p-Adic Dynamics and Sullivan's No Wandering Domains Theorem

In this paper we study dynamics on the Fatou set of a rational function . Using a notion of components of the Fatou set defined by Benedetto, we state and prove an analogue of Sullivan's No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points.     

Remarks on Nonlinear Neumann Problems in Periodic Domains

We study a semilinear elliptic equation Au = f(x, u) with nonlinear Neumann boundary condition Bu = φ(ξ, u) in an unbounded domain Ω ? ?ⁿ, the boundary of which is defined by periodic functions. We assume that f and φ and the coefficients of the operators are asymptotically periodic in the space variables. Our main result states the existence of an asymptotically decaying, nontrivial solution of this problem with minimal energy. The proof is based on the concentration-compactness principle.     

Consensus of a Two-Agent System with Nonlinear Dynamics and Time-Varying Delay

Applications of Mathematics - To explore the impacts of time delay on nonlinear dynamics of consensus models, we incorporate time-varying delay into a two-agent system to study its long-time...     

Metastable Dynamics and Spatially Inhomogeneous Equilibria in Dumbbell-Shaped Domains

The motion of internal layers for three singularly perturbed reaction diffusion problems, including the Allen–Cahn equation, is studied in a two-dimensional dumbbell-shaped domain. The channel region that connects the two attachments, or lobes, of the dumbbell is taken to be rectangular. The motion of straight-line internal layers in the channel region is analyzed by using an asymptotic projection method. It is shown that this motion is metastable and highly dependent on the local convexity properties of the boundary near the contact region between the ends of the channel and the two attachments. When the domain is nonconvex it is shown that the metastable internal layers dynamics in the channel tends, as t →∞, to a limiting, stable, spatially inhomogeneous equilibrium solution.     

Some Problems in Operator Theory on Bounded Symmetric Domains

We discuss some problems concerning the operators K _m (Z,Z ^*), where K _m is the reproducing kernel of the Peter–Weyl space with signature m and Z stands for the commuting tuple of operators of multiplication by the coordinate variables on a bounded symmetric domain. These problems have applications for construction of operator models, as well as for a possible generalization of the recent dilation theory for row contractions due to Arveson.     

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge–Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.     

在时间尺度上研究带有脉冲的泄漏时滞细胞神经网络反周期解的动态特征

在时间尺度上,通过使用叠合度理论中的连续定理和微分不等式技巧,研究带有脉冲的时变泄漏时滞细胞神经网络模型的反周期解,获得了一些使带有脉冲的时变泄漏时滞细胞神经网络模型的反周期解存在和全局指数稳定的充分条件,并将以前的结论在时间尺度上做了扩展.     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

Dissipative Elliptic Problems in Domains with Cylindrical Ends, Scattering Matrices, and Radiation Conditions

In a domain with cylindrical ends at infinity, we consider a general elliptic dissipative boundary value problem. The coefficients of the imaginary part of the operator of the problem vanish as The asymptotic behavior of the solutions is expressed in terms of incoming and outgoing waves (the amplitudes of such waves can grow at infinity). We introduce an (augmented) scattering matrix and, in terms of this matrix, we compute the number of linearly independent solutions to the homogeneous problem vanishing at infinity with a given rate. We discuss the statement of a problem with the so-called radiation conditions. The natural radiation conditions (only outgoing waves occur in asymptotic formulas for solutions) can be applied in any case. Other admissible radiation conditions for the problem under consideration are connected with the natural ones via scattering matrices. Bibliography: 12 titles.     

Dirichlet Problems in Polyhedral Domains I: Regularity of the Solutions

The solution of the Dirichlet problem relative to an elliptic operator in a polyhedron has a complex singular behaviour near edges and vertices. Here we show that this solution and its conormal derivative have a global regularity in appropriate weighted Sobolev spaces. We also investigate some compact embeddings of these spaces. The present results will be applied in a forthcoming work to the constructive treatment of the problem by optimal convergent finite clement method and boundary element method.     

Epi/Hypo-Convergence of Bifunctions on General Domains and Approximations of Quasi-Variational Problems

Set-Valued and Variational Analysis - Epi/hypo-convergence is extended to the case of bifunctions defined on general domains. Its basic characterizations are established. Variational properties...     

强Z-连续domain和Z-代数domain的刻画定理

给出强Z-连续domain和Z-代数domain的一个刻画及一个范畴性质--余反射性质.     

Local Hardy Spaces and Inhomogeneous Dirichlet Problems in Exterior Regular Domains

In this paper, firstly we give an atomic decomposition of the local Hardy spaces h^p_r(Ω)(O < p ≤ 1) and their dual spaces. where the domain Ω is exterior regular in R^n(n ≥ 3). Then for given data f ∈ h^p_r(Ω), we discuss the inhomogeneous Dirichlet problems: {Lu = f quad in Ω u = 0 qquad on ∂ Ω where the operator L is uniformly elliptic. Also we obtain the estimation of Green potential in the local Hardy spaces h^p_r(Ω).     

Riemann and Dirichlet Type Problems for Hyperanalytic Functions in Fractal Domains

In this paper we are interested in finding solvability conditions for Riemann and Dirichlet type boundary value problems with generalized Hölder-continuous boundary data in the case of hyperanalytic functions defined in a domain of the complex plane bounded by an h-summable closed curve.     

Curves, Domains and Picard's Theorem

Based on computing spherical lengths of polygonal curves andspherical areas of domains bounded by polygonal curves on thecomplex plane, this paper proves a property of holomorphic functionsand, as an application, gives a very brief proof of a famousinequality obtained by L. V. Ahlfors which easily implies Picard'stheorem. Taken together with the arguments given by Alhfors,therefore, this paper in fact gives an elementary, brief andgeometrical proof of Picard's theorem.     

Existence Results for Singular Boundary Value Problems on Unbounded Domains in Banach Spaces

In this paper, the Krasnosels’kii fixed point theorem in cones for strict set-contractions is used to investigate the existence of single and twin positive solutions for a class of a two-point boundary value problem of second-order nonlinear differential equations posed on an infinite interval. The nonlinearity, which may have a time-singularity, takes values in a general Banach space and has at most polynomial growth with respect to the unknown.