Comprehensive G?bner Basis Theory for a Parametric Polynomial Ideal and the Associated Completion Algorithm

Gr?bner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gr?bner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gr?bner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Gr?bner basis of the associated specialized polynomial ideal.For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to Gr?bner basis theory are reexamined and/or further developed for the parametric case:(i) Definition of a comprehensive Gr?bner basis,(ii) test for a comprehensive Gr?bner basis,(iii) parameterized rewriting,(iv) S-polynomials among parametric polynomials,(v) completion algorithm for directly computing a comprehensive Gr?bner basis from a given basis of a parametric ideal. Elegant properties of Gr?bner bases in the classical ideal theory, such as for a fixed admissible term ordering,a unique Gr?bner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Gr?bner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.     

Comprehensive Gröbner basis theory for a parametric polynomial ideal and the associated completion algorithm

Gröbner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gröbner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gröbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Gröbner basis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to Gröbner basis theory are reexamined and/or further developed for the parametric case: (i) Definition of a comprehensive Gröbner basis, (ii) test for a comprehensive Gröbner basis, (iii) parameterized rewriting, (iv) S-polynomials among parametric polynomials, (v) completion algorithm for directly computing a comprehensive Gröbner basis from a given basis of a parametric ideal. Elegant properties of Gröbner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique Gröbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Gröbner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.     

ON THE ALGEBRAIC VARIETY DECOMPOSITION

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THE MINIMAL CHARACTERISTIC BASIS OF A POLYNOMIAL IDEAL

The concept of minimal characteristic basis(abbr.char basis) of a polynomial ideal ispresened to give the uniqueness of the char basis of an ideal.For the prime ideal,we give an algorithmto construct the minimal char basis from generators of the given ideal.     

SOME STUDIES ON GRBNER BASES FOR MODULES AND APPLICATIONS

Hong's theorem for Grobner bases under polynomial composition is generalized to module case with more general cosideration. Similar generalization is also considered when the coefficient field is replaced by a Dedekind domain.     

ON FIRST INTEGRALS OF POLYNOMIAL AUTONOMOUS SYSTEMS

Using Carleman linearization procedure, this paper investigates the problem of first integrals of polynomial autonomous systems and proposes a procedure to find the first integrals of polynomial family for the systems. A generalized eigenequation is obtained and then the problem is reduced to the solvability of the eigenequation. The result is a generalization of some known results.     

基于II型最优正规基的串行乘法器

研究有限域(定义在GF(2)上的m维向量空间)的正规基乘法器。基于域元素的正规基表示和多项式基表示可以相互转换,通过分析多项式基下的乘法运算公式,得到利于串行乘法器设计的有效性质。利用该性质,提出了一个新的串行正规基乘法器,该乘法器要求(2m-2)个二值输入的异或门,m个二值输入的与门。在进行具体的硬件实现时,所需要的存储空间为3m+1。     

A criterion for testing whether a difference ideal is prime

This paper presents a criterion for testing the irreducibility of a polynomial over an algebraic extension field. Using this criterion and the characteristic set method, the authors give a criterion for testing whether certain difference ascending chains are strong irreducible, and as a consequence, whether the saturation ideals of these ascending chains are prime ideals.     

COHERENT, REGULAR AND SIMPLE SYSTEMS IN ZERO DECOMPOSITIONS OF PARTIAL DIFFERENTIAL SYSTEMS

1.IntroductionWeareconcernedwithalgebraicdifferentialpolynomials(abbreviatedas'd-pcis)inafinitenumberofdifferentialindeterminatesoveradifferentialfieldofcharacteristic0.Givenafinitesetofsuchd-pcis,wewouldliketostudyd-Zero(),thedifferentialalgebraicsetdefinedbyPoverauniversaldifferentialfield.Wemayaskwhetherd-Zero()isempty,whetherad-pcivanishesond-Zero()(differentialradicalidealmembershipproblem),whatthedimensionofd-Zero()is,howtodecomposed-Zero()intoirreduciblealgebraicdifferentialsets,andso…     

ON ALGEBRICO-DIFFERENTIAL EQUATIONS-SOLVING

The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations. As an illustration of the method, the Devil's Problem of Pommaret is solved in details.     

