Análisis Funcional y AplicacionesNo Descriptionhttps://hdl.handle.net/10784/4394https://repository.eafit.edu.co/retrieve/ea05f714-c745-4dff-8c4b-574065b6dade/2024-09-08T17:18:34Z2024-09-08T17:18:34Z91A Riemannian Geometry in the q-Exponential Banach Manifold Induced by q-DivergencesLoaiza Ossa, Gabriel IgnacioQuiceno Echavarría, Héctor Románhttps://hdl.handle.net/10784/74112021-11-03T17:41:21Z2013-01-01T00:00:00Zdc.title: A Riemannian Geometry in the q-Exponential Banach Manifold Induced by q-Divergences
dc.contributor.author: Loaiza Ossa, Gabriel Ignacio; Quiceno Echavarría, Héctor Román
dc.description.abstract: For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built. In this paper, the geometry induced on this manifold is characterized by q-divergence functionals. This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Moreover, the classical Amari’s α-connections appears as special case of the q −connections ∇ (q). The main result is the expected one, namely the zero curvature of the manifold. Loading... Geometric Science of InformationGeometric Science of Information Look Inside Chapter Metrics Downloads1K Provided by Bookmetrix Other actions Export citation About this Book Reprints and Permissions Add to Papers Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn
2013-01-01T00:00:00ZWavelet-Petrov-Galerkin Method for the Numerical Solution of the KdV EquationVillegas Gutiérrez, Jairo AlbertoCastaño B., JorgeDuarte V., JulioFierro Y., Esperhttps://hdl.handle.net/10784/74102021-09-24T21:44:19Z2012-01-01T00:00:00Zdc.title: Wavelet-Petrov-Galerkin Method for the Numerical Solution of the KdV Equation
dc.contributor.author: Villegas Gutiérrez, Jairo Alberto; Castaño B., Jorge; Duarte V., Julio; Fierro Y., Esper
dc.description.abstract: The development of numerical techniques for obtaining approximate solutions of partial differential equations has very much increased in the last decades. Among these techniques are the finite element methods and finite difference. Recently, wavelet methods are applied to the numerical solution of partial differential equations, pioneer works in this direction are those of Beylkin, Dahmen, Jaffard and Glowinski, among others. In this paper, we employ the Wavelet-Petrov-Galerkin method to obtain the numerical solution of the equation Korterweg-de Vries (KdV).; The development of numerical techniques for obtaining approximate solutions of partial differential equations has very much increased in the last decades. Among these techniques are the finite element methods and finite difference. Recently, wavelet methods are applied to the numerical solution of partial differential equations, pioneer works in this direction are those of Beylkin, Dahmen, Jaffard and Glowinski, among others. In this paper, we employ the Wavelet-Petrov-Galerkin method to obtain the numerical solution of the equation Korterweg-de Vries (KdV).
