Physics YANG Yi-song Chinese Quarterly Journal of Mathematics    2023, 38 (3): 221-253.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.001 Abstract （240）      PDF（pc） （496KB）（104）       Knowledge map    Save In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered. Related Articles | Metrics

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The Least Squares {P,Q,k+1}-Reflexive Solution to a Matrix Equation DONG Chang-zhou, LI Hao-xue Chinese Quarterly Journal of Mathematics    2023, 38 (2): 210-220.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.008 Abstract （76）      PDF（pc） （383KB）（102）       Knowledge map    Save  Let P ∈C m×m and Q∈C n×n be Hermitian and {k +1 } -potent matrices, i.e., P k+1 = P = P ∗ , Q k+1 = Q = Q ∗ , where ( · ) ∗ stands for the conjugate transpose of a matrix. A matrix X ∈C m×n is called {P,Q,k +1 } -reflexive (anti-reflexive) if PXQ = X ( PXQ = −X ). In this paper, the least squares solution of the matrix equation AXB = C subject to {P,Q,k +1 } -reflexive and anti--reflexive constraints are studied by converting into two simpler cases: k=1 and k=2. Related Articles | Metrics

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A Note on Preemptive Scheduling with Multiple Maintenance Activities to Minimize the Total Late Work HE Ru-yan, YUAN Jin-jiang, ZHANG Yuan Chinese Quarterly Journal of Mathematics    2022, 37 (4): 331-342.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.001 Abstract （93）      PDF（pc） （374KB）（97）       Knowledge map    Save We study the single-machine preemptive scheduling problem with multiple maintenance activities to minimize the total late work, in which the jobs must be processed in the time space not occupied by the maintenance intervals. For this problem, we present a polynomial algorithm to determine the optimal schedule and establish a formula expression to the optimal value. Moreover, our result is used to correct some minor errors in the literature related to the single-machine (preemptive or non-preemptive) scheduling with one maintenance activity to minimize the total late work.  Related Articles | Metrics

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The Semi-Convergence Properties of the Generalized Shift-Splitting Methods for Singular Saddle Point Problems HUANG Zhuo-Hong Chinese Quarterly Journal of Mathematics    2023, 38 (2): 145-156.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.004 Abstract （74）      PDF（pc） （300KB）（93）       Knowledge map    Save Recently, some authors (Shen and Shi, 2016) studied the generalized shift- splitting (GSS) iteration method for singular saddle point problem with nonsymmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. In this paper, we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite (1,1)-block and symmetric positive semidefinite (2,2)-block, prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix. Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use. Related Articles | Metrics

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A New Existence Theorem for Global Attractors and its Application to a Non-Classical Diffusion Equation QIN Yu-ming, CHEN Jia-le, JIANG Hui-te Chinese Quarterly Journal of Mathematics    2023, 38 (3): 276-289.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.003 Abstract （72）      PDF（pc） （364KB）（90）       Knowledge map    Save In this paper, we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems, then establish a new existence theorem of global attractors which will be applied to a nonclassical diffusion equation for the norm-to-weak continuous, weakly compact semigroup on H01(Ω) and H2(Ω)∩H01(Ω) respectively. As an application of this new existence theorem of global attractors, we obtain the existence of the global attractors onH01(Ω) and H2(Ω)∩ H01(Ω) respectively for a nonclassical-diffusion equation. Related Articles | Metrics

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Hopf-Rinow Theorem on Convex Complex Finsler Manifolds LI Hong-jun Chinese Quarterly Journal of Mathematics    2024, 39 (1): 31-45.   DOI: 10.13371/j.cnki.chin.q.j.m.2024.01.003 Abstract （87）      PDF（pc） （365KB）（90）       Knowledge map    Save Suppose (M,F) is a convex complex Finsler manifold. We prove that geodesics of (M,F) are locally minimizing. Hence, F introduces a distance function d such that (M,d) is a metric space from topology. Next, we prove the classical Hopf-Rinow Theorem holds on (M,F). Related Articles | Metrics

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A Note on the Girth of 3-Regular Hamiltonian Graph ZHAO Qiu-lan, YUAN Jin-jiang Chinese Quarterly Journal of Mathematics    2022, 37 (4): 430-431.   DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.011 Abstract （90）      PDF（pc） （226KB）（88）       Knowledge map    Save It is well-known that the Petersen graph is nonhamiltonian. A very short proof for this result was presented in [2] due to D. B. West. In this note, by extending the proof technique in [2], we briefly show that the girth of every 3-regular hamiltonian graph on n≥10 vertices is at most (n+ 4)/3. Related Articles | Metrics

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On the Stokes Phenomenon and Representation Theory of Quantum Groups XU Xiao-meng Chinese Quarterly Journal of Mathematics    2023, 38 (3): 311-330.   DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.005 Abstract （59）      PDF（pc） （458KB）（87）       Knowledge map    Save This is a survey paper that lists our research works in the study of Stokes phenomenon of meromorphic ordinary differential equations and its relation with representation theory of quantum groups Related Articles | Metrics