A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over C[z(1),., z(d)] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely gen...

The conditions for the existence and stability of cosmological power-law scaling solutions are established when the Einstein-Hilbert action is modified by the inclusion of a function of the Gauss-Bonnet curvature invariant. The general form of the action that leads to such solutions is determined for the case where the universe is sourced by a barotropic perfect fluid. It is shown by employing ...

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature.We dis...

Following are the abstracts of contributions (i.e., talks and posters) to the 13th Annual Meeting of the IMS held at the National University of Ireland Maynooth, 6–8 September 2000. The abbreviations after the names designate: ‘M’ for main speaker, ‘S’ for speaker, ‘RS’ for research student, and ‘P’ for poster. The abstracts are included as provided by the contributors for this volume of the Bu...

Let D be a domain in the complex in-plane and let as: D —• R be a regular minimal surface. Let M(K) be the set of points tu 0. The components of M(K) are of three types: isolated points; simple analytic arcs terminating nowhere in D; analytic Jordan curves in D. Components of the third type are r...

We prove that a 3–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2π is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem ...

We discuss the junction conditions in the context of the Randall-Sundrum model with the Gauss-Bonnet interaction. We consider the Z2 symmetric model where the brane is embedded in an AdS5 bulk, as well as a model without Z2 symmetry in which the brane (in this case called by tradition “shell”) separates two metrically different AdS5 regions. We show that the Israel junction conditions across th...

The aim of this paper is to define a new class surfaces in Euclidean space using the concept osculating circle. Given regular curve C, surface circles generated by C set all at points C. It proved that these contain one-parametric family planar lines curvature. A classification given canal surfaces, Weingarten with constant Gauss curvature and mean