RBF-Volterra级数非线性系统建模

提出一种RBF-Volterra级数非线性系统建模方法。利用基函数多项式（BFP）完备的非线性系统逼近能力,在基函数族单一尺度和平移联动约束下,导出RBF-Volterra级数模型;RBF-Volterra级数具有＂截尾不截维＂特性,对非线性强弱变化适应能力强,且结构紧凑,数值稳定性好,能效避免维数灾难问题。实例仿真说明了该方法的有效性。     

THE UNIVERSAL GROEBNER BASIS UNDER COMPOSITION

Composition is the operation of replacing variables in a polynomial with other polynomials. The main question in this paper is: when does composition commute with universal Groebner basis computation? We prove that this happens iff the composition is single variable. This has a natural application in the computation of universal Groebner bases of composed polynomials.     

THE λ-GROBNER BASES UNDER POLYNOMIAL COMPOSITION

Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. λ-Grgbner basis is an especial Grobner basis. The main problem in the paper is： when does composition commute with λ-Grobner basis computation？ We shall answer better the above question. This has a natural application in the computation of λ-Grobner bases.     

Discovering non-terminating inputs for multi-path polynomial programs

This paper investigates the termination problems of multi-path polynomial programs （MPPs） with equational loop guards. To establish sufficient conditions for termination and nontermination simultaneously, the authors propose the notion of strong/weak non-termination which under/over- approximates non-termination. Based on polynomial ideal theory, the authors show that the set of all strong non-terminating inputs （SNTI） and weak non-terminating inputs （WNTI） both correspond to tile real varieties of certain polynomial ideals. Furthermore, the authors prove that the variety of SNTI is computable, and under some sufficient conditions the variety of WNTI is also computable. Then by checking the computed SNTI and WNTI varieties in parallel, termination properties of a consid- ered MPP can be asserted. As a consequence, the authors establish a new framework for termination analysis of MPPs.     

雷达系统发现与跟踪目标仿真元模型方法

雷达系统仿真模型复杂、计算量大,是导弹攻防对抗仿真实验复杂度的主要组成部分。为了降低仿真实验的复杂度,依据雷达系统工作原理,为雷达系统仿真模型构建了四种形式的元模型:线性元模型、多项式元模型、径向基函数元模型和Krging元模型。通过实验比较看出,二阶多项式元模型具有拟合精度高、计算速度快、模型简洁等特点。在攻防对抗仿真中用它代替雷达系统仿真模型,可以大大提高仿真实验的效率。     

QUANTITATIVE PROPERTIES OF GREEN EQUIVALENCES ON SPECIAL MONOIDS

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The Invertibility of Rational Univariate Representations

In this paper, the so-called invertibility is introduced for rational univariate representations, and a characterization of the invertibility is given. It is shown that the rational univariate representations, obtained by both Rouillier’s approach and Wu’s method, are invertible. Moreover, the ideal created by a given rational univariate representation is defined. Some results on invertible rational univariate representations and created ideals are established. Based on these results, a new approach is presented for decomposing the radical of a zero-dimensional polynomial ideal into an intersection of maximal ideals.

     

STABILITY OF SWITCHED POLYNOMIAL SYSTEMS

This paper investigates the stability of （switched） polynomial systems. Using semi-tensor product of matrices, the paper develops two tools for testing the stability of a （switched） polynomial system. One is to convert a product of multi-variable polynomials into a canonical form, and the other is an easily verifiable sufficient condition to justify whether a multi-variable polynomial is positive definite. Using these two tools, the authors construct a polynomial function as a candidate Lyapunov function and via testing its derivative the authors provide some sufficient conditions for the global stability of polynomial systems.     

ON SUFFICIENT AND NECESSARY OF EXISTENCE FOR A CLASS OF SINGULAR OPTIMAL STOCHASTIC CONTROL

We study a class of discounted models of singular stochastic control. In this kind of models, not only the structure of cost function has been extended to some general type, but also the state can be represented as the solution of a class of stochastic differential equations with nonlinear drift and diffusion term. By the various methods of stochastic analysis, we derive the sufllcient and necessary conditions of the existence of optimal control.     

GLOBAL PHASE-PORTRAITS OF PLANEHOMOGENEOUS POIYNOMIAL VECTOR FIELDSAND STABILITY OF THE ORIGIN

1.IntroductionT.Dated]obtainedanalgebraicclassificationofplanehomogeneousquadraticvectorfields,A.Cima[2]gaveanalgebraicandtopologicalclassificationofhomogeneouscubicvectorfieldsintheplane.Theirmethodswerebasedontheclassificationofforth-orderbinaryformsdev…