Rank 2 arithmetically Cohen‐Macaulay bundles on a general quintic surface

Rank 2 arithmetically Cohen‐Macaulay vector bundles on a general quintic hypersurface of the three‐dimensional projective space are classified (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noncrossing trees are almost conditioned Galton–Watson trees

A noncrossing tree (NC‐tree) is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments that do not cross. In this article, we show that NC‐trees with size n are conditioned Galton–Watson trees. As corollaries, we give the limit law of depth‐first traversal processes and the limit profile of NC‐trees. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 115–125, 2002     

Cohen–Lenstra heuristic and roots of unity

We report on computational results indicating that the well-known Cohen–Lenstra–Martinet heuristic for class groups of number fields may fail in many situations. In particular, the underlying assumption that the frequency of groups is governed essentially by the reciprocal of the order of their automorphism groups, does not seem to be valid in those cases. The phenomenon is related to the presence of roots of unity in the base field or in intermediate fields. For all the examples considered, we propose alternative predictions which agree closely with the data, and which are inspired by results of Gerth.     

On the notion of Cohen–Macaulayness for non-Noetherian rings

There exist many characterizations of Noetherian Cohen–Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non-Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster–Eagon result on Cohen–Macaulayness of invariant rings, in the context of non-Noetherian rings.     

On symmetric algebras which are Cohen Macaulay

We give some necessary and sufficient conditions for S_A(m) being Cohen-Macaulay, where S_A(m) is the Symmetric algebra of the maximal ideal of an homomorphic image A of a regular local ring.This paper was supported by C.N.R. (Consiglio Nazionale delle Ricerche)     

Exponential stability of reaction–diffusion generalized Cohen–Grossberg neural networks with time-varying delays

The stability property of reaction–diffusion generalized Cohen–Grossberg neural networks (GDCGNNs) with time-varying delay are considered. Without assuming the monotonicity and differentiability of activation functions, nor symmetry of synaptic interconnection weights, delay independent and easily verifiable sufficient conditions to guarantee the exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, by employing the method of variational parameter and inequality technique. One example is given to illustrate the theoretical results.     

Stability analysis of Cohen–Grossberg neural networks with discontinuous neuron activations

In this paper, we consider the dynamical behavior of delayed Cohen–Grossberg neural networks with discontinuous activation functions. Some sufficient conditions are derived to guarantee the existence, uniqueness and global stability of the equilibrium point of the neural network. Convergence behavior for both state and output is discussed. The constraints imposed on the interconnection matrices, which concern the theory of M-matrices, are easily verifiable and independent of the delay parameter. The obtained results improve and extend the previous results. Finally, we give an numerical example to illustrate the effectiveness of the theoretical results.     

Crowding games are sequentially solvable

A sequential-move version of a given normal-form game Γ is an extensive-form game of perfect information in which each player chooses his action after observing the actions of all players who precede him and the payoffs are determined according to the payoff functions in Γ. A normal-form game Γ is sequentially solvable if each of its sequential-move versions has a subgame-perfect equilibrium in pure strategies such that the players' actions on the equilibrium path constitute an equilibrium of Γ.  A crowding game is a normal-form game in which the players share a common set of actions and the payoff a particular player receives for choosing a particular action is a nonincreasing function of the total number of players choosing that action. It is shown that every crowding game is sequentially solvable. However, not every pure-strategy equilibrium of a crowding game can be obtained in the manner described above. A sufficient, but not necessary, condition for the existence of a sequential-move version of the game that yields a given equilibrium is that there is no other equilibrium that Pareto dominates it. Received July 1997/Final version May 1998     

Global dynamic analysis of general Cohen–Grossberg neural networks with impulse

In this paper, a class of general Cohen–Grossberg neural networks with impulse is studied. Based on the method of Lyapunov functional, sufficient conditions on global exponential stability are given. Furthermore, many corollaries are obtained. Our results improve some of the earlier findings, and are suitable for many applications.     

Simplicial groups that are models for algebraic K-theory

In this paper, we present and compare some simplicial groups, functorially associated to a ring R, whose homotopy groups are Quillens K-groups of R. The first such simplicial group is the group (NQPR), where is the loop space construction of Clemens Berger, applied to the simplicial set NQPR (the nerve of Quillens category QPR). The second is a subgroup GR of the simplicial group (NQPR). This second group is compared to Kans construction [12] of a loop group for a connected simplicial set, and shown to be isomorphic to it as a simplicial group. Other simplicial groups that are models for algebraic K-theory are also presented; in particular, the subgroup G(s.PR) of (s.PR); here, s.PR is Waldhausens simplicial set [25], [26]. We initially give an exposition of Bergers construction in general; then, we present the construction of GR and a summary of Kans construction. Next, we point out that GR is an infinite loop object in the category of simplicial groups, and draw some corollaries. We then compare directly the homotopy groups thus constructed with the classical K-theory in degrees 0 and 1. The final section compares various models.     

Some criteria for robust stability of Cohen–Grossberg neural networks with delays

This paper considers the problem of robust stability of Cohen–Grossberg neural networks with time-varying delays. Based on the Lyapunov stability theory and linear matrix inequality (LMI) technique, some sufficient conditions are derived to ensure the global robust convergence of the equilibrium point. The proposed LMI conditions can be checked easily by recently developed algorithms solving LMIs. Comparisons between our results and previous results admits our results establish a new set of stability criteria for delayed Cohen–Grossberg neural networks. Numerical examples are given to illustrate the effectiveness of our results.     

Analysis of stability for impulsive fuzzy Cohen–Grossberg BAM neural networks with delays

In this paper, based on the topological degree theory, Lyapunov functional method and inequality analysis technique, the existence and global exponential stability of equilibrium of impulsive fuzzy Cohen–Grossberg bi‐directional associative memory neural networks with delays, are investigated. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential p-stability of delayed Cohen–Grossberg-type BAM neural networks with impulses

An impulsive Cohen–Grossberg-type bidirectional associative memory (BAM) neural networks with distributed delays is studied. Some new sufficient conditions are established for the existence and global exponential stability of a unique equilibrium without strict conditions imposed on self regulation functions. The approaches are based on Laypunov–Kravsovskii functional and homeomorphism theory. When our results are applied to the BAM neural networks, our results generalize some previously known results. It is believed that these results are significant and useful for the design and applications of Cohen–Grossberg-type bidirectional associative memory networks.     

Stability and Hopf bifurcation of a delayed Cohen–Grossberg neural network with diffusion

In this paper, a delayed Cohen–Grossberg neural network with diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation at the trivial steady state are established, respectively. By using the normal form theory and the center manifold reduction of partial function differential equations, formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.     

Novel stability criteria for uncertain delayed Cohen–Grossberg neural networks using discretized Lyapunov functional

This paper deals with the stability analysis of delayed uncertain Cohen–Grossberg neural networks (CGNN). The proposed methodology consists in obtaining new robust stability criteria formulated as linear matrix inequalities (LMIs) via the Lyapunov–Krasovskii theory. Particularly one stability criterion is derived from the selection of a parameter-dependent Lyapunov–Krasovskii functional, which allied with the Gu’s discretization technique and a simple strategy that decouples the system matrices from the functional matrices, assures a less conservative stability condition. Two computer simulations are presented to support the improved theoretical results.     

Stability analysis in Lagrange sense for a non-autonomous Cohen–Grossberg neural network with mixed delays

The paper discusses the global exponential stability in the Lagrange sense for a non-autonomous Cohen–Grossberg neural network (CGNN) with time-varying and distributed delays. The boundedness and global exponential attractivity of non-autonomous CGNN with time-varying and distributed delays are investigated by constructing appropriate Lyapunov-like functions. Moreover, we provide verifiable criteria on the basis of considering three different types of activation function, which include both bounded and unbounded activation functions. These results can be applied to analyze monostable as well as multistable biology neural networks due to making no assumptions on the number of equilibria. Meanwhile, the results obtained in this paper are more general and challenging than that of the existing references. In the end, an illustrative example is given to verify our results.     

Nonlinear measure approach for the robust exponential stability analysis of interval inertial Cohen–Grossberg neural networks

This article is concerned with the existence and robust stability of an equilibrium point that related to interval inertial Cohen–Grossberg neural networks. Such condition requires the existence of an equilibrium point to a given system, so the existence and uniqueness of the equilibrium point are emerged via nonlinear measure method. Furthermore, with the help of Halanay inequality lemma, differential mean value theorem as well as inequality technique, several sufficient criteria are derived to ascertain the robust stability of the equilibrium point for the addressed system. The results obtained in this article will be shown to be new and they can be considered alternative results to previously results. Finally, the effectiveness and computational issues of the two models for the analysis are discussed by two examples. © 2016 Wiley Periodicals, Inc. Complexity 21: 459–469, 2016