Explicit isogeny descent on elliptic curves

In this note, we consider an l-isogeny descent on a pair of elliptic curves over Q. We assume that l > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finite- dimensional F_l-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the l-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor.Comment: 17 pages, accepted for publication in Mathematics of Computatio

Nanoindentation and incipient plasticity

This paper presents a large-scale atomic resolution simulation of nanoindentation into a thin aluminum film using the recently introduced quasicontinuum method. The purpose of the simulation was to study the initial stages of plastic deformation under the action of an indenter. Two different crystallographic orientations of the film and two different indenter geometries (a rectangular prism and a cylinder) were studied. We obtained both macroscopic load versus indentation depth curves, as well as microscopic quantities, such as the Peierls stress and density of geometrically necessary dislocations beneath the indenter. In addition, we obtain detailed information regarding the atomistic mechanisms responsible for the macroscopic curves. A strong dependence on geometry and orientation is observed. Two different microscopic mechanisms are observed to accommodate the applied loading: (i) nucleation and subsequent propagation into the bulk of edge dislocation dipoles and (ii) deformation twinning

Application of thermal radiation data to fishery oceanography

Work on Project Little Window-1 and -2, DRIR data conversion, improvements on the APT receiver, equatorial upwelling, and the topic of thermal fronts is discussed

Analytical investigation of rotor wake formation and geometry

A number of refinements in the computer code were worked out and tested. Three codes have been written to date. One program is for an isolated wing and is being used to compare with data for the vortex wake (Weston). The second code is for an isolated wing with a streamwise vortex passing above it. This program is being used to validate the computational procedure for incorporating the vortex into the Euler equation calculations. The third program is the hovering rotor code which is the overall objective of the research. The optimization calculations for a hovering helicopter rotor have been completed

Quasicontinuum simulation of fracture at the atomic scale

We study the problem of atomic scale fracture using the recently developed quasicontinuum method in which there is a systematic thinning of the atomic-level degrees of freedom in regions where they are not needed. Fracture is considered in two distinct settings. First, a study is made of cracks in single crystals, and second, we consider a crack advancing towards a grain boundary (GB) in its path. In the investigation of single crystal fracture, we evaluate the competition between simple cleavage and crack-tip dislocation emission. In addition, we examine the ability of analytic models to correctly predict fracture behaviour, and find that the existing analytical treatments are too restrictive in their treatment of nonlinearity near the crack tip. In the study of GB-crack interactions, we have found a number of interesting deformation mechanisms which attend the advance of the crack. These include the migration of the GB, the emission of dislocations from the GB, and deflection of the crack front along the GB itself. In each case, these mechanisms are rationalized on the basis of continuum mechanics arguments

Degree spectra for transcendence in fields

We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis