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An asymmetric ArzelàAscoli theorem
 Topology and Its Applications
"... An ArzelàAscoli theorem for asymmetric metric spaces (sometimes called quasimetric spaces) is proved. One genuinely asymmetric condition is introduced, and it is shown that several classic statements fail in the asymmetric context if this assumption is dropped. ..."
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An ArzelàAscoli theorem for asymmetric metric spaces (sometimes called quasimetric spaces) is proved. One genuinely asymmetric condition is introduced, and it is shown that several classic statements fail in the asymmetric context if this assumption is dropped.
Gradient flows in asymmetric metric spaces
 In preparation
"... This article is concerned with gradient flows in asymmetric metric spaces, that is, spaces with a topology induced by an asymmetric metric. Such asymmetry appears naturally in many applications, e.g., in mathematical models for materials with hysteresis. A framework of asymmetric gradient flows is e ..."
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This article is concerned with gradient flows in asymmetric metric spaces, that is, spaces with a topology induced by an asymmetric metric. Such asymmetry appears naturally in many applications, e.g., in mathematical models for materials with hysteresis. A framework of asymmetric gradient flows is established under the assumption that the metric is weakly lower semicontinuous in the second argument (and not necessarily on the first), and an existence theorem for gradient flows defined on an asymmetric metric space is given. 1
GRADIENT FLOWS AS A SELECTION PROCEDURE FOR EQUILIBRIA OF NONCONVEX ENERGIES∗
"... Abstract. For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on ..."
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Abstract. For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on the Lennard–Jones potential, is presented. As an application of this technique, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated in the case of elastic deformation and a simple damaged state.
Contents lists available at ScienceDirect Neural Networks
"... journal homepage: www.elsevier.com/locate/neunet ..."
EQUILIBRIA OF NONCONVEX ENERGIES
"... Gradient flows as a selection procedure for equilibria ..."
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Monotonicity of transport plans
, 2006
"... We study monotonicity properties for minimizers of transport problems. In the onedimensional case, we present an algorithm to construct minimizing monotone transport plans by “monotonizing ” a given minimizing transport plan. This method applies in particular to the case of the L 1Wasserstein metr ..."
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We study monotonicity properties for minimizers of transport problems. In the onedimensional case, we present an algorithm to construct minimizing monotone transport plans by “monotonizing ” a given minimizing transport plan. This method applies in particular to the case of the L 1Wasserstein metric where we prove the existence of monotone minimizers for arbitrary marginals. We find that monotone transport plans are in a certain sense close to monotone transport maps.
Report no. 05/11 Continuum Limit of a OneDimensional Atomistic Energy Based on Local Minimization
"... For atomistic energies, global minimization gives the wrong qualitative behaviour and therefore continuum limits should be formulated in terms of local minimization. In this paper, a possible process is suggested, to describe local minimization for a simple onedimensional problem with body and su ..."
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For atomistic energies, global minimization gives the wrong qualitative behaviour and therefore continuum limits should be formulated in terms of local minimization. In this paper, a possible process is suggested, to describe local minimization for a simple onedimensional problem with body and surface energy. It is shown that an atomistic gradient flow evolution converges to a continuum gradient flow as the spacing between the atoms tends to zero. In addition, the convergence of local minimizers is investigated, in the case of both elastic deformation and fracture. Key words and phrases: continuum limit, atomistic models, local minimization, gradient flows
Gradient Flows in Asymmetric Metric Spaces
"... flows in asymmetric metric spaces. Nonlinear Analysis: Theory ..."
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