Semi inherited bivariate interpolation

The bivariate interpolation in two dimensional space R2 is more complicated than that in one dimensional space R, because there is no Haar space of continuous functions in R2. Therefore, the bivariate interpolation has not a unique solution for a set of arbitrary distinct pairwise points. In this work, we suggest a type of basis which depends on the points such that the bivariate interpolation has the unique solution for any set of distinct pairwise points. In this case, the matrix of bivariate interpolation has the semi inherited factorization.     

QUARTIC CONVEX TYPE PIECEWISE POLYNOMIAL

In the present paper we consider quartic piecewise polynomial for approximation to the function f ∈ C 2 [0, 1]. A convex type condition has been imposed in the partition so that the matrix involved for the computation of pp functions is of lower band. This reduces the computation for constructions of the pp functions for the approximation.     

修正的 Thiele-Werner型有理插值

Through adjusting the order of interpolation nodes, we gave a kind of modified Thiele-Werner rational interpolation. This interpolation method not only avoids the infinite value of inverse differences in constructing the Thiele continued fraction interpolation, but also simplifies the interpolating polynomial coefficients with constant coefficients in the Thiele-Werner rational interpolation. Unattainable points and determinantal expression for this interpolation are considered. As an extension, some bivariate analogy is also discussed and numerical examples are given to show the validness of this method.     

构造二元向量值有理插值函数的一种新方法

At present, the methods of constructing vector valued rational interpolation function in rectangular mesh are mainly presented by means of the branched continued fractions. In order to get vector valued rational interpolation function with lower degree and better approximation effect, the paper divides rectangular mesh into pieces by choosing nonnegative integer parameters d1 （0 〈 dl ≤ m） and d2 （0 ≤ d2≤ n）, builds bivariate polynomial vector interpolation for each piece, then combines with them properly. As compared with previous methods, the new method given by this paper is easy to compute and the degree for the interpolants is lower.     

On convergence analysis of a derivative-free trust region algorithm for constrained optimization with separable structure

In this paper,we propose a derivative-free trust region algorithm for constrained minimization problems with separable structure,where derivatives of the objective function are not available and cannot be directly approximated.At each iteration,we construct a quadratic interpolation model of the objective function around the current iterate.The new iterates are generated by minimizing the augmented Lagrangian function of this model over the trust region.The filter technique is used to ensure the feasibility and optimality of the iterative sequence.Global convergence of the proposed algorithm is proved under some suitable assumptions.     

On the order of approximation for the rational interpolation to |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper.For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are xi=(i/n)r(i=1,2, … ,n;r＞0), it is proved that the exact order of approximation is O(1/n), O(1/nlogn) and O(1/nr), respectively, corresponding to 0＜r＜1, r=1 and r＞1.     

UNIFORM APPROXIMATION OF ENTIRE FUNCTIONS OF SLOW GROWTH ON COMPACT SETS

In the present paper,we study the polynomial approximation of entire functions of several complex variables.The characterizations of generalized order and generalized type of entire functions of slow growth are obtained in terms of approximation and interpolation errors.     

球面上的几何连续插值

In this paper, a new method for geometrically continuous interpolation in spheres is proposed. The method is entirely based on the spherical B′ezier curves defined by the generalized de Casteljau algorithm. Firstly we compute the tangent directions and curvature vectors at the endpoints of a spherical B′ezier curve. Then, based on the above results, we design a piecewise spherical B′ezier curve with G 1 and G 2 continuity. In order to get the optimal piecewise curve according to two different criteria, we also give a constructive method to determine the shape parameters of the curve. According to the method, any given spherical points can be directly interpolated in the sphere. Experimental results also demonstrate that the method performs well both in uniform speed and magnitude of covariant acceleration.     

基于修正的第二类Chebyshev结点的Newman型有理插值函数对|x|的逼近

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.     

A Remark on Pl Type Interpolation on Non-Uniformly Distributed Nodes on the Unit Circle

In this paper we study the problem of explicit representation and convergence of P′al type(0;1) interpolation and its converse, with some additional conditions,on the non-uniformly distributed nodes on the unit circle obtained by projecting the interlaced zeros of Pn(x) and P′n(x) on the unit circle. The motivation to this problem can be traced to the recent studies on the regularity of Birkhoff interpolation and P′al type interpolations on non-uniformly distributed zeros on the unit circle.     

