Preprints
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Preprints from members of the UM School of Mathematics, September 2008 - present.
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Item Dynamics for Ginzburg-Landau vortices under a mixed flow(2008-11-07) Kurzke, Matthias; Melcher, Christof; Moser, Roger; Spirn, DanielWe consider a complex Ginzburg-Landau equation that contains a Schrodinger term and a damping term that is proportional to the time derivative. Given well-prepared initial conditions that correspond to quantized vortices, we establish the vortex motion law until collision time.Item The Foppl-von Karman equations for plates with incompatible strains(2010-02-24) Lewicka, Marta; Mahadevan, L.; Pakzad, RezaWe provide a derivation of the Foppl-von Karman equations for the shape of and stresses in an elastic plate with residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials.Item Lectures on moving frames(2009-01-21) Olver, Peter J.This article surveys the equivariant method of moving frames, along with a variety of applications to geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant curve flows.Item A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry(2008-11-26) Lewicka, MartaWe prove that the critical points of the $3$d nonlinear elasticity functional over a thin shell of arbitrary geometry and of thickness $h$, as well as the weak solutions to the static equilibrium equations (formally the Euler Lagrange equations associated to the elasticity functional) converge, in the limit of vanishing thickness $h$, to the critical points of the generalized von Karman functional on the mid-surface, recently derived in [14]. This holds provided the elastic energy of the $3$d deformations scale like $h^4$ and the magnitude of the body forces scale like $h^3$.Item Resolutions, relation modules and Schur multipliers for categories(2008-09-15) Webb, PeterWe show that the construction in group cohomology of the Gruenberg resolution associated to a free presentation and the resulting relation module can be copied in the context of representations of categories. We establish five-term exact sequences in the cohomology of categories and go on to show that the Schur multiplier of the category has properties which generalize those of the Schur multiplier of a group.Item Stratifications and Mackey functors II: globally defined Mackey functors(2008-09-16) Webb, PeterWe describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.