SUFFICIENT CONDITIONS FOR LOCAL SOLVABILITY OF LEFT INVARIANT LPDOS ON THE GROUPS OF TYPE H

Some sufficient conditions are established for local solvability of inhomogeneous left invariant partial differential operators on the groups of type H. First some general sufficient conditions are derived for local solvablity of left invariant operators on the groups of type H. Then by asing these results the author discusses the local solvability of a special class of inhomogeneous left invariant operators on this type of groups.     

ON THE INVARIANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLD

In this paper, the vertical and horizontal distributions of an invariant submanifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.     

具有变时滞Hopfield型神经网络的正不变集与吸引集

研究了具有变时滞Hopfield型神经网络的正不变集与吸引集，获得了正不变集与吸引集存在性的充分判据。     

Analytic properties of Rényi’s invariant density

Under certain conditions a many-to-one transformation of the unit interval onto itself possesses a finite invariant ergodic measure equivalent to Lebesgue measure. The purpose of this paper is to investigate these conditions and to show how differentiable and analytic properties of the invariant density are inherited from the original transformation.     

Noninvariant Boundary Conditions

It is generally believed that in order to solve initial and boundary value problems using Lie symmetry methods, the boundary and initial conditions need to be left invariant by the infinitesimal symmetry generator which admits the invariant solution. In this article we give less restrictive conditions on the imposed initial and boundary values in order that they be recoverable with a particular symmetry generator.     

Fibonacci迹映射的不变锥面

该文利用［１］中关于可反转映射的ＫＡＭ定理,证明一定条件下Ｒ~３上的可反转映射在下动点附近存在不变锥面．并验证Ｆｉｂｏｎａｃｃｉ迹映射中存在这种不变锥面．     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

Two-dimensional invariant tori in the neighborhood of an elliptic equilibrium of Hamiltonian systems

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.     

A CHARACTERIZATION OF LOCALLY COMPACT INNER AMENABLE GROUPS

For locally compact groups G, Kuan Yuan studied a notion of inner amenability groups, that is, if there exists an inner invariant mean on G. In this article, among other things, the author investigates the inner amenability on a locally compact group G. The author gives sufficient conditions and some necessary conditions about G to have an inner invariant mean.     

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we obtain a characterization of invariant measures of stochastic evolution equations and stochastic partial differential equations of pure jump type. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.     

Krein空间中无穷维Hamilton算子的极大确定不变子空间的存在性问题

本文研究了不定度规空间空间中的无穷维Hamilton算子.利用Plus算子存在极大不变子空间的性质,获得了无穷维Hamilton算子在Krein空间中存在极大确定不变子空间的充分条件.     

Nonsymmetric ground states of symmetric variational problems

In this paper we study a minimization problem which is invariant by rotation. The corresponding Euler-Lagrange equations are semilinear elliptic equations in an exterior domain with Neumann boundary conditions. We prove that this minimization problem has at least one solution. Yet all its solutions are shown not to be rotationally invariant. Furthermore we describe how the radial symmetry is broken.     

Invariant and stably invariant sets for differential inclusions

We discuss conditions, in terms of Lyapunov functions, under which a given set in the extended phase space of a nonautonomous differential inclusion becomes positively invariant, invariant, stably invariant, or asymptotically stably invariant. We also derive conditions under which the integral funnel of a differential inclusion is recurrent in time. A series of examples are considered. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 262, pp. 202–221.     

Conditions for Permanence and Ergodicity of Certain SIR Epidemic Models

In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results.

     

Localization of periodic orbits of the Rössler system under variation of its parameters

The localization problem of compact invariant sets of the Rössler system is considered in this paper. The main interest is attracted to a localization of periodic orbits. We establish a number of algebraic conditions imposed on parameters under which the Rössler system has no compact invariant sets contained in half-spaces z > 0; z < 0 and in some others. We prove that if parameters (a, b, c) of the Rössler system are such that this system has no equilibrium points then it has no periodic orbits as well. In addition, we give localization conditions of compact invariant sets by using linear functions and one quadratic function.     

Analytically invariant spectral resolvents of closed operators

The concept of analytically invariant subspace, in the framework of spectral resolvents, is extended to unbounded closed operators. One obtains necessary and sufficient conditions for a spectral resolvent to be analytically invariant. An application, concerning a strong type of spectral decomposition for closed operators with the spectra on the real line, concludes this study.     

On projective invariant subgroups of Abelian groups

The properties of projective invariant subgroups are studied. The structure of these subgroups in nonreduced groups is described. The conditions under which projective invariant subgroups are fully invariant are considered.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

Bounds for a domain containing all compact invariant sets of the system describing the laser–plasma interaction

In this paper we consider the localization problem of compact invariant sets of the system describing the laser–plasma interaction. We establish that this system has an ellipsoidal localization for simple restrictions imposed on its parameters. Then we improve this localization by applying other localizing functions. In addition, we give sufficient conditions under which the origin is the unique compact invariant set.