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2021, Vol. 41 No. 4   Published date: 30 December 2021
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• Peakon, Pseudo-Peakon, Periodic Peakon and Compacton Determined by Exact Solutions of Singular Nonlinear Traveling Wave Systems

  Li Jibin

  2021, 41(4): 1. doi:

  Abstract ( 1666 )   PDF (26699KB) ( 305 )   　　

  In this paper, we first study the exact peakon, periodic peakon, pseudo-peakon as well as the compacton solutions for the generalized Camassa-Holm equation and the Degasperis-Procesi equation. Based on the method of dynamical systems and the theory of singular traveling wave equations, the exact explicit parametric representations of the above mentioned solutions are derived. These solutions tell us that peakon is a limit solution of a family of periodic peakons or a limit solution of a family of pseudo-peakons under two classes of limit senses. The pseudo-peakon and pseudo-periodic peakon family are smooth classical solutions with two time scales.

  
  

  Second, we use some nonlinear wave equation models to show that there exist various exact explicit peakon solutions, which are different from the peakon solutions given by the generalized Camassa-Holm equation and the Degasperis-Procesi equation.

  
  

  Third, we point out that the so called ``peakon equations'' in some references have no peakons. Corresponding to these ``peakon equations'', their traveling systems are the singular traveling systems of the second kind, which can not have the peakon solution.

• On the Geodesic-transitivity of Finite Graphs

  2021, 41(4): 32. doi:

  Abstract ( 1376 )   PDF (210KB) ( 471 )   　　

  The symmetry problem for finite graphs has been extensively investigated over the past

  few decades. This article is devoted to giving an introduction in one particular family of symmetric

  graphs, namely (locally) $s$-geodesic-transitive graphs. Recently, substantial progress has been made on the study of this family of graphs, many

  open problems have been solved, and many new research problems have arisen.

  The methods used in this area range

  from deep group theory, including the finite simple group classification, through to combinatorial

  techniques.

• Extended Binding Number Results on Fractional (g, f, n, m)­critical Deleted Graphs

  Lan Meihui, Gao Wei

  2021, 41(4): 50. doi:

  Abstract ( 1237 )   PDF (153KB) ( 327 )   　　

  As an extension of the factor, the fractional factor allows each edge to give a real number in the range of 0 to 1, and degree of fraction of each vertex to be controlled within a certain range (determined by the values of functions g and f, corresponding to the upper and lower fractional degree boundary). The score factor has a wide range of applications in communication networks, and the score critical deleted graph can be used to measure the feasibility of transmission when the network is damaged at a certain moment. In this short note, we mainly present some extended binding number conclusions on fractional (g, f, n, m)­critical deleted graphs.

• Higher Accuracy Shape­preserving Modeling Based on the Two­level Fitting Method

  Yang Dangfu, Liu Shengjun, Liu Pingbo , Liu Xinru

  2021, 41(4): 57. doi:

  Abstract ( 1291 )   PDF (24680KB) ( 231 )   　　

  Compactly supported radial basis function (CSRBF) has been widely used in surface modeling methods to interpolate or approximate the given data, which avoids solving a large dense linear system with a proper supported radius. The surfaces reconstructed by the CSRBF-based method usually are not shape preserving, while the multivariate multiquadric quasi-interpolation results the lower approximation accuracy. In this paper, we introduce a two-level fitting method to conduct the shape-preserving modelling with a higher accuracy. An initial shape-preserving model is constructed by using the lower accuracy quasi-interpolation, and then a CSRBF-based networks interpolation is performed to compensate the errors between the initial fitting model and the given data, then the higher accuracy shape-preserving model can be obtained. Moreover, we discuss the choice of the smoothing factor in quasi-interpolation and the supported radius in CSRBF-based networks, and an empirical formula between them is constructed. The numerical examples demonstrate the performance of our method.

