The Experts below are selected from a list of 5220 Experts worldwide ranked by ideXlab platform
Serge Tabachnikov - One of the best experts on this subject based on the ideXlab platform.
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the pentagram map a discrete integrable system
Communications in Mathematical Physics, 2010Co-Authors: Valentin Ovsienko, Richard Evan Schwartz, Serge TabachnikovAbstract:The pentagram map is a Projectively natural Transformation defined on (twisted) polygons. A twisted polygon is a map from \({\mathbb Z}\) into \({{\mathbb{RP}}^2}\) that is periodic modulo a Projective Transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].
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The Pentagram Map: A Discrete Integrable System
Communications in Mathematical Physics, 2010Co-Authors: Valentin Ovsienko, Serge TabachnikovAbstract:The pentagram map is a Projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the Projective plane that is periodic modulo a Projective Transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.
Valentin Ovsienko - One of the best experts on this subject based on the ideXlab platform.
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the pentagram map a discrete integrable system
Communications in Mathematical Physics, 2010Co-Authors: Valentin Ovsienko, Richard Evan Schwartz, Serge TabachnikovAbstract:The pentagram map is a Projectively natural Transformation defined on (twisted) polygons. A twisted polygon is a map from \({\mathbb Z}\) into \({{\mathbb{RP}}^2}\) that is periodic modulo a Projective Transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].
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The Pentagram Map: A Discrete Integrable System
Communications in Mathematical Physics, 2010Co-Authors: Valentin Ovsienko, Serge TabachnikovAbstract:The pentagram map is a Projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the Projective plane that is periodic modulo a Projective Transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.
Jose Alberto Orejuela - One of the best experts on this subject based on the ideXlab platform.
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a non trivial connection for the metric affine gauss bonnet theory in d 4
Physics Letters B, 2019Co-Authors: Bert Janssen, Alejandro Jimenezcano, Jose Alberto OrejuelaAbstract:Abstract We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the Projective invariance of general Lovelock actions and show that all connections constructed by acting with a Projective Transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of Transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.
Bert Janssen - One of the best experts on this subject based on the ideXlab platform.
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a non trivial connection for the metric affine gauss bonnet theory in d 4
Physics Letters B, 2019Co-Authors: Bert Janssen, Alejandro Jimenezcano, Jose Alberto OrejuelaAbstract:Abstract We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the Projective invariance of general Lovelock actions and show that all connections constructed by acting with a Projective Transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of Transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.
Fabienne Betting - One of the best experts on this subject based on the ideXlab platform.
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3D-2D Projective registration of free-form curves and surfaces
Computer Vision and Image Understanding, 1997Co-Authors: Jacques Feldmar, Nicholas Ayache, Fabienne BettingAbstract:Some medical interventions require knowing the correspondence between an MRI/CT image and the actual position of the patient. Examples occur in neurosurgery and radiotherapy, but also in video surgery (laparoscopy). We present in this paper three new techniques for performing this task without artificial markers. To do this, we find the 3D-2D Projective Transformation (composition of a rigid displacement and a perspective projection) which maps a 3D object onto a 2D image of this object. Depending on the object model (curve or surface), and on the 2D image acquisition system (X-Ray, video), the techniques are different but the framework is common: Results are presented on a variety of real medical data to demonstrate the validity of our approach.
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3d 2d Projective registration of free form curves and surfaces
International Conference on Computer Vision, 1995Co-Authors: Jacques Feldmar, Nicholas Ayache, Fabienne BettingAbstract:Some medical interventions require knowing the correspondence between an MRI/CT pre-operative image and the actual position of the patient. Examples occur in neurosurgery, radiotherapy, interventional radiology, but also in video surgery (laparoscopy). We present in this article three new techniques for performing this task without artificial markers. We find the 3D-2D Projective Transformation (composition of a rigid displacement and a perspective projection) which maps a 3D object onto a 2D image of this object. Depending on the object model (curve or surface), and on the 2D image acquisition system (X-Ray, video), the techniques are different but the framework is common. It does not depend on the initial relative positions of the objects and deals with the occlusions and the outliers. Results are presented on real medical data to demonstrate the validity of our approach. >
