圆锥曲线的三次有理多项式参数化

圆锥曲线重新参数化可以提高曲线参数的均匀性,且增强在拼接点处的光滑性.常用的参数化方法是采用一次有理多项式或二次有理多项式.采用三次有理多项式对圆锥曲线重新参数化,使曲线的次数由二次升到六次.以圆弧为例所得的实验结果袁明,在两段圆弧的公共点处的连续性为C~3,而且三次有理多项式参数化与弧长参数化的弦长偏差相比二次有理多项式参数化减小两个数量级.     

泰勒级数的发现及其思想方法

研究了泰勒级数的发现过程,分析了其中的数学思想方法,指出了数学思想的继承性以及合情推理在数学发现中的作用.在数学教学中应重视合情推理的教学,它是培养学生创新能力的有效手段.     

A product integration method for the approximation of the early exercise boundary in the American option pricing problem

Integral equation methods are now becoming well‐established tools in the study of financial models used in theory and practice. In this paper, we investigate the fully nonlinear weakly singular nonstandard Volterra integral equations representing the early exercise boundary of American option contracts, which gained popularity in recent years. We propose a product integration approach based on linear barycentric rational interpolation to solve the problem. The price of the option will then be computed using the obtained approximation of the early exercise boundary and a barycentric rational quadrature. The convergence of the approximation scheme will also be analyzed. Finally, some numerical experiments based on the introduced method are presented and compared with some exiting approaches.     

On dynamical behavior for optical solitons sustained by the perturbed Chen–Lee–Liu model

This study investigates the perturbed Chen–Lee–Liu model that represents the propagation of an optical pulse in plasma and optical fiber. The generalized exponential rational function method is used for this purpose. As a result, we obtain some non-trivial solutions such as the optical singular, periodic, hyperbolic, exponential, trigonometric soliton solutions. We aim to express the pulse propagation of the generated solutions, by taking specific values for the free parameters existed in the obtained solutions. The obtained results show that the generalized exponential rational function technique is applicable, simple and effective to get the solutions of nonlinear engineering and physical problems. Moreover, the acquired solutions display rich dynamical evolutions that are important in practical applications.     

Soliton solution to the complex modified Korteweg–de Vries equation on both zero and nonzero background via the robust inverse scattering method

In this paper, based on the robust inverse scattering method, we construct two kinds of solutions to the focusing modified Korteweg–de Vries equation. One is the classical soliton solution under the zero background condition and the other one is given through the nonzero background. Especially, for the nonzero background case, we choose a special spectral parameter such that the nonzero background solution is changed into the rational travelling waves. Finally, we also give a simple analysis of the soliton as the time $t$ is large, then we give the comparison between the exact solution and the asymptotic solution.     

A class of two-dimensional rational maps with self-excited and hidden attractors

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan—Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.     

矩形网格上二元有理插值的存在性问题

In this paper, making use of bivariate polynomial Lagrange iterpolation formula on rectangular grids, we set up the existence criterion of bivariate rational interpolants problem and its representation formula. Numerical examples are given.     

Lump and Stripe Soliton Solutions to the Generalized Nizhnik-Novikov-Veselov Equation

With the aid of the truncated Painlevé expansion, a set of rational solutions of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov (GNNV) equation with the quadratic function which contains one lump soliton is derived. By combining this quadratic function and an exponential function, the fusion and fission phenomena occur between one lump soliton and a stripe soliton which are two kinds of typical local excitations. Furthermore, by adding a corresponding inverse exponential function to the above function, we can derive the solution with interaction between one lump soliton and a pair of stripe solitons. The dynamical behaviors of such local solutions are depicted by choosing some appropriate parameters.     

离散的Painlevé方程Ⅱ的有理解

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＮｏｎｌｉｎｅａｒｉｎｔｅｇｒａｂｌｅｄｉｓｃｒｅｔｅｓｙｓｔｅｍｓａｒｅｎｏｗａｔｔｒａｃｔｉｎｇｍｕｃｈａｔｔｅｎｔｉｏｎ．Ａｍｏｎｇｔｈｅｍ,ｔｈｅｄｉｓｃｒｅｔｅＰａｉｎｌｅｖéｅｑｕａｔｉｏｎｓａｒｅｅｘｐｅｃｔｅｄｔｏｂｅｔｈｅｍｏｓｔｆｕｎｄａｍｅｎｔａｌｏｎｅｓｉｎａｎａｌｏｇｙｔｏｔｈｅｃｏｎｔｉｎｕｏｕｓｃａｓｅ．Ｃｌｏｓｅｒｓｔｕｄｉｅｓａｒｅｈｏｗｒｅｖｅａｌｉｎｇｔｈｅｉｒｒｉｃｈｍａｔｈｅｍａｔｉｃａｌｓｔｒｕｃｔｕｒｅｓ,ｓｕｃｈａｓ…     

关于模的最大有理扩张

ｉ）设Ｍ_ｉ为一族左Ｒ－模,记Ｍ_ｉ为Ｍ_ｉ的最大有理扩张,本文首先考察了∑⊕Ｍ_ｉ和∑⊕Ｍ_ｉ之间的关系,并证明了如对环Ｒ上任何一族模上述两者相等的充要条件是环Ｒ上每一个模都有理完全．ｉ）利用ｉ）研究了无零因子环上模的最大有理扩张,得到了一些结果．