Singular Integral Equations

The integral equationP∫cK(ζ′,ζ)ζ′−ζφ(ζ′)dζ′=h(ζ)φ(ζ)+f(ζ)is shown to have simple solutions obtained by standard and elementary methods if h and K have appropriate analytic properties.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70799/2/JMAPAQ-7-12-2121-1.pd

An Inverse Scattering Transform for the Lattice Potential KdV Equation

The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include discrete soliton solutions, Backlund transformations and an associated linear problem, called a Lax pair, for which it provides the compatibility condition. In this paper, we solve the initial value problem for the LKdV equation through a discrete implementation of the inverse scattering transform method applied to the Lax pair. The initial value used for the LKdV equation is assumed to be real and decaying to zero as the absolute value of the discrete spatial variable approaches large values. An interesting feature of our approach is the solution of a discrete Gel'fand-Levitan equation. Moreover, we provide a complete characterization of reflectionless potentials and show that this leads to the Cauchy matrix form of N-soliton solutions

Existence and Uniqueness Theorems for the Neutron Transport Equation

In an attempt to understand the conditions under which the neutron transport equation has solutions, and the properties of those solutions, a number of existence and uniqueness theorems are proved. One finds that the properties of the solution are closely related to the boundedness of the source as well as to certain velocity‐space integrals of the scattering kernel. Both time‐dependent and time‐independent equations are considered as are also the time‐dependent and time‐independent adjoint equations. Although only a very few of all possible existence and uniqueness theorems for these equations are considered here, the work may serve as a guide to the treatment of similar problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70329/2/JMAPAQ-4-11-1376-1.pd

Stochastic evolution of four species in cyclic competition

We study the stochastic evolution of four species in cyclic competition in a well mixed environment. In systems composed of a finite number $N$ of particles these simple interaction rules result in a rich variety of extinction scenarios, from single species domination to coexistence between non-interacting species. Using exact results and numerical simulations we discuss the temporal evolution of the system for different values of $N$, for different values of the reaction rates, as well as for different initial conditions. As expected, the stochastic evolution is found to closely follow the mean-field result for large $N$, with notable deviations appearing in proximity of extinction events. Different ways of characterizing and predicting extinction events are discussed.Comment: 19 pages, 6 figures, submitted to J. Stat. Mec

How ADP students navigate enablements and constraints of the programme: An exploration of structure and agency

Academic Development Programmes (ADPs), or Extended Curriculum Programmes (ECPs), continue to play a central role in increasing access to previously marginalised students in higher education in South Africa. Using Archer’s morphogenetic approach, this study examines how a group of ADP students “made their way” through their engineering undergraduate studies. Twelve students in their fourth year of study were interviewed three times and selected university documents were analysed. The authors found that the fragmented curriculum, shortened consolidation and examination periods, and unfavourable examination timetables potentially constrained the students’ aspirations. In addition, the mainstream students and lecturers’ ideas about ADP students worsened their experience of marginalisation and exception. We also found that students experienced the mainly black student enrolment of the ADP as racial discrimination. The findings indicate that students found themselves in enormously constrained circumstances, but they also exhibited what Archer calls “corporate agency” and different “modes of reflexivity” to overcome some of these constraints. We argue that the establishment of Academic Development Programmes as separate from mainstream curricula, while enabling access to some extent, may have unintended consequences of also constraining the students for whom they are designed

E-beam generated holographic masks for optical vector-matrix multiplication

An optical vector matrix multiplication scheme that encodes the matrix elements as a holographic mask consisting of linear diffraction gratings is proposed. The binary, chrome on glass masks are fabricated by e-beam lithography. This approach results in a fairly simple optical system that promises both large numerical range and high accuracy. A partitioned computer generated hologram mask was fabricated and tested. This hologram was diagonally separated outputs, compact facets and symmetry about the axis. The resultant diffraction pattern at the output plane is shown. Since the grating fringes are written at 45 deg relative to the facet boundaries, the many on-axis sidelobes from each output are seen to be diagonally separated from the adjacent output signals

General flux to a trap in one and three dimensions

The problem of the flux to a spherical trap in one and three dimensions, for diffusing particles undergoing discrete-time jumps with a given radial probability distribution, is solved in general, verifying the Smoluchowski-like solution in which the effective trap radius is reduced by an amount proportional to the jump length. This reduction in the effective trap radius corresponds to the Milne extrapolation length.Comment: Accepted for publication, in pres

Elastic Radiation in a Half‐Space

A Green's function for the elastic wave equation, which satisfies certain boundary conditions on the surface of a homogeneous half‐space, is derived by means of the Fourier transformation. This half‐space Green's function is then applied to the computation of radiative effects due to the earth's surface when a radiating source is located on or within that surface. The results obtained are to be taken as an extension of a previous and similar formulation for the infinite medium due to Case and Colwell.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70190/2/JMAPAQ-11-8-2546-1.pd

Renormalization in Quantum Mechanics

We implement the concept of Wilson renormalization in the context of simple quantum mechanical systems. The attractive inverse square potential leads to a \b function with a nontrivial ultraviolet stable fixed point and the Hulthen potential exhibits the crossover phenomenon. We also discuss the implementation of the Wilson scheme in the broader context of one dimensional potential problems. The possibility of an analogue of Zamolodchikov's $C$ function in these systems is also discussed.Comment: 16 pages, UR-1310, ER-40685-760. (Additional references included.