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Dynamical Analysis of a Toxic-phytoplankton-zooplankton Model with Chemotaxis and Allee Effects Dong Yuqin, Chen Shaoyu, Dai Binxiang Mathematical Theory and Applications    2023, 43 (4): 1-28.   DOI: 10.3969/j.issn.1006-8074.2023.04.001 Abstract （1328）      PDF（pc） （357KB）（162）       Knowledge map    This paper demonstrates the global existence and boundedness of the solutions of a toxic- phytoplankton-zooplankton model with chemotaxis and Allee effects in a smooth bounded domian with no-flux boundary condition. This result holds for arbitrary spatial dimension and small chemotaxis coefficients. It is also proved that the positive homogeneous steady state loses its stability when the chemotactic coefficient surpasses a threshold value, and the nonhomogeneous steady states bifurcate from the homogeneous steady state. Finally a numerical simulation is performed. Related Articles | Metrics

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On the Anti-concentration Functions of Some Familiar Families of Distributions Hu Zechun, Song Renming, Tan Yuan Mathematical Theory and Applications    2024, 44 (1): 1-15.   DOI: 10.3969/j.issn.1006-8074.2024.01.001 Abstract （916）      PDF（pc） （194KB）（153）       Knowledge map    Let $\{X_{\alpha}\}$ be a family of random variables following a certain type of distributions with finite expectation $\E[X_{\alpha}]$ and finite variance $\Var(X_{\alpha})$, where $\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \infty)\ni y\to \inf_{\alpha}\P\left(|X_{\alpha}-\E[X_{\alpha}]|\geq y \sqrt{\Var(X_{\alpha})}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including the uniform, exponential, non-degenerate Gaussian and student's $t$-distributions, the anti-concentration function is not identically zero, which means that the corresponding families of random variables have some sort of anti-concentration property; while for some other familiar families of distributions, including the binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero. Related Articles | Metrics

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Lp-convergence Rate of the Tamed Euler Scheme for SDEs with Piecewise Continuous Drift Coefficient Hu Huimin, Gan Siqing Mathematical Theory and Applications    2024, 44 (2): 1-19.   DOI: 10.3969/j.issn.1006-8074.2024.02.001 Abstract （821）      PDF（pc） （214KB）（109）       Knowledge map    In this paper we study the $L^p$-convergence rate of the tamed Euler scheme for scalar stochastic differential equations (SDEs) with piecewise continuous drift coefficient. More precisely, under the assumptions that the drift coefficient is piecewise continuous and polynomially growing and that the diffusion coefficient is Lipschitz continuous and non-zero at the discontinuity points of the drift coefficient, we show that the SDE has a unique strong solution and the $L^p$-convergence order of the tamed Euler scheme is at least 1/2 for all $p \in [1,\infty)$. Moreover, a numerical example is provided to support our conclusion. Related Articles | Metrics

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A Semi-implicit Collocation Scheme for the Allen-Cahn Equation Tong Yanlei, Yao Xiaozhen, Li Mengting, Li Yiwen, Weng Zhifeng Mathematical Theory and Applications    2023, 43 (4): 106-122.   DOI: 10.3969/j.issn.1006-8074.2023.04.007 Abstract （619）      PDF（pc） （960KB）（200）       Knowledge map    In this paper, a semi-implicit collocation scheme and the corresponding discrete linear equation system are firstly derived for the Allen-Cahn equation by discretizing it in spatial direction with the barycentric Lagrange interpolation collocation method, in time direction with the backward Euler scheme and the Crank-Nicolson scheme respectively and treating its nonlinear term with an explicit scheme. Then, the consistency of the one-dimensional spatial semi-discrete schemes and the two-dimensional full discrete schemes are analyzed. Finally, the high accuracy and energy decreasing law of the semi-implicit collocation scheme are verified by numerical examples. Comparing with the two kinds of classical difference schemes, the numerical results show that the proposed scheme can achieve higher accuracy with fewer nodes. Related Articles | Metrics

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Numerical Simulation Algorithms for Stochastic Differential Equations in Systems Biology Niu Yuanling, Chen Lin, Chen luonan Mathematical Theory and Applications    2023, 43 (4): 76-92.   DOI: 10.3969/j.issn.1006-8074.2023.04.005 Abstract （608）      PDF（pc） （508KB）（332）       Knowledge map    Many phenomena in systems biology, such as the biochemical reaction process, the evolution of ecosystems, the spread of infectious diseases, can be described by stochastic differential equations (SDEs). Considering the influence of randomness, stochastic differential equation models can describe the evolution of variables over time more accurately than deterministic differential equation models. However, the analytical solutions of most stochastic differential equations cannot be obtained. Even though some of them can be obtained, the forms of the solutions are usually extremely complex. One therefore requires proper numerical methods to approximate their solutions on computers. These stochastic differential equation models in systems biology usually have the properties of high dimension, high nonlinearity, and the solutions being located in a specified region. It is difficult to simulate them numerically. This paper reviews the numerical simulation algorithms of several typical models in systems biology (biochemical reaction models, ecosystem models, infectious disease models, population genetics models, cell differentiation models), and briefly introduces their advantages and disadvantages. Related Articles | Metrics

