鲁棒稳定性分析的值映射与参数化方法

本文运用值映射与参数化的方法研究了鲁棒D稳定性问题。文中给出了主要结果——参数空间鲁棒分析的边界定理,作为该结果的应用,一些现代鲁棒分析的知名结果,例如多项式族的稜边定理,Kharitonov定理和菱形定理都被简洁地证明出来;给出了一种判定双参数多项式族D稳定的新方法。用该方法可以方便地确定H稳定多项式按一定摄动模式(区间方式和菱形方形)的最大摄动界。     

多项式函数族的零点的区域稳定性

本文证明,对于某些类型的复系数多项式函数族P_x,只要族中特定的几个多项式的零点均落在复平面C的某一区域U上,则P_x中所有的多项式的零点便全都落在U上。本文考虑了U为开圆盘、扇形、半平面、1/4平面等多种情形。相应的多项式族P_x的系数向量空间X可以是多种多样的斜方体或凸集。本文得到的结果推广了目前在控制理论界中引起了热烈讨论的Kharitonov定理。     

实多项式有纯虚零点的代数判据

多项式零点分布的研究,在数学的许多分枝及工程应用中都有重要意义.其中人们最关心的问题之一是,判定一个多项式是否为 Hurwitz 多项式,即它的零点是否均具有负实部.对此著名的 Routh-Hurwitz (代数)判据和 Nyquist (频域)判据已给出了完全的解答.近年来,鲁棒反馈镇定等问题的研究,又提出了判定系数在某一范围内变动的一簇多项式的稳定性问题.这方面最引人注目的是 Kharitonov 的结果及随后人们所作的各     

复多项式族鲁棒稳定性的边界定理

依据文献[1]的思想和理论,我们曾得出了复系数多项式值映射,D等价等概念和边界定理,棱边定理等结论,这些结论是可以改进的.本文利用文献[1]的思想和方法给出了多项式边界定理的一般形式,改进了文献[2]中的某些结果.利用这一边界定理可以给出已有的关于多项式族鲁棒稳定性定理简洁的证明,同时为有关问题的进一步研究提供了有效的工具.     

多变量多项式矩阵素分解的结果和问题

多变量矩阵素分解问题是多维系统和信号处理学科中的基本问题,从提出以来,已经经过众多学者的研究.近年来随着符号代数计算的快速发展,情况有了根本的改变.概述近几年来该领域的研究进展情况和一些相关未解决的问题,从中可以看出该领域的研究密切依赖于机械化数学的研究进展.作为多变多项式矩阵分解理论的应用,给出两个变量多项式矩阵分解存在性定理的新证明.从中可以看出,2个变量多项式矩阵和3个以上变量矩阵的本质区别.     

结构摄动下多项式的鲁棒稳定性

本文对与Ｋｈａｒｉｔｏｎｏｖ多项多族相对偶的菱形族式项式的一类结构摄动下的鲁棒稳定性进行了详细的研究，提出了在此结构摄动下，菱形族形族多项式稳定当且仅当检验有限个顶点多项式或有限个边多项式的稳定性定理，而菱形族定量只是本定理的推论，另外，对低阶多项式族进行了讨论。     

多项式族的根分布不变性

本文研究多项式族的根分布不变性问题.我们首先提出了多项式族根分布的广义剔零原理,给出了参数空间中鲁棒稳定性的复边界定理和复棱边定理,并基于广义剔零原理得到了参数空间和系数空间中关于根分布的相应结论.另外,对于系数空间中鲁棒稳定性中实棱边定理,我们证明了它对稳定区域的要求还可放宽.对于一些更具几何特征的凸多面体和特定的稳定区域,棱边定理还可进一步改进,使所需检验的棱边数目与凸多面体的棱边数目无关.最后,我们给出了检验棱边根分布的Nyquist型图示方法.     

代数基本定理的一个新证明

§1.引言关于复数域的代数封闭性的定理(即通常所谓“代数基本定理”)自高斯开始已有了不少的证明;直到现在,尽管由于函数论和拓扑学的发展此定理所肯定的事实已变得十分明显,但数学家们对寻求新的证明仍未完全丧失与趣。如所周知,这一定理肯定任一复系数多项式均可在复数域上分解为线性因子的乘积.本文的目的是要给这个定理提供一个新的证明.在证明中将对多项式的次数作归纳法.证明所依据的都是分析和     

多项式簇的稳定性分析

本文利用值映射的概念在系数空间里讨论了多项式簇的Hurwitz稳定性的鲁棒性,给出了判别多项式簇是Hurwitz稳定的原象定理,同时应用这个定理证明了边界定理,棱边定理,Харитонов定理和菱形簇定理.     

