Vandermonde行列式对Lagrange插值公式的应用

给出了广义Vandermonde行列式的一种求法,并运用它给出了Lagrange插值公式的几个证明.     

某种行列式的新型解法

通过寻找拉格朗日插值公式和范德蒙行列式之间的关系。探讨某类”阶行列式求解方法．在教学中充分运用研究探索的方法来引导学生的学习,对学生的成长将起到重要的作用．     

关于Vandermonde矩阵及其行列式的注记

给出Vandermonde矩阵及其行列式的若干应用,揭示它在高等代数和矩阵分析等方面的重要地位.具体来说,运用Vandermonde行列式来计算几个与之相关的行列式,运用线性方程组来证明组合恒等式,给出两个特殊的Vandermonde矩阵的应用,特别是用Schur矩阵给出了樊畿不等式的一个证明,给出了Vandermonde矩阵与Cauchy矩阵的一个恒等式.     

Lagrange插值公式与Hermite插值公式的注记

给出一种基于商的形式的Lagrange与Hermite插值公式及其证明,同时还给出了两个相关的不等式.     

矩阵基础上的行列式理论

首先证明矩阵的伴随矩阵存在性定理,其次利用递推定义方法建立行列式概念,由此可构建矩阵基础上的行列式理论。     

Lagrange插值公式的几种构造性证明

利用中国剩余定理、行列式以及线性方程组理论给出了Lagrange插值公式的几种构造性证明,得到了Vandermonde矩阵的逆矩阵的一种算法.     

一类中值问题的行列式解法

通过行列式的形式构造辅助函数,用来求解与微分中值定理有关的一类存在性问题.     

反对称线性函数与行列式

通过对反对称线性函数及其性质的探讨,给出了有关行列式传统结论的一种新的表述,将几何直观与行列式的运算有机结合起来,以揭示行列式的一些更直观、具体的内涵.     

一类二元有理插值的存在性问题

利用二元Lagrange插值公式对一类二元有理插值函数的存在性给出了一个判别方法,并在判别出该二元有理插值函数存在时,给出了它的表现公式。此外,对导致二元有理插值函数不存在的不可达点,本文给出了一种处理方法,使之由不可达点变成可达点。文章的最后还给出若干数值例子说明了本方法的有效性.     

拉普拉斯（Laplace）定理的简化证明

文[1]P90-93提出并证明了拉普拉斯定理,但其证明很繁琐,本文给出一个简明易懂的证明,供读者参考。引理设α_1,α_2,……,α_n及β_1,β_2,……β_n分别是1,2,……n的一个排列,则     

Multivariate Vandermonde Determinants and General Birkhoff Interpolation

In this paper, we consider the Straight Line Type Node Configuration C (SLTNCC) in multivariate polynomial interpolation as the result of different kinds of transformations of lines (such as parallel translations, rotations). Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.     

Algebraic Numbers Close to 1 in Non-Archimedean Metrics

Let 1 be an algebraic number with relatively small height. Recently, many authors, including Amoroso, Dubickas, Mignotte and Waldschmidt, stated sharp lower bounds for the quantity | – 1|. Here, we provide a p-adic analogue of their results. For instance, we give an upper bound for the absolute value of the norm of – 1, and we show that our estimate is rather sharp in terms of the degree of . Further, we discuss a generalization in several variables of our result.     

Determinants in Triangulated Categories

For a small triangulated category , Bass's K ₁ group is described, and the theorem of the heart is proved. We define the determinant map from to Neeman's , and we compute this map when is the derived category of an Abelian category .     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Classification of Bivariate Configurations with Simple Lagrange Interpolation Formulae

In 1977 Chung and Yao introduced a geometric characterization in multivariate interpolation in order to identify distributions of points such that the Lagrange functions are products of real polynomials of first degree. We discuss and describe completely all these configurations up to degree 4 in the bivariate case. The number of lines containing more nodes than the degree is used for classifying these configurations.