An Inequality About the Largest Roots of a Polynomial

 We present an upper bound for the product of the k largest roots of a monic polynomial of degree n with complex coefficients. For that, we use the theory of compound matrices and the localization of the eigenvalues of matrices. Received: February 26, 1996; revised version: March 20, 1997     

利用Zernike多项式对湍流波前进行波前重构

分析了利用Ｚｅｒｎｉｋｅ多项式进行波前重构时模式耦合与混淆产生的原因，以及模式耦合与混淆对受大气湍流影响的光学波前进行波前重构的精度和稳定性的影响。取除去平移的前６５项Ｚｅｒｎｉｋｅ多项式构造湍流波前。针对６１单元自适应光学系统的波前传感器子孔径布局进行仿真，通过比较Ｎｓ不同取值时的湍流波前重构精度和稳定性，得出了该条件下Ｎｓ的最佳取值。     

Polynomial chaos expansion for sensitivity analysis

In this paper, the computation of Sobol's sensitivity indices from the polynomial chaos expansion of a model output involving uncertain inputs is investigated. It is shown that when the model output is smooth with regards to the inputs, a spectral convergence of the computed sensitivity indices is achieved. However, even for smooth outputs the method is limited to a moderate number of inputs, say 10-20, as it becomes computationally too demanding to reach the convergence domain. Alternative methods (such as sampling strategies) are then more attractive. The method is also challenged when the output is non-smooth even when the number of inputs is limited.     

Minimal Homogeneous Bases for Polynomial Ideals

The minimal homogeneous basis is defined. It is shown that every minimal homogeneous basis of an ideal has the same number of elements. An algorithm for finding a minimal homogeneous basis is developed. The main idea is to modify the classical Buchbergers algorithm for finding a Gröbner basis so that a minimal homogeneous basis can be obtained as a subset of a Gröbner basis.     

Stochastic Polynomial Interpolation for Uncertainty Quantification With Computer Experiments

Multivariate polynomials are increasingly being used to construct emulators of computer models for uncertainty quantification. For deterministic computer codes, interpolating polynomial metamodels should be used instead of noninterpolating ones for logical consistency and prediction accuracy. However, available methods for constructing interpolating polynomials only provide point predictions. There is no known method that can provide probabilistic statements about the interpolation error. Furthermore, there are few alternatives to grid designs and sparse grids for constructing multivariate interpolating polynomials. A significant disadvantage of these designs is the large gaps between allowable design sizes. This article proposes a stochastic interpolating polynomial (SIP) that seeks to overcome the problems discussed above. A Bayesian approach in which interpolation uncertainty is quantified probabilistically through the posterior distribution of the output is employed. This allows assessment of the effect of interpolation uncertainty on estimation of quantities of interest based on the metamodel. A class of transformed space-filling design and a sequential design approach are proposed to efficiently construct the SIP with any desired number of runs. Simulations demonstrate that the SIP can outperform Gaussian process (GP) emulators. This article has supplementary material online.     

Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality

Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A=k[X ₁ ,,X _n ]/, where is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space . An important feature of our algorithms is that we do not require to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 ^{nD 5/2} ) and O(n2 ^{nD 5/2} ) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.     

Polynomial parametrization of nonsingular complete intersection curves

In this paper we give a new method to compute a polynomial parametrization, in case it exists, of an affine nonsingular complete intersection curve. Our method is based on the study of vector fields on nonsingular algebraic curves. In contrast to the existing methods, our algorithm does not use any projection, and no sample point on the curve is needed. It is also important to stress the fact that the algorithm produces a parametrization with coefficients in the ground field.     

罗茨真空泵主轴输入端动密封问题探讨

分析了罗茨真空泵轴封漏油的原因，介绍了提高轴封有效性的措施。     

基于多项式混沌理论的不确定度评定与分析

针对基于ISO GUM提供的测量不确定度评估方法所存在的局限,提出了采用多项式混沌方法进行测量结果不确定度评估的新方法。分析了GUM测量不确定度评估方法所隐含的假设条件,给出了多项式混沌方法的测量不确定度评估原理。通过GUM与多项式混沌方法扩展不确定度包含因子估计误差的数值计算表明,当随机变量的PDF不满足高斯分布时,多项式混沌估计方法所获得的扩展不确定度更能反映实际情况。     

