广义幂级数环上Morita Context环的稳定条件

In this paper, we show that if rings A and B are （s, 2）-rings, then so is the ring of a Morita Context（[[A^S,≤]],[[B^S,≤]],[[M^S,≤]],[[N^S,≤]],ψ^S,Ф^S）of generalized power series. Also we get analogous results for unit 1-stable ranges, GM-rings and rings which have stable range one. These give new classes of rings satisfying such stable range conditions.     

Morita Contexts with Many Units

In this paper, we investigate stable range conditions over Morita contexts. As applications, we calculate the Whitehead groups of Morita contexts for exchange rings with primitive factors artinian.     

关于正则模的自同态环

本文研究正则模Ｐ的自同态环Ｓ，主要给出使Ｓ的稳定秩为１的Ｐ的几个充分条件．     

形式三角矩阵环的可分性和稳定度

In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely generated projec-tive T-modules to have n in the stable range is given under the assumption that A and B are exchange rings.     

Extensions of Rings with Many Units

Abstract

In this paper, we investigate stable range conditions over extensions of matrix rings. It is shown that a ring R satisfies (s, 2)-stable range if and only if R has a complete orthogonal set {e ₁,…, e _n } of idempotents such that all e _i Re _i satisfy (s, 2)-stable range. Also we extend this result to (s, 2)-rings and rings satisfying unit 1-stable range.     

Global analysis in delayed ratio-dependent Gause-type predator-prey models

A class of three-dimensional delayed Gause-type predator-prey model with ratio-dependent is considered. Firstly, we present some results, including the boundedness of solutions and the permanence of system. Secondly, we construct a Lyapunov function to give the global asymptotically stable of the positive equilibrium under some parameter conditions. Finally, we analyed the influence of the time delay on the system and showed that the occurrence of small range of periodic motion.     

New global stability conditions for a class of difference equations

In this paper we consider some classes of difference equations, including the well-known Clark model, and study the stability of their solutions. In order to do that we introduce a property, namely semicontractivity, and study relations between ‘semi-contractive’ functions and sufficient conditions for the solution of the difference equation to be globally asymptotically stable. Moreover, we establish new sufficient conditions for the solution to be globally asymptotically stable, and we improve the ‘3/2 criteria’ type stability conditions.      

Computation of k1 Via mennicke symbols

For any ring A, the group K₁A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke’s symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK₁A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK₁ is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).     

一类Kolmogorov模型的全局稳定性

本文对如下的一类Kolmogorov模型进行了定性分析,得到了系统(Eqs.)的唯一正平衡点全局稳定的充分条件和给出了生态解释。     

Homology stability for linear groups

Summary LetR be a commutative finite dimensional noetherian ring or, more generally, an associative ring which satisfies one of Bass' stable range conditions. We describe a modified version of H. Maazen's work [18], yielding stability for the homology of linear groups overR. Applying W.G. Dwyer's arguments (cf. [9]) we also get stability for homology with twisted coefficients. For example,H ₂(GL _n (R), R ⁿ ) takes on a stable value whenn becomes large.     

Asymmetric Spotty Patterns for the Gray–Scott Model in R2

In this paper, we rigorously prove the existence and stability of asymmetric spotty patterns for the Gray–Scott model in a bounded two-dimensional domain. We show that given any two positive integers   k ₁, k ₂  , there are asymmetric solutions with   k ₁  large spots (type A) and   k ₂  small spots (type B). We also give conditions for their location and calculate their heights. Most of these asymmetric solutions are shown to be unstable. However, in a narrow range of parameters, asymmetric solutions may be stable.     

On endomorphism rings of regular modules

In this paper, we study the endomorphism rings of regular modules. We give sufficient conditions on a regular projective moduleP such that End_R (P) has stable range one. Dedicated to Professor Zhou Boxun for his 80'th Birthday The author is supported by the NNSF of China (No. 19601009)     

Relaxation oscillations in a class of predator-prey systems

We consider a class of three-dimensional, singularly perturbed predator-prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical systems techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model will be discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameters values of the model.     

A convergence analysis of the Landweber iteration for nonlinear ill-posed problems

Summary. In this paper we prove that the Landweber iteration is a stable method for solving nonlinear ill-posed problems. For perturbed data with noise level we propose a stopping rule that yields the convergence rate ) under appropriate conditions. We illustrate these conditions for a few examples. Received February 15, 1993 / Revised version received August 2, 1994     

Quasi-duo rings and stable range descent

In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. Using some of these characterizations, we prove that, if a quasi-duo ring R has stable range ?n, the same is true for any semisplit corner ring of R. This contrasts with earlier results of Vaserstein and Warfield, which showed that the stable range can increase unboundedly upon descent to (even) Peirce corner rings.     

THE VOLTERRA MODEL FOR THREE SPACIES PREDATOR-PREY SYSTEMS IN LOOPS

We consider the three species predator-prey model with the same intrinsic growth rates, where species 3 feeds on species 2, species 2 feeds on species 1, species 1 feeds on species 3. An open question raised by Nishan Krikorian is answered: We obtain the necessary and sufficient conditions for all the orbits to be unbounded. We also obtain the necessary and sufficient conditions for the positive equilibrium to be globally stable. It is shown that there exists a family of neutrally stable periodic orbits, in which we extend Darboux method to three-species models for the first time.     

Stable multi-sets

In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets.  For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined.  The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks. Received: February 14, 2001/Revised version: September 7, 2001     

On Maximal Inequalities for Stable Stochastic Integrals

Sharp maximal inequalities in large and small range are derived for stable stochastic integrals. In order to control the tail of a stable process, we introduce a truncation level in the support of its Lévy measure: we show that the contribution of the compound Poisson stochastic integral is negligible as the truncation level is large, so that the study is reduced to establish maximal inequalities for the martingale part with a suitable choice of truncation level. The main problem addressed in this paper is to give upper bounds which remain bounded as the parameter of stability of the underlying stable process goes to 2. Applications to estimates of first passage times of symmetric stable processes above positive continuous curves complete this work.      

Generalized Stable Exchange Rings

In this paper, we investigate diagonal reductions of matrices over generalized stable exchange rings. We show that every regular matirx over generalized stable exchange rings with stable range 2 admits power diagonal reduction.AMS Subject Classification (1991): 16A30 16E50This work was supported by the National Natural Science Foundation of China.     

Exchange Rings Satisfying the <Emphasis Type="Italic">n</Emphasis>-Stable Range Condition,II

This is a continuation of the paper [14]. It is shown that any finite subdirect product of exchange rings satisfying the n-stable range condition is still an exchange ring satisfying the n-stable range condition. Furthermore, we give necessary and sufficient conditions on matrices over an exchange ring R, under which R satisfies the n-stable range condition. This generalizes the corresponding results for unit-regular rings and the stable range one condition.2000 Mathematics Subject Classification: 19B10, 16E50This work was supported by the National Natural Science Foundation of China (Grant No. 19801012) and the Ministry of Education of China.