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.

An egalitarian network model for the emergence of simple and complex cells in visual cortex

Abstract: We explain how simple and complex cells arise in a large-scale neuronal network model of the primary visual cortex of the macaque. Our model consists of ≈4,000 integrate-and-fire, conductance-based point neurons, representing the cells in a small, 1-mm2 patch of an input layer of the primary visual cortex. In the model the local connections are isotropic and nonspecific, and convergent input from the lateral geniculate nucleus confers cortical cells with orientation and spatial phase preference. The balance between lateral connections and lateral geniculate nucleus drive determines whether individual neurons in this recurrent circuit are simple or complex. The model reproduces qualitatively the experimentally observed distributions of both extracellular and intracellular measures of simple and complex response.

On the competitiveness of on-line real-time task scheduling

Abstract: With respect to on-line scheduling algorithms that must direct the service of sporadic task requests we quantify the benefit of clairvoyancy, i.e., the power of possessing knowledge of various task parameters of future events. Specifically, we consider the problem of preemptively sheduling sporadic task requests in both uni- and multi-processor environments. If a task request is successfuly scheduled to completion, a value equal to the task's execution time is obtained; otherwise no value is obtained. We prove that no on-line scheduling algorithm can guarantee a cumulative value greater than 1/4th the value obtainable by a clairvoyant scheduler; i.e., we prove a 1/4th upper bound on the competitive factor of on-line real-time schedulers. We present an online uniprocessor scheduling algorithm TD 1 that actually has a competitive factor of 1/4; this bound is thus shown to be tight. We further consider the effect of restricting the amount of overloading permitted (the loading factor), and quantify the relationship between the loading factor and the upper bound on the competitive factor. Other results of a similar nature deal with the effect of value densities (measuring the importance of type of a task). Generalizations to dual-processor on-line scheduling are also considered. For the dual-processor case, we prove an upper bound of 1/2 on the competitive factor. This bound is shown to be tight in the special case when all the tasks have the same density and zero laxity.

Algorithms for constrained molecular dynamics

Abstract: In molecular dynamics simulations, the fastest components of the potential field impose severe restrictions on the stability and hence the speed of computational methods. One possibility for treating this problem is to replace the fastest components with algebraic length constraints. In this article the resulting systems of mixed differential and algebraic equations are studied. Commonly used discretization schemes for constrained Hamiltonian systems are discussed. The form of the nonlinear equations is examined in detail and used to give convergence results for the traditional nonlinear solution technique SHAKE iteration and for a modification based on successive overrelaxation (SOR). A simple adaptive algorithm for finding the optimal relaxation parameter is presented. Alternative direct methods using sparse matrix techniques are discussed. Numerical results are given for the new techniques, which have been implemented in the molecular modeling software package CHARMM and show as much as twofold improvement over SHAKE iteration. © 1995 John Wiley & Sons, Inc.

A Survey of Procedural Noise Functions

Abstract: Procedural noise functions are widely used in Computer Graphics, from off-line rendering in movie production to interactive video games. The ability to add complex and intricate details at low memory and authoring cost is one of its main attractions. This survey is motivated by the inherent importance of noise in graphics, the widespread use of noise in industry, and the fact that many recent research developments justify the need for an up-to-date survey. Our goal is to provide both a valuable entry point into the field of procedural noise functions, as well as a comprehensive view of the field to the informed reader. In this report, we cover procedural noise functions in all their aspects. We outline recent advances in research on this topic, discussing and comparing recent and well established methods. We first formally define procedural noise functions based on stochastic processes and then classify and review existing procedural noise functions. We discuss how procedural noise functions are used for modeling and how they are applied to surfaces. We then introduce analysis tools and apply them to evaluate and compare the major approaches to noise generation. We finally identify several directions for future work.

Stochastic homogenization of Hamilton‐Jacobi‐Bellman equations

Abstract: We study the homogenization of some Hamilton-Jacobi-Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton-Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment. c � 2006 Wiley Periodicals, Inc.