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Thomas Hudson

Publications and source records attributed to Thomas Hudson.

47 records · Page 3Linked to original sources
arXiv2016

Asymptotic analysis of boundary layers in a repulsive particle system

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. Our setting covers pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via Gamma-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.

math.AP↗
arXiv2016

Upscaling a model for the thermally-driven motion of screw dislocations

We formulate and study a stochastic model for the thermally-driven motion of interacting straight screw dislocations in a cylindrical domain with a convex polygonal cross-section. Motion is modelled as a Markov jump process, where waiting times for transitions from state to state are assumed to be exponentially distributed with rates expressed in terms of the potential energy barrier between the states. Assuming the energy of the system is described by a discrete lattice model, a precise asymptotic description of the energy barriers between states is obtained. Through scaling of the various physical constants, two dimensionless parameters are identified which govern the behaviour of the resulting stochastic evolution. In an asymptotic regime where these parameters remain fixed, the process is found to satisfy a Large Deviations Principle. A sufficiently explicit description of the corresponding rate functional is obtained such that the most probable path of the dislocation configuration may be described as the solution of Discrete Dislocation Dynamics with an explicit anisotropic mobility which depends on the underlying lattice structure.

math.AP↗
arXiv2015

Generalised symplectic Schubert classes

Under the assumption that the base field k has characteristic 0, we compute the algebraic cobordism fundamental classes of a family of Schubert varieties isomorphic to full and symplectic flag bundles. We use this computation to prove a formula for the push-forward class of the Bott-Samelson resolutions associated to the symplectic flag bundle. We specialise our formula to connective K-theory and the Grothendieck ring of vector bundles, hence obtaining a description for all Schubert classes.

math.AG↗
arXiv2014

Analysis of stable screw dislocation configurations in an anti--plane lattice model

We consider a variational anti-plane lattice model and demonstrate that at zero temperature, there exist locally stable states containing screw dislocations, given conditions on the distance between the dislocations, and on the distance between dislocations and the boundary of the crystal. In proving our results, we introduce approximate solutions which are taken from the theory of dislocations in linear elasticity, and use the inverse function theorem to show that local minimisers lie near them. This gives credence to the commonly held intuition that linear elasticity is essentially correct up to a few spacings from the dislocation core.

math.AP↗
arXiv2013

Fourier-Mukai transformation on algebraic cobordism

We define a notion of Fourier-Mukai transform on algebraic cobordism cycles with $\mathbb{Q}$-coefficients on an abelian variety. We use this to produce a Beauville decomposition of algebraic cobordism and study its consequences, including a decomposition of the cobordism motive of an abelian variety.

math.AG↗
arXiv2013

A Thom-Porteous formula for connective K-theory using algebraic cobordism

We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by computing the fundamental classes of Schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.

math.AG↗
arXiv2013

Existence and stability of a screw dislocation under anti-plane deformation

We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove existence of a global energy minimiser and thus demonstrate that dislocations are globally stable equilibria within our model.

math.AP↗
arXiv2012

Thom-Porteous formulas in algebraic cobordism

We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We then provide a geometric interpretation to the double beta-polynomials of Fomin and Kirillov by specializing our formula to the case of connected K-theory.

math.AG↗
arXiv2012

Gamma-expansion for a 1D Confined Lennard-Jones model with point defect

We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the Γ-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a Γ-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.

math.AP↗