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Notes on Systems of Linear Congruence Equations Shi Wenzhang, Liu Heguo Mathematical Theory and Applications    2023, 43 (4): 29-47.   DOI: 10.3969/j.issn.1006-8074.2023.04.002 Abstract （591）      PDF（pc） （206KB）（179）       Knowledge map    The Chinese Remainder Theorem is a fundamental principle in number theory. In this paper we start with the canonical form of integer matrix under modular transformations and give another proof to the Chinese Remainder Theorem. Then by using the invariant factors we give some sufficient and necessary conditions for the existence of solutions for two classes of systems of linear congruence equations with multiple variables, and further more, find the number of solutions to the systems. Related Articles | Metrics

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The Mordell-Weil Groups of Cubic Pencils Mo Jiali Mathematical Theory and Applications    2023, 43 (4): 48-58.   DOI: 10.3969/j.issn.1006-8074.2023.04.003 Abstract （549）      PDF（pc） （175KB）（147）       Knowledge map    In this paper we study the influences of the base points of cubic pencils on the Mordell-Weil groups. Specifically, we investigate and classify the cubic pencils with 8, 7 and 6 base points in general position, and give some applications. Related Articles | Metrics

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Field-Split Preconditioners for Solving Three-dimensional Reservoir Flows in Fractured Porous Media Yang Nian, Yang Haijian, Shao Baiqiang Mathematical Theory and Applications    2023, 43 (4): 123-140.   DOI: 10.3969/j.issn.1006-8074.2023.04.008 Abstract （536）      PDF（pc） （1756KB）（126）       Knowledge map    With the applications of Newton-Krylov method in solving large sparse nonlinear equations, the design of linear preconditioners plays a vital role in the whole solver. In this paper, we study the application of different combinations of field-split (FS) preconditioners based on physical and domain decomposition methods to the unsteady flow problem of fractured porous media. Under the framework of domain decomposition technology, several new FS preconditioners are considered: the additive FS preconditioner, the multiplicative FS preconditioner, the Schur-complement FS preconditioner and the constrained pressure residual (CPR) preconditioner. The corresponding subsystems are approximately solved by the restricted additive Schwarz (RAS) algorithm. In order to further improve the performance of the FS preconditioner, we designe a two-level FS preconditioner. Numerical experiments on Tianhe-2 supercomputer show that the proposed preconditioner has a good robustness and parallel scalability. Related Articles | Metrics

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An Unconditionally Energy Stable Numerical Scheme for the Modified Phase Field Crystal Equation Liang Yihong, Jia Hongen Mathematical Theory and Applications    2023, 43 (4): 59-75.   DOI: 10.3969/j.issn.1006-8074.2023.04.004 Abstract （530）      PDF（pc） （1434KB）（147）       Knowledge map    This paper constructs a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme for the modified phase field crystal equation with periodic boundary conditions. The unique solvability, unconditionally energy stability and unconditionally temporal convergence of order 2 of the numerical scheme are showed by introducing a Lagrange multiplier to deal with the nonlinear terms and adopting the second-order Crank-Nicolson method to discrete time. Numerical experiments are given in the last section to validate the efficiency of the proposed scheme. Related Articles | Metrics

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A Predictor-corrector Smoothing Newton Method for Solving the Special Weighted Linear Complementarity Problem He Xiaorui, Tang Jingyong Mathematical Theory and Applications    2023, 43 (4): 93-105.   DOI: 10.3969/j.issn.1006-8074.2023.04.006 Abstract （501）      PDF（pc） （220KB）（116）       Knowledge map    In this paper, we study the method for solving the special weighted linear complementarity problem. Based on a weighted smoothing function, we reformulate the problem as a system of smooth nonlinear equations and then propose a predictor-corrector smoothing Newton method to solve it. Under some suitable conditions, we show that the algorithm has the global and local quadratic convergence properties. In particular, when the solution set is nonempty we show that the merit function sequence converges to zero. Numerical experiments demonstrate that our algorithm is effective. Related Articles | Metrics