中国剩余定理在非标准数论中的推广

我们推广了中国剩余定理,用于研究数论模型,并在非标准数论模型中对比分析若干数论定理.     

The Generalization of the Chinese Remainder Theorem

We generalize the Chinese Remainder Theorem, use it to study number theory models, compare and analyse several number theory theorems in non-standard number theory models. Received August 18, 1999, Accepted February 5, 2001     

关于陷门背包公开钥密码系统的注记

利用不定方程理论及中国剩余定理,我们设计了一类陷门背包公开钥密码系统,它们具有更好的安全性.     

Independence of vectors in codes over rings

We study codes over Frobenius rings. We describe Frobenius rings via an isomorphism to the product of local Frobenius rings and use this decomposition to describe an analog of linear independence. Special attention is given to codes over principal ideal rings and a basis for codes over principal ideal rings is defined. We prove that a basis exists for any code over a principal ideal ring and that any two basis have the same number of vectors. Hongwei Liu is supported by the National Natural Science Foundation of China (10571067).     

On Remainder Equations

The remainder equations were introduced by S. O. Hansson. In this paper, we will give an exhaustive characterization of the set of solutions for remainder equations. Moreover, solutions to some unsolved problems proposed by Hansson are reported.     

Some Solved and Unsolved Remainder Equations

The remainder set A?B of a set of sentences A modulo a set of sentences B is the set of all maximal subsets of A not implying any element of B. A remainder equation is an expression containing remainder sets, such as {A} = B?X, in which at least one set is unknown. Solutions to some classes of remainder equations are reported, and some unsolved problems are listed.     

Optimal estimates of common remainder for the robust Chinese Remainder Theorem

Common remainder is significant to the estimation of the robust Chinese Remainder Theorem (CRT). This paper presents two optimal estimates of common remainder for the robust CRT. The two different optimal estimates are obtained based on different definitions of circular distance. Both of the two estimations are more effective with lower computational complexity than the existing searching method. Simulation results show that the two estimations have nearly the same performance, however, the second optimal estimation has less computation than the first one. These optimal estimates can improve the performance of the estimation of the robust CRT.     

带多余参数的回归方程系统的两步估计

研究了带多余参数的回归方程系统，得到了参数的两步估计是某一特定形式的一个充要条件，并讨论了两步估计的有限样本性质。     

Finding -strong pseudoprimes

Let be odd primes and . Put

Then we call the kernel, the triple the signature, and the height of , respectively. We call a -number if it is a Carmichael number with each prime factor . If is a -number and a strong pseudoprime to the bases for , we call a -spsp . Since -numbers have probability of error (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of (the smallest strong pseudoprime to all the first prime bases). If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement.

In this paper, we first describe an algorithm for finding -spsp(2)'s, to a given limit, with heights bounded. There are in total -spsp's with heights . We then give an overview of the 21978 - spsp(2)'s and tabulate of them, which are -spsp's to the first prime bases up to ; three numbers are spsp's to the first 11 prime bases up to 31. No -spsp's to the first prime bases with heights were found. We conjecture that there exist no -spsp's to the first prime bases with heights and so that

which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those -spsp's is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates of -spsp's and their prime factors to Miller's tests, and obtain the desired numbers. At last we speed our algorithm for finding larger -spsp's, say up to , with a given signature to more prime bases. Comparisons of effectiveness with Arnault's and our previous methods for finding -strong pseudoprimes to the first several prime bases are given.

     

N=2^t点DFT的快速卷积算法

从所周知,循环卷积和离散富里叶变换(DFT)可以互相计算,只要得到其中一个的快速算法就可导出另一个的快速算法。循环卷积目前已有乘法量为O(N)的最佳算法(特别是当N较小时),为此关键是如何将DFT转化为循环卷积,当DFT的长度N=p(p为素数),Rader利用有限域GF(p)的乘法群是循环群就成功地将p点DFT转化为Q(p)(F(p)为户的Euler函数)点循环卷积;当N=p~e时,由于商环Z/(p~e)存在F(p~c)阶元素,人们也成功地将p~c点DFT转化为P(p~(c-1))一系列循环卷积,即一个y(p~c)点循环卷积,二个P(p~(c-1))点