Artificial neural system modeling of Monte Carlo simulations of polymers

In this work, a neural network was used to learn features in potential energy surfaces and relate those features to conformational properties of a series of polymers. Specifically, we modeled Monte Carlo simulations of 20 polymers in which we calculated the characteristic ratio and the temperature coefficient of the characteristic ratio for each polymer. We first created 20 rotational potential energy surfaces using MNDO procedures and then used these energy surfaces to produce 10000 chains, each chain 100 bonds long. From these results we calculated the mean-square end-to-end distance, the characteristic ratio and its corresponding temperature coefficient. A neural network was then used to model the results of these Monte Carlo calculations. We found that artificial neural network simulations were highly accurate in predicting the outcome of the Monte Carlo calculations for polymers for which it was not trained. The overall average error for prediction of the characteristic ratio was 4,82%, and the overall average error for prediction of the temperature coefficient was 0,89%.     

Solutions to the nonlinear Schrödinger equation with sequences of initial data converging to a Dirac mass

We study the nonlinear Schrödinger equation with sequences of initial data that converge to a Dirac mass, and study the asymptotic behaviour of solutions. In doing so we find a connection to previously known long time asymptotics. We demonstrate a type of universality in the behaviour of solutions for real initial data, and we also show how this universality breaks down for examples of initial data that are not purely real.     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

A cyclic hexacopper(II) fluoro complex that encapsulates two fluoride anions

The complex [[Cu3(HpztBu)4(mu-pztBu)2(mu-F)2(mu 3-F)]2]F2 (HpztBu = 3[5]-tert-butylpyrazole) has a cyclic, C2v-symmetric hexacopper core. The two non-coordinated F- anions are encapsulated within cavities formed by three HpztBu ligands.     

Supramolecular stabilization of N(2)H(7)(+)

The cation N(2)H(7)(+) has been stabilized in a largely hydrophobic supramolecular environment and characterized in the solid state. The cation is situated in the bowl-shaped cavity of calix[4]arene. All of the hydrogen atoms are clearly discernible owing to high-quality X-ray data as well as lack of disorder and symmetry-imposed ambiguity. It appears that electrostatic interactions play a critical role in stabilizing the structure.     

Rational Functions with a General Distribution of Poles on the Real Line Orthogonal with Respect to Varying Exponential Weights: I

Orthogonal rational functions are characterized in terms of a family of matrix Riemann–Hilbert problems on ?, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of ‘model’ matrix Riemann–Hilbert problems which are amenable to asymptotic analysis via the Deift–Zhou non-linear steepest-descent method.     

Three-dimensional blow-up solutions of the Navier-Stokes equations

In this paper we extend the plane blow-up results of Grundy& McLaughlin (1997) to the three-dimensional Navier-Stokes equations.Using a solution structure originally due to Lin we first providenumerical evidence for the existence of blow-up solutions on- < x, z < , 0 y 1 with boundary conditions on y = 0and y = 1 involving derivatives of the velocity components.The formulation enables us to consider plane and radial flowas special cases. Various features of the computations are isolatedand are used to construct a formal asymptotic solution closeto blow-up. We show that the numerical and asymptotic analysesprovide a mutually consistent global picture which supportsthe conclusion that, for the family of problems we considerhere, blow-up in fact can take place in three dimensions butat an inverse linear rate rather than the faster inverse squareof the plane case.