2012-01-01T00:00:00ZA q-exponential statistical Banach manifoldQuiceno Echavarría, Héctor RománLoaiza Ossa, Gabriel Ignaciohttps://hdl.handle.net/10784/52452021-09-24T21:44:19Z2013-02-01T00:00:00Zdc.title: A q-exponential statistical Banach manifold
dc.contributor.author: Quiceno Echavarría, Héctor Román; Loaiza Ossa, Gabriel Ignacio
dc.description.abstract: Letµbe a given probability measure andMµ the set ofµ-equivalent strictly positive probability densities -- In this paper we construct a Banach manifold on Mµ, modeled on the space L∞(p · µ) where p is a reference density, for the non-parametric q-exponential statistical models (Tsallis’s deformed exponential), where 0 < q < 1 is any real number -- This family is characterized by the fact that when q → 1, then the non-parametric exponential models are obtained and the manifold constructed by Pistone and Sempi is recovered, up to continuous embeddings on the modeling space -- The coordinate mappings of the manifold are given in terms of Csiszár’s Φ-divergences; the tangent vectors are identified with the one-dimensional q-exponential models and q-deformations of the score function; Letµbe a given probability measure andMµ the set ofµ-equivalent strictly positive probability densities -- In this paper we construct a Banach manifold on Mµ, modeled on the space L∞(p · µ) where p is a reference density, for the non-parametric q-exponential statistical models (Tsallis’s deformed exponential), where 0 < q < 1 is any real number -- This family is characterized by the fact that when q → 1, then the non-parametric exponential models are obtained and the manifold constructed by Pistone and Sempi is recovered, up to continuous embeddings on the modeling space -- The coordinate mappings of the manifold are given in terms of Csiszár’s Φ-divergences; the tangent vectors are identified with the one-dimensional q-exponential models and q-deformations of the score function
2013-02-01T00:00:00ZA Riemannian geometry in the q-Exponential Banach manifold induced by q-DivergencesLoaiza Ossa, Gabriel IgnacioQuiceno Echavarría, Héctor Románhttps://hdl.handle.net/10784/52442021-09-24T21:44:19Z2013-01-01T00:00:00Zdc.title: A Riemannian geometry in the q-Exponential Banach manifold induced by q-Divergences
dc.contributor.author: Loaiza Ossa, Gabriel Ignacio; Quiceno Echavarría, Héctor Román
dc.description.abstract: For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built -- In this paper, the geometry induced on this manifold is characterized by q-divergence functionals -- This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections -- Moreover, the classical Amari´s α-connections appears as special case of the q−connections (q) -- The main result is the expected one, namely the zero curvature of the manifold; For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built -- In this paper, the geometry induced on this manifold is characterized by q-divergence functionals -- This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections -- Moreover, the classical Amari´s α-connections appears as special case of the q−connections (q) -- The main result is the expected one, namely the zero curvature of the manifold
2013-01-01T00:00:00ZLimits of quotients of bivariate real analytic functionsMolina, SergioCadavid Moreno, Carlos AlbertoVélez Caicedo, Juan Diegohttps://hdl.handle.net/10784/50662021-09-24T21:44:18Z2013-03-01T00:00:00Zdc.title: Limits of quotients of bivariate real analytic functions
dc.contributor.author: Molina, Sergio; Cadavid Moreno, Carlos Alberto; Vélez Caicedo, Juan Diego
dc.description.abstract: Necessary and sufficient conditions for the existence of limits of the form lim(x,y)→(a,b) f (x, y)/g(x, y) are given, under the hypothesis that f and g are real analytic functions near the point (a, b), and g has an isolated zero at (a, b) -- The given criterion uses a constructive version of Hensel’s Lemma which could be implemented in a computer algebra system in the case where f and g are polynomials with rational coefficients, or more generally, with coefficients in a real finite extension of the rationals -- A high level
description of an algorithm for determining the existence of the limit as well as its computation is provided; Necessary and sufficient conditions for the existence of limits of the form lim(x,y)→(a,b) f (x, y)/g(x, y) are given, under the hypothesis that f and g are real analytic functions near the point (a, b), and g has an isolated zero at (a, b) -- The given criterion uses a constructive version of Hensel’s Lemma which could be implemented in a computer algebra system in the case where f and g are polynomials with rational coefficients, or more generally, with coefficients in a real finite extension of the rationals -- A high level description of an algorithm for determining the existence of the limit as well as its computation is provided
2013-03-01T00:00:00ZOptimization problem with interval-valued random functionsPuerta Yepes, María Eugeniahttps://hdl.handle.net/10784/46182020-08-15T14:37:31Z2012-12-10T00:00:00Zdc.title: Optimization problem with interval-valued random functions
dc.contributor.author: Puerta Yepes, María Eugenia
dc.description: This paper proposes optimization with interval-valued random functions as a very natural theoretical tool for posing and solving optimization problems arising in various fields such as oil reservoir exploration and structural design, since the input data for these problems consist of large sets of inequalities, obtained by repeating a particular measurement many times. The technique of weighted sums, the Aumann’s integral and properties of interval arithmetic are used to establish necessary and sufficient conditions for the solution of these optimization problems. Moreover, we compare the numerical results obtained by applying the stochastic approach and the interval approach.
