Chinese Quarterly Journal of Mathematics

Harvesting in a Toxic Predator-Prey Model with Carrying Capacity and Maturation Double Delays

ZHONG Ying, WEI Yu-ming

2023, 38(4): 331-348. doi:10.13371/j.cnki.chin.q.j.m.2023.04.001

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In this paper, a model of predator-prey with dual delay in maturation and
carrying capacity is discussed, in which the past activity of the prey should have an
impact on the carrying capacity, the mature prey initiates defense mechanisms to release
toxins when subjected to predation, and a commercial harvest of the prey is performed.
The stability of the equilibrium of the system in the absence of delay is examined and
the optimal harvesting strategy of the model is proven. By investigating the roots of the
characteristic equation and applying normalized theory, the properties of the coexistence
equilibrium of the system and the conditions for the occurrence of the Hopf bifurcation in
the neighborhood of the positive equilibrium are described for various combinations of
delays. In the end, numerical simulations are used to verify theoretical analysis results.

Singularity of Two Kinds of Four Cycle Graphs

YOU Xiao-jie, MA Hai-cheng, ZHANG Bin, LI Ya-lan

2023, 38(4): 349-359. doi:10.13371/j.cnki.chin.q.j.m.2023.04.002

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Let G be a finite simple graph and A(G) be its adjacency matrix. Then
G is singular if A(G) is singular. The graph obtained by bonding the starting vertices and ending vertices of three paths P_a1, P_a2, P_a3is called θ-graph, represented
by θ(a1,a2,a3). The graph obtained by bonding the two end vertices of the path
P_s to the vertices of the θ(a₁,a₂,a₃) and θ(b₁,b₂,b₃) of degree three, respectively,
is denoted by α(a₁,a₂,a₃,s,b₁,b₂,b₃) and called α-graph. β-graph is denoted when
β(a₁,a₂,a₃,b₁,b₂,b₃) =α(a₁,a₂,a₃,1,b₁,b₂,b₃). In this paper, we give the necessary and
sufficient conditions for the singularity of α-graph and β-graph, and prove that the
probability that a random given α-graph and β-graph is a singular graph is equal to 1423/2048
and 733/1024 , respectively.

Initial Boundary Value Problem for Pseudo-Parabolic p-Laplacian Type Equation with Logarithmic Nonlinearity

PAN Jia-hui, FANG Shao-mei

2023, 38(4): 360-369. doi:10.13371/j.cnki.chin.q.j.m.2023.04.003

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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.

Construction of a Class of Gerstenhaber Algebras

HOU Bo, KOU Wen

2023, 38(4): 370-378. doi:10.13371/j.cnki.chin.q.j.m.2023.04.004

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For any K-algebra A, based on Hochschild complex and Hochschild cohomology of A, we construct a new Gerstenhaber algebra, and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.

Subordination and Superordination Results for a Certain of Integral Operator Involving Generalized Mittag-Leffler Functions

WANG Xiao-yuan, LIU Ya-juan

2023, 38(4): 379-391. doi:10.13371/j.cnki.chin.q.j.m.2023.04.005

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In the paper, by making use of the principle of subordination between analytic
functions with the multiplier transforms defined by generalized Mittag-Leffler function, the
authors investigate subclasses of univalent analytic functions, such as starlike functions,
convex functions, close-to-convex functions and quasiconvex functions. Several inclusion
relationships, inequality properties, subordination and superordination results associated
with the multiplier transforms are proved and the sandwich-type results are also obtained.

Space-Time Legendre Spectral Collocation Methods for Korteweg-De Vries Equation

WANG Chuan, QIAO Yan

2023, 38(4): 392-400. doi:10.13371/j.cnki.chin.q.j.m.2023.04.006

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A Legendre-Legendre spectral collocation scheme is constructed for Kortewegde Vries (KdV) equation on bounded domain by using the Legendre collocation method
in both time and space, which is a nonlinear matrix equation that is changed to a
nonlinear systems and can be solved by the usual fixed point iteration. Numerical results
demonstrate the efficiency of the method and spectral accuracy.

The Explicit Formula for the Moore-Penrose Inverse of a 2×2 Block Matrix

ZENG Min, YUAN Yong-xin

2023, 38(4): 401-409. doi:10.13371/j.cnki.chin.q.j.m.2023.04.007

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The representation for the Moore-Penrose inverse of the matrix
A B
C D
is derived by using the solvability theory of linear equations, where A∈C^m×n, B∈C^m×p, C∈C^q×nand D∈C^q×p, with which some special cases are discussed.

The Existence of Ground State Solutions for a Class of Sublinear Kirchhoff Equations

XU Na

2023, 38(4): 410-414. doi:10.13371/j.cnki.chin.q.j.m.2023.04.008

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In this paper, we study a class of sublinear Kirchhoff equations:
- (a+ b \int_{\mathbb{R}^N} |\nabla u|^2dx)\Delta u+ V(x)u =f(x,u) \ \ &\text{in}\ \mathbb{R}^N,
where a,b>0, V :R^N→R can be sign-changing, and f :R^N×R→R. Under some conditions on V and f, we verify that the
problem possesses at least one energy solution by using
variational method.

Competitive Equilibrium of Central Bank Digital Currency and Private Cryptocurrency: A Perspective of Regulatory

SUN Xiao-lei

2023, 38(4): 415-423. doi:10.13371/j.cnki.chin.q.j.m.2023.04.009

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This paper constructs a searching model to study the impact of monetary
policy, fiscal policy, transaction costs of private cryptocurrency and regulatory uncertainty
on the competitive equilibrium between Central Bank Digital Currency and private
cryptocurrency. The results show: (1) Central Bank Digital Currency and private cryptocurrency can conditionally coexist. (2) The level of economic welfare is related to monetary
policy and fiscal policy but independent from the issuance and transaction of private
cryptocurrency. (3) Under complete competition of private cryptocurrency mining, the
mining cost becomes deadweight loss for whole society.

Exact Boundary Controllability for a 1-D Second-Order Quasilinear Hyperbolic System

WANG Ke, PAN Pan-pan

2023, 38(4): 424-440. doi:10.13371/j.cnki.chin.q.j.m.2023.04.010

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In this paper, we propose a second-order quasilinear hyperbolic system. By
means of the theory on semi-global C¹solution to first-order quasilinear hyperbolic
systems, we establish the existence and uniqueness of semi-global C²solution to this
second-order quasilinear hyperbolic system. After then, the general constructive framework
is utilized to obtain the local exact boundary controllability for this second-order system.
Keywords: First-order quasilinear hyperbolic system; Second-order quasilinear hyperbolic system; Semi-global solution; Exact boundary controllability

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