关于Kalman的标准系统分解

本文给出一个反例,指出Kalman关于他本人提出的标准系统分解的原始证明以及后来孙承启先生在《自动化学报》1984年第3期上发表的证明均是错误的。事实上,如何严格地证明(或反证)这一重要结论尚是一个未有彻底解决的问题。     

On the Canonical Decomposition of Quiver Representations

Kac introduced the notion of the canonical decomposition for a dimension vector of a quiver. Here we will give an efficient algorithm to compute the canonical decomposition. Our study of the canonical decomposition for quivers with three vertices gives us fractal-like pictures.     

Decomposition Numbers and Canonical Bases

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an nth root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space representation of . It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible subrepresentation generated by the vacuum vector. As a consequence, using the works of Ariki and Varagnolo and Vasserot, it is possible to give a new proof of Lusztig"s character formula for the simple U _v (sl _r )-modules at roots of unity, which does not involve representations of of negative level.     

Jordan标准形理论解析方法的难点分解

Jordan标准形理论是线性代数中的核心内容之一.运用解析方法证明Jordan标准形的存在性也是线性代数的教学难点.结合教学实践,通过分析其数学思想和数学方法,对Jordan标准形的解析方法进行难点分解.     

线性系统标准分解的解耦问题

本文解决了线性系统标准分解在其分解的四个子空间上解耦为四个独立子系统的问题。     

The Numbers of Shared Upper Bounds Determine a Poset

We prove that a finite poset P=(V,≤) is determined up to some permutation of its elements by the function defined by m_P({u,v})=|{w∈V∣u≤w and v≤w}|.     

Birkhoff Centre of a Poset

In this paper, it is proved that the Boolean centre of a semigroup S with sufficiently many commuting idempotents is isomorphic to the inverse limit of the directed family of Birkhoff centres (or Boolean centres) of a class of bounded semigroups. The Birkhoff centre is defined for any poset and proved that it is a relatively complemented distributive lattice whenever it is nonempty. It is observed that for a semilattice S, the Birkhoff centres as a semigroup and as a poset coincide. Also it is observed that for a Lattice (L, , ), the Birkhoff centres of the semilattices (L, ) and (L, ) coincide with the Birkhoff centre of L. Finally it is proved that for a lattice (L, , ), the Boolean centres of the semilattices (L, ) and (L, ) coincide with the Boolean centre of L.AMS Subject classification (1991): 06A12, 20M15     

On a Poset of Trees II

In this article, we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the so‐called generalized tree shift which seems to be a powerful tool to attack the problems of the above‐mentioned kind. We show that the generalized tree shift increases the largest eigenvalue of the adjacency matrix and Laplacian matrix, decreases the coefficients of the characteristic polynomials of these matrices in absolute value. We will prove similar theorems for the independence polynomial and the edge cover polynomial. The generalized tree shift induces a partially ordered set on trees having fixed number of vertices. The smallest element of this poset is the path, largest element is the star. Hence, the above‐mentioned results imply the extremality of the path and the star for these parameters.     

Triangular Decomposition of Composition Algebras of Domestic Canonical Algebras

Let A be a domestic canonical algebra over a finite field. In this article, we prove that the composition algebra C(A) of A has a triangular decomposition 𝒫·𝒯·? corresponding to the division of the indecomposable modules into the preprojectives, the regulars, and the preinjectives.     

Invariant Computation in a Poset

Kong and Ribemboim (1994) define for every poset P a sequence P = D⁰(P), D(P), D²(P), D³(P)… of posets, where Dⁱ(P) = D(D^i??1(P)) consists of all maximal antichains of D^i??1(P). They prove that for a finite poset P, there exists an integer i ≥?0 such that Dⁱ(P) is a chain. In this paper, for every finite poset P, we show how to calculate the smallest integer i for which Dⁱ(P) is a chain.

     

线性定常系统规范结构分解问题的研究

单输入单输出线性定常系统的状态空间按能控性和能观测性进行结构分解,能全面、完整的反映系统的各个子系统特性,理论上揭示状态空间的本质特征.通过规范性分解,不完全能控和不完全能观测的连续时间线性时不变系统可分解为"能控能观测","能控不能观测","不能控能观测","不能控不能观测"四个子空间,但是其分解后子空间的维度受非奇异变换矩阵P_c的影响.提出了一种约束变量的方法,用来选取P_c中列向量的规则,以控制原系统分解后子空间的表现形式.经实例验算,方法简单有效,具有一定的教学和理论意义.     

Canonical Decomposition of Polyhedral Functions and Scissors Congruence in the Projective Space

A combinatorial decomposition of polyhedral functions on the projective space is obtained, and the notion of equidecomposability for such functions is introduced. As an application, a criterion of equidecomposability of polyhedral functions on the even-dimensional projective space is established. Bibliography: 4 titles.     

Explicit Expressions of Smith's Determinant on a Poset

We introduce the meet-function and join-function on a subset of a poset and use them to give some explicit expressions of Smith's determinant. Received September 28, 1998, Accepted February 4, 1999     

Collection of Finite Lattices Generated by a Poset

It is proved that the collection of all finite lattices with the same partially ordered set of meet-irreducible elements can be ordered in a natural way so that the obtained poset is a lattice. Necessary and sufficient conditions under which this lattice is Boolean, distributive and modular are given.     

Loopless Generation of Linear Extensions of a Poset

Canfield and Williamson gave the first loopless algorithm for generating all linear extensions of a poset. It elegantly generates all signed extensions, resulting in each extension appearing somewhere with each sign, but retains only every other one independent of its sign. It uses an array for the extension. In this paper we give another loopless algorithm for generating all the linear extensions. It generates each extension only once and uses a list for the extensions.     

Poset embeddings of Hilbert functions

For a standard graded algebra $R$ , we consider embeddings of the poset of Hilbert functions of $R$ -ideals into the poset of $R$ -ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements–Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.     

The Automorphism Group of the Subsequence Poset B m,n

In this paper we are interested in the automorphism group of the poset B _{m, n} . B _{m, n} constitutes the words obtained from the cyclic word of length n on an alphabet of m letters in by deleting on all possible ways and their natural order. We prove: Résumé: Le but de ce papier est la détermination du groupe d"automorphismes des ordres B _{m, n} . Il s"agit des mots obtenus à partir du mot cyclique de longeur n sur un alphabet de m lettres par suppression successive de lettres et ordonnés naturellement. On prouve: AutB _{m, n} = {S _n for 1 n m, S ₂ S _2m-n for m + 1n 2m - 1, S ₂for 2m n.     

复向量空间的分解与复线性变换的Jordan标准形*

本文引入表征复线性变换结构的新对象。这些新对象给出复向量空间关于复线性变换分解的新结果且给出构造Jordan标准形的所有Jordan基。因而,它们能够直接导致着名的Jordan定理及空间的第三分解定理,且能给出对Jordan形精致微妙结构的新的深刻洞察。后者表明,复线性变换的Jordan标准形是一种在双重任意选择下具有不变性的结构。     

可图序列偏序集的极小元

ｎ项非增非负整数序列是可图的,若是某个阶简单图的度序列．所有项和为２ｍ、迹为ｆ的ｎ项可图序列的集合Ｇ_ｎ,ｍ,ｆ在优超关系下是一个偏序集．本文刻划了偏序集Ｇ_ｎ,ｍ,ｆ的极小元,并确定各种可图序列偏序集中极小元的个数．