A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.

We investigate the use of the concept of indiscernibilityand vagueness in spatial information systems. To representindiscernibility and vagueness we use rough sets, respectivelyfuzzy sets. We introduce a theoretical model to supportapproximate queries in information systems and we show howthose queries can be used to perform uncertain classi.cations.We also explore how to assess quality of uncertainclassi.cations and ways to compare those classi.cations to eachother in order to assess accuracies. We implement the querylanguage in an SQL relational language to demonstrate thefeasibility of approximate queries and we perform an experimenton real data using uncertain classi.cations.

<p>We investigate the use of the concept of indiscernibilityand vagueness in spatial information systems. To representindiscernibility and vagueness we use rough sets, respectivelyfuzzy sets. We introduce a theoretical model to supportapproximate queries in information systems and we show howthose queries can be used to perform uncertain classi.cations.We also explore how to assess quality of uncertainclassi.cations and ways to compare those classi.cations to eachother in order to assess accuracies. We implement the querylanguage in an SQL relational language to demonstrate thefeasibility of approximate queries and we perform an experimenton real data using uncertain classi.cations.</p>

This dissertation proposes a novel theoretical framework for the data partitioning problem in computer vision and machine learning. The framework is based on level set methods that are derived from variational calculus and involve a curve-based objective function which integrates both boundary and region based information in a generic form. The proposed approaches within the framework provide original solutions to two important problems in variational methods, namely parameter tuning and information fusion, collectively termed Learning Level Sets in this thesis. Moreover, a novel pattern classification algorithm, namely Level Learning Sets, is proposed to classify any general dataset, including sparse and non sparse data. It is based on the same optimisation process of the objective function directly related to the curve propagation theory used in level set theory. The proposed approach learns the knowledge required for parameter tuning and information fusion in level set methods using machine learning techniques. It uses acquired knowledge to automatically perform parameter tuning and information fusion in level set methods. In the case of pattern classification, variational methods using level set theory optimise decision boundary construction in feature space. Consequently, the optimised values of the objective level set function over the feature space represent the model for pattern classification. The proposed automatic parameter tuning and information fusion method embedded in the level set method framework has been employed to provide original solutions to image segmentation and object extraction in computer vision. On the other hand, the Level Learning Set has been extended and applied to a variety of pattern classification problems". Several experimental results for each of the above methods are provided, demonstrating the effectiveness of the proposed solutions and indicating the potential of the automatic and dynamic tuning and fusion approaches as well as the Level Learning Set model.

A function is dened to be hardly e-open provided that theinverse image of each e-dense subset of the codomain that is contained in a proper open set is e-dense in the domain. Characterizations and properties of hardly e-open functionsare presented.

In the study of sets of points certain sets are found to be negligible, especially when applied to the theory of functions. The purpose of this paper is to discuss three of these "negligible" types, namely, exhaustible sets, denumerable sets, and sets of Lebesgue measure zero. We will present a complete existential theory in q-space for the three set properties mentioned above, followed by a more restricted discussion in the linear continuum by use of interval properties.

This paper considers the concept of inner and outer content, which was introduced by Camille Jordan and Giuseppe Peano near the end of the nineteenth century.

A vertex υ in a set S is said to be cost effective if it is adjacent to at least as many vertices in V\S as it is in S and is very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is (very) cost effective if every vertex in S is (very) cost effective. The minimum cardinality of a (very) cost effective dominating set of G is the (very) cost effective domination number of G. Our main results include a quadratic upper bound on the very cost effective domination number of a graph in terms of its domination number. The proof of this result gives a linear upper bound for hereditarily sparse graphs which include trees. We show that no such linear bound exists for graphs in general, even when restricted to bipartite graphs. Further, we characterize the extremal trees attaining the bound. Noting that the very cost effective domination number is bounded below by the domination number, we show that every value of the very cost effective domination number between these lower and upper bounds for trees is realizable. Similar results are given for the cost effective domination number.

This thesis is concerned with problems of variable selection, influence of sample size and related issues in the applications of various techniques of exploratory multivariate analysis (in particular, correspondence analysis, biplots and canonical correspondence analysis) to archaeology and ecology. Data sets (both published and new) are used to illustrate these methods and to highlight the problems that arise - these practical examples are returned to throughout as the various issues are discussed. Much of the motivation for the development of the methodology has been driven by the needs of the archaeologists providing the data, who were consulted extensively during the study. The first (introductory) chapter includes a detailed description of the data sets examined and the archaeological background to their collection. Chapters Two, Three and Four explain in detail the mathematical theory behind the three techniques. Their uses are illustrated on the various examples of interest, raising data-driven questions which become the focus of the later chapters. The main objectives are to investigate the influence of various design quantities on the inferences made from such multivariate techniques. Quantities such as the sample size (e.g. number of artefacts collected), the number of categories of classification (e.g. of sites, wares, contexts) and the number of variables measured compete for fixed resources in archaeological and ecological applications. Methods of variable selection and the assessment of the stability of the results are further issues of interest and are investigated using bootstrapping and procrustes analysis. Jack-knife methods are used to detect influential sites, wares, contexts, species and artefacts. Some existing methods of investigating issues such as those raised above are applied and extended to correspondence analysis in Chapters Five and Six. Adaptions of them are proposed for biplots in Chapters Seven and Eight and for canonical correspondence analysis in Chapter Nine. Chapter Ten concludes the thesis.

It may be said of certain pairs of elements of a set that one element precedes the other. If the collection of all such pairs of elements in a given set exhibits certain properties, the set and the collection of pairs is said to constitute a partially ordered set. The purpose of this paper is to explore some of the properties of partially ordered sets.