Arnau Notari, Andrés Roger
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- PublicationApproximation of Almost Diagonal Non-linear Maps by Lattice Lipschitz Operators(Springer-Verlag, 2024-03) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; Erdogan, Ezgi; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; Agencia Estatal de Investigación; Universitat Politècnica de València[EN] Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction.
- PublicationRepresentation of Lipschitz Maps and Metric Coordinate Systems(MDPI AG, 2022-10) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; AGENCIA ESTATAL DE INVESTIGACION; UNIVERSIDAD POLITECNICA DE VALENCIA[EN] Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze ¿Pietsch Theorem inspired factorizations" through subspaces of `¿ and L1, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane¿Whitney formulas, are also given.
- PublicationEccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs(MDPI AG, 2024-11) Arnau Notari, Andrés Roger; Sánchez Pérez, Enrique Alfonso; Sanjuan Silvestre, Sergi; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; European Commission; Agencia Estatal de Investigación; Universitat Politècnica de València[EN] The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens-Eells space, along with the duality between M and the metric dual space M# defined by the real-valued Lipschitz functions on M. However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of M# for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.
- PublicationA Bellman-Ford Algorithm for the Path-Length-Weighted Distance in Graphs(MDPI AG, 2024-08) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; García Raffi, Luis Miguel; Sánchez Pérez, Enrique Alfonso; Sanjuan Silvestre, Sergi; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; European Commission; Agencia Estatal de Investigación; Universitat Politècnica de València[EN] Consider a finite directed graph without cycles in which the arrows are weighted by positive weights. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. It has the distinct characteristic that, when calculated along the same path, it may result in a shorter distance between far-apart vertices than between adjacent ones. This property can be particularly useful for modeling scenarios where the connections between vertices are obscured by numerous intermediate vertices, such as in cases of financial fraud. For example, to hide dirty money from financial authorities, fraudsters often use multiple institutions, banks, and intermediaries between the source of the money and its final recipient. Our distance would serve to make such situations explicit. Thus, although our algorithm is based on arguments similar to those at work for the Bellman-Ford and Dijkstra methods, it is in fact essentially different, since the calculation formula contains a weight that explicitly depends on the number of intermediate vertices. This fact totally conditions the algorithm, because longer paths could provide shorter distances-contrary to the classical algorithms mentioned above. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.
- PublicationMeasure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression(MDPI AG, 2023-04-07) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; AGENCIA ESTATAL DE INVESTIGACION; UNIVERSIDAD POLITECNICA DE VALENCIA[EN] This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p-average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For p = 2, explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given.
- PublicationEnseñanza del aprendizaje por refuerzo con un sencillo ejemplo de minimización de funciones(Editorial Universitat Politècnica de València, 2023-10-06) Arnau Notari, Andrés Roger; García Raffi, Luis Miguel; Calabuig Rodriguez, Jose Manuel; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; Polish National Agency for Strategic Partnership; Universitat Politècnica de València; Agencia Estatal de Investigación[ES] En este trabajo se presenta una sesión a modo de taller orientada al estudiantado universitarios para que entiendan los fundamentos del aprendizaje por refuerzo (RL). Esta técnica de inteligencia artificial no es comúnmente estudiada por su dificultad, por ello se expone una simplificación del RL, que se aplica a la resolución de un problema de optimización. Además se analizará la manera de abordar el problema de optmización como un juego, puesto que este es una aplicación natural del RL.
- PublicationExtension procedures for lattice Lipschitz operators on Euclidean spaces(Springer-Verlag, 2023-04) Arnau Notari, Andrés Roger; Calabuig Rodriguez, Jose Manuel; Erdogan, Ezgi; Sánchez Pérez, Enrique Alfonso; Departamento de Matemática Aplicada; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería de Caminos, Canales y Puertos; Escuela Técnica Superior de Ingeniería Industrial; Universitat Politècnica de València[EN] We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions.
- PublicationUna Aplicación Práctica de Cómo Trabajar los Objetivos de Desarrollo Sostenible en una asignatura de Cálculo(Editorial Universitat Politècnica de València, 2024-12-20) Arnau Notari, Andrés Roger; Burgos Simón, Clara; Ortigosa Araque, Nuria; Departamento de Matemática Aplicada; Escuela Técnica Superior de Ingeniería Geodésica, Cartográfica y Topográfica; Instituto Universitario de Matemática Multidisciplinar; Instituto Universitario de Matemática Pura y Aplicada; Escuela Técnica Superior de Ingeniería Industrial[ES] El alumnado universitario suele preguntarse por el propósito final de lo que están aprendiendo, especialmente en asignaturas de matemáticas. El objetivo de este trabajo es presentar una aplicación del mundo real en relación con los Objetivos de Desarrollo Sostenible (ODSs) en la que los alumnos de la asignatura de Cálculo del Grado en Ingeniería en Geomática y Topografía de la Universitat Politècnica de València deben resolver un problema de optimización. Tras la realización de una encuesta, queda patente la buena aceptación que reciben este tipo de sesiones por el alumnado, resaltando la importancia de integrar los ODS en la educación universitaria.