Evolutionary dynamics of a system with periodic coefficients

We investigate a problem in evolutionary game theory based on replicator equations with periodic coefficients. This approach to evolution combines classical game theory with differential equations. The RPS (Rock-Paper-Scissors) system studied has application to the population biology of lizards and to bacterial dynamics. The presence of periodic coefficients models variations in the environment due to seasonal effects and results in parametric excitation which is studied through the use of perturbation series and numerical integration.     

The smooth solutions of a class of coupled KdV equations.

This paper is devoted to the study of the periodic initial boundary value problem and Cauchy problem for the coupled KdV equations. By the Galerkin method and sequential approximation, we get a seies of priori estimates and establish the existence of classical local solution to the periodic problem for the system. Then we obtain the existence and uniqueness of global smooth solution when the coefficients of the system satisfy certain conditions by energy menthod, conserved quantities and nonconservative quatity I(u,v).     

Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization

In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem.     

Homogenization of a Coupled System with Periodic Oscillating Coefficients

We study the homogenization of a coupled system with periodic oscillating coefficients in bounded non-homogeneous media. The system couples the Navier–Stokes and a classical parabolic diffusive equation. To do that, we introduce a generalized compensate compactness result and a suitable class of test function to this problem. By passing the limit, we obtain the homogenized model of this problem.     

On the periodic KdV equation in weighted Sobolev spaces

We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients.     

Multipoint problem for typeless factorized differential operators

We analyze the well-posedness of a problem with multipoint conditions in the time variable and periodic conditions in the spatial coordinates for differential operators decomposable into operators of the first order with complex coefficients. We establish conditions for the existence and uniqueness of the classical solution of the problem under consideration and prove metric theorems for the lower estimates of small denominators appearing in the process of construction of the solution.     

Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ ^N with isolated holes of size ɛ² (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.     

A boundary-value problem for parabolic equations with general nonlocal conditions

We study a boundary-value problem with general two-point conditions with respect to the time coordinate, and periodic conditions on the spatial coordinates for Shilov-parabolic equations with constant coefficients. We construct the solution in the form of a Fourier series. We establish conditions for existence and uniqueness of a classical solution of the problem. We prove quantitative theorems on a lower bound for the small denominators that arise in solving the problem. Translated fromMatematichni Methody i Fiziko-mekhanichni Polya, Vol. 38, 1995.     

Proof of the Razumov-Stroganov conjecture

The Razumov-Stroganov conjecture relates the ground-state coefficients in the periodic even-length dense O(1) loop model to the enumeration of fully-packed loop configurations on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points.Here we prove this conjecture, by means of purely combinatorial methods. The main ingredient is a generalization of the Wieland proof technique for the dihedral symmetry of these classes, based on the ‘gyration’ operation, whose full strength we will investigate in a companion paper.     

A model of immune system with time-dependent immune reactivity

In this paper we deal with the Marchuk model of an immune system. Among the main parameters of the model are the coefficients which describe the state of infected organism and the rate of production of antibodies. In the classical model these coefficients are constants. We consider the case when these coefficients are time-dependent. In particular, we are interested in the case of periodic coefficients which can describe periodic changes of the immune reactivity due to periodic changes of the environment. We examine the asymptotic behaviour of solutions. Under some assumptions we prove that the solutions tend to periodic functions. We also present the results of numerical simulations to illustrate the behaviour of solutions.     

The Krzyż conjecture revisited

The Krzy? conjecture concerns the largest values of the Taylor coefficients of a non-vanishing analytic function bounded by one in modulus in the unit disk. It has been open since 1968 even though information on the structure of extremal functions is available. The purpose of this paper is to collect various conditions that the coefficients of an extremal function (and various other quantities associated with it) should satisfy if the conjecture is true and to show that each one of these properties is equivalent to the conjecture itself. This may provide several possible starting points for future attempts at solving the problem.     

Nonlocal boundary-value problem for parabolic equations

We study the problem for Shilov parabolic equations of arbitrary order with constant coefficients with conditions nonlocal in time and periodic in space variables. We establish conditions for the existence and uniqueness of a classical solution of the problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1621–1626, December, 1994.The work was supported by the Foundation for Fundamental Studies of Ukrainian State Committee on Science and Technology.     

Bochner’s theorem and Stepanov almost periodic functions

Motivated by a renewed interest in generalizations of classical almost periodicity (originally due to Harald Bohr), we develop a theorem of Bochner within the framework of almost periodic functions in the sense of Stepanov. As a result we establish some conditions that guarantee the existence of Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients. Finally, we extend a now classic theorem of Favard originally stated for classical almost periodic functions to the Stepanov almost periodic case.      

Resz means of bounded states: semi-classical asymptotic

The semi-classical asymptotic behaviour of the Riesz means of a distribution of eigenvalues is investigated, at a non-critical energy level. For Schrödinger type operators, we obtain the second term related to the periodic trajectories of the classical Hamiltonian. This oscillating term is explained for Riesz means of order one with the aim of discussing a Lieb-Thirring conjecture.     

On the detection of singularities of a periodic function

We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also help us to analyze the capabilities of periodic spline wavelets, trigonometric polynomial wavelets, and some of the classical summability methods in the theory of Fourier series. This revised version was published online in June 2006 with corrections to the Cover Date.     

源于DNA的非线性波动方程组的Cauchy问题(英文)

本文首先证明源于DNA的非线性波动方程组的周期边值问题局部古典解的存在性和唯一性.其次通过周期边值问题序列证明这个方程组的Cauchy问题存在唯一的局部古典解.     

Approximate periodic solutions of a nonlinear parabolic system and an identification problem

This work continues our study in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005] on the identification problem for the coefficients for the lower order terms in a parabolic system, through its approximate periodic solutions. Different from the work in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005], our system now is nonlinear and the coefficients to be detected are from the first order term. From the application point of view, we now try to determine the diffusion coefficients for the system by the observation over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper.     

A finite element method for elliptic problems with rapidly oscillating coefficients

In this paper, we consider solving second-order elliptic problems with rapidly oscillating coefficients. Under the assumption that the oscillating coefficients are periodic, on the basis of classical homogenization theory, we present a finite element method whose key is to combine a numerical approximation of the 1-order approximate solution of those equations and a numerical approximation of the classical boundary corrector of those equations from different meshes exploiting the need for different levels of resolution. Numerical experiments are included to illustrate the competitive behavior of the proposed finite element method.     

Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.