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Universal metric spaces
Raška, Martin ; Hušek, Miroslav (advisor) ; Vejnar, Benjamin (referee)
The thesis covers the properties of isometric embeddings of metric spaces into the Urysohn universal space U (P.S. Urysohn, 1927) and its generalizations (M. Katětov, 1988). The examination of various metric properties of the space U leads to the question of extendability of the embedding ϕ: M → U from a subspace M of a space P onto an embedding Φ: P → U. We approach to this question in situation P = M ∪ {p} in finer form. If ϕ denotes an embedding M → U, let Rϕ denotes the set of images of the point p in U under all possible isometric extensions of the embedding ϕ (we call Rϕ the space of realizations). The main objective of this thesis is answering the following question: Which forms do the spaces Rϕ assume, if ϕ passes all embeddings of the space M into the space U? Corollary 1 and theorem 3 in the II. part of the thesis metrically characterize the family {Rϕ|ϕ: M → U}. We use previous results in part III in order to determine the number of classes of metrically equivalent embeddings of the space M into the space U. As a consequence, we obtain the result of J. Melleray about the homogeneity of the space U.
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Complexity of sets
Hartman, Juraj ; Zelený, Miroslav (advisor) ; Spurný, Jiří (referee)
In this thesis we first introduce Borel hierarchy of sets in metric spaces and prove some of its properties. Then for special Borel subsets of special metric spaces (Euclidean space of real numbers and the hyperspace of compact subsets of a Polish space with Vietoris topology) we find out where they are in Borel hierarchy, i. e. we find out the class of Borel hierarchy, in which they are, and such that they are in no smaller class with respect to inclusion, which can be understood as an expression of its complexity. Finally we give an example of a coanalytic subset of the hyperspace of compact subsets of a Polish space, which is not Borel, with the proof of its coanalyticity. Powered by TCPDF (www.tcpdf.org)
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Index Suitable for Similar Search in High-dimensional Spaces
Krejčová, Martina ; Kopecký, Michal (advisor) ; Skopal, Tomáš (referee)
In this paper, we focus on indexing and searching in high-dimensional data. To achieve the target we implemented the Metric Index, a model of the similarity search based on the metric spaces, that employs many of known principles of partitioning and filtering. The metric space is a general model of similarity, which enables the usage of implemented index for various data. With this index, stored data could be searched effectively. The internal structure of data is hidden, we just require an implementation of the function for feature extraction, which produces a vector representing data, and the metric function applicable to the given data. The Metric Index was implemented as a data cartridge, the mechanism for extending the capabilities of the Oracle server. This data cartridge enables indexing of large unstructured data in the Oracle server known as LOBs.
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Vlastnosti metrických prostorů pomocí konvergence
Pokorný, Robin ; Hušek, Miroslav (advisor) ; Simon, Petr (referee)
In this thesis we deal with generalisation of the structure of convergence in metric spaces and characterisation of some properties using sequences. On basic of behaviour of convergent sequences in metric spaces which we alongside with selected properties in metric spaces remind we introduce the general structures. The first of them - sequential spaces - includes the information about the limit of a sequence which we consider to be unique. The second - uniformly sequential spaces - generalises the relation of adjacency of two sequences. We show that continuity of a mapping, topology, compactness, connectedness and separability can be induced from a sequential structure. In addition, we show total boundedness and completeness can be characterised using the term of a Cauchy sequence which we can define in a uniformly sequential structure. Boundedness is shown to be independent on both of those structures.
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Multimodal Database Search
Krejčíř, Tomáš ; Stryka, Lukáš (referee) ; Chmelař, Petr (advisor)
The field that deals with storing and effective searching of multimedia documents is called Information retrieval. This paper describes solution of effective searching in collections of shots. Multimedia documents are presented as vectors in high-dimensional space, because in such collection of documents it is easier to define semantics as well as the mechanisms of searching. The work aims at problems of similarity searching based on metric space, which uses distance functions, such as Euclidean, Chebyshev or Mahalanobis, for comparing global features and cosine or binary rating for comparing local features. Experiments on the TRECVid dataset compare implemented distance functions. Best distance function for global features appears to be Mahalanobis and for local features cosine rating.
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