Óscar Valero | Universitat de les Illes Balears (original) (raw)

Papers by Óscar Valero

Research paper thumbnail of On aggregation of normed structures

Mathematical and Computer Modelling, 2011

In 1981, Borsik and Doboš studied in depth the problem of how to merge, by means of a function, a... more In 1981, Borsik and Doboš studied in depth the problem of how to merge, by means of a function, a collection (not necessarily finite) of distance spaces in order to obtain a single one as a result [J. Borsik, J. Doboš, On a product of metric spaces, Math. Slovaca 31 (1981) 193-205]. Later on, Herburt and Moszyńska studied the same problem for the case of normed linear spaces, inspired by the fact that every norm induces in a natural way a distance on a linear space, and analyzed the relationship between the both aforenamed problems [I. Herburt, M. Moszyńska, On metric products, Colloq. Math. 62 (1991) 121-133]. More recently, Romaguera and Schellekens introduced a mathematical approach, based on the notions of asymmetric distance and asymmetric normed linear space, which is suitable for the complexity analysis of programs and algorithms in Computer Science [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311-322]. In this paper, motivated by the importance of the information fusion techniques in Artificial Intelligence and by the utility of asymmetric distances and asymmetric norms in Computer Science, we study the Herburt and Moszyńska problem for asymmetric normed linear spaces. In particular we give a general description of how to combine a collection (not necessarily finite) of asymmetric normed linear spaces in order to obtain a single one as output and, in addition, we clear up the relationship between this problem and its analogous of combining asymmetric distance spaces which has been already explored by Mayor and Valero [G. Mayor, O. Valero, Aggregation of asymmetric distances in computer science, Inform. Sci. 180 (2010) 803-812]. Furthermore, it is shown that the asymmetric norms employed, in the spirit of Romaguera and Schellekens, in complexity analysis can be retrieved as a particular case of the developed theory. The last fact opens the possibility of applying a wide range of properties from the general aggregation theory in Artificial Intelligence to the complexity analysis of programs and algorithms in Computer Science.

Research paper thumbnail of On aggregation of metric structures: the extended quasi-metric case

International Journal of Computational Intelligence Systems, 2013

Research paper thumbnail of Aggregation of asymmetric distances in Computer Science

Information Sciences, 2010

In this paper we provide a general description of how to combine a collection (not necessarily fi... more In this paper we provide a general description of how to combine a collection (not necessarily finite) of asymmetric distances in order to obtain a single one as output. To this end we introduce the notion of asymmetric distance aggregation function that generalizes the well-known one for distance spaces given by Borsik and Doboš [J. Borsik, J. Doboš, On a product of metricspaces, Math. Slovaca 31 (1981) 193-205]. Among other results, a characterization of such functions is obtained in terms of monotony and subadditivity. Finally, we relate our results to Computer Science. In particular we show that the mathematical formalism based on complexity distances, which has been introduced by Romaguera and Schellekens [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topol. Appl. 98 (1999) 311-322] for the complexity analysis of programs and algorithms, can be obtained as a particular case of our new framework using appropriate asymmetric aggregation distance functions.

Research paper thumbnail of A General Approach to Midpoint Theory and Aggregation of Quasimetrics

International Journal of Intelligent Systems, 2013

ABSTRACT Many fields in applied sciences, like Artificial Intelligence and Computer Science, use ... more ABSTRACT Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the mipoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of “compromise” or “middle way” between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved.

Research paper thumbnail of On Banach fixed points theorems for partial metric spaces

Applied General Topology, 2005

In this paper we prove several generalizations of the Banach fixed point theorem for partial metr... more In this paper we prove several generalizations of the Banach fixed point theorem for partial metric spaces (in the sense of O'Neill) given in [14], obtaining as a particular case of our results the Banach fixed point theorem of Matthews ([12]), and some well-known classical fixed point theorems when the partial metric is, in fact, a metric. 2000 AMS Classification: 54H25, 54E50, 54E99, 68Q55.

