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.

Complementarity and nondegeneracy in semidefinite programming

Abstract: Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions $X$ and $Z$. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks of $X$ and $Z$ which are consistent with the nondegeneracy conditions.

Ab initio molecular dynamics: basic concepts, current trends and novel applications

Abstract: The field of ab initio molecular dynamics (AIMD), in which finite temperature molecular dynamics (MD) trajectories are generated with forces obtained from accurate 'on the fly' electronic structure calculations, is a rapidly evolving and growing technology that allows chemical processes in condensed phases to be studied in an accurate and unbiased way. This article is intended to present the basics of the AIMD method as well as to provide a broad survey of the state of the art of the field and showcase some of its capabilities. Beginning with a derivation of the method from the Born–Oppenheimer approximation, issues including the density functional representation of electronic structure, basis sets, calculation of observables and the Car–Parrinello extended Lagrangian algorithm are discussed. A number of example applications, including liquid structure and dynamics and aqueous proton transport, are presented in order to highlight some of the current capabilities of the approach. Finally, advanced topics such as inclusion of nuclear quantum effects, excited states and scaling issues are addressed.