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Reflected Stochastic Partial Differential Equations with Elliptic Operators Qian Hongchao, Li Ruizhi, Gui Yewei, Peng Jun Mathematical Theory and Applications    2024, 44 (1): 16-30.   DOI: 10.3969/j.issn.1006-8074.2024.01.002 Abstract （477）      PDF（pc） （214KB）（131）       Knowledge map    This paper is concerned with a class of multi-dimensional reflected stochastic partial differential equations with elliptic operators, whose solutions are constrained on a bounded convex domain. The aim of this paper is to establish the existence and uniqueness theorem of solutions for the reflected stochastic partial differential equations with the penalization method. Related Articles | Metrics

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Oscillations in a Predator-prey Model with Cooperative Breeding of Predators Zhang Yun, Su Juan, Zou Lan Mathematical Theory and Applications    2024, 44 (1): 31-44.   DOI: 10.3969/j.issn.1006-8074.2024.01.003 Abstract （447）      PDF（pc） （1077KB）（114）       Knowledge map    The behavior of cooperation is common in animals. A lot of work showed that cooperative predation in predator-prey systems will induce complicated dynamics such as complicated coexistence of equilibria and population oscillations. In this paper, we consider a predator-prey system with cooperative breeding of predators, and focus on the oscillations related to the Hopf bifurcations of the model. Firstly, we discuss the number of internal equilibrium and the qualitative properties of internal equilibrium, and show that the equilibrium ~$E_{1}$~ is of center type under certain parameter conditions. Secondly, we analyze the Hopf bifurcation at $E_{1}$, and give the corresponding bifurcation conditions. Finally, we give an example of a weak focus of order 3, and study the complex oscillation behaviors of the model through this example. Because of the coexistence of multiple limit cycles, even under the same parameter conditions, the periods and amplitudes vary with initial values. It also reflects the high sensitivity of solutions on initial values. Related Articles | Metrics

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Embeddedness, Compactness and Uniform Convexity of Anisotropic Banach Space Valued Musielak-Orlicz Spaces He Yal, Xu Jingshi Mathematical Theory and Applications    2024, 44 (1): 45-61.   DOI: 10.3969/j.issn.1006-8074.2024.01.004 Abstract （445）      PDF（pc） （185KB）（139）       Knowledge map    In this paper we give the necessary and sufficient conditions for the embedding of an anisotropic Banach space valued Musielak-Orlicz space into another anisotropic Banach space valued Musielak-Orlicz space, and the necessary and sufficient conditions for a subset of an anisotropic Banach space valued Musielak-Orlicz space to be relatively compact, and characterize the uniform convexity of anisotropic Banach space valued Musielak-Orlicz space. Related Articles | Metrics

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Identifying and Locating Codes on Some  Cayley  Graphs Lu Qiming, Song Shujiao Mathematical Theory and Applications    2024, 44 (1): 78-92.   DOI: 10.3969/j.issn.1006-8074.2024.01.006 Abstract （445）      PDF（pc） （2576KB）（119）       Knowledge map    In 2019, Junnila, Laihonen and Paris studied the identifying codes and locating codes on the circulant graphs $C_{n}(1,d)$, $C_{n}(1,d-1,d)$ and $C_{n}(1,d-1,d,d+1)$. In this paper we study the identifying codes and locating codes of Cayley graphs on the Abelian groups of order $p^{2}$ and order $2n$ within 8 degrees, determine their optimal bounds, and give some examples that reach the optimal bounds. Our results generalize multiple results on identifying and locating codes. Related Articles | Metrics

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The Gerschgorin Disk Theorem and Regularity Conditions for Complex Interval Matrices Cheng Long, Xia Dandan, Li Yaotang Mathematical Theory and Applications    2024, 44 (1): 109-121.   DOI: 10.3969/j.issn.1006-8074.2024.01.008 Abstract （426）      PDF（pc） （252KB）（118）       Knowledge map    In this paper, the Gerschgorin disk theorem on eigenvalues of complex matrices is generalized to complex interval matrices, in which the Gerschgorin disk regions of eigenvalues of complex interval matrices are presented. It is showed that the Gerschgorin disk regions are contained in the Gerschgorin square regions for eigenvalues of complex interval matrices. Then, two new sufficient conditions for the regularity of complex interval matrices are obtained by applying the Gerschgorin disk theorem of complex interval matrices. Related Articles | Metrics

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The Transition Probability and Instantaneous Distribution of the Output Process of  M/M/1  Queueing System Li Junping, Cheng Lan Mathematical Theory and Applications    2024, 44 (1): 93-108.   DOI: 10.3969/j.issn.1006-8074.2024.01.007 Abstract （420）      PDF（pc） （209KB）（100）       Knowledge map    In this paper, we study the output process of M/M/1 queueing system, obtain its transition probability at arrival times and instantaneous distribution at any time. Related Articles | Metrics

