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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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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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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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Semigroup-valued metric spaces
Konečný, Matěj ; Hubička, Jan (advisor)
The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework - semigroup-valued metric spaces - for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for Λ-ultrametric...
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Semigroup-valued metric spaces
Konečný, Matěj ; Hubička, Jan (advisor)
The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework - semigroup-valued metric spaces - for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for Λ-ultrametric...
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Semigroup-valued metric spaces
Konečný, Matěj ; Hubička, Jan (advisor) ; Pultr, Aleš (referee)
The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework - semigroup-valued metric spaces - for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for Λ-ultrametric...
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Combinatorial Properties of Metrically Homogeneous Graphs
Konečný, Matěj ; Hubička, Jan (advisor) ; Nešetřil, Jaroslav (referee)
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory com- bines these two fields and is concerned with Ramsey-type questions about certain model-theoretic structures. In 2005, Nešetřil initiated a systematic study of the so-called Ramsey classes of finite structures. This thesis is a contribution to the programme; we find Ramsey expansions of the primitive 3-constrained classes from Cherlin's catalogue of metrically homogeneous graphs. A key ingredient is an explicit combinatorial algorithm to fill-in the missing distances in edge-labelled graphs to obtain structures from Cherlin's classes. This algorithm also implies the extension property for partial automorphisms (EPPA), another combinatorial property of classes of finite structures. 1
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Topological entropy
Češík, Antonín ; Vejnar, Benjamin (advisor) ; Pražák, Dalibor (referee)
In this thesis we study topological entropy as an invariant of topological dynamical systems. The first chapter contains basic definitions and examples of topological dynamical systems. In the second chapter we introduce the definition of topological entropy on a compact metric space. We study its properties, in particular the fact that it is invariant under conjugacy. The chapter concludes with calculation of topological entropy for the examples introduced in the first chapter. The last chapter deals with generalizing the notion of topological entropy to noncompact metric spaces. The case of piecewise affine maps on the real line is studied in more detail.
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