<p>U prvoj glavi dat je prikaz osnovnih pojmova koji se koriste u tezi, sa naglaskom na osnovne pojmove teorije grafova. U drugoj glavi daju se grafovski dokazi nekih geometrijskih problema. Na primer, dajemo dokaz poznate teoreme Silvestra. iz Kombinatorne geometrije, koristeći grafove. Treće poglavlje sadrži neke geometrijske dokaze u grafovima. Na primer, poznatu Turanovu teoremu, Pikovu teoremu i drugo. U četvrtom poglavlju dajemo metodičku transformaciju prethodnih problema. Neki od problema razloženi su na delove koji se mogu prezentirati od osnovne &scaron;kole do poslediplomskih studija. Peto poglavlje sadrži komentare prethodnog, kao i preporuku UNESCO-a iz 1992. godine o obrazovanju učenika u novom milenijumu.</p> / <p>Chapter 1 contains a shoot review of the basic notions which are used in the thesis with emphasis on the basic notions of graph theory. In Chapter&nbsp; 2we present graph proofs of some geometrical problems. For example, we present some proofs of the well known theorem of Sylvester, from Combinatorial geometry, using graph methods. In Chapter 3 we present some proofs in graphs which use geometry. For example, famous Turan&rsquo;s theorem, Pick&rsquo;s theorem and some others. Chapter 4 contains methodology transformations i.e. we apply the results from the previous chapters to the situations in the clossroom. In Chapter 5 we have some comments of UNESCO, from 1992, on education of children.</p>

Sequences of event intervals occur in several application domains, while their inherent complexity hinders scalable solutions to tasks such as clustering and classification. In this thesis, we propose a novel spectral embedding representation of event interval sequences that relies on bipartite graphs. More concretely, each event interval sequence is represented by a bipartite graph by following three main steps: (1) creating a hash table that can quickly convert a collection of event interval sequences into a bipartite graph representation, (2) creating and regularizing a bi-adjacency matrix corresponding to the bipartite graph, (3) defining a spectral embedding mapping on the bi-adjacency matrix. In addition, we show that substantial improvements can be achieved with regard to classification performance through pruning parameters that capture the nature of the relations formed by the event intervals. We demonstrate through extensive experimental evaluation on five real-world datasets that our approach can obtain runtime speedups of up to two orders of magnitude compared to other state-of-the-art methods and similar or better clustering and classification performance. / Sekvenser av händelsesintervall förekommer i flera applikationsdomäner, medan deras inneboende komplexitet hindrar skalbara lösningar på uppgifter som kluster och klassificering. I den här avhandlingen föreslår vi en ny spektral inbäddningsrepresentation av händelsens intervallsekvenser som förlitar sig på bipartitgrafer. Mer konkret representeras varje händelsesintervalsekvens av en bipartitgraf genom att följa tre huvudsteg: (1) skapa en hashtabell som snabbt kan konvertera en samling händelsintervalsekvenser till en bipartig grafrepresentation, (2) skapa och reglera en bi-adjacency-matris som motsvarar bipartitgrafen, (3) definiera en spektral inbäddning på bi-adjacensmatrisen. Dessutom visar vi att väsentliga förbättringar kan uppnås med avseende på klassificeringsprestanda genom beskärningsparametrar som fångar arten av relationerna som bildas av händelsesintervallen. Vi demonstrerar genom omfattande experimentell utvärdering på fem verkliga datasätt att vår strategi kan erhålla runtime-hastigheter på upp till två storlekar jämfört med andra modernaste metoder och liknande eller bättre kluster- och klassificerings- prestanda.

