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The dissipation distance for a 2D single crystal with two symmetric slip systems

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arxiv math/0207198 v1 pith:HKB5FNMJ submitted 2002-07-22 math.OC math.MG

classification math.OCmath.MG
keywords distanceproblemslipassociatedbackwardscrystaldissipationhyperbolic
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We solve a model problem from single crystal plasticity.We consider 4 slip systems in the plane with orthogonal slip-directions and equal slip rates, forward as well as backwards. We compute the associated dissipation distance by solving an optimal control problem. It turns out that from a computational point of view computing the distance is inexpensive. We put special emphasis on visualization of the metric spheres and the associated length-minimizing curves. As a byproduct we also solve a related problem, optimal path planning for a car driving forwards and backwards with limited turning radius in the hyperbolic plane. This is a hyperbolic version of the Reeds-Shepp-Car-Problem first discussed in (Pac. J. Math. 145:367-393, 1990).

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  1. Generating Sets of Stochastic Matrices

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    For stochastic matrices of size 2x2 and 3x3, explicit generating sets are constructed, with worst-case factor count exactly 4 in dimension 2 and at most 20 in dimension 3.

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