McCoy环的扩张(英文)

A ring R is said to be right McCoy if the equation f(x)g(x)=0,where f(x)and g(x)are nonzero polynomials of R[x],implies that there exists nonzero s∈R such that f(x)s=0.It is proven that no proper(triangular)matrix ring is one-sided McCoy.It is shown that for many polynomial extensions,a ring R is right McCoy if and only if the polynomial extension over R is right McCoy.     

谱任意的符号模式矩阵

一个n阶符号模式矩阵A称为是谱任意的,如果对任意的实系数n次首1多项式r(x),在A的定性矩阵类Q(A)中至少存在一个实矩阵B,使得B的特征多项式是r(x),文中证明了当n为奇数时n阶谱任意符号模式矩阵是存在的。     

Matrix fractions and full system equivalence

Following the definition by Hayton et al. (1990) of full systemequivalence for linear multivariables systems, the concept offull unimodular equivalence is defined for matrix fraction descriotions(MFDs).Full unimodular equivalence is an extension of unimodular equivalence,which has been proposed by Kailath (1980) and Smith (1981) forsystems described by polynomial matrix models in either leftor right MFD form. Full unimodular equivalence has the propertyof leaving invariant both the finite and infinite structureof the system. It is shown that full unimodular equivalencefor MFDs and full system equivalence coincide.     

To solving problems of algebra for two-parameter matrices. III

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem of solving an equation F(λ, μ)x = 0 with a polynomial two-parameter matrix F(λ, μ) to solving an equation of the form D(λ, μ)y = 0, where D(λ, μ) = A(μ)-λB(μ) is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix F(λ, μ) are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 166–207.     

Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].     

多项式矩阵代数性质讨论

本文利用系统与控制论中有关多项式矩阵的结果,对多项式矩阵代数性质进行讨论,得到的主要结果有多项式方阵环是主理想环,也是主单侧理想环。     

最小多项式矩阵与线性多变量系统(Ⅰ)

本文分为两部分:(Ⅰ)为关于最小多项式矩阵的理论:(Ⅱ)为最小多项式矩阵理论在线性多变量系统中的应用.在(Ⅰ)中,我们给出了线性变换在向量组的消失多项式矩阵与最小多项式矩阵的概念,给出了不变子空间的生成组与最小生成组的概念.在讨论了这些概念的基本性质之后,我们研究了它们与线性变换在任何不变子空间上诱导算子对应的特征矩阵之间的关系,给出了向量组的最小多项式矩阵类的特征,并给出了有相同生成空间的生成组之间的充分必要条件.利用这些结果,对于给定的矩阵A,给出了能使系统x=Ax+Bu完全可控的矩阵B的全体的集合的表达式.     

线性时不变系统两种描述的等价性

自从Kalman提出了用状态空间方法描述系统后,Rosenbrock与Wolovich等又提出了用微分算子描述系统的方法。这两种描述方法,利用各自表示方法的特点,在多变量系统理论的研究上,都取得了很大的进展。 本文利用Yokoyama标准形与多项式矩阵之间的关系,把上述两种描述联系起来,给出了它们之间等价的转换形式。这样就可以把在一种描述方法上得到的结果,等价地搬到另一种描述方法的系统上去。     

A CLASS OF SPECTRALLY ARBITRARY RAY PATTERNS

An n × n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f(x) with coefficients from C, there is a complex matrix in the ray pattern class of A such that its characteristic polynomial is f(x). In this paper, a family ray patterns is proved to be spectrally arbitrary by using Nilpotent-Jacobian method.     

Tauber-Bedingungen für Verfahren Mit Abschnittskonvergenz

We consider a summability matrix A with sectional convergence (in the null-domain) and use sectional submethods B. If (I - B)s has small oscillation (for a suitable B), then s also has small oscillation. Employing a set of such B, we obtain Tauberian results yielding convergence (also with one-sided conditions). In particular this analysis applies to weighted means. Thus we have a new approach to results given by Móricz and Rhoades [3].     

