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Matthias Schwarz - One of the best experts on this subject based on the ideXlab platform.
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the role of the legendre transform in the study of the floer complex of cotangent bundles
Communications on Pure and Applied Mathematics, 2015Co-Authors: Alberto Abbondandolo, Matthias SchwarzAbstract:Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM → R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional that is associated to L to the Floer complex that is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder that on the boundary satisfy half of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a Heuristic Argument which, independently of any Fredholm and compactness analysis, explains why the spaces of maps that are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our Heuristic Arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pullback of the second Stiefel-Whitney class of TM on 2-tori in M.© 2015 Wiley Periodicals, Inc.
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the role of the legendre transform in the study of the floer complex of cotangent bundles
arXiv: Symplectic Geometry, 2013Co-Authors: Alberto Abbondandolo, Matthias SchwarzAbstract:Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Phi from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Psi of Phi. Contrary to other previously defined maps going in the same direction, Psi is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a Heuristic Argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Phi and Psi are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our Heuristic Arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of TM on 2-tori in M.
Karamanos, Spyros A. - One of the best experts on this subject based on the ideXlab platform.
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Stability of long transversely-isotropic elastic cylindrical shells under bending
Elsevier Ltd., 2010Co-Authors: Houliara Sotiria, Karamanos, Spyros A.Abstract:AbstractThe present paper investigates buckling of cylindrical shells of transversely-isotropic elastic material subjected to bending, considering the nonlinear prebuckling ovalized configuration. A large-strain hypoelastic model is developed to simulate the anisotropic material behavior. The model is incorporated in a finite-element formulation that uses a special-purpose “tube element”. For comparison purposes, a hyperelastic model is also employed. Using an eigenvalue analysis, bifurcation on the prebuckling ovalization path to a uniform wrinkling state is detected. Subsequently, the postbuckling equilibrium path is traced through a continuation arc-length algorithm. The effects of anisotropy on the bifurcation moment, the corresponding curvature and the critical wavelength are examined, for a wide range of radius-to-thickness ratio values. The calculated values of bifurcation moment and curvature are also compared with analytical predictions, based on a Heuristic Argument. Finally, numerical results for the imperfection sensitivity of bent cylinders are obtained, which show good comparison with previously reported asymptotic expressions
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Stability of long transversely-isotropic elastic cylindrical shells under bending
'Elsevier BV', 2010Co-Authors: Houliara S., Karamanos, Spyros A.Abstract:The present paper investigates buckling of cylindrical shells of transversely-isotropic elastic material subjected to bending, considering the nonlinear prebuckling ovalized configuration. A large-strain hypoelastic model is developed to simulate the anisotropic material behavior, The model is incorporated in a finite-element formulation that uses a special-purpose "tube element". For comparison purposes, a hyperelastic model is also employed. Using an eigenvalue analysis, bifurcation on the prebuckling ovalization path to a uniform wrinkling state is detected. Subsequently, the postbuckling equilibrium path is traced through a continuation arc-length algorithm. The effects of anisotropy on the bifurcation moment, the corresponding curvature and the critical wavelength are examined, for a wide range of radius-to-thickness ratio values. The calculated values of bifurcation moment and curvature are also compared with analytical predictions, based on a Heuristic Argument. Finally, numerical results for the imperfection sensitivity of bent cylinders are obtained, which show good comparison with previously reported asymptotic expressions. (C) 2009 Elsevier Ltd. All rights reserved
Bruce Reed - One of the best experts on this subject based on the ideXlab platform.