多项式正交滤波器与共轭正交滤波器组

主要讨论了多项式正交滤波器和共轭正交滤波器组的构造方法,首先利用Riesz引理和特殊的余弦三角形多项式,给出了一种多项式正交滤波器的构造算法,该算法可以构造出一系列特性各异的紧支撑正交小波基;还给出了由一个矩阵CQFs派生多个新的矩阵CQFs的共轭正交滤波器组算法,包括由低阶矩阵CQFs构造高阶矩阵CQFs。     

Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions

An algorithm is developed for the simultaneous optimization of several response functions that depend on the same set of controllable variables and are adequately represented by polynomial regression models of the same degree. The data are first checked for linear dependencies among the responses. If such dependencies exist, a basic set of responses among which no linear functional relationships exist is chosen and used in developing a function that measures the distance of the vector of estimated responses from the estimated “ideal” optimum. This distance function permits the user to account for the variances and covariances of the estimated responses and for the random error variation associated with the estimated ideal optimum. Suitable operating conditions for the simultaneous optimization of the responses are specified by minimizing the prescribed distance function over the experimental region. An extension of the optimization procedure to mixture experiments is also given and the method is illustrated by two examples.     

罗茨鼓风机房通风特性研究

分析某化工企业罗茨鼓风机房通风系统流体动力学特性,探索了利用CFD方法研究通风系统的方法。结果表明：对于功率比较大的电机和罗茨风机房,采用从一个墙面进气,而在靠近对面墙的屋顶排气的通风系统具有比较好的通风效果;不同轴流风机功率对厂房内的空气流动特性影响不大;通风系统设计中还要尽量避免出现气流短路的情况。     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

Improvement of Polynomial Methods for Reproducing Functional Dependences

A comparative analysis is made of polynomial methods of reproducing functions. Recommendations on reducing the errors and/or increasing the interpolation interval for Newton polynomials are proposed in a general form and also using the example of the sin function.     

罗茨真空泵特性指标及测试方法研究

本文分析阐明了罗茨泵的特征性能为漏率、零流量压缩比、最大容许压差和噪声。提出了新概念—基础压力的定义:即泵按规定条件工作,在不引入气体的情况下,标准测试罩内趋向稳定的压力。并建议以漏率取代极限压力作为罗茨泵的性能考核指标,使罗茨泵性能的测量方法更接近、更符合实际运行状态。根据实验数据,提出了对零流量压缩比测量装置中的前级管路规定的改进,对最大容许压差测量方法提出了修改以及对噪声的测量方法提出一些新的见解。     

高氧气调对鲜切莲藕包装质量的影响

试验研究了高氧气调(O2的体积分数＞70%)对鲜切莲藕包装质量的影响,并与低氧气调(体积分数为2%的O2 体积分数为6%的CO2)包装鲜切莲藕的比较.结果表明:高氧气调包装对鲜切莲藕有保鲜作用,适当的高氧气调包装能够抑制鲜切莲藕的软化、营养成分损失等.氧气含量对鲜切莲藕表面亮度、酸含量和Vc含量影响不明显,但氧气含量越高,鲜切莲藕的重量、硬度保持越好.当包装内氧气浓度下降到70%的时候,其保鲜效果也随之明显下降.     

A Posteriori Error Estimates of a Weakly Over-Penalized Symmetric Interior Penalty Method for Elliptic Eigenvalue Problems

A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.     

On Short Multiplications and Divisions

Computing only the low degree terms of the product of two univariate polynomials is called a short multiplication. By decomposition into subproblems, a short multiplication can be reduced to appropriate addition of the results of a number of full multiplications. In this paper a new way of choosing the size of the subproblems is proposed. Computing the quotient of two polynomials is called a short division. The ideas used in the short multiplication algorithm are transferred to an algorithm for short divisions. Finally, several applications of short multiplications and divisions are pointed out. Received: July 9, 1998; revised version: July 6, 1999     

伪抛物型积分微分方程的混合有限元误差估计

基于Raviart-Thomas空间,本文对伪抛物型积分微分方程初边值问题提出了混合有限元方法.与通常的有限元方法相比,该方法可以同时高精度逼近未知函数及未知函数的梯度.通过引入广义混合椭圆投影,证明了其存在唯一性,并得到了其一系列性质,利用其性质给出了平方模范数下的最优误差估计;利用广义混合椭圆投影和正则Green函数得到了最大模范数下的拟最优误差估计.