It is shown that certain natural maps between the ideal, Gromov, and end boundaries of a complete CAT(0) space can fail to be either injective or surjective. Additionally the natural map from the Gromov boundary to the end boundary of a complete CAT(−1) space can fail to be either injective or surjective.

This paper presents a method for interactive design by means of extending the PDE based approach for surface generation. The governing partial differential equation is generalized to arbitrary order allowing complex shapes to be designed as single patch PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating the geometry of shape that satisfying an arbitra...

In the one-loop approximation for Euclidean quantum gravity, the boundary conditions which are completely invariant under gauge transformations of metric perturbations involve both normal and tangential derivatives of the metric perturbations h 00 and h 0i , while the h ij perturbations and the whole ghost one-form are set to zero at the boundary. The corresponding one-loop divergency for pure ...

For an external problem in IR d (d = 2; 3) such that the unknown function satisses the wave equation outside a nite domain, we generate artiicial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artiicial boundary which is a circle (d = 2) or a sphere (d = 3): The questions of numerica...

The quantum theory of a free particle on a portion of two-dimensional Euclidean space bounded by a circle and subject to non-local boundary conditions gives rise to bulk and surface states. Starting from this well known property, a counterpart for gravity is here considered. In particular, if spatial components of metric perturbations are set to zero at the boundary, invariance of the full set ...

— We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in ∩s<1/2H (Θ). We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations w...

We consider the problem of enumerating spanning trees on lattices. Closed-form expressions are obtained for the spanning tree generating function for a hypercubic lattice of size N1×N2×· · ·×Nd in d dimensions under free, periodic, and a combination of free and periodic boundary conditions. Results are also obtained for a simple quartic net embedded on two non-orientable surfaces, a Möbius stri...

For an external problem in IR d (d = 2; 3) such that the unknown function satisses the wave equation outside a nite domain, we generate artiicial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artiicial boundary which is a circle (d = 2) or a sphere (d = 3): The questions of numerica...

In the one-loop approximation for Euclidean quantum gravity, the boundary conditions which are completely invariant under gauge transformations of metric perturbations cannot be written in terms of complementary projection operators. By contrast, they express the h00 and h0i perturbations at the boundary as integrals at the boundary of the action of a set of di erential operators on metric pert...

The differentiation matrix for a Daubechies-based wavelet basis defined on an interval will be constructed. It will be shown that the differentiation matrix based on the currently available boundary constructions does not maintain the superconvergence encountered under periodic boundary conditions. BJgy .. .. . .. .A0000 w__ •t!ar ~'..,-