To Solving Multiparameter Problems of Algebra. 4. The AB-Algorithm and its Applications

The paper continues the investigation of methods for factorizing q-parameter polynomial matrices and considers their applications to solving multiparameter problems of algebra. An extension of the AB-algorithm, suggested earlier as a method for solving spectral problems for matrix pencils of the form A - λB, to the case of q-parameter (q ≥ 1) polynomial matrices of full rank is proposed. In accordance with the AB-algorithm, a finite sequence of q-parameter polynomial matrices such that every subsequent matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain rule for selecting specific basis matrices is described. Applications of the AB-algorithm to computing complete polynomials of a q-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial solutions of an arbitrary q-parameter polynomial matrix are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 127–143.     

Decomposability of matrix polynomials with coefficients possessing quasi-simple structure

The decomposability of matrix polynomials with commuting coefficients of quasi-simple structure into regular factors and the solvability of the corresponding matrix polynomial one-sided equations are investigated.Translated from Ukrainskii Matematicheskie Zhurnal, Vol. 44, No. 12, pp. 1716–1718, December, 1992.     

A new partial regularity proof for solutions of nonlinear elliptic systems

We prove partial regularity for weak solutionsu of the fully nonlinear elliptic system divA (x, u, Du) +B (x, u, Du)=0, whereA andB have natural polynomial growth, and whereB satisfies a two-sided or a one-sided condition. We do not employ a reverse Hölder inequality.     

Projektive Moduln über Polynomringen A[T1,...,Tm] mit einem regulären Grundring A

We are concerned with a question of H. Bass and D. Quillen which asks whether the following is true:If A is a regular noetherian ring, then every finitely generated projective module P over a polynomial extension A[T] of A is extended from A. (cf. [1], § 4, (IX) and [7]).We give an affirmative answer, either if (i) A equals a ring of fractions of a polynomial ring over a regular noetherian ring B with dim B2, or if (ii) A equals a ring of fractions of a polynomial extension of a power series ring over a complete regular local ring B with dim B2.(ii) implies the case that A is an unramified complete regular local ring. This generalizes the result in [4], which has been proved independently in [5]. (i) spezializes to the known theorem of Quillen and Suslin if A=B (cf. [2]).     

重模剩余类环的矩阵表示

对模m的剩余类环Zm上的多项式环Zm[x]中的任一n次（n≥1）首一多项式P,给出了重模剩余类环Zm[x]/（P）到Zm上的n阶全矩阵环Mn（Zm）的一类单同态,从而实现了Zm[x]/（P）的矩阵表示.若A为P的友矩阵,则Zm[x]/（P）的矩阵表示为{an-1An-1＋…＋a1A＋a0E｜ai∈Zm,0≤i≤n-1}...     

On Semiscalar and Quasidiagonal Equivalences of Matrices

For a certain class of polynomial matrices A(x), we consider transformations SA(x)R(x) with invertible matrices S and R(x), i.e., the so-called semiscalarly equivalent transformations. We indicate necessary and sufficient conditions for this type of equivalence of matrices. We introduce the notion of quasidiagonal equivalence of numerical matrices. We establish the relationship between the semiscalar and quasidiagonal equivalences and the problem of matrix pairs.     

Skew-prime polynomial matrices: The polynomial-model approach

We examine the concept of skew-primeness of polynomial matrices in terms of the associated polynomial model. It is shown that skew-primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism. This basic result is then applied to the special case of nonsingular polynomial matrices. We investigate the nonuniqueness of skew-complements of a skew-prime pair. It is shown that the space of equivalence classes of skew-complements is in bijective correspondence with a finite-dimensional linear space. Finally, the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly.     

单边单项式序下的SAGBI基

设K是一个域,R是具有SM-基B的一个K-代数,且是B上一个单边（即左或右）单项式序.那么关于交换多项式代数和非交换自由代数的子代数在双边单项式序下经典的SAGBI基理论可完整地推广到R的子代数上来.特别地,对于一类N-分次代数,存在计算有限n-截断SAGBI基的有效算法,并且第一次阐明了在单边单项式序下讨论SAGBI...     

Some algebras similar to the 2×2 Jordanian matrix algebra

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2×2 matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.     

素GPI-环的幂零多项式

Ｒ是素ＧＰＩ－环,若多项式ｆ（ｘ_１,…,ｘ_d）在Ｒ上是幂零的,则或ｆ（ｘ_１…,ｘ_d）是Ｒ的恒 等式,或Ｒ是有限域上的有限矩阵环