O problema de reconhecimento de expressões matemáticas manuscritas envolve três subproblemas importantes: segmentação de símbolos, reconhecimento de símbolos e análise estrutural de expressões. Para avaliar métodos e técnicas de reconhecimento, eles precisam ser testados sobre conjuntos de amostras representativos do domínio de aplicação. Uma das preocupações que tem sido apontada ultimamente é a quase inexistência de base de dados pública de expressões matemáticas, o que dificulta o desenvolvimento e comparação de diferentes abordagens. Em geral, os resultados de reconhecimento apresentados na literatura restringem-se a conjuntos de dados pequenos, não disponíveis publicamente, e muitas vezes formados por dados que visam avaliar apenas alguns aspectos específicos do reconhecimento. No caso de expressões online, para treinar e testar reconhecedores de símbolos, as amostras são em geral obtidas solicitando-se que as pessoas escrevam uma série de símbolos individualmente e repetidas vezes. Tal tarefa é monótona e cansativa. Uma abordagem alternativa para obter amostras de símbolos seria solicitar aos usuários a transcrição de expressões modelo previamente definidas. Dessa forma, a escrita dos símbolos seria realizada de forma natural, menos monótona, e várias amostras de símbolos poderiam ser obtidas de uma única expressão. Para evitar o trabalho de anotar manualmente cada símbolo das expressões transcritas, este trabalho propõe um método para casamento de expressões matemáticas manuscritas, no qual símbolos de uma expressão transcrita por um usuário são associados aos correspondentes símbolos (previamente identificados) da expressão modelo. O método proposto é baseado em uma formulação que reduz o problema a um problema de associação simples, no qual os custos são definidos em termos de características dos símbolos e estrutura da expressão. Resultados experimentais utilizando o método proposto mostram taxas médias de associação correta superiores a 99%. / The problem of recognizing handwritten mathematical expressions includes three important subproblems: symbol segmentation, symbol recognition, and structural analysis of expressions. In order to evaluate recognition methods and techniques, they should be tested on representative sample sets of the application domain. One of the concerns that are being repeatedly pointed recently is the almost non-existence of public representative datasets of mathematical expressions, which makes difficult the development and comparison of distinct approaches. In general, recognition results reported in the literature are restricted to small datasets, not publicly available, and often consisting of data aiming only evaluation of some specific aspects of the recognition. In the case of online expressions, to train and test symbol recognizers, samples are in general obtained asking users to write a series of symbols individually and repeatedly. Such task is boring and tiring. An alternative approach for obtaining samples of symbols would be to ask users to transcribe previously defined model expressions. By doing so, writing would be more natural and less boring, and several symbol samples could be obtained from one transcription. To avoid the task of manually labeling the symbols of the transcribed expressions, in this work a method for handwritten expression matching, in which symbols of a transcribed expression are assigned to the corresponding ones in the model expression, is proposed. The proposed method is based on a formulation that reduces the matching problem to a linear assignment problem, where costs are defined based on symbol features and expression structure. Experimental results using the proposed method show that mean correct assignment rate superior to 99% is achieved.

給定一個圖形G，以及集合M，M為一描述圖形G中各點擁有訊息之情形的集合。圖形G相對於M的的傳遞數是指，於最短時間內，讓圖形中全部點皆獲得所有種類之訊息，並將符號記為t(G;M) 。傳遞過程中每個時間單位將受到下列限制: (1)圖形上的每個點只能與自己相鄰的點交換訊息。 (2)兩個相鄰的點在每個單位時間裡至多只能交換一個訊息。 我們希望可以找到在最短的時間裡完成傳遞的方法，也就是讓圖形G中的每一個點都獲得所有種類之訊息，我們稱此類型問題為訊息傳遞問題。 在本論文中，給定一個圖形G，且圖形G中每個點的訊息只有一個，G中任兩點的訊息都不會相同，符號t(G)代表完成傳遞所需最少的時間單位。我們給定圖形的傳遞數的上界與下界，並且定出一套公式計算樹圖、完全二部圖及雙環網路圖的傳遞數。 / Given a graph G together with a set M , the transmission number of G corresponding to M , denoted by t(G;M), is the minimum number of time needed to complete the transmission , that is, to let all the vertices in G know all the messages in M , subject to the constraints that at each time unit, each vertex can interchange messages with all its neighbors, but the number of messages that two vertices can interchange at each time unit is at most one. We want to find the minimum number of time units required to complete the transmission, that is, to let all the vertices in G know all the messages. We call such a problem the message transmission problem. Given a graph G, the transmission number of G, denoted t(G), is the minimum number of time units required to complete the transmission, under the condition that |m(v)|=1 for all v in V(G). In this thesis, we give upper and lower bounds for the transmission number of G, and give formulas to compute the transmission numbers of trees, complete bipartite graphs and double loop networks.

