5,455 research outputs found
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.Comment: 44 page
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.Comment: Major revision, including overhaul of notation, additions to Section
6 on Large Deviations, and resolution of conjecture in original version. 45
pages, 2 figures, 1 tabl
On the K-theoretic fundamental classes of Deligne-Lusztig varieties
In this paper we express the class of the structure sheaves of the closures
of Deligne--Lusztig varieties as explicit double Grothendieck polynomials in
the first Chern classes of appropriate line bundles on the ambient flag
variety. This is achieved by viewing such closures as degeneracy loci of
morphisms of vector bundles.Comment: 8 pages, minor change
Properties of screw dislocation dynamics: time estimates on boundary and interior collisions
In this paper, the dynamics of a system of a finite number of screw
dislocations is studied. Under the assumption of antiplane linear elasticity,
the two-dimensional dynamics is determined by the renormalised energy. The
interaction of one dislocation with the boundary and of two dislocations of
opposite Burgers moduli are analysed in detail and estimates on the collision
times are obtained. Some exactly solvable cases and numerical simulations show
agreement with the estimates obtained.Comment: 25 pages, 4 figure
Analysis of an atomistic model for anti-plane fracture
We develop a model for an anti-plane crack defect posed on a square lattice
under an interatomic pair-potential with nearest-neighbour interactions. In
particular, we establish existence, local uniqueness and stability of solutions
for small loading parameters and further prove qualitatively sharp far-field
decay estimates. The latter requires establishing decay estimates for the
corresponding lattice Green's function, which are of independent interest
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