We extend the trace formula recently proven for general one-dimensional Schrödinger operators which obtains the potential V (x) from a function ξ(x,λ) by deriving trace relations computing moments of ξ(x,λ)dλ in terms of polynomials in the derivatives of V at x. We describe the relation of those polynomials to KdV invariants. We also discuss trace formulae for analogs of ξ associated with bound...

2014 We study the influence of boundary conditions on the existence and coexistence of distortions (textures, defects) in ordered media. The standard topological classification is based on mappings from spheres into the manifold of internal states (characterizing the order). For periodic, or partly periodic boundary conditions, the spheres have to be replaced by tori, or quasitori. The modifica...

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

Let S be a closed orientable surface with genus g ≥ 2. For a sequence σi in the Teichmüller space of S, which converges to a projective measured lamination [λ] in the Thurston boundary, we obtain a relation between λ and the geometric limit of pants decompositions whose lengths are uniformly bounded by a Bers constant L. We also show that this bounded pants decomposition is related to the Gromo...

Periodic boundary conditions are applied to a ferromagnetic spin lattice. A symmetrical lattice and its contributions all over space are being used. Results, for the Ising model with ferromagnetic interaction that decays as a 1/r law, are discussed in the mean field approximation.

An approach for ­ nding the e¬ective properties of two-dimensional periodic rectangular checkerboard media is presented; it is demonstrated that one can take a relatively simple model problem, an appropriate system of joined quarter planes, and that after appropriate conformal transformation one can use this to solve for either periodic or doubly periodic structures. Illustrative calculations i...

We consider the Navier-Stokes equations on thin 3D domains Qε = Ω×(0, ε), supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous ...

We consider the one-dimensional steady fin problem with the Dirichlet boundary condition at one end and the Neumann boundary condition at the other. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for whic...

Near-linear evolution in the Korteweg-de-Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction. Mathematics Subject Classification: 35Q53

An analytic theory of periodically perturbed magneto-inductive waveguides is developed and used to determine the mode spectrum of distorted MI ring resonators. A set of coupled equations is first established from the recurrence equations for an infinite set of coupled elements, and used to determine the dispersion equation. Solutions for ring resonators are then found using periodic boundary co...