We give a strongly polynomial time combinatorial algorithm to minimise the largest eigenvalue of the weighted Laplacian of a bipartite graph. This is accomplished by solving the dual graph embedding problem which arises from a semidefinite programming formulation. In particular, the problem for trees can be solved in time cubic in the number of vertices.

Les travaux présentés dans ce manuscrit s’inscrivent dans le contexte de l’inspection automatisée d’assemblages mécaniques aéronautiques par vision artificielle. Il s’agit de décider si l’assemblage mécanique a été correctement réalisé (assemblage conforme). Les travaux ont été menés dans le cadre de deux projets industriels. Le projet CAAMVis d’une part, dans lequel le capteur d’inspection est constitué d’une double tête stéréoscopique portée par un robot, le projet Lynx© d’autre part, dans lequel le capteur d’inspection est une caméra Pan/Tilt/Zoom (vision monoculaire). Ces deux projets ont pour point commun la volonté d’exploiter au mieux le modèle CAO de l’assemblage (qui fournit l’état de référence souhaité) dans la tâche d’inspection qui est basée sur l’analyse de l’image ou des images 2D fournies par le capteur. La méthode développée consiste à comparer une image 2D acquise par le capteur (désignée par « image réelle ») avec une image 2D synthétique, générée à partir du modèle CAO. Les images réelles et synthétiques sont segmentées puis décomposées en un ensemble de primitives 2D. Ces primitives sont ensuite appariées, en exploitant des concepts de la théorie de graphes, notamment l’utilisation d’un graphe biparti pour s’assurer du respect de la contrainte d’unicité dans le processus d’appariement. Le résultat de l’appariement permet de statuer sur la conformité ou la non-conformité de l’assemblage. L’approche proposée a été validée à la fois sur des données de simulation et sur des données réelles acquises dans le cadre des projets sus-cités. / The work presented in this manuscript deals with automated inspection of aeronautical mechanical parts using computer vision. The goal is to decide whether a mechanical assembly has been assembled correctly i.e. if it is compliant with the specifications. This work was conducted within two industrial projects. On one hand the CAAMVis project, in which the inspection sensor consists of a dual stereoscopic head (stereovision) carried by a robot, on the other hand the Lynx© project, in which the inspection sensor is a single Pan/Tilt/Zoom camera (monocular vision). These two projects share the common objective of exploiting as much as possible the CAD model of the assembly (which provides the desired reference state) in the inspection task which is based on the analysis of the 2D images provided by the sensor. The proposed method consists in comparing a 2D image acquired by the sensor (referred to as "real image") with a synthetic 2D image generated from the CAD model. The real and synthetic images are segmented and then decomposed into a set of 2D primitives. These primitives are then matched by exploiting concepts from the graph theory, namely the use of a bipartite graph to guarantee the respect of the uniqueness constraint required in such a matching process. The matching result allows to decide whether the assembly has been assembled correctly or not. The proposed approach was validated on both simulation data and real data acquired within the above-mentioned projects.