Research paper thumbnail of On aggregation of normed structures

Mathematical and Computer Modelling, 2011

In 1981, Borsik and Doboš studied in depth the problem of how to merge, by means of a function, a... more In 1981, Borsik and Doboš studied in depth the problem of how to merge, by means of a function, a collection (not necessarily finite) of distance spaces in order to obtain a single one as a result [J. Borsik, J. Doboš, On a product of metric spaces, Math. Slovaca 31 (1981) 193-205]. Later on, Herburt and Moszyńska studied the same problem for the case of normed linear spaces, inspired by the fact that every norm induces in a natural way a distance on a linear space, and analyzed the relationship between the both aforenamed problems [I. Herburt, M. Moszyńska, On metric products, Colloq. Math. 62 (1991) 121-133]. More recently, Romaguera and Schellekens introduced a mathematical approach, based on the notions of asymmetric distance and asymmetric normed linear space, which is suitable for the complexity analysis of programs and algorithms in Computer Science [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311-322]. In this paper, motivated by the importance of the information fusion techniques in Artificial Intelligence and by the utility of asymmetric distances and asymmetric norms in Computer Science, we study the Herburt and Moszyńska problem for asymmetric normed linear spaces. In particular we give a general description of how to combine a collection (not necessarily finite) of asymmetric normed linear spaces in order to obtain a single one as output and, in addition, we clear up the relationship between this problem and its analogous of combining asymmetric distance spaces which has been already explored by Mayor and Valero [G. Mayor, O. Valero, Aggregation of asymmetric distances in computer science, Inform. Sci. 180 (2010) 803-812]. Furthermore, it is shown that the asymmetric norms employed, in the spirit of Romaguera and Schellekens, in complexity analysis can be retrieved as a particular case of the developed theory. The last fact opens the possibility of applying a wide range of properties from the general aggregation theory in Artificial Intelligence to the complexity analysis of programs and algorithms in Computer Science.

Research paper thumbnail of On aggregation of metric structures: the extended quasi-metric case

International Journal of Computational Intelligence Systems, 2013

Research paper thumbnail of Aggregation of asymmetric distances in Computer Science

Information Sciences, 2010

In this paper we provide a general description of how to combine a collection (not necessarily fi... more In this paper we provide a general description of how to combine a collection (not necessarily finite) of asymmetric distances in order to obtain a single one as output. To this end we introduce the notion of asymmetric distance aggregation function that generalizes the well-known one for distance spaces given by Borsik and Doboš [J. Borsik, J. Doboš, On a product of metricspaces, Math. Slovaca 31 (1981) 193-205]. Among other results, a characterization of such functions is obtained in terms of monotony and subadditivity. Finally, we relate our results to Computer Science. In particular we show that the mathematical formalism based on complexity distances, which has been introduced by Romaguera and Schellekens [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topol. Appl. 98 (1999) 311-322] for the complexity analysis of programs and algorithms, can be obtained as a particular case of our new framework using appropriate asymmetric aggregation distance functions.

Research paper thumbnail of A General Approach to Midpoint Theory and Aggregation of Quasimetrics

International Journal of Intelligent Systems, 2013

ABSTRACT Many fields in applied sciences, like Artificial Intelligence and Computer Science, use ... more ABSTRACT Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the mipoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of “compromise” or “middle way” between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved.

Research paper thumbnail of On Banach fixed points theorems for partial metric spaces

Applied General Topology, 2005

In this paper we prove several generalizations of the Banach fixed point theorem for partial metr... more In this paper we prove several generalizations of the Banach fixed point theorem for partial metric spaces (in the sense of O'Neill) given in [14], obtaining as a particular case of our results the Banach fixed point theorem of Matthews ([12]), and some well-known classical fixed point theorems when the partial metric is, in fact, a metric. 2000 AMS Classification: 54H25, 54E50, 54E99, 68Q55.