A semi-algebraic approach for asymptotic stability analysis

This paper deals with the problem of computing Lyapunov functions for asymptotic stability analysis of autonomous polynomial systems of differential equations. We propose a new semi-algebraic approach by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first constructing a semi-algebraic system and then solving this semi-algebraic system in an adaptive way. Experiment results show that our semi-algebraic approach is more efficient in practice, especially for low-order systems.     

分块平方和分解

本文定义了分块平方和可分解多项式的概念.粗略地说,它是这样一类多项式,只考虑其支撑集(不考虑系数)就可以把它的平方和分解问题等价地转换为较小规模的同类问题(换句话说,相应的半正定规划问题的矩阵可以分块对角化).本文证明了近年文献中提出的两类方法—分离多项式(split polynomial)和最小坐标投影(minimal coordinate projection)—都可以用分块平方和可分解多项式来描述,证明了分块平方和可分解多项式集在平方和多项式集中为零测集.     

A New Method for Real Root Isolation of Univariate Polynomials

A new algorithm for real root isolation of univariate polynomials is proposed, which is mainly based on exact interval arithmetic and bisection method. Although exact interval arithmetic is usually supposed to be inefficient, our algorithm is surprisingly fast because the termination condition of our algorithm is different from those of existing algorithms which are mostly based on Descartes’ rule of signs or Vincent’s theorem and we decrease the times of Taylor shifts in some cases. We test our algorithm on a large number of examples from the literature and report the performance.      

Relative purity over noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality λ ≥ |R|, then its injective hull E(M) is of the same size. Further, if M is an injective module with |M| ≥ (2^λ)⁺ and K ≤ M is its submodule such that |M/K| ≤ λ, then K contains an injective submodule L with |M/L| ≤ 2^λ. These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author’s previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3–6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105–107], [Pure subgroups, Math. Bohem. 126 (2001), 649–652]. This research has been partially supported by the Grant Agency of the Charles University, grant #GAUK 301-10/203115/B-MAT/MFF and also by the institutional grant MSM 113 200 007.     

On torsionfree classes which are not precover classes

In the class of all exact torsion theories the torsionfree classes are cover (pre-cover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented. This research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/06/0510 and also by the institutional grant MSM 0021620839.     

On the Existence of Relative Injective Covers

Let be a hereditary torsion theory for the category -mod of unital left -modules over an associative ring with an identity element. The purpose of this note is to prove that if the associated Gabriel filter consists of finitely presented left ideals, then every module has a -injective cover and if contains a cofinal subset of finitely presented left ideals, then every module has a -torsionfree -injective cover. The methods used working with pure submodules contained in ``large" submodules also allow to unify the proofs of some previously known results.     

Precovers

Let be an abstract class (closed under isomorpic copies) of left R-modules. In the first part of the paper some sufficient conditions under which is a precover class are given. The next section studies the -precovers which are -covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left R-modules. Especially, several sufficient conditions for the existence of -torsionfree and -torsionfree -injective covers are presented.     

Butler groups as smooth ascending unions

Fuchs and Rangaswamy [8] proved that a smooth ascending union G = U α<λ Gα of pure B₂-subgroups is a B₂-group provided either λ = ω_o or λ = ω₁,/G/ = N₁ and all the subgroups Gα are decent in G. In this paper, we extend this result to arbitrary cardinal numbers λ under suitable hypotheses on the factor-groups G _α+1/G _α.