ALGEBRA OPERATORS

Author: TOMEO PERUCHA VENANCIO.
Year: 2003.
University: POLITÉCNICA DE MADRID [www.upm.es].
Place of defense: FACULTAD DE INFORMÁTICA.
Place of preparation: FACULTAD DE INFORMATICA.

Summary: The map auto-organizativo (MAO) is a type of artificial neural network and competitive no-supervisada. It has been used traditionally in engineering tasks as a tool for automatic classification (clustering) and especially in tasks related to the exploratory data analysis and data mining, as its primary purpose is visualization of multidimensional data nonlinear relationships. However, despite the importance of the task of visualization, graphics techniques to analyze MAO is not abundant in the literature. This thesis presents several new techniques that complement, enhance and facilitate the visual analysis of MAO Kohonen, both from the standpoint of exploratory data analysis, and from the point of view of understanding the process of adaptation to a distribution of MAO data. The motivation to develop new visualization techniques arises for the following reasons: the relative lack of methods to the important task of viewing, the need to analyze MAO with different methods, the need to improve several methods described in the literature and the possibility of innovate developing new strategies visualization. This menera emphasis has been placed on developing techniques not generally used in the past in an attempt to overcome limitations of various methods described in the literature. The first nuveo method called "method like triangles" is a strategy where interpolation geometric patterns of distribution of entry are projected to an area of continuous observation. This based on the preservation of geometric similarity between several triangles formed by an employer and two vectors reference MAO in the space of the data, and a candidate and the two corresponding neurons in space observation. The method is the projection minimizing a function of cost measuring distances between several errors or triangles. The method overcomes significantly to other strategies interpolation described in the literature. You can plan all data on non-linear way, it is appropriate when the size of MAO is small, it is robust and can adequately describe certain types of distributions are difficult to visualize with most methods of visualization. Several visualization methods MAO generate monochrome images which are analyzed individually and provide specific information on the data. It proposes a strategy to facilitate the work of the analyst in terms of combining information from several methods by simple overlapping images based on an additive color model. Images are defined with different colors and combined by a simple sum of its color components. The resulting images are more comprehensive and robust, especially when the images combine to provide the same kind of information. The study conducted focuses mainly on the combination of distance matrices with histograms of data. An alternative to the matrix of distances, which generate images are monochrome and the most popular methods to visualize the structure of clusters of data, is to employ strategies that illustrate the different clusters with different colors. One of these strategies is to use models shrinkage of neurons. It presents an efficient method contraccióhn, "algorithm grouping of neurons", the structure and philosophy is similar to that hdel algorithm training of MAO, where the concepts have been spent to update the positions of neurons in a continuous mapping instead of vectors own reference MAO. This menera, neurons are drawn on a map depending on the distance between their vectors in the space of reference data. Its main advantage is its low cost computational which enables to analyze MAO-sized high. Finally, the paper proposes a techniques 8 ca alter 465 native based on explicit display on a map or graph space observation that link neurons whose vectors of reference are coming in the space of data, such as minimum or tree generator "graph Hebbiano" created with the principle of learning Hebbiano competitive. The resulting images help to analyze the intrinsic dimension of the data in each area of the map and provide a visual and intuitive measure of preserving the topology of MAO.

Author: FERNÁNDEZ POLO FRANCISCO JOSÉ.
Year: 2003.
University: GRANADA [www.ugr.es].
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: The purpose of this report is to contribute to development and knowledge of a particular class of Banach spaces that are called JB * -triples complex or real. The unifying theme is the study of geometric properties of these spaces with the aim of showing the relationship between these properties and algebraic properties inherent in the aforementioned areas. The JB * -triples complex Banach spaces are complex product with a triple that meet certain conditions on the rule. These structures introduced by Kaup in 1983, mainstream, among others, the classic C * -álgebras and JB * -álgebras. The JB * -triples real were introduced by Isidro, Kaup and Rodriguez in 1995 and are merely subriples real closed JB * -triples complex. The work carried out in this report is structured in four chapters. The first is a required introductory chapter where introduces the basic facts of the theory of genetic JB * -triples complex and real, also showing a large number of examples. Among these examples, paying particular attention to factors Cartan known as complex and real. The second chapter is dedicated to thorough knowledge of the elements tripotentes (elements that match your product riple) factors Cartan complex and real. This study allows us to rediscover that Cartan complex factors are generated by grills, families tripotentes that meet certain algebraic properties. Moreover tested that the vast majority of factors are generated by real Cartan grills whose elements satisfy some suitable properties algebracias. The third chapter is divided into two sections. In the first section we get a geometric characterization of tripotentes a JB * -triple real or complex solely in terms of the standard underlying Banach space. The second section is devoted entirely to the theorem Banach-Stone for JB * -triples. First, we use the characterization cited above to get an easy demonstration alternative theorem Banach-Stone for JB * -triples complex, which states that isometrías linear sobreyectivas between JB * -triples complexes are isomorfismos. We also get, among others, a theorem Type Banachs-Stone for Cartan real factors, a result which responds to a problem opened established in 1997 by Professor Kaup, and a theorem Banach-Stone for JB * -triples real. Score stating that the isometrías linear sobreyectivas between two JB * -triples real such that the bidual the first one does not contain factors Cartan real or complex rank one are isomorfismos of tirples. In the fourth Chapter obtain an extension of the classic theorem Lusin Theory of the measure to the most general atmosphere of JB * -triples real and complex. To obtain this result we need to conduct a thorough study of the geometry of the subspace Peirce associated with a tirpotente which has enabled us to obtain a new geometric inequality even in the atmosphere of the JB * -álgebras. We had been necessary to obtain a theorem Egoroff for JB * -triples complex and real.