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2024, Vol. 44 No. 2   Published date: 28 June 2024
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• L^p-convergence Rate of the Tamed Euler Scheme for SDEs with Piecewise Continuous Drift Coefficient

  Hu Huimin, Gan Siqing

  2024, 44(2): 1-19. doi: 10.3969/j.issn.1006-8074.2024.02.001

  Abstract ( 1208 )   PDF (214KB) ( 189 )   　　

  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.

• Multiple Solutions for Nonhomogeneous Kirchhoff-Schrödinger-Poisson System with  p-Laplacian

  Wang Mei, Suo Hongmin, Zhang Xuemei, Wang Chenxi

  2024, 44(2): 20-35. doi: 10.3969/j.issn.1006-8074.2024.02.002

  Abstract ( 406 )   PDF (192KB) ( 212 )   　　

  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.

• Normalized Solutions for a Class of Kirchhoff Equations

  Liang Yanxia

  2024, 44(2): 36-50. doi: 10.3969/j.issn.1006-8074.2024.02.003

  Abstract ( 398 )   PDF (180KB) ( 181 )   　　

  In this paper we study the existence of normalized solutions to a class of Kirchhoff equations with the variational method, and obtain the normalized ground state solutions when $\mu>0$, and the local minimizers and the mountain pass type critical points for the energy functionals when $\mu<0$.

• Solvabilities of Interval Linear Inequalities over Max-min Algebra

  Li Haohao, Wang Lulu, Jia Shengnan

  2024, 44(2): 51-64. doi: 10.3969/j.issn.1006-8074.2024.02.004

  Abstract ( 377 )   PDF (165KB) ( 140 )   　　

  The characteristics of interval systems under the framework of extremum algebra is a meaningful research direction. In this paper we study the characterizations of various kinds of solvabilities of interval linear inequalities in the max-min algebra, including the weak solvability, strong solvability, tolerance solvability, strongly tolerance solvability, control solvability and strongly control solvability. Furthermore, we analyze the existence of solutions and the solvabilities, and show these two concepts are equivalent in certain situations. In addition, we find the maximum solutions corresponding to different kinds of solvabilities.

• Existence of Distinct Positive Integer Solutions to a Generalized Form of Erdös-Straus Conjecture

  You Lihua, Li Jiayin, Yuan Pingzhi

  2024, 44(2): 65-79. doi: 10.3969/j.issn.1006-8074.2024.02.005

  Abstract ( 406 )   PDF (192KB) ( 151 )   　　

  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 $t\geq k\geq 2$.

• Block-transitive Automorphism Groups of 3- (v,7,λ) Designs

  Chen Shihan, Dai Shaojun, Zhao Kun

  2024, 44(2): 80-91. doi: 10.3969/j.issn.1006-8074.2024.02.006

  Abstract ( 410 )   PDF (162KB) ( 138 )   　　

  Let ${\cal S}$=$({\cal P},{\cal B})$

  be a nontrivial $3$-$(v,7,\lambda)$ design with $\lambda\geq 2$. In this paper we show that if $G$ is a block-transitive automorphism group of $\cal S$, then $G$ is point-primitive of affine

  or almost simple type. In addition, the case that $G$ is a finite almost simple group with an alternating group socle $A_n$ is considered as well.

• Genus of Cartesian Product of a Complete Regular Tripartite Graph and a Bipartite Graph

  Guo Ting

  2024, 44(2): 92-102. doi: 10.3969/j.issn.1006-8074.2024.02.007

  Abstract ( 417 )   PDF (208KB) ( 154 )   　　

  Let $K_{m,m,m}$ ($m\geq1$) be a complete regular tripartite graph, and $G$ be a bipartite graph with girth greater than 4. In this paper, the genus of cartesian product of $K_{m,m,m}$ and $G$ with $\Delta(G)\leq 2m$ is determined. It generalizes the result by Bonnington and Pisanski, which gives the genus of cartesian product of $K_{m,m,m}$ and an even cycle. Moreover, the nonorientable genera of cartesian products of $K_{m,m,m}$ and some non-bipartite graphs are obtained.

• 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

  2024, 44(2): 103-125. doi: 10.3969/j.issn.1006-8074.2024.02.008

  Abstract ( 424 )   PDF (2097KB) ( 166 )   　　

  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.