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how to determine if a random graph with a fixed degree sequence has a giant component
Foundations of Computer Science, 2016Co-Authors: Felix Joos, Dieter Rautenbach, Guillem Perarnau, Bruce ReedAbstract:The traditional Erdos-Renyi model of a random network is of little use in modelling the type of complex networks which modern researchers study. In this graph, every pair of vertices is equally likely to be connected by an edge. However, 21st century networks are of diverse nature and usually exhibit inhomogeneity among their nodes. This motivates the study, for a fixed degree sequence D=(d1, ..., dn), of a uniformly chosen simple graph G(D) on {1, ..., n} where the vertex i has degree di. In this paper, we study the existence of a giant component in G(D). A Heuristic Argument suggests that a giant component in G(D) will exist provided that the sum of the squares of the degrees is larger than twice the sum of the degrees. In 1995, Molloy and Reed essentially proved this to be the case when the degree sequence D under consideration satisfies certain technical conditions [Random Structures & Algorithms, 6:161-180]. This work has attracted considerable attention, has been extended to degree sequences under weaker conditions and has been applied to random models of a wide range of complex networks such as the World Wide Web or biological systems operating at a sub-molecular level. Nevertheless, the technical conditions on D restrict the applicability of the result to sequences where the vertices of high degree play no important role. This is a major problem since it is observed in many real-world networks, such as scale-free networks, that vertices of high degree (the so-called hubs) are present and play a crucial role. In this paper we characterize when a uniformly random graph with a fixed degree sequence has a giant component. Our main result holds for every degree sequence of length n provided that a minor technical condition is satisfied. The typical structure of G(D) when D does not satisfy this condition is relatively simple and easy to understand. Our result gives a unified criterion that implies all the known results on the existence of a giant component in G(D), including both the generalizations of the Molloy-Reed result and results on more restrictive models. Moreover, it turns out that the Heuristic Argument used in all the previous works on the topic, does not extend to general degree sequences.
Alberto Abbondandolo - One of the best experts on this subject based on the ideXlab platform.
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the role of the legendre transform in the study of the floer complex of cotangent bundles
Communications on Pure and Applied Mathematics, 2015Co-Authors: Alberto Abbondandolo, Matthias SchwarzAbstract:Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM → R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional that is associated to L to the Floer complex that is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder that on the boundary satisfy half of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a Heuristic Argument which, independently of any Fredholm and compactness analysis, explains why the spaces of maps that are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our Heuristic Arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pullback of the second Stiefel-Whitney class of TM on 2-tori in M.© 2015 Wiley Periodicals, Inc.
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the role of the legendre transform in the study of the floer complex of cotangent bundles
arXiv: Symplectic Geometry, 2013Co-Authors: Alberto Abbondandolo, Matthias SchwarzAbstract:Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Phi from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Psi of Phi. Contrary to other previously defined maps going in the same direction, Psi is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a Heuristic Argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Phi and Psi are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our Heuristic Arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of TM on 2-tori in M.
Felix Joos - One of the best experts on this subject based on the ideXlab platform.
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how to determine if a random graph with a fixed degree sequence has a giant component
Foundations of Computer Science, 2016Co-Authors: Felix Joos, Dieter Rautenbach, Guillem Perarnau, Bruce ReedAbstract:The traditional Erdos-Renyi model of a random network is of little use in modelling the type of complex networks which modern researchers study. In this graph, every pair of vertices is equally likely to be connected by an edge. However, 21st century networks are of diverse nature and usually exhibit inhomogeneity among their nodes. This motivates the study, for a fixed degree sequence D=(d1, ..., dn), of a uniformly chosen simple graph G(D) on {1, ..., n} where the vertex i has degree di. In this paper, we study the existence of a giant component in G(D). A Heuristic Argument suggests that a giant component in G(D) will exist provided that the sum of the squares of the degrees is larger than twice the sum of the degrees. In 1995, Molloy and Reed essentially proved this to be the case when the degree sequence D under consideration satisfies certain technical conditions [Random Structures & Algorithms, 6:161-180]. This work has attracted considerable attention, has been extended to degree sequences under weaker conditions and has been applied to random models of a wide range of complex networks such as the World Wide Web or biological systems operating at a sub-molecular level. Nevertheless, the technical conditions on D restrict the applicability of the result to sequences where the vertices of high degree play no important role. This is a major problem since it is observed in many real-world networks, such as scale-free networks, that vertices of high degree (the so-called hubs) are present and play a crucial role. In this paper we characterize when a uniformly random graph with a fixed degree sequence has a giant component. Our main result holds for every degree sequence of length n provided that a minor technical condition is satisfied. The typical structure of G(D) when D does not satisfy this condition is relatively simple and easy to understand. Our result gives a unified criterion that implies all the known results on the existence of a giant component in G(D), including both the generalizations of the Molloy-Reed result and results on more restrictive models. Moreover, it turns out that the Heuristic Argument used in all the previous works on the topic, does not extend to general degree sequences.
