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.

Maximum differentiation (MAD) competition: a methodology for comparing computational models of perceptual quantities.

Abstract: We propose an efficient methodology for comparing computational models of a perceptually discriminable quantity. Rather than comparing model responses to subjective responses on a set of pre-selected stimuli, the stimuli are computer-synthesized so as to optimally distinguish the models. Specifically, given two computational models that take a stimulus as an input and predict a perceptually discriminable quantity, we first synthesize a pair of stimuli that maximize/minimize the response of one model while holding the other fixed. We then repeat this procedure, but with the roles of the two models reversed. Subjective testing on pairs of such synthesized stimuli provides a strong indication of the relative strengths and weaknesses of the two models. Specifically, the model whose extremal stimulus pairs are easier for subjects to discriminate is the better model. Moreover, careful study of the synthesized stimuli may suggest potential ways to improve a model or to combine aspects of multiple models. We demonstrate the methodology for two example perceptual quantities: contrast and image quality.

The Nonlinear Interaction of Barotropic and Equatorial Baroclinic Rossby Waves

Abstract: Simplified asymptotic equations are developed for the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a significant midlatitude projection in the presence of suitable horizontally and vertically sheared zonal mean flows. The simplified equations allow for nonlinear energy exchange between the barotropic Rossby waves and the baroclinic equatorial waves for nonzero zonal mean vertical shear through wave‐wave interactions. Idealized examples in the model demonstrate that midlatitude Rossby wave trains in a baroclinic mean shear can transfer their energy to localized equatorially trapped baroclinic Rossby waves through a nonlinear ‘‘westerly wind burst’’ mechanism. Conversely, equatorially trapped baroclinic Rossby wave trains in the idealized model can transfer substantial energy to the midlatitude barotropic Rossby waves. From the viewpoint of applied mathematics, the asymptotic equations derived here have several novel features.

Well-Posedness for the FENE Dumbbell Model of Polymeric Flows

Abstract: We prove local and global well-posedness for the FENE dumbbell model for a very general class of potentials. Indeed, in prior local or global well-posedness results, conditions on the strength of the singularity (or on the parameter b) were made. Here we give a proof in the general case. We also prove global existence results if the data is small or if we restrict to the co-rotational model in dimension 2. © 2008 Wiley Periodicals, Inc.

A core library for robust numeric and geometric computation

Abstract: Nonrobustness is a well-known problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (CORE) for robust numeric and geometric computation based on the principles of Exact Geometric Computation (EGC). Through our library, for the first time, any programmer can write robust and efficient algorithms. The Core Library is based on a novel numerical core that is powerful enough to support EGC for algebraic problems. This is coupled with a simple delivery mechanism which transparently extends conventional C/C++ programs into robust codes. We are currently addressing efficiency issues in our library: (a) at the compiler and language level, (b) at the level of incorporating EGC techniques, as well as the (c) the system integration of both (a) and (b). Pilot experimental results are described. The basic library is availableathttp://cs.nyu.edu/exact/core/andthe C++-to-C compiler is under development.

Fredholm determinants, Jimbo‐Miwa‐Ueno τ‐functions, and representation theory

Abstract: The authors show that a wide class of Fredholm determinants arising in the representation theory of “big” groups, such as the infinite-dimensional unitary group, solve Painleve equations. Their methods are based on the theory of integrable operators and the theory of Riemann-Hilbert problems. © 2002 Wiley Periodicals, Inc.