We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several p...

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...

We define the Dirichlet to Neumann operator on exterior differential forms for a compact Riemannian manifold with boundary and prove that the real additive cohomology structure of the manifold is determined by the DN operator. In particular, an explicit formula is obtained which expresses Betti numbers of the manifold through the DN operator. We express also the Hilbert transform through the DN...

For a d-dimensional convex lattice polytope P , a formula for the boundary volume vol(∂P ) is derived in terms of the number of boundary lattice points on the first bd/2c dilations of P . As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f -vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to...

This paper studies the existence and uniqueness of solutions for a type of second-order two-point boundary-value problem depending on the firstorder derivative through a non-linear term. By constructing a special cone and using the upper and lower solutions method, we obtain the sufficient conditions of the existence and uniqueness of solutions, and a monotone iterative sequence solving the bou...

In this paper, the natural boundary element method (NBEM) for an anisotropic hyperbolic problem in an exterior elliptic domain is investigated. By the theory of the natural boundary reduction (NBR), the natural integral equation (NIE) and the Poisson integral formula of the problem considered are obtained, and the numerical method of the NIE is given. Finally, some numerical examples are presen...