基于指标形式张量的微分几何定理机器证明

提出了一个基于指标形式张量的微分几何定理的机器证明算法.该算法将微分几何定理转化成带指标的张量多项式的计算问题,然后通过利用重写规则,挖掘等价条件和分次选取条件等方法大大减少了这个多项式系统的方程个数.再利用这个多项式系统本身和关于哑元的方程三角化这个多项式系统,将所得到的首项代入结论, 从而得到了该定理的机器证明.该算法不仅能够证明基于指标形式张量的微分几何定理,也可以用于张量方程的求解.     

矩阵对策鞍点定理的归纳法证明

Many proofs have been published for the minimax theorem, and all the published inductive proofs have been indirect ones. It has been pointed out that a direct inductive proof is needed, especially for instructional purposes, since indirect proofs are more or less implicit in nature. Such a direct proof is given in [4]. Now the minimax theorem can be stated equivalently in terms of saddle point; And it is the object of the present paper to give a direct inductive proof for the saddle point version of this theorem.     

The equational part of proofs by structural induction

We study proofs by structural induction and the equational subproofs that emerge from inductive proofs. Due to the role these subproofs play in the proof process, existing theorem provers tend to use ad hoc methods rather than existing semi-decision methods for equational proofs. We present a formalized version of these ad hoc methods, and prove some results on its completeness. Finally we present some examples on the formalized method used in fully automatized proofs by structural induction.This research was partly supported by The Research Council of Norway, division NAVF.     

Combinatorial proofs of some determinantal identities

In this paper, we give combinatorial proofs of some determinantal identities. In fact, we give a combinatorial proof of a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs along with some interesting applications. We also give an almost visual proof of a recent result of Oliver Knill, namely ‘The generalized Cauchy–Binet Theorem.’     

On quantitative versions of theorems due to F.E. Browder and R. Wittmann

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.     

A short proof of two shuffling theorems for tilings and a weighted generalization

Recently, Lai and Rohatgi discovered a shuffling theorem for lozenge tilings of doubly-dented hexagons, which generalized the earlier work of Ciucu. Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized some other previous work of Ciucu. In this paper, we give a unified proof of these two shuffling theorems, which also covers the weighted case. Unlike the original proofs, our arguments do not use the graphical condensation method but instead rely on a well-known tiling enumeration formula due to Cohn, Larsen, and Propp. Fulmek independently found a similar proof of Lai and Rohatgi's original shuffling theorem. Our proof also gives a simple explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings.     

Other Proofs of Old Results

We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.     

Der invariante Kontinuitätssatz für meromorphe Funktionen

There are several proofs of the general version of the Kontinuitätssatz for meromorphic functions which is invariant under biholomorphic mappings. They are considerably more complicated than the proof of the analogous theorem for holomorphic functions. We present a method of proof which is as simple as the one for holomorphic functions and which allows to extend the theorem to infinite dimensions.     

On pairs of multidimensional matrices

Recently Bapat applied a topological theorem of Kronecker and generalized a theorem of Sinkhorn on positive matrices. Here we give an alternative proof of a slightly stronger version of his generalization. This proof combines Kakutani's fixed point theorem and the duality theorem of linear programming and gives yet another proof of a theorem of Bacharach and Menon on pairs of nonnegative matrices.     

A formalization of Sambins's normalization for GL

Sambin [6] proved the normalization theorem (Hauptsatz) for GL, the modal logic of provability, in a sequent calculus version called by him GLS. His proof does not take into account the concept of reduction, commonly used in normalization proofs. Bellini [1], on the other hand, gave a normalization proof for GL using reductions. Indeed, Sambin's proof is a decision procedure which builds cut-free proofs. In this work we formalize this procedure as a recursive function and prove its recursiveness in an arithmetically formalizable way, concluding that the normalization of GL can be formalized in PA. MSC: 03F05, 03B35, 03B45.     

A new proof of the strong duality theorem for semidefinite programming

Semidefinite programs are convex optimization problems arising in a wide variety of applications and are the extension of linear programming. Most methods for linear programming have been generalized to semidefinite programs. Just as in linear programming, duality theorem plays a basic and an important role in theory as well as in algorithmics. Based on the discretization method and convergence property, this paper proposes a new proof of the strong duality theorem for semidefinite programming, which is different from other common proofs and is more simple.