Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.     

Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n?10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.     

基于切比雪夫-高斯网格的奇摄动问题的近似解的构造

研究基于切比雪夫-高斯网格的二阶奇摄动反应扩散问题.根据奇摄动问题边界层的特性,将该问题的边界层区域和正则区域分别通过作变换再使用切比雪夫多项式逼近,得到两个边值问题,并通过设置过渡边值将这两个边值问题转化为两个代数方程组,获得其近似解的构造.最后给出该方法的误差估计.     

基于Chebyshev多项式的身份认证方案的研究

基于有限域的Chebyshev多项式由于其良好的单向性和半群特性,被广泛应用到公钥密码算法、密钥协商及身份认证方案中。在安全性方面对已有的基于Chebyshev多项式的身份认证方案进行了分析,并提出了一个新的身份认证方案。通过对其性能的研究,新的方案是安全有效的。     

余弦函数型最佳一致逼近多项式

依据最佳一致逼近的基本理论,围绕切比雪夫多项式的特征方程,参照余弦函数的变化图形,建立了余弦函数型最佳一致逼近Cn(x)多项式.@@文中介绍了Cn(x)多项式在被逼近函数y(x)=0条件下依据的微分方程、相关定义、有关性质、数学表式、递推公式;讨论了它与切比雪夫Tn(x)多项式之间的关系及转化;提供了在y(x)≠0条件下Cn(x)多项式转化成c(x)所具有的特征和特点;给出了关于c(x)多项式得以实现的具体算法.应用实例表明,在减少多项式的摆动性、提高逼近精度、增大预测范围方面都有较大的改善和提高.     

Object representation and recognition in shape spaces

In this paper, we describe a shape space based approach for invariant object representation and recognition. In this approach, an object and all its similarity transformed versions are identified with a single point in a high-dimensional manifold called the shape space. Object recognition is achieved by measuring the geodesic distance between an observed object and a model in the shape space. This approach produced promising results in 2D object recognition experiments: it is invariant to similarity transformations and is relatively insensitive to noise and occlusion. Potentially, it can also be used for 3D object recognition.     

一种新型非线性偏最小二乘方法研究及应用--基于Chebyshev多项式改进的PLS方法

在实际生产过程中，过程变量之间往往存在大量相关关系，甚至非线性相关关系。过程变量间存在线性相关时，可采用偏最小二乘方法(Partial Least Squares，PLS)计算模型参数，但由于PLS方法采用线性关系来联系输入和输出因子，因而并不能有效地应用于非线性较强的过程。在这种情况下要对变量进行有效的压缩维数．需要采用非线性PLS方法。基于Chebyshev多项式改进的多元多项式PLS方法，是一种新的非线性PLS方法。该方法利用Chebyshev多项式的正交性质和递推性质，将过程输入变量正交化、线性化后，再应用PLS方法计算模型参数并还原，从而得到比较精确的模型。由于该方法综合考虑了输入变量的自相关和输入变量间的协相关关系。能够更有效地表达过程变量间的非线性关系，因此其对非线性过程的研究提供了新的思路。     

Numerical inversion of multidimensional Laplace transforms by the Laguerre method

Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace transforms to multidimensional Laplace transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ε-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.     

On Pattern Frequency Occurrences in a Markovian Sequence

Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for moments (including the variance), and probability of r pattern occurrences for three different regions of r , namely: (i) r=O(1) , (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct certain language expressions that characterize pattern occurrences which are later translated into generating functions. We then use analytical methods to extract asymptotic behaviors of the pattern frequency from the generating functions. These findings are of particular interest to molecular biology problems (e.g., finding patterns with unexpectedly high or low frequencies, and gene recognition), information theory (e.g., second-order properties of the relative frequency), and pattern matching algorithms (e.g., q -gram algorithms).     

一种新的谱随机有限元方法

提出了一种新的谱随机有限元分析方法——递推求解方法。讨论了已有的几种随机场的谱展式，提出将随机结构的随机响应表示成非正交多项式混沌展式，并证明了这个展式的收敛性。在此基础上将随机微分方程表示成了和摄动法类似的一系列确定的递推方程，这些递推方程可用确定性有限元方法求解。该方法比在正交多项式混沌基上投影并取期望的方法更简洁实用，更适合大规模有限元的求解。该方法同样能解决有较大随机涨落的力学问题。