• A Note on the Minimal Nonnegative Solution for Regular M-­matrix Algebraic Riccati Equations

  Guan Jinrui, Ren Fujiao

  2021, 41(4): 77. doi:

  Abstract ( 1333 )   PDF (3232KB) ( 456 )   　　

  Research on the theories and efficient numerical methods of M-­matrix algebraic Riccati equations (MARE) has become a hot topic in recent years. In this paper, we study the existence of minimal nonnegative solution for MARE and give a new proof to the existence of minimal nonnegative solution for the MARE associated with a regular M-­matrix, which is much simpler than the original proof. In addition, we give a wider condition to guarantee the existence of minimal nonnegative solution for the MARE, which is an extension of the existing results.

• Extension of a p²-­dimensional Fusion Category with Applications

  Chen Yashu, Dong Jingcheng

  2021, 41(4): 83. doi:

  Abstract ( 1261 )   PDF (196KB) ( 425 )   　　

  This paper studies the extension of a p²-dimensional fusion category, obtains all possible category types, and applies it to the classification of semisimple Hopf algebras. In addition, this paper also studies the cases when the grading group is Zq and S3.

• Syzygies of Points in the Projective Plane

  Mo Jiali Yu Qi

  2021, 41(4): 92. doi:

  Abstract ( 1210 )   PDF (248KB) ( 546 )   　　

  In this paper, we study the question about the syzygies of points in the projective plane. Firstly, let $X$ be a finite set consists of 7 distinct points in projective plane $\mathbb{P}^2$. We give the representation of syzygies of $X$, and the minimal free resolutions of corresponding saturated homogeneous ideal $I_X$. According to the number and position of the base points of the linear system, all planar cubic linear systems are classified, and 11 different planar cubic linear systems are obtained.

• Well-posedness for Fractional Nonclassical Diffusion Equations with Time-dependent Diffusion Coefficients

  2021, 41(4): 100. doi:

  Abstract ( 1445 )   PDF (200KB) ( 572 )   　　

  This paper discusses the well-posedness problem of fractional nonclassical diffusion equations with time-dependent dissipation coefficients. Using the nonclassical Faedo-Galerkin method, the interpolation inequality and the control convergence principle, the existence, uniqueness and continuous dependence on initial values of the global weak solution in $\mathcal{H}^{\theta} (0 < \theta \leq 1) $ for the equations are obtained, where the nonlinearity $f$ satisfies the polynomial growth of arbitrary order.

• Research on Algorithm of Minimum Edge Covering Problem on Hypergraphs

  2021, 41(4): 109. doi:

  Abstract ( 1386 )   PDF (1350KB) ( 894 )   　　

  In this paper, the minimum edge covering problem of general hypergraphs and a class of special hypergraphs is investigated. The minimum edge covering problem of hypergraphs is a NP-hard problem. We design a layering algorithm to solve the problem, which the approximate ratio is reached $f$ and the time complexity is $O(km)$; then we provide two tight examples. The minimum edge covering problem of the special hypergraphs is solvable in polynomial time, and the strategy to solve the problem is: a corresponding root tree is firstly constructed, and then the tree traverses from down to up according to the level of each node with the dynamic programming, which based on the particularity of the root tree; MEC algorithm is designed for the minimum edge covering of special hypergraphs and the time complexity is $O({m^3})$.

• Bank Loan Portfolio Optimization Model Based on SQP Algorithm

  2021, 41(4): 119. doi:

  Abstract ( 1290 )   PDF (227KB) ( 556 )   　　

  Existing models for portfolio optimization of commercial banks use the distribution of returns as a normal distribution, which does not conform to the characteristics of the actual yield, and most studies do not consider the impact of existing loans on yield and risk, making the portfolio risk assessment improperly. The portfolio optimization model considering the stock loan under the stable distribution can reflect the characteristics of the actual yield, correctly assess the loan portfolio risk, and control the bias to a certain extent, so that the loan portfolio obtains excess returns. Considering the requirements of risk dispersion, the introduction of risk concentration to restrict the allocation proportion of incremental loans of the loan portfolio to avoid the additional risks caused by a certain loan, thus establishing a new bank loan portfolio optimization model. By analyzing and optimizing the model characteristics, target function form and number of variables, SQP algorithm is selected to allocate the final incremental loan ratio. After testing with the actual data, the new model is simple and feasible, so it can be actually used in the selection of bank loans.