We place the Thomas-Fermi model of the quantum theory of atoms, molecules, and solids on a firm mathematical footing. Our results include: (1) A proof of existence and uniqueness of solutions of the nonlinear Thomas-Fermi equations as well as the fact that these solutions minimize the Thomas-Fermi energy functional, (2) a proof that in a suitable large nuclear charge limit, the quantum mechanic...

By variational methods, we provide a simple proof of existence of a heteroclinic orbit to the Hamiltonian system u�� = ∇W (u) that connects the two global minima of a double-well potential W . Moreover, we consider several inhomogeneous extensions.

A proof of existence of stationary dark soliton solutions of the cubic-quintic nonlinear Schrödinger equation with a periodic potential is given. It is based on the interpretation of the dark soliton as a heteroclinic on the Poincaré map.

This paper concerns the problems of quadratic hedging and pricing, and mean-variance portfolio selection in an incomplete market setting with continuous trading, multiple assets, and Brownian information. In particular, we assume throughout that the parameters describing the market model may be random processes. We approach these problems from the perspective of linear-quadratic (LQ) optimal co...

We present a technique for the rigorous computation of periodic orbits in certain ordinary differential equations. The method combines set oriented numerical techniques for the computation of invariant sets in dynamical systems with topological index arguments. It not only allows for the proof of existence of periodic orbits but also for a precise (and rigorous) approximation of these. As an ex...

Given a nite set of non-collinear points in the plane there exists a line that passes through exactly two points. Such a line is called an ordinary line. An e cient algorithm for computing such a line was proposed by Mukhopadhyay et al [MAH97]. In this note we extend this result in two directions. We rst show how to use this algorithm to compute an ordinary conic, that is, a conic passing throu...

The uniqueness has been proved by Piatetskiî-Sapiro and Safarevic. The proof of existence is outlined below. Let o be the closed point of A = Spec C[[t] ]. Let An be the finite covering of A obtained by extracting an nth. root of t; let on be the closed point of An. A family of surfaces over An is a flat, projective map ƒ: Xn —> An such that the generic fiber is smooth, connected, and two dimen...

This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point source. In one space dimension, solutions of these problems with zero initial data are known to approach the stationary solution in an asymptotically self-similar...

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d ≥ 1 dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, [19], and used in our earlier work, [6].

where T1, . . . , Tn, S1, . . . , Sn : [a, b] → R. (2) Suppose K 6= 0. Then we may consider each of the systems {T1, . . . , Tn}, {S1, . . . , Sn} linearly independent. Indeed, starting from an expression of kind (1), we consequently reduce the number of items while it is needed. Assume that we need to express the functions (2) in terms of K. In order to do it, we find such points (a proof of e...