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.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Finding strong pseudoprimes to several bases

Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pomerance et al. and Jaeschke, are known for . Upper bounds for were given by Jaeschke.

In this paper we tabulate all strong pseudoprimes (spsp's) to the first ten prime bases which have the form with odd primes and There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp's to both bases 31 and 37. As a result the upper bounds for and are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for to be a strong pseudoprime to one or to several prime bases. Comparisons of effectiveness with both Jaeschke's and Arnault's methods are given.

     

On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality

where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,

where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:

and the corresponding eigenfunction spaces are

     

On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions

Gauss-Lobatto quadrature formulae associated with symmetric weight functions are considered. The kernel of the remainder term for classes of analytic functions is investigated on elliptical contours. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with either the real or the imaginary axis. The results obtained here are an analogue of some recent results of T. Schira concerning Gaussian quadratures.

     

一类拟线性椭圆型方程的奇摄动

本文讨论了一类二阶拟线性椭圆型方程的奇摄动问题,给出了外部解和边界层项的N阶递推方程,并对余项进行了估计,从而导得了解的渐近展开式和摄动问题解的存在唯一性.     

一种求解Burgers方程的高精度紧致差分格式

针对Burgers方程,采用余项修正法和欧拉公式,推导了一种新的四层高精度紧致差分隐格式,其截断误差为O(τ~2+τh~2+h~4),即当τ=O(h~2)时,格式空间具有四阶精度;然后通过数值实验验证了格式的精确性和可靠性.     

On the Sauer-Xu formula for the error in multivariate polynomial interpolation

Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree in variables, at a `correct' point set.

     

一种基于小波系数动态量化的鲁棒数字水印算法

提出了一种改进的基于动态量化小波系数的鲁棒水印算法.水印通过动态量化嵌入到载体图像小波变换的高频子带上,混沌优化用于搜索嵌入水印的最大可用块数,中国剩余定理用于安全地产生动态量化参数.该算法和Lin等人提出的数字水印方案相比,不仅具有良好的隐蔽性、安全性和不可感知性,同时,算法也具有良好的抗压缩攻击等鲁棒性能.仿真结果验证了算法的有效性.     

单位群阶为2pqr的模n剩余类环

应用中国剩余定理与模n剩余类环 及其单位群 的分解定理, 讨论并确定了当群 的阶为 的部分情形时, 的群结构与n的取值, 其中 , , 为素数(不必完全不同).