O problema de reconhecimento de expressões matemáticas manuscritas envolve três subproblemas importantes: segmentação de símbolos, reconhecimento de símbolos e análise estrutural de expressões. Para avaliar métodos e técnicas de reconhecimento, eles precisam ser testados sobre conjuntos de amostras representativos do domínio de aplicação. Uma das preocupações que tem sido apontada ultimamente é a quase inexistência de base de dados pública de expressões matemáticas, o que dificulta o desenvolvimento e comparação de diferentes abordagens. Em geral, os resultados de reconhecimento apresentados na literatura restringem-se a conjuntos de dados pequenos, não disponíveis publicamente, e muitas vezes formados por dados que visam avaliar apenas alguns aspectos específicos do reconhecimento. No caso de expressões online, para treinar e testar reconhecedores de símbolos, as amostras são em geral obtidas solicitando-se que as pessoas escrevam uma série de símbolos individualmente e repetidas vezes. Tal tarefa é monótona e cansativa. Uma abordagem alternativa para obter amostras de símbolos seria solicitar aos usuários a transcrição de expressões modelo previamente definidas. Dessa forma, a escrita dos símbolos seria realizada de forma natural, menos monótona, e várias amostras de símbolos poderiam ser obtidas de uma única expressão. Para evitar o trabalho de anotar manualmente cada símbolo das expressões transcritas, este trabalho propõe um método para casamento de expressões matemáticas manuscritas, no qual símbolos de uma expressão transcrita por um usuário são associados aos correspondentes símbolos (previamente identificados) da expressão modelo. O método proposto é baseado em uma formulação que reduz o problema a um problema de associação simples, no qual os custos são definidos em termos de características dos símbolos e estrutura da expressão. Resultados experimentais utilizando o método proposto mostram taxas médias de associação correta superiores a 99%. / The problem of recognizing handwritten mathematical expressions includes three important subproblems: symbol segmentation, symbol recognition, and structural analysis of expressions. In order to evaluate recognition methods and techniques, they should be tested on representative sample sets of the application domain. One of the concerns that are being repeatedly pointed recently is the almost non-existence of public representative datasets of mathematical expressions, which makes difficult the development and comparison of distinct approaches. In general, recognition results reported in the literature are restricted to small datasets, not publicly available, and often consisting of data aiming only evaluation of some specific aspects of the recognition. In the case of online expressions, to train and test symbol recognizers, samples are in general obtained asking users to write a series of symbols individually and repeatedly. Such task is boring and tiring. An alternative approach for obtaining samples of symbols would be to ask users to transcribe previously defined model expressions. By doing so, writing would be more natural and less boring, and several symbol samples could be obtained from one transcription. To avoid the task of manually labeling the symbols of the transcribed expressions, in this work a method for handwritten expression matching, in which symbols of a transcribed expression are assigned to the corresponding ones in the model expression, is proposed. The proposed method is based on a formulation that reduces the matching problem to a linear assignment problem, where costs are defined based on symbol features and expression structure. Experimental results using the proposed method show that mean correct assignment rate superior to 99% is achieved.

Moderní výzkum v environmentální mikrobiologii využívá k popisu mikrobiálních komunit genomická data, především sekvenaci DNA. Oblast, která zkoumá veškerý genetický materiál přítomný v environmentálním vzorku, se nazývá metagenomika. Tato doktorská práce se zabývá metagenomikou z pohledu bioinformatiky, která je nenahraditelná při výpočetním zpracování dat. V teoretické části práce jsou popsány dva základní přístupy metagenomiky, včetně jejich základních principů a slabin. První přístup, založený na cíleném sekvenování, je dobře rozpracovanou oblastí s velkou řadou bioinformatických technik. Přesto mohou být metody pro porovnávání vzorků z několika prostředí podstatně vylepšeny. Přístup představený v této práci používá unikátní transformaci dat do podoby bipartitního grafu, kde je jedna partita tvořena taxony a druhá vzorky, případně různými prostředími. Takový graf plně reflektuje kvalitativní i kvantitativní složení analyzované mikrobiální sítě. Umožňuje masivní redukci dat pro jednoduché vizualizace bez negativních vlivů na automatickou detekci komunit, která dokáže odhalit shluky podobných vzorků a jejich typických mikrobů. Druhý přístup využívá sekvenace celého metagenomu. Tato strategie je novější a příslušející bioinformatické nástroje jsou méně propracované. Hlavní výzvou přitom zůstává rychlá klasifikace sekvencí, v metagenomice označovaná jako „binning“. Metoda představená v této práci využívá přístupu zpracování genomických signálů. Tato unikátní metodologie byla navržena na základě podrobné analýzy redundance genetické informace uložené v genomických signálech. Využívá transformace znakových sekvencí do několika variant fázových signálů. Navíc umožňuje přímé zpracování dat ze sekvenace nanopórem v podobě nativních proudových signálů.

This diploma thesis deals with optimization models with design of a new waste management infrastructure in the Czech Republic, such that combustible waste, which is not utilized by the material recovering, can be used by energy recovering. This task is handled by optimization models, including trac and mixing problems. First of all, the concepts of graph theory and optimization are presented in this paper. Subsequently, some of the GAMS functions are discussed, and later the VBA programming language used to handle the larger data quickly is presented. In the main part, three gradually expanding models are developed. At the end the data from the waste management information system are implemented into them.

We give a strongly polynomial time combinatorial algorithm to minimise the largest eigenvalue of the weighted Laplacian of a bipartite graph. This is accomplished by solving the dual graph embedding problem which arises from a semidefinite programming formulation. In particular, the problem for trees can be solved in time cubic in the number of vertices.