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Inclusion of Musielak-Orlicz Type Spaces Liu Kaituo, Yu Qian Mathematical Theory and Applications    2024, 44 (1): 62-77.   DOI: 10.3969/j.issn.1006-8074.2024.01.005 Abstract （405）      PDF（pc） （220KB）（91）       Knowledge map    In this paper, the inclusion of Musielak-Orlicz spaces is proved, which is a generalization of the inclusion of Orlicz spaces. Due to the fact that a variable exponent space is a special case of the Musielak-Orlicz space, we can obtain the equivalent relations of inclusion conditions between the spaces $L^{p(\cdot)}$ and $L^{q( \cdot)}$. At the same time, we get the inclusion of weighted Orlicz spaces. In addition, we also prove the inclusion of weak Musielak-Orlicz spaces, which is a generalization of weak variable exponent spaces. As applications, the equivalent relations of inclusion conditions between weak variable exponent spaces $wL^{p(\cdot)}$ and $wL^{q(\cdot)}$ and the inclusion of weak Orlicz spaces are established. Related Articles | Metrics

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Multiple Solutions for Nonhomogeneous Kirchhoff-Schrödinger-Poisson System with  p-Laplacian Wang Mei, Suo Hongmin, Zhang Xuemei, Wang Chenxi Mathematical Theory and Applications    2024, 44 (2): 20-35.   DOI: 10.3969/j.issn.1006-8074.2024.02.002 Abstract （344）      PDF（pc） （192KB）（123）       Knowledge map    In this paper we consider the existence of solutions to the nonhomogeneous quasilinear Kirchhoff- Schr\"{o}dinger-Poisson system with $ p $-Laplacian: \begin{equation*}\label{1ip} \begin{cases}\displaystyle - \Big(a-b \int_{\mathbb{R}^3}|\nabla u|^p{\rm d}x \Big)\Delta_p u+|u|^{p-2}u+\lambda\phi_uu= |u|^{q-2}u+h(x), &\text{ }x\in \mathbb{R}^3,\\ -\Delta\phi=u^2, &\text{ }x\in \mathbb{R}^3,\\ \end{cases} \end{equation*} where $ a,b>0 $, $ \frac{4}{3} < p < \frac{12}{5} $, $ p < q < p^* =\frac{3p}{3-p} $, $ \lambda > 0 $. Under suitable assumptions on $ h(x) $, the existence of multiple solutions of the system is obtained by using the Ekeland variational principle and the Mountain pass theorem. Related Articles | Metrics

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Prediction of Active Value of Anti-breast Cancer Drug Candidates Based on GA-BP Neural Network Model Shang Yaxin , Lei Xiaojie, Fang Ziniu , Zhang Hongwei Mathematical Theory and Applications    2024, 44 (2): 103-125.   DOI: 10.3969/j.issn.1006-8074.2024.02.008 Abstract （340）      PDF（pc） （2097KB）（97）       Knowledge map    Screening for anti-breast cancer drug candidates is of great significance in the treatment of breast cancer. Antihormonal therapy for breast cancer is often used in breast cancer patients with ERα expression, and the higher the anti-ERα activity value, the more effective the drug is for treatment. In this paper, 729 molecular descriptors of the compound are filtered firstly using the gradient boost model XGBoost and the distance correlation coefficient matrix, then based on the filtered 20 molecular descriptors with their activity values and the genetic algorithm, a GA-BP neural network model is established, which has a mean squared error MSE=0. 105 and a coefficient of determination $\mathrm{R}^2=0. 946$, and therefore is a high-precision model for screening potential drugs based on data mining techniques.  Related Articles | Metrics

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Existence of Distinct Positive Integer Solutions to a Generalized Form of Erdös-Straus Conjecture You Lihua, Li Jiayin, Yuan Pingzhi Mathematical Theory and Applications    2024, 44 (2): 65-79.   DOI: 10.3969/j.issn.1006-8074.2024.02.005 Abstract （328）      PDF（pc） （192KB）（83）       Knowledge map    In this paper, we study the (distinct) positive integer solution of the equation \begin{equation*}\label{eq12}\frac{k}{n} = \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_t}\end{equation*} with $n>k\geq 2$ and $ t\geq 2$. We show that the above equation has at least one distinct positive integer solution if it has a positive integer solution. When $k=5$, we show the above equation has at least one distinct positive integer solution for all $n\geq 3$ except possibly when $n\equiv 1, 5041, 6301, 8821, 13861, 15121(\mbox{mod } 16380)$ with $t=3$, and for all $n\geq 3$ except possibly when $n\equiv 1, 81901(\mbox{mod } 163800)$ with $t=4$. Furthermore, we point out that the above equation has at least one distinct positive integer solution for all $n(>k)$ when $t\geq k\geq 2$. Related Articles | Metrics