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2022, Vol. 42 No. 1   Published date: 31 March 2022
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• Integer Vectors Arising From Cluster Algebras

  Fu Changjian, Geng Shengfei, Liu Pin

  2022, 42(1): 1-15. doi:

  Abstract ( 1809 )   PDF (7707KB) ( 515 )   　　

  There are four kinds of integer vectors arising from the structure theory of cluster algebras: $c$-vector, $d$-vector, $f$-vector and $g$-vector, which have played important roles in the study of cluster algebras. This paper is devoted to reviewing the concepts of those vectors and introducing their properties, research problems and research progress.

• A General Framework to Construct High­order Unconditionally Structure­preserving Parametric Methods

  Zhang Hong, Liu Lele , Qian Xu, Song Songhe

  2022, 42(1): 16-50. doi:

  Abstract ( 1153 )   PDF (22585KB) ( 485 )   　　

  High-order accurate and stable explicit methods are powerful in solving differential equations efficiently. In this work, we propose a systematic framework to trade off accuracy for stability, especially the unconditional preservation of strong stability, positivity, range boundedness and contractivity. The whole algorithm consists of three steps: (1) Introducing a stabilizing term in the continuous system; (2) Integrating the system using an explicit exponential method; (3) Substituting the exponential functions with suitable approximations. We first show that a class of first- and second-order exponential time difference Runge-Kutta schemes are capable to preserve structures unconditionally when suitable stabilization parameter is chosen. Then by adopting the integrating factor approach with high-order Runge-Kutta and multi-step schemes as underlying schemes, three different approximation techniques are developed to make high-order schemes unconditionally structure-preserving, i.e., (1) a Taylor polynomial approximation; (2) a recursive approximation; (3) an approximation using combinations of exponential and linear functions. The proposed parametric schemes can be deployed to stiff problems straightforwardly by treating the stiff linear term as an integrating factor. The resulting time integration methods retain the explicitness and convergence orders of underlying time-marching schemes, yet with unconditional preservation of structures. The proposed framework using the second and third approximations has relatively mild requirement on underlying schemes, i.e., all coefficients are non-negative. Thus the parametric Runge-Kutta schemes can reach up to the fourth-order, and there is no order barrier in parametric multi-step schemes. The only free parameter--the stablization parameter in the framework can be determined a priori based on the forward Euler conditions. Unlike implicit methods, the parametric methodology allows for solving nonlinear problems stably and explicitly. As an alternative to conditionally structure-preserving methods, the proposed schemes are promising for the efficient computation of stiff and nonlinear problems. Numerical tests on benchmark problems with different stiffness are carried out to assess the performance of parametric methods.

• Decoupling Analysis and Numerical Solution of Thermal/Acoustic Coupling Equations

  Zhu Liyan, Deng Youjun, Duan Chaohua, Li Tao

  2022, 42(1): 51-64. doi:

  Abstract ( 1346 )   PDF (5683KB) ( 503 )   　　

  In this paper we consider the equation coupled by a thermoelastic wave equation and a heat conduction equation in a homogeneous isotropic medium, and give the decoupling analysis and numerical implementation with the finite difference method. Inside solids, the governing equation of acoustic wave propagation consists of coupled heat conduction equation and thermoelastic dynamic equation due to the temperature effect and elastic deformation of medium acoustic parameters, which makes it very difficult to solve numerically. The bidirectional coupling is decoupled into sequential coupling according to the different characteristic time advances of the two equations. Omitting the influence of strain displacement on the heat conduction equation, we first solve the heat conduction equation, and then the thermoelastic wave equation is solved by taking the temperature field as an additional thermal load to obtain the strain displacement field of the structure. The heat conduction equation is solved by the classical finite difference method. We investigate the application of finite difference method to thermoelastic wave equation. Because the hyperbolic equation has high demand for stability of the algorithm, the ordinary explicit and implicit difference methods cannot achieve ideal effect. We apply the principle of numerical viscosity correction and the five points CDD8 format to finite difference method in the elastic wave equation. The program is coded in FORTRAN. The numerical results show that the accuracy and efficiency are satisfactory.

