数学季刊

李永凤, 朱城志, 刘艳伟

2023, 38(1): 1-19. doi:10.13371/j.cnki.chin.q.j.m.2023.01.001

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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.

Allen-Cahn方程的虚拟元方法

王培珍, 田旭

2023, 38(1): 20-29. doi:10.13371/j.cnki.chin.q.j.m.2023.01.002

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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.

有向图的加权解析挠率

任世全, 王冲

2023, 38(1): 30-49. doi:10.13371/j.cnki.chin.q.j.m.2023.01.003

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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.

具有勢壘的Sturm-Liouville算子的特徵值和特徵函數

SARWAR Qanitah 黄振友 ZAHID Abdul Hannan

2023, 38(1): 50-61. doi:10.13371/j.cnki.chin.q.j.m.2023.01.004

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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.

一类非凸优化问题的邻近拟牛顿方法的复杂性

金玲子

2023, 38(1): 62-84. doi:10.13371/j.cnki.chin.q.j.m.2023.01.005

摘要 ( 68 )

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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.

收益率椭球分布不确定下的均值-CVaR优化研究

卿乃侨

2023, 38(1): 85-96. doi:10.13371/j.cnki.chin.q.j.m.2023.01.006

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The article explores a mean-CVaR ratio model with returns distribution
uncertainty. To describe the uncertainty of returns distribution, a mixture ellipsoidal
distribution absorbing some typical distributions such as the mixture distribution and
and ellipsoidal distribution is introduced. Then, by using robust technique with some
assumptions, the original robust mean-CVaR ratio model can be formulated as a second-
order cone optimization model where the underlying random returns have a mixture
ellipsoidal distribution. As an illustration, the corresponding robust optimization models
are applied to allocations of assets in securities market. Numerical simulations are
presented to illustrate the relation between robustness and optimality and to compare
mixture ellipsoidal distribution to some typical distributions as well.

I型左删失下Topp–Leone分布的参数及可靠度函数的估计

龙沁怡, 徐丽平

2023, 38(1): 97-110. doi:10.13371/j.cnki.chin.q.j.m.2023.01.007

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Firstly, the maximum likelihood estimate and asymptotic confidence interval
of the unkown parameter for the Topp-Leone distribution are obtained under Type-I left
censored samples, furthermore, the asymptotic confidence interval of reliability function
is obtained based on monotonicity. Secondly, under different loss functions, the Bayesian
estimates of the unkown parameter and reliability function are obtained, and the expected
mean square errors of Bayesian estimates are calculated. Monte-Carlo method is used to
calculate the mean values and relative errors of the estimates. Finally, an example of life
data is analyzed by using the statistical method in this paper.

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