2012-12-10T00:00:00ZComputational methods for solving multi-objective uncertain optimization problemsPuerta Yepes, María EugeniaCano Cadavid, Andrés Felipehttps://hdl.handle.net/10784/45572020-02-14T21:15:01Z2011-05-12T00:00:00Zdc.title: Computational methods for solving multi-objective uncertain optimization problems
dc.contributor.author: Puerta Yepes, María Eugenia; Cano Cadavid, Andrés Felipe
dc.description.abstract: In recent years, there has been an increasing interest in the multi-objective uncertain optimization, discussed
in the framework of the interval-valued optimization, as a consequence theoretical developments have
achieved significant results as theorems analogous to the conditions of Karush Kunt Tucker, but computational
developments are still incipient. This paper makes an extension of Strength Pareto Evolutionary Algorithm 2
- SPEA2 - and Multi-objective Particle Swarm Optimization - MOPSO -, which ones are traditionally used in
multi-objective optimization, these are modified to the case of multi-objective uncertain optimization, where
the model uses the interval-valued optimization as shown by Wu [?, ?, ?], these new algorithms have arithmetic advantage in the image set of the objective function. At the end, numerical examples are shown where they applied the algorithms implemented.
2011-05-12T00:00:00ZDuality in Multi-Objective Optimization Under UncertaintyPuerta Yepes, María EugeniaGaviria, C.Fernández Gutiérrez, Juan Pablohttps://hdl.handle.net/10784/45562020-03-12T17:25:36Z2014-02-20T00:00:00Zdc.title: Duality in Multi-Objective Optimization Under Uncertainty
dc.contributor.author: Puerta Yepes, María Eugenia; Gaviria, C.; Fernández Gutiérrez, Juan Pablo
dc.description: In this paper we extend to a multi-objective optimization with a interval-valued function and real valued constraints, the concepts of Wolfes duality elaborated on [1] and [2], for the interval-valued mono-objective case with real constraints or intervals-valued. First of all, being supported on the Wolfes duality theory valued set [3], we perform an extension to optimize a deterministic multi-objective function, and with real valued constraints of a proposal made by Wolfe [4] for a duality in a optimization with objective function and real valued constraints. The theorems 3.1 and 3.3 are theorems of duality in a weak and strong sense, respectively; i.e, they guarantee that all objective value of a dual problem are lesser than all the objective value of the primal problem, and, under some conditions, the furthest values of the primal and dual problem are the same. Secondly, being supported on [3] and, in the Wolfes duality for a deterministic multi-objective case, we developed Wolfes duality concepts for an optimization under uncertainty with a interval-valued function criteria with real valued constraints. Lem-mas 4.3, 4.4 and proposition 4.5 constitute a duality in a weak sense, and theorems 4.8 and 4.11 constitute a duality in a strong sense.
2014-02-20T00:00:00ZA Riemannian Geometry in the q-Exponential Banach Manifold induced by q-Divergences.Quiceno, H. R.Loaiza, Gabrielhttps://hdl.handle.net/10784/44012014-11-07T20:32:58Z2013-01-01T00:00:00Zdc.title: A Riemannian Geometry in the q-Exponential Banach Manifold induced by q-Divergences.
dc.contributor.author: Quiceno, H. R.; Loaiza, Gabriel
dc.description.abstract: For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built. In this paper, the geometry induced on this manifold is characterized by q-divergence functionals. This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections. Moreover, the classical Amari’s α-connections appears as special case of the q −connections ∇ (q). The main result is the expected one, namely the zero curvature of the manifold.
2013-01-01T00:00:00Z