• The Difference of Mostar Index and Irregularity of Unicyclic and Bicyclic Graphs with Small Diameter

  Wu Tingzeng, Zeng Xiaolin

  2022, 42(1): 65-84. doi:

  Abstract ( 1280 )   PDF (8977KB) ( 292 )   　　

  Let $G$ be a connected graph. Gao et al. first introduced the invariant of $G$: $\Delta M(G)=M_{o}(G)-irr(G)$, and raised a problem: how to determine the extremal graphs among all connected graphs of order $n$ with respect to $\Delta M(G)$, where $M_{o}(G)$ and $irr(G)$ stand for the Mostar index and irregularity of $G$, respectively. In this paper, we characterize the upper bounds of $\Delta M(G)$ over all unicyclic graphs and bicyclic graphs with diameter 3, and determine the extremal graphs.

• The Greatest Common Divisor of Certain Set of Binomial Coefficients

  Xiao Jiaqi, Yuan Pingzhi, Lin Xucan

  2022, 42(1): 85-91. doi:

  Abstract ( 1226 )   PDF (3506KB) ( 328 )   　　

  In this paper, we prove that if $n\geq4$ and $a\ge 0$ are integers satisfying $a<\frac{n}{3}$, then

  $$\gcd\left(\left\{\binom{n}{k}:a<k<n-a\right\}\right)=\prod_{n=p^{m}+b(n,p),\ 0\le b(n,p)\leq a,} p,$$ where $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, and the product in the right hand side runs through all primes $p$ such that $ n=p^{m}+b(n,p), m\in\mathbb{N}$ and $0\le b(n,p)\leq a$.

  As an application of our result, we give an answer to a problem in Hong [16].

  
  

• A Sampled Second­order Stochastic Algorithm

  Wang Jing, Wang Xiangmei

  2022, 42(1): 92-103. doi:

  Abstract ( 1243 )   PDF (5879KB) ( 356 )   　　

  In this paper a sampled second-order stochastic algorithm (SSN-Lissa) is presented by combining the Lissa and SSN algorithms for large-scale machine learning optimizations. The linear convergence of the algorithm is established under the assumption that the objective functions are smooth and strongly convex. Numerical experiments show that the new algorithm is more effective than the Lissa and SSN.

• An Algorithm on Uniform Machine Scheduling Under Vertex Cover Constraints

  2022, 42(1): 104-110. doi:

  Abstract ( 1123 )   PDF (3959KB) ( 242 )   　　

  Given $m$ uniform machines and $n$ jobs, let the speed of the $j$th machine be $s_{j}$, the processing time of the $i$th job be $p_ {i}$, which involves that the load on the $j$th machine is $\frac{p_i}{s_j}$. Construct a weighted undirected graph $\hat{G} = (V, E; W) $, where the $n$ vertices of the graph represent the $n$ jobs and the vertex weight represents the processing time of the corresponding job. This paper studies the uniform machine scheduling problem under some vertex cover constraints, which is a combinatorial problem of two combinatorial optimization problems. The goal is to firstly determine a vertex cover of the graph $\hat{G}$, that is, a vertex subset of the graph, such that the subset contains at least one endpoint of each edge and then to minimize the makespan when all the jobs corresponding to the subset are put to process on the $m$ uniform machines. The problem is NP-hard. We design a $(2+{(m-1)\cdot s_m\over \sum_{j=1}^m s_j})$- approximate algorithm based on the layering algorithm and the LSPT algorithm for it, and it is realized that the approximation ratio is relatively good when the speeds of all machines are not much different.

• Topological Pressure and Factor Mapping of Sub­additive Potential Function

  Wang Wei, Gao Xiaoyan

  2022, 42(1): 111-116. doi:

  Abstract ( 1100 )   PDF (2512KB) ( 256 )   　　

  In this paper, under the standing hypothesis, an upper bound estimation of the topological pressure for sub

  additive potential functions is obtained by introducing the factor mapping.

• Construction of Robust ARMA Residual Control Chart with Applications in Financial Markets

  2022, 42(1): 117-129. doi:

  Abstract ( 1173 )   PDF (5557KB) ( 225 )   　　

  Traditional ARMA residual control charts are often sensitive to outliers, which easily leads to failures in monitoring. In order to solve this problem, this article uses the idea of robust statistics to revise the traditional ARMA residual control charts, and constructs a robust ARMA residual control chart algorithm to overcome the influence of outliers on the model. From the simulation and empirical results, it is known that when there are no outliers in the data, the monitoring results obtained by the traditional and robust ARMA residual control charts are basically the same; when there are outliers in the data, compared to the traditional ARMA residual control charts the robust ARMA residual control charts can more effectively resist the influence of outliers, and have better anti-interference and high tolerance.