一类Turan型多项式序列的构造及其性质

应用实系数多项式的性质构造了一类满足Turan型不等式的多项式序列,证明了该多项式序列的几个性质,并给出了一些应用.     

实系数三、四次多项式是von Neumann多项式的充要条件

本文用初等方法证明了实系数三、四次多项式是von Neumann多项式的便于应用的充要条件。     

一元复系数多项式实根的判定

对于一元实系数多项式的实根问题,运用斯图姆(Sturm)方法不仅可以确定其实根的个数以及正负根的个数,而且对于任意给定的区间(a,b)可以确定这个多项式在此区间内实根的个数,但是对于一元复系数多项式呢?本文给出一般方法把一元复系数多项式的实根问题归     

多项式函数的次数与极值点的关系

本文给出并严格证明了"一元实系数偶数(≥2)次多项式函数有极值"这一定理,并对多项式函数的次数及最高次项系数的符号与极值点的关系进行了归纳.     

实多项式方程有解的非标准判定 *

对于一个系数在可计算序域上的多元多项式方程 ,给出了该方程有实解的两个判别定理 .在此基础上研究了二元多项式 ,从而给出了判定二元多项式的实零点存在性以及半定性的有效方法 .此外 ,藉助于计算机 ,处理了几个有关实例 .处理手段是 :通过无限小量的引进 ,将问题所涉及的系数域扩充为一个可计算的非Archimedes序域 .     

一元多项式的根

对于1’>O,如果了(劣)~‘扩+‘一:扩一’十…十al劣+a0是复系数一元:次多项式,那么方程f(:)二。,即 a·‘,+‘一、‘,一‘+…+a,:+a。=0②①叫做复系数一元二次方程.方程②的根,也称多项式① 的根. 一27一中学数学(湖北)1992.12 类似地,如果j(,)是实系数(或有理系数、整系数等)一元:(、>0)次多项式,那么方程j(幻二O叫做实系数(或有理系数、整系数等)一元:次方程. 关于多项式的根的个数有以下重要定理: l代数墓本定理一元:次多项式在复数集中至少有一个根. :799年伟大的数学家高斯证明了这一重要定理· 2根的个数定理一元二次多项式有且仅有…     

实系数二元二次多项式可实分解的条件及其操作

实系数二元二次多项式可实分解的条件及其操作４３００６２湖北大学林六十安系数二元二次多项式可实分解的条件已有不少论述，但一般都是判定定理．如何分解？未加阐述．本文给出一个简易的又便于操作的判定定理．定理当且仅当方程ａｌ’一ｈｉ＋ｃ。０（ａ一０）和ａｓ’...     

用特征方程解二元二次方程组

用特征方程解二元二次方程组胡圣团（津市职业中专学校４１５４００）为了下面的需要，我们先讨论一下实系数二元二次多项式的复分解问题．设是一实系数二元二次多项式．由解析几何理论知识，如何ｇ（ｘ，ｙ）能解，则二次曲线ｇ（ｘ，ｙ）＝０是退化的，即曲线上有奇点，...     

关于实系数多项式实根的定限方法

本文用一种严谨的方法讨论有实用价值的实系数多项式的实根的定限方法     

关于多项式求根的一个并行算法的收敛性

Ⅰ 引言 设是首项系数为1的实系数或复系数的n次多项式Durand和Kerner独立地提出了求(1)的所有根r_1,…,r_n的并行算法:     

Construction of polynomial solutions to some boundary value problems for Poisson’s equation

A polynomial solution of the inhomogeneous Dirichlet problem for Poisson’s equation with a polynomial right-hand side is found. An explicit representation of the harmonic functions in the Almansi formula is used. The solvability of a generalized third boundary value problem for Poisson’s equation is studied in the case when the value of a polynomial in normal derivatives is given on the boundary. A polynomial solution of the third boundary value problem for Poisson’s equation with polynomial data is found.     

Apparent singular points of factors of reducible generalized hypergeometric equations

We consider a reducible generalized hypergeometric equation, whose sub‐equation possesses apparent singular points. We determine the polynomial whose roots are these points. We show that this polynomial is a generalized hypergeometric polynomial.     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

Construction of polynomial solutions of a linear second-order differential equation

For given particular polynomial solutions, we construct polynomial coefficients of a second-order differential equation. We obtain a criterion for the equation in question to have two solutions of the form \(u_i (t) = a_i t^{\varrho _i } + t^{\varrho _i - 1} \). Under certain restrictions on the parameters, polynomial solutions of the Heun equation are given.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

Kalman particle swarm optimized polynomials for data classification

Data classification is an important area of data mining. Several well known techniques such as decision tree, neural network, etc. are available for this task. In this paper we propose a Kalman particle swarm optimized (KPSO) polynomial equation for classification for several well known data sets. Our proposed method is derived from some of the findings of the valuable information like number of terms, number and combination of features in each term, degree of the polynomial equation etc. of our earlier work on data classification using polynomial neural network. The KPSO optimizes these polynomial equations with a faster convergence speed unlike PSO. The polynomial equation that gives the best performance is considered as the model for classification. Our simulation result shows that the proposed approach is able to give competitive classification accuracy compared to PNN in many datasets.     

Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems

We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.     

Existence of closed solutions for a polynomial first order differential equation

We find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the Abel equation as a particular case. Such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields.     

Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.     

Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial boundary data in the unit ball.