We construct examples, that log Hölder continuity of the integrated density of states cannot be improved. Our examples are limit-periodic.

In two-dimensional lattice fermion model a determinant representation for the twopoint correlation function of the twist field in the disorder phase is obtained. This field is defined by twisted boundary conditions for lattice fermion field. The large distance asymptotics of the correlation function is calculated at the critical point and in the scaling region. The result is compared with the v...

We consider the convective Cahn–Hilliard equation with periodic boundary conditions as an infinite dimensional dynamical system and establish the existence of a compact attractor and a finite dimensional inertial manifold that contains it. Moreover, Gevrey regularity of solutions on the attractor is established and used to prove that four nodes are determining for each solution on the attractor...

A large number of di erential equation problems which admit traveling waves have very large (typically in nite) naturally de ned domains, with boundary conditions de ned at the domain boundary. To be able to numerically solve these problems in smaller subdomains of the original domain, arti cial boundary conditions must be de ned for these subdomains. One such arti cal boundaryconditionswhich c...

In this chapter we focus on what can be called the purest turbulence problem, as well as the classical one. It is in fact so pure that, strictly speaking, it cannot exist in nature, although experience shows that natural turbulence often has behavior remarkably close to the pure form. This is the case where rotational and gravitational forces are negligible (i.e., g = f = 0); b or c can be cons...

Dynamics of a chaotic spiking neuron model are being studied mathematically and experimentally. The Nonlinear Dynamic State neuron (NDS) is analysed to further understand the model and improve it. Chaos has many interesting properties such as sensitivity to initial conditions, space filling, control and synchronization. As suggested by biologists, these properties may be exploited and play vita...

The laminar–turbulent boundary Σ is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognizing approaches to these structures and then for identifyi...

The complex Ginzburg± Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of diþ erent symmetries. Solutions of the CGL equation may or may not be ® xed by the action of these symmetries. We investigate the stability of chaotic solutions with some re ̄ ectional symmetry to perturbations which break that symmetry. This can be achieved by considering th...

In the present work, we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems. More specifically, we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration. In our study, the singularities ...

We investigate the application of windowed Fourier frames to the numerical solution of partial differential equations, focussing on elliptic equations. The action of a partial differential operator (PDO) on a windowed plane wave is close to a multiplication, where the multiplication factor is given by the symbol of the PDO evaluated at the wave number and central position of the windowed plane ...