On Factorization of Molecular Wavefunctions

Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions \{$\Phi$\} of the molecular Schr\"odinger equation as originally proposed by Hunter in the 1970s. The idea is to represent $\Phi$ in the form $\varphi\chi$ where $\chi$ is \textit{purely} a function of the nuclear coordinates, while $\varphi$ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an `exact' representation of $\Phi$ can be achieved in this form with $\varphi$ and $\chi$ interpretable as `electronic' and `nuclear' wavefunctions respectively. We offer a mathematical analysis of these proposals that identifies ambiguities stemming mainly from the singularities in the Coulomb potential energy.Comment: Manuscript submitted to Journal of Physics A: Mathematical and Theoretical, May 2015. Accepted for Publication August 24 201

Analysis and simulation of a magnetic bearing suspension system for a laboratory model annular momentum control device

A linear analysis and the results of a nonlinear simulation of a magnetic bearing suspension system which uses permanent magnet flux biasing are presented. The magnetic bearing suspension is part of a 4068 N-m-s (3000 lb-ft-sec) laboratory model annular momentum control device (AMCD). The simulation includes rigid body rim dynamics, linear and nonlinear axial actuators, linear radial actuators, axial and radial rim warp, and power supply and power driver current limits

User's guide to a system of finite-element supersonic panel flutter programs

The utilization and operation of a set of six computer programs for the prediction of panel flutter at supersonic speeds by finite element methods are described. The programs run individually to determine the flutter behavior of a flat panel where the finite elements which model the panel each have four degrees of freedom (DOF), a curved panel where the finite elements each have four DOF, and a curved panel where the finite elements each have six DOF. The panels are assumed to be of infinite aspect ratio and are subjected to either simply-supported or clamped boundary conditions. The aerodynamics used by these programs are based on piston theory. Application of the program is illustrated by sample cases where the number of beam finite elements equals four, the in-plane tension parameter is 0.0, the maximum camber to panel length ratio for a curved panel case is 0.05, and the Mach number is 2.0. This memorandum provides a user's guide for these programs, describes the parameters that are used, and contains sample output from each of the programs

Description of a digital computer simulation of an Annular Momentum Control Device (AMCD) laboratory test model

A description of a digital computer simulation of an Annular Momentum Control Device (AMCD) laboratory model is presented. The AMCD is a momentum exchange device which is under development as an advanced control effector for spacecraft attitude control systems. The digital computer simulation of this device incorporates the following models: six degree of freedom rigid body dynamics; rim warp; controller dynamics; nonlinear distributed element axial bearings; as well as power driver and power supply current limits. An annotated FORTRAN IV source code listing of the computer program is included

Comment on `On the Quantum Theory of Molecules' [J. Chem.Phys. {\bf 137}, 22A544 (2012)]

In our previous paper [J. Chem.Phys. {\bf 137}, 22A544 (2012)] we argued that the Born-Oppenheimer approximation could not be based on an exact transformation of the molecular Schr\"{o}dinger equation. In this Comment we suggest that the fundamental reason for the approximate nature of the Born-Oppenheimer model is the lack of a complete set of functions for the electronic space, and the need to describe the continuous spectrum using spectral projection.Comment: 2 page

Non-linear effects on Turing patterns: time oscillations and chaos.

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space, produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, Turing patterns oscillate in time, a phenomenon which is expected to occur only in a three morphogen system. When varying a single parameter, a series of bifurcations lead to period doubling, quasi-periodic and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examined the Turing conditions for obtaining a diffusion driven instability and discovered that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. All this results demonstrates the limitations of the linear analysis for reaction-diffusion systems

In Fed meetings, decision making is free – but not equal.

With its ability to influence interest rates globally, the Federal Open Market Committee (FOMC) of the US Federal Reserve is arguably one of the most important decision making bodies on the planet. But how does it come to its decisions? In new research which analyses transcripts of FOMC deliberations over nearly 30 years, Joseph Gardner and John T. Woolley find that women speak less than men for nearly their entire tenure on the FOMC. While women are free to speak, they write, they do not participate equally in FOMC deliberations, and this could be influencing policy choices

Enhancing the Fed’s transparency didn’t hurt its deliberations

For more than two decades, transcripts of the US Federal Reserve’s Open Market Committee meetings have been made available to the public. But has the move to greater transparency about monetary policymaking hurt committee deliberations? In new research which examines committee meeting transcripts from 1978 to 2007, Joseph Gardner and John T. Woolley find that leadership – not transparency – had the greatest effect on how members deliberated during meetings

Influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise

Numerous mathematical models exploring the emergence of complexity within developmental biology incorporate diffusion as the dominant mechanism of transport. However, self-organizing paradigms can exhibit the biologically undesirable property of extensive sensitivity, as illustrated by the behavior of the French-flag model in response to intrinsic noise and Turing’s model when subjected to fluctuations in initial conditions. Domain growth is known to be a stabilizing factor for the latter, though the interaction of intrinsic noise and domain growth is underexplored, even in the simplest of biophysical settings. Previously, we developed analytical Fourier methods and a description of domain growth that allowed us to characterize the effects of deterministic domain growth on stochastically diffusing systems. In this paper we extend our analysis to encompass stochastically growing domains. This form of growth can be used only to link the meso- and macroscopic domains as the “box-splitting” form of growth on the microscopic scale has an ill-defined thermodynamic limit. The extension is achieved by allowing the simulated particles to undergo random walks on a discretized domain, while stochastically controlling the length of each discretized compartment. Due to the dependence of diffusion on the domain discretization, we find that the description of diffusion cannot be uniquely derived. We apply these analytical methods to two justified descriptions, where it is shown that, under certain conditions, diffusion is able to support a consistent inhomogeneous state that is far removed from the deterministic equilibrium, without additional kinetics. Finally, a logistically growing domain is considered. Not only does this show that we can deal with nonmonotonic descriptions of stochastic growth, but it is also seen that diffusion on a stationary domain produces different effects to diffusion on a domain that is stationary “on average.

Stochastic reaction & diffusion on growing domains: understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it