Isometric embeddings in cosmology and astrophysics

Recent interest in higher-dimensional cosmological models has prompted some significant work on the mathematical technicalities of how one goes about embedding spacetimes into some higher-dimensional space. We survey results in the literature (existence theorems and simple explicit embeddings); briefly outline our work on global embeddings as well as explicit results for more complex geometries; and provide some examples. These results are contextualized physically, so as to provide a foundation for a detailed commentary on several key issues in the field such as: the meaning of ‘Ricci equivalent’ embeddings; the uniqueness of local (or global) embeddings; symmetry inheritance properties; and astrophysical constraints.     

Constrained Optimal Design of Circular Plates Against Buckling∗

Several problems are investigated in which the buckling loads of structures are maximized, subject to the restriction that the volume of structural material is specified. In addition, a constraint is placed upon either the maximum allowable prebuckling stress, or the minimum allowable thickness. The structures analyzed are flat, axisymmetric circular plates, loaded by uniform radial thrust, and assumed to buckle axisymmetrically. The plates are investigated for both sandwich and solid cross-sectional geometries.     

A NOVEL METHOD OF FITTING SPATIO‐TEMPORAL MODELS TO DATA,WITH APPLICATIONS TO THE DYNAMICS OF MOUNTAIN PINE BEETLES

Abstract We develop a modular landscape model for the mountain pine beetle (Dendroctonus ponderosae Hopkins) infestation of a stage‐structured forest of lodgepole pine (Pinus contorta Douglas). Beetle attack dynamics are modeled using response functions and beetle movement using dispersal kernels. This modeling technique yields four model candidates. These models allow discrimination between four broad possibilities at the landscape scale: whether or not beetles are subject to an Allee effect at the landscape scale and whether or not host selection is random or directed. We fit the models with aerial damage survey data to the Sawtooth National Recreation Area using estimating functions, which allows for more rapid and complete parameter determination. We then introduce a novel model selection procedure based on facial recognition technology to compliment traditional nonspatial selection metrics. Together with these we are able to select a best model and draw inferences regarding the behavior of the beetle in outbreak conditions.     

Spurious Behaviour of Numerically Computed Fluid Flow

We investigate the stability to aliasing errors of numericalschemes for hydrodynamics, taking the viscous Burgers' equationas a model for systems with a term that is quadratic in thevelocity. Considering wavelengths equal to three times the mesh-spacing,and arbitrary mean flow, we are able to demonstrate explicitlyfor common schemes (a) a sufficient criterion for stabilityand (b) blow-up of solutions in a finite time when (a) is violated.Singular behaviour is shown to persist at all wavelengths: studiesof wavelengths up to thirty times mesh-spacing make it clearthat a profile with a single region of strong convergent flowis most conducive to instability. In contrast, spectral (Galerkin)and upwind schemes are shown to be stable for all flows andperiods.     

A Variational Approach to Upper and Lower Solutions

Traditional proofs of the upper and lower solution existenceresults for nonlinear elliptic equations use monotone iterationschemes based on the maximum principle. In this paper, a directvariational method is employed which characterizes the solutionas the global minimum of a functional on a closed convex set.     

Multiple groups in the symmetry classification of adiabatic electronic wavefunctions

It was originally shown by Longuet-Higgins and colleagues that when the electronic Schrödinger equation is solved as a function of the nuclear coordinates the adiabatic electronic wavefunction can undergo a change of sign after completing a closed circuit. This geometric phase occurs for a circuit around a conical intersection, and in particular around a conical intersection corresponding to a linear Jahn—Teller effect. The adiabatic wavefunctions are classified here under a group called the adiabatic multiple group, which is a generalization of the ‘vibronic double group’ of C _3v introduced by Hougen, and is distinct from the familiar electron-spin double group. Although the real electronic wavefunctions can be only double-valued, the groups can have higher multiplicity because of the possibility of different circuits. For a number of symmetric- and spherical-top point groups, the adiabatic multiple group is shown to be the direct product of the point group with a phase group. The adiabatic multiple group can be applied to individual adiabatic orbitals, and so to configurations built from these orbitals. This leads to the rule that the linear Jahn—Teller effect vanishes in the single-configuration approximation for configurations containing non-degenerate electrons plus an even number of e electrons. There does not appear to be any cancellation effect for electron configurations of cubic molecules containing f electrons.     

Shear stress correlations in hard and soft sphere fluids

The shear stress autocorrelation function has been studied by molecular dynamics simulation using the q^?n potential for very large n. The results are analysed and interpreted here by comparing them with the shear stress response function for hard spheres. It is shown that the hard sphere response function has a singular contribution, and that this is reproduced accurately by the simulations for large n. A simple model for the stress autocorrelation function at finite n is proposed, based on the required hard sphere limiting form.     

DEVELOPMENT OF AN OPTIMAL WATER ALLOCATION DECISION TOOL FOR THE MAJOR CROPS DURING THE WATER DEFICIT PERIOD IN THE SOUTHEAST UNITED STATES

ABSTRACT. We developed a dynamic economic model to optimize irrigation water allocations during water deficit periods for three major crops grown in the humid southeastern United States. Analysis involved the use of crop simulation models to capture (a) the yield water relationship and (b) soil moisture dynamics from one week to another week. A hy‐drological model was used to find the water supply; combinations of hydrological and simulation models were used to find the optimal water allocation during each week in corn, cotton and peanuts. Results indicated that farmers should irrigate the most valuable crop first (peanuts) before applying water to other crops (corn and cotton). Results also showed that, because of restriction on total water supply, an increase in crop acreage did not increase the net revenue of the farm in a proportionate amount. Results should provide guidelines to water managers, engineers, policy makers, and farmers regarding an optimal amount of water allocation that will maximize net returns when water shortage is a serious concern.