Interval Arithmetic in Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition requires many very time consuming operations, including resultant computation, polynomial factorization, algebraic polynomial gcd computation and polynomial real root isolation. We show how the time for algebraic polynomial real root isolation can be greatly reduced by using interval arithmetic instead of exact computation. This substantially reduces the overall time for cylindrical algebraic decomposition.     

基于她分解与多项式的无线传感器网络密钥算法

提出了基于QR分解与二元多项式的密钥建立与分配方案。该方案以二元多项式的计算结果作为无线传感器网络的密钥。二元多项式的其中一个参数由对称矩阵进行QR分解生成,节点部署后交换Q矩阵的行信息再与R矩阵的列信息相乘生成多项式的参数。多项式的另一个参数由各自生成的随机数确定。分析结果表明：该方案可以提高存储效率、网络连通性、抗捕获性能,并能提供额外的通信链路验证。     

Multirate sampled-data systems analysis via vector operators

The primary difficulties of both the time-domain switch decomposition method and the frequency-domain decomposition method are overcome by introducing certain matrix operators and performing spectral factorization of resulting matrices of polynomials in thez-transform variable. Topological operations of the switch-decomposition method are simplified. This new approach eliminates the need to solve a system of equations with rational polynomial coefficients such as arises in the frequency-decomposition method. The determination of a multirate sampled-data system's characteristic polynomial no longer requires the evaluation of a determinant of rational polynomial elements. New results on obtaining modifiedz-transforms from standardz-transforms at a faster rate and vice versa are presented.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

Decompositions of algebras over ℝ and ℂ

We consider the boolean complexity of the decomposition of matrix algebras over and with bases consisting of matrices over a number field. Deterministic polynomial time algorithms for the decomposition of semi-simple algebras over these fields and Las Vegas polynomial time algorithms for the decomposition of simple algebras are obtained.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统,提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵,从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器,由多项式全基构造的控制器将有很多项单项式. 为克服这一问题,文中算法给出含最少单项式的简约型控制器设计方法,并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明,文中所给的控制方法取得良好性能.     

二维多项式非线性系统的镇定控制研究

针对二维多项式非线性系统,提出了基于特征根负定配置的镇定控制方法.引入自由多项式,克服系统状态矩阵描述的不惟一性,进而降低控制综合问题求解的保守性.将特征根负定配置问题转化成多项式正定性验证问题,控制器设计问题通过多项式分解最终可由半定规划工具数值求解.在所提出的处理方法的基础上,讨论了4类二维多项式非线性系统的镇定控制问题.仿真结果验证了所提出方法的有效性.     

Robust Stability Analysis of Nonlinear Hybrid Systems

We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems.      

Decomposition of algebras over finite fields and number fields

We consider the boolean complexity of the decomposition of semi-simple algebras over finite fields and number fields.We present new polynomial time algorithms for the decomposition of semi-simple algebras over these fields. Our algorithms are somewhat simpler than previous algorithms, and provide parallel reductions from semi-simple decomposition to the factorization of polynomials. As a consequence we obtain efficient parallel algorithms for the decomposition of semi-simple algebras over small finite fields. We also present efficient sequential and parallel algorithms for the decomposition of a simple algebra from a basis and a primitive idempotent. These will be applied in a subsequent paper to obtain Las Vegas polynomial time algorithms for the decomposition of matrix algebras over and .     

Squarefree Decomposition of Univariate Polynomials Depending on a Parameter. Application to the Integration of Parametric Rational Functions

In this paper, we describe the application of a new version of Barnett’s method to the squarefree decomposition of a univariate polynomial with coefficients in K [ x ], x being a parameter andK a characteristic zero field. This new version of Barnett’s method uses Bezoutian matrices instead of matrices obtained from evaluating polynomials in a companion matrix and allows the determination of the squarefree decomposition parametrizing the gcd of the polynomial and its successive derivatives with respect to the main variable. The application of this parametric squarefree decomposition to the integration of parametric rational functions is also presented.     

SOS-based solution approach to polynomial LPV system analysis and synthesis problems

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial linear parameter-varying (LPV) system analysis and control synthesis problems. Instead of solving matrix variables over positive cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size, and is computationally attractive. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach.     

On the verification of polynomial system solvers

We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high efficiency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.     

Generalized neurofuzzy network modeling algorithms usingBezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.     

Decomposing Polynomial Systems into Simple Systems

A simple system is a pair of multivariate polynomial sets (one set for equations and the other for inequations) ordered in triangular form, in which every polynomial is squarefree and has non-vanishing leading coefficient with respect to its leading variable. This paper presents a method that decomposes any pair of polynomial sets into finitely many simple systems with an associated zero decomposition. The method employs top-down elimination with splitting and the formation of subresultant regular subchains as basic operation.     

The basic element method

The basic element method (BEM) for decomposition of the algebraic polynomial via one cubic and three quadratic parabolas (basic elements) is developed within the four-point transformation technique. Representation of the polynomial via basic elements gives a lever for solving various tasks of applied mathematics. So, in the polynomial approximation and smoothing problems, the BEM allows one to reduce the computational complexity of algorithms and increase their resistance to errors by choosing an internal relationship structure between variable and control parameters.     

Stability,instability and aperiodicity tests for linear discrete systems

The system characteristic polynomial is replaced by its inverse which is decomposed as a combination of lower-degree and lower-degree inverse polynomials. A sequence of polynomials of descending degree is determined by successive decomposition.A necessary and sufficient condition of stability as well as a sufficient condition of instability, depending on the coefficients of decomposition, are given. The test for stability or instability is proposed.When testing aperiodicity, the transformation mapping the real segment (0, 1) onto the periphery of the unit circle, is used. The system characteristic polynomial whose roots are to be tested for aperiodicity is replaced by another one whose roots should be tested for stability.     

Galois theory,splitting fields and computer algebra

We provide some algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra.