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.     

非自治Schrdinger-KdV型藕合方程组的一致吸引子及其维数估计

本文研究了非自治Schr■dinger-KdV型藕合方程组的非线性动力学行为．运用具有两个参数的算子簇来描述非自治无穷维动力系统的方法,证明了该系统的一致吸引子的存在性．进一步,对其Hausdorff维数进行了估计．     

Calculating the first nontrivial 1-cocycle in the space of long knots

For spaces of knots in ℝ³, the Vassiliev theory defines the so-called cocycles of finite order. The zero-dimensional cocycles are finite-order invariants. The first nontrivial cocycle of positive dimension in the space of long knots is one-dimensional and is of order 3. We apply the combinatorial formula given by Vassiliev in his paper     

Persistence,extinction and global asymptotical stability of a non-autonomous predator–prey model with random perturbation

A two-species stochastic non-autonomous predator–prey model is investigated. Sufficient criteria for extinction, non-persistence in the mean and weak persistence in the mean are established. The critical value between persistence and extinction is obtained for each species in many cases. It is also shown that the system is globally asymptotically stable under some simple conditions. Some numerical simulations are introduced to illustrate the main results.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg–Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by additive noise with α∈(0, 1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.     

Representation of certain infinitely divisible probability measures on Banach spaces

Subclasses L₀ ? L₁ ? … ? L_∞ of the class L₀ of self-decomposable probability measures on a Banach space are defined by means of certain stability conditions. Each of these classes is closed under translation, convolution and passage to weak limits. These subclasses are analogous to those defined earlier by K. Urbanik on the real line and studied in that context by him and by the authors. A representation is given for the characteristic functionals of the measures in each of these classes on conjugate Banach spaces. On a Hilbert space it is shown that L_∞ is the smallest subclass of L₀ with the closure properties above containing all the stable measures.     

Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

     

On extension of cocycles to normalizer elements, outer conjugacy, and related problems

Let be an ergodic automorphism of a Lebesgue space and a cocycle of with values in an Abelian locally compact group . An automorphism from the normalizer of the full group is said to be compatible with if there is a measurable function such that at a.e. . The topology on the set of all automorphisms compatible with is introduced in such a way that becomes a Polish group. A complete system of invariants for the -outer conjugacy (i.e. the conjugacy in the quotient group is found. Structure of the cocycles compatible with every element of is described.

     

Relative difference sets fixed by inversion (III)—Cocycle theoretical approach

In this paper, we give a characterization of a group G which contains a semiregular relative difference set R relative to a central subgroup N containing the commutator subgroup [G,G] of G such that 1∈R and rRr=R for all r∈R. In particular, these relative difference sets are fixed by inversion and inner automorphisms of the group are multipliers. We also present a construction of such relative difference sets.