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Luis Vega - One of the best experts on this subject based on the ideXlab platform.
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Vortex Filament Equation for a Regular Polygon in the hyperbolic plane.
arXiv: Numerical Analysis, 2020Co-Authors: Francisco De La Hoz, Sandeep Kumar, Luis VegaAbstract:The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a Regular planar Polygon in the hyperbolic space. Unlike in the Euclidean space, the planar Polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar Polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar Polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.
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vortex filament equation for a Regular Polygon
Nonlinearity, 2014Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we study the evolution of the localized induction approximation (LIA), also known as vortex lament equation, Xt = Xs^ Xss; for X(s; 0) a Regular planar Polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that X(s;t) is also a Polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau sum. We also study the fractal behavior of X(0;t), relating it with the socalled Riemann’s non-dierentiable function, that, as proved by S. Jaard, ts with the multifractal model of U. Frisch and G. Parisi, for fully developed turbulence.
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the evolution of the local induction approximation for a Regular Polygon
Esaim: Proceedings, 2014Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we consider the so-called local induction approximation (LIA): Xt = Xs ^ Xss, where ^ is the usual cross product, and s denotes the arc-length parametrization. We study its evolution, taking planar Regular Polygons of M sides as initial data. Assuming uniqueness and bearing in mind the invariances and symmetries of the problem, we are able to fully characterize, by algebraic means, X(s,t) and its derivative, the tangent vector T(s,t), at times t which are rational multiples of 2�/M 2 . We show that the values at those instants are intimately related to the generalized quadratic Gaus sums. Resume. Dans cette courte note, nous considerons l'approximation d'induction locale (communement appelee LIA, par ses initiales en anglais): Xt = Xs ^ Xss, ou ^ symbolise le produit croise habituel et s est le parametrage par longueur d'arc. Nous etudions son evolution, en considerant des Polygones plans reguliers a M cotes comme donnees initiales. En supposant l'unicite et en prenant en compte les invariances et les symetries du probleme, nous sommes capables de caracteriser completement, par des techniques algebriques, X(s,t) et sa derivee, le vecteur tangent T(s,t), a des instants t qui sont des multiples rationnels de 2�/M 2 . Nous montrons que les valeurs a ces instants sont intimement liees aux sommes quadratiques generalisees de Gaus.
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vortex filament equation for a Regular Polygon
arXiv: Analysis of PDEs, 2013Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a Regular planar Polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf X(s, t)$ is also a Polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau{\ss} sum. We also study the fractal behavior of $\mathbf X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal.
Huannming Chou - One of the best experts on this subject based on the ideXlab platform.
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heat transfer characteristics of an insulated Regular polyhedron by using a Regular Polygon top solid wedge thermal resistance model
Energy Conversion and Management, 2003Co-Authors: Kingleung Wong, Huannming ChouAbstract:The relation of the critical and the neutral thickness and heat transfer characteristics for an insulated Regular polyhedron are investigated by using a new Regular Polygon top solid wedge thermal resistance (RPSWT) model obtained from a solid angle concept. It is found that the dimensionless heat transfer characteristics of the insulated polyhedrons are the same as those of an insulated sphere even though their actual heat transfer rates are different. It is also found that the conventional constant surface area plate model cannot be used to analyze the insulated polyhedrons, especially in situations of small to medium body size and/or with a greater insulation thickness. The single RPSWT model should also be applied to the heat transfer problem of the wedge block with a Regular Polygon top.
Nana Odishelidze - One of the best experts on this subject based on the ideXlab platform.
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general case of contact problems for a Regular Polygon weakened with full strength hole
Zeitschrift für Angewandte Mathematik und Physik, 2015Co-Authors: Nana Odishelidze, Francisco Criadoaldeanueva, Francisco Criado, J M SanchezAbstract:The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a Regular Polygon and the internal boundary is the required full-strength hole including the origin of coordinates. The full-strength hole is cycle symmetric. It is assumed that to every link of the broken line conforming the outer boundary of the given body are applied absolutely smooth rigid punches with rectilinear bases, which are under the action of the force P that applies to their middle points. There is no friction between the surface of the given elastic body and the punches. The uniformly distributed normal stress Q is applied to the hole boundary. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and the shape of the hole’s contour are determined under the condition that the tangential normal stress arising at it takes a constant value. A similar problem is considered for a square and an equilateral triangle, which are weakened with full-strength holes. Using the method developed here, the partially unknown boundary value problems under consideration is reduced to known boundary value problems of the theory of analytic functions. The solutions are presented in quadratures, and full-strength contours are constructed.
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Solution of one mixed problem of plate bending for a domain with partially unknown boundary
Acta Mechanica, 2011Co-Authors: Nana Odishelidze, Francisco Criado, F. Criado-aldeanueva, J M SanchezAbstract:The paper addresses the problem of finding a full-strength contour in the problem of plate bending for a cycle-symmetric doubly connected domain. An isotropic elastic plate, bounded by a Regular Polygon, is weakened by a required full-strength hole whose symmetry axes are the Regular Polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. An unknown part of the boundary is free from external forces. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterising an elastic equilibrium of the body) and of an unknown full-strength contour are determined. A numerical analysis is performed and the corresponding plots are obtained by means of the Mathcad system.
