Hydrodynamical limit for a Hamiltonian system with weak noise

Starting from a general hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prolve that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisfy the Euler equation of conservation laws up to a fixed timet provided that the Euler equation has a smooth solution with a given initial data up to timet. The strength of the noise term is chosen to be very small (but nonvanishing) so that it disappears in the scaling limit.Research partially supported by U.S. National Science Foundation grants DMS 89001682, DMS 920-1222 and a grant from ARO, DAAL03-92-G-0317Research partially supported by U.S. National Science Foundation grants DMS-9101196, DMS-9100383, and PHY-9019433-A01, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship     

Flower-shaped ZnO nanoparticles as an efficient, heterogeneous and reusable catalyst in the synthesis of N-arylhomophthalimides and benzannelated isoquinolinones

A simple and green protocol, developed utilizing an efficient, heterogeneous and recyclable catalyst, i.e. zinc oxide nanoparticles (ZnO NPs)-mediated synthesis of N-arylhomophthalimides and benzannelated isoquinolinones, is reported. The structures of the desired products were characterized by FTIR, ¹H NMR, ¹³C NMR, and HRMS techniques. The ZnO NPs exhibited excellent catalytic activity and the proposed methodology is capable of providing the desired products in good yield and purity.     

KOtBu‐Catalyzed Michael Addition Reactions Under Mild and Solvent‐Free Conditions

Designed transition metal complexes predominantly catalyze Michael addition reactions. Inorganic and organic base‐catalyzed Michael addition reactions have been reported. However, known base‐catalyzed reactions suffer from the requirement of solvents, additives, high pressure and also side‐reactions. Herein, we demonstrate a mild and environmentally friendly strategy of readily available KO^tBu‐catalyzed Michael addition reactions. This simple inorganic base efficiently catalyzes the Michael addition of underexplored acrylonitriles, esters and amides with (oxa‐, aza‐, and thia‐) heteroatom nucleophiles. This catalytic process proceeds under solvent‐free conditions and at room temperature. Notably, this protocol offers an easy operational procedure, broad substrate scope with excellent selectivity, reaction scalability and excellent TON (>9900). Preliminary mechanistic studies revealed that the reaction follows an ionic mechanism. Formal synthesis of promazine is demonstrated using this catalytic protocol.     

Approximating dominant singular triplets of large sparse matrices via modified moments

A procedure for determining a few of the largest singular values and corresponding singular vectors of large sparse matrices is presented. Equivalent eigensystems are solved using a technique originally proposed by Golub and Kent based on the computation of modified moments. The asynchronicity in the computations of moments and eigenvalues makes this method attractive for parallel implementations on a network of workstations. Although no obvious relationship between modified moments and the corresponding eigenvectors is known to exist, a scheme to approximate both eigenvalues and eigenvectors (and subsequently singular values and singular vectors) has been produced. This scheme exploits both modified moments in conjunction with the Chebyshev semi-iterative method and deflation techniques to produce approximate eigenpairs of the equivalent sparse eigensystems. The performance of an ANSI-C implementation of this scheme on a network of UNIX workstations and a 256-processor Cray T3D is presented.This research was supported in part by the National Science Foundation under grant numbers NSF-ASC-92-03004 and NSF-ASC-94-11394.     

Behavior of the solution of a random semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a random source at the origin. We study the solution, which describes the equilibrium of this system, and prove that, as the space variable tends to infinity, the solution becomes a.s. asymptotic to a steady state. We also study the fluctuations of the solution around the steady state.     

Symmetric Simple Exclusion Process:¶Regularity of the Self-Diffusion Coefficient

We prove that the self-diffusion coefficient of a tagged particle in the symmetric exclusion process in Z ^d , which is in equilibrium at density α, is of class C ^∞ as a function of α in the closed interval [0,1]. The proof provides also a recursive method to compute the Taylor expansion at the boundaries. Received: 6 December 2000 / Accepted: 6 April 2001     

Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.Supported by NSF Grant MCS-8301364, ONR Contract N00014-81-K-0012 and a Fellowship from John S. Guggenheim Memorial Foundation     

Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

Roland and Varadhan (Appl. Numer. Math., 55:215–226, 2005) presented a new idea called “squaring” to improve the convergence of Lemaréchal’s scheme for solving nonlinear fixed-point problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, , 2004) noted that Lemaréchal’s scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations. Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first- and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority compared to the classical, “unsquared” vector polynomial methods.