Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Cloaking via change of variables for the Helmholtz equation

Abstract: The transformation optics approach to cloaking uses a singular change of coordinates, which blows up a point to the region being cloaked. This paper examines a natural regularization, obtained by (1) blowing up a ball of radius ρ rather than a point, and (2) including a well-chosen lossy layer at the inner edge of the cloak. We assess the performance of the resulting near-cloak as the regularization parameter ρ tends to 0, in the context of (Dirichlet and Neumann) boundary measurements for the time-harmonic Helmholtz equation. Since the goal is to achieve cloaking regardless of the content of the cloaked region, we focus on estimates that are uniform with respect to the physical properties of this region. In three space dimensions our regularized construction performs relatively well: the deviation from perfect cloaking is of order ρ. In two space dimensions it does much worse: the deviation is of order 1/|log ρ|. In addition to proving these estimates, we give numerical examples demonstrating their sharpness. Some authors have argued that perfect cloaking can be achieved without losses by using the singular change-of-variable-based construction. In our regularized setting the analogous statement is false: without the lossy layer, there are certain resonant inclusions (depending in general on ρ) that have a huge effect on the boundary measurements. © 2010 Wiley Periodicals, Inc.

Assessing and understanding the impact of stratospheric dynamics and variability on the earth system

Abstract: New modeling efforts will provide unprecedented opportunities to harness our knowledge of the stratosphere to improve weather and climate prediction.

Growth rates for the linearized motion of fluid interfaces away from equilibrium

Abstract: We consider the motion of a two-dimensional interface separating an inviscid, incompressible, irrotational fluid, influenced by gravity, from a region of zero density. We show that under certain conditions the equations of motion, linearized about a presumed time-dependent solution, are wellposed; that is, linear disturbances have a bounded rate of growth. If surface tension is neglected, the linear equations are well-posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity, similar to the criterion formulated by G. I. Taylor. If surface tension is included, the linear equations are well-posed without qualifications, whether the fluid is above or below the interface. An interesting qualitative structure is found for the linear equations. A Lagrangian approach is used, like that of numerical work such as [3], except that the interface is assumed horizontal at infinity. Certain integral equations which occur, involving double layer potentials, are shown to be solvable in the present case. © 1993 John Wiley & Sons, Inc.

Climate Stability for a Sellers-Type Model

Abstract: We study a diffusive energy-balance climate model governed by a nonlinear parabolic partial differential equation Three positive steady-state solutions of this equation are found; they correspond to three possible climates of our planet: an interglacial (nearly identical to the present climate), a glacial, and a completely ice-covered earth We consider also models similar to the main one studied, and determine the number of their steady states All the models have albedo continuously varying with latitude and temperature, and entirely diffusive horizontal heat transfer The diffusion is taken to be nonlinear as well as linear We investigate the stability under small perturbations of the main model's climates A stability criterion is derived, and its application shows that the 'present climate' and the 'deep freeze' are stable, whereas the model's glacial is unstable A variational principle is introduced to confirm the results of this stability analysis For a sufficient decrease in solar radiation (about 2%) the glacial and interglacial solutions disappear, leaving the ice-covered earth as the only possible climate

The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected

Abstract: This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3. This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). The lamination theorem is stated in the global case where the lamination is, in fact, a foliation. The first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph. This is done by showing that if the curvature is large at some point (and hence the surface is not a graph), then it is a double spiral staircase like the helicoid. To prove that such a disk is a double spiral staircase, we showed in the first three papers of the series that it is built out of N-valued graphs where N is a fixed number. In this paper we will deal with how the multi-valued graphs fit together and, in particular, prove regularity of the set of points of large curvature - the axis of the double spiral staircase. The first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase.