Papers by Juan-pablo Ortega
Lecture Notes in Mathematics, 2007
Lectures on Mechanics, 1992
This encyclopedia article briefly reviews without proofs some of the main results in cotangent bu... more This encyclopedia article briefly reviews without proofs some of the main results in cotangent bundle reduction. The article recalls most the necessary prerequisites to understand the main results.
Reservoir computing is a recently introduced machine learning paradigm that has already shown exc... more Reservoir computing is a recently introduced machine learning paradigm that has already shown excellent performances in the processing of empirical data. We study a particular kind of reservoir computers called time-delay reservoirs that are constructed out of the sampling of the solution of a time-delay differential equation and show their good performance in the forecasting of the conditional covariances associated to multivariate discrete-time nonlinear stochastic processes of VEC-GARCH type as well as in the prediction of factual daily market realized volatilities computed with intraday quotes, using as training input daily log-return series of moderate size. We tackle some problems associated to the lack of task-universality for individually operating reservoirs and propose a solution based on the use of parallel arrays of time-delay reservoirs.
For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surroundin... more For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given.
1 Overview of the Field Geometry, as learned by students of grade school, is the business of line... more 1 Overview of the Field Geometry, as learned by students of grade school, is the business of lines, angles, circles and triangles. It's useful because, locally, the concepts of geometry are similar to objects of our world. Using geometry, we can compute areas, heights, and angles. Almost all of us know some geometry. Many of us need it, from time to time.
Stefan Berceanu A Holomorphic Representation of the Semidirect Sum of Symplectic and Heisenberg L... more Stefan Berceanu A Holomorphic Representation of the Semidirect Sum of Symplectic and Heisenberg Lie Algebras............................................................... 5-13 ... Alberto Enciso and Daniel Peralta-Salas Electric Fields Created by Point Charges: Some Geometrical and Topological Results........ ...
We give a few examples of how, using the knowledge available on the geometry of Hamiltonian dynam... more We give a few examples of how, using the knowledge available on the geometry of Hamiltonian dynamics with symmetry, standard critical point theory can be adapted to this setup in order to obtain predictions on the existence of various dynamical elements and, moreover, it can be used to provide estimates on the number of these solutions. The proofs of these results, as well as additional information, can be found in
ABSTRACT In this chapter we present the most elementary version of symplectic reduction using sta... more ABSTRACT In this chapter we present the most elementary version of symplectic reduction using standard momentum maps. The symplectic reduction method represents a generalization and synthesis of various techniques of elimination of variables from classical mechanics that are based on the existence of conserved quantities. Early specific examples are the reduction to the center of mass frame in the n-body problem using translational invariance and Jacobi’s elimination of the node that allows the elimination of four variables using rotational invariance.
ABSTRACT One of the main themes of this book is the notion of symmetry. The main goal in the chap... more ABSTRACT One of the main themes of this book is the notion of symmetry. The main goal in the chapters at the core of this book is explaining to the reader how the symmetries of a Hamiltonian dynamical system can be used to simplify or reduce the study of that system. From the mathematical point of view the description of symmetries is implemented via the use of Lie group actions and, more generally, pseudogroups and groupoids. In the following two chapters we review all the material concerning these topics that will be needed in the rest of the book.
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Papers by Juan-pablo Ortega