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a mixed problem of plate bending for a Regular Polygon weakened with a required full strength hole
Acta Mechanica, 2009Co-Authors: Nana Odishelidze, Francisco Criadoaldeanueva, Francisco CriadoAbstract:The paper addresses a problem of bending of an isotropic elastic plate, bounded by a Regular Polygon weakened with a required full-strength hole whose symmetric axes are the Regular Polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. This plate bends under the action of concentrated moments applied to the middle points of the bars. Unknown part of the boundary is free from external forces. Using the methods of complex analysis the plate deflection and required full-strength contours are determined. Numerical analysis is performed and the corresponding graphs are constructed.
Francisco Criado - One of the best experts on this subject based on the ideXlab platform.
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general case of contact problems for a Regular Polygon weakened with full strength hole
Zeitschrift für Angewandte Mathematik und Physik, 2015Co-Authors: Nana Odishelidze, Francisco Criadoaldeanueva, Francisco Criado, J M SanchezAbstract:The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a Regular Polygon and the internal boundary is the required full-strength hole including the origin of coordinates. The full-strength hole is cycle symmetric. It is assumed that to every link of the broken line conforming the outer boundary of the given body are applied absolutely smooth rigid punches with rectilinear bases, which are under the action of the force P that applies to their middle points. There is no friction between the surface of the given elastic body and the punches. The uniformly distributed normal stress Q is applied to the hole boundary. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and the shape of the hole’s contour are determined under the condition that the tangential normal stress arising at it takes a constant value. A similar problem is considered for a square and an equilateral triangle, which are weakened with full-strength holes. Using the method developed here, the partially unknown boundary value problems under consideration is reduced to known boundary value problems of the theory of analytic functions. The solutions are presented in quadratures, and full-strength contours are constructed.
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Solution of one mixed problem of plate bending for a domain with partially unknown boundary
Acta Mechanica, 2011Co-Authors: Nana Odishelidze, Francisco Criado, F. Criado-aldeanueva, J M SanchezAbstract:The paper addresses the problem of finding a full-strength contour in the problem of plate bending for a cycle-symmetric doubly connected domain. An isotropic elastic plate, bounded by a Regular Polygon, is weakened by a required full-strength hole whose symmetry axes are the Regular Polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. An unknown part of the boundary is free from external forces. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterising an elastic equilibrium of the body) and of an unknown full-strength contour are determined. A numerical analysis is performed and the corresponding plots are obtained by means of the Mathcad system.
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a mixed problem of plate bending for a Regular Polygon weakened with a required full strength hole
Acta Mechanica, 2009Co-Authors: Nana Odishelidze, Francisco Criadoaldeanueva, Francisco CriadoAbstract:The paper addresses a problem of bending of an isotropic elastic plate, bounded by a Regular Polygon weakened with a required full-strength hole whose symmetric axes are the Regular Polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. This plate bends under the action of concentrated moments applied to the middle points of the bars. Unknown part of the boundary is free from external forces. Using the methods of complex analysis the plate deflection and required full-strength contours are determined. Numerical analysis is performed and the corresponding graphs are constructed.
Francisco De La Hoz - One of the best experts on this subject based on the ideXlab platform.
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Vortex Filament Equation for a Regular Polygon in the hyperbolic plane.
arXiv: Numerical Analysis, 2020Co-Authors: Francisco De La Hoz, Sandeep Kumar, Luis VegaAbstract:The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a Regular planar Polygon in the hyperbolic space. Unlike in the Euclidean space, the planar Polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar Polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar Polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.
-
vortex filament equation for a Regular Polygon
Nonlinearity, 2014Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we study the evolution of the localized induction approximation (LIA), also known as vortex lament equation, Xt = Xs^ Xss; for X(s; 0) a Regular planar Polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that X(s;t) is also a Polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau sum. We also study the fractal behavior of X(0;t), relating it with the socalled Riemann’s non-dierentiable function, that, as proved by S. Jaard, ts with the multifractal model of U. Frisch and G. Parisi, for fully developed turbulence.
-
the evolution of the local induction approximation for a Regular Polygon
Esaim: Proceedings, 2014Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we consider the so-called local induction approximation (LIA): Xt = Xs ^ Xss, where ^ is the usual cross product, and s denotes the arc-length parametrization. We study its evolution, taking planar Regular Polygons of M sides as initial data. Assuming uniqueness and bearing in mind the invariances and symmetries of the problem, we are able to fully characterize, by algebraic means, X(s,t) and its derivative, the tangent vector T(s,t), at times t which are rational multiples of 2�/M 2 . We show that the values at those instants are intimately related to the generalized quadratic Gaus sums. Resume. Dans cette courte note, nous considerons l'approximation d'induction locale (communement appelee LIA, par ses initiales en anglais): Xt = Xs ^ Xss, ou ^ symbolise le produit croise habituel et s est le parametrage par longueur d'arc. Nous etudions son evolution, en considerant des Polygones plans reguliers a M cotes comme donnees initiales. En supposant l'unicite et en prenant en compte les invariances et les symetries du probleme, nous sommes capables de caracteriser completement, par des techniques algebriques, X(s,t) et sa derivee, le vecteur tangent T(s,t), a des instants t qui sont des multiples rationnels de 2�/M 2 . Nous montrons que les valeurs a ces instants sont intimement liees aux sommes quadratiques generalisees de Gaus.
-
vortex filament equation for a Regular Polygon
arXiv: Analysis of PDEs, 2013Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a Regular planar Polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf X(s, t)$ is also a Polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau{\ss} sum. We also study the fractal behavior of $\mathbf X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal.
