Spline analysis of Hydrographic data

AbstractThe basic problem involved in determining where the ship can not go is an attempt to reconstruct the sea bed. The interpolation of points necessary to reconstruct the sea bed was done using a bicubic spline. This method was chosen because of the similarities between the boundary conditions believed to be characteristic of the modeling problem and those of the natural spline. These include the continuity of the first and second derivatives, and the minimum curvature exhibited by the spline method which is characteristic of the sea bottom. The major problem faced in modeling the sea bed was selecting the extra data points needed in order to find a meaningful solution. This selection was done both by intuition and by constructing splines to model the possible behavior along a straight line. The results were two different models: a ridge model, characterized by a single shallow ridge in the center of the region; and a hill model, characterized by two smaller ridges. By varying one of these extra data points (called critical points), several models of both these extremes as well as intermediate models were generated. However, it was found that the number of given points did not permit a definitive model. Data was needed inside the region, especially at the critical points and at the exterior points in order to better define the boundary. The boundary could not be reliably determined since our spline model does not allow for accurate extrapolation. Thus, the model, although close to what is believed to be the correct model, is not good enough to allow for navigation because of the limited number of given data points

A Reilly formula and eigenvalue estimates for differential forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page

Thermo-elasticity for anisotropic media in higher dimensions

In this note we develop tools to study the Cauchy problem for the system of thermo-elasticity in higher dimensions. The theory is developed for general homogeneous anisotropic media under non-degeneracy conditions. For degenerate cases a method of treatment is sketched and for the cases of cubic media and hexagonal media detailed studies are provided.Comment: 33 pages, 5 figure

Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region larger than the one provided by the Cauchy-Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the apriori estimates for the Weyl equations, associated to the "Bel-Robinson norms". In particular if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a "geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.Comment: 68 page

On a computer-aided approach to the computation of Abelian integrals

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four

First-order quasilinear canonical representation of the characteristic formulation of the Einstein equations

We prescribe a choice of 18 variables in all that casts the equations of the fully nonlinear characteristic formulation of general relativity in first--order quasi-linear canonical form. At the analytical level, a formulation of this type allows us to make concrete statements about existence of solutions. In addition, it offers concrete advantages for numerical applications as it now becomes possible to incorporate advanced numerical techniques for first order systems, which had thus far not been applicable to the characteristic problem of the Einstein equations, as well as in providing a framework for a unified treatment of the vacuum and matter problems. This is of relevance to the accurate simulation of gravitational waves emitted in astrophysical scenarios such as stellar core collapse.Comment: revtex4, 7 pages, text and references added, typos corrected, to appear in Phys. Rev.