Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey's Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number.

Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3. The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

The aging brain is characterized by alteration in synaptic contacts, which leads to decline of motor and cognitive functions. These changes are reflected in the age related shifts in power spectrum of electroencephalogram (EEG) signals in both wakefulness and sleep. Various non-linear measures have been used to obtain more insights from EEG analysis compared to the conventional spectral analysis. In our study we used Sample Entropy to quantify regularity of the EEG signal. Because elderly subjects arouse from sleep more often than younger subjects, we hypothesized that Entropy of EEG signals from elderly subjects would be higher than that from middle aged subjects, within a sleep stage. We also hypothesized that the entropy increases during and following an arousal and does not return to background levels immediately after an arousal. Our results show that Sample Entropy varies systematically with sleep state in healthy middle-aged and elderly female subjects, reflecting the changing regularity in the EEG. Sample Entropy is significantly higher in elderly in sleep Stage 2 and REM, suggesting that in these two sleep stages the cortical state is closer to wake than in middle-aged women. Sample Entropy is higher in post-arousal compared to the pre-arousal and stays high for a 30 sec period.

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In graph theory, as in many fields of mathematics, one is often interested in finding the maxima or minima of certain functions and identifying the points of optimality. We consider a variety of functions on graphs and hypegraphs and determine the structures that optimize them. A central problem in extremal (hyper)graph theory is that of finding the maximum number of edges in a (hyper)graph that does not contain a specified forbidden substructure. Given an integer n, we consider hypergraphs on n vertices that do not contain a strong simplex, a structure closely related to and containing a simplex. We determine that, for n sufficiently large, the number of edges is maximized by a star. We denote by F(G, r, k) the number of edge r-colorings of a graph G that do not contain a monochromatic clique of size k. Given an integer n, we consider the problem of maximizing this function over all graphs on n vertices. We determine that, for large n, the optimal structures are (k − 1)2-partite Turán graphs when r = 4 and k ∈ {3, 4} are fixed. We call a graph F color-critical if it contains an edge whose deletion reduces the chromatic number of F and denote by F(H) the number of copies of the specified color-critical graph F that a graph H contains. Given integers n and m, we consider the minimum of F(H) over all graphs H on n vertices and m edges. The Turán number of F, denoted ex(n, F), is the largest m for which the minimum of F(H) is zero. We determine that the optimal structures are supergraphs of Tur´an graphs when n is large and ex(n, F) ≤ m ≤ ex(n, F)+cn for some c > 0.

Avec la loi du 26 juillet 2005, le sort des créanciers postérieurs a subi de profondes modifications. Répartis en deux catégories distinctes par l’effet d’un critère téléologique, leur traitement par la procédure collective n’est plus identique. Les créanciers postérieurs non éligibles au traitement préférentiel subissent les règles contraignantes de la procédure collective, tandis que seuls les créanciers postérieurs dits « méritants » bénéficient d’un paiement à l’échéance et d’un paiement par privilège. Toutefois, par de nombreux aspects, les créanciers postérieurs élus sont également confrontés à la rigueur de la procédure, altérant leurs possibilités réelles de paiement. La situation des créanciers postérieurs, dans leur ensemble, contraste avec celle conférée au débiteur qui n’a plus à craindre l’ouverture d’une procédure collective, celle-ci étant devenue une technique de gestion mise à sa disposition et particulièrement protectrice de ses droits. Dès lors, le salut des créanciers postérieurs semble se situer à l’extérieur de la procédure. Instrumentalisé par le droit des entreprises en difficulté, le droit des garanties de paiement, droit par nature égoïste, leur offre de nombreuses opportunités d’échapper à l’emprise de la procédure. Si le sujet impose une approche technique de la situation des créanciers postérieurs, il a surtout pour ambition de s’inscrire dans une perspective d’ensemble afin de mettre en lumière les nombreuses contradictions qui irriguent la matière. / With the law of July 26th, 2005, the situation of posterior creditors has changed. Divided into two distinct categories by the effect of a teleological criterion, their treatment by insolvency procedures are not identical. Posterior creditors who are not eligible for the preferential treatment suffer the binding rules of insolvency law, while only posterior creditors called "deserving" receive a payment due date and a privilege. However, in several aspects, these posterior creditors also face the rigor of the procedure, altering their chances to obtain payment. The situation of posterior creditors contrast with the situation conferred to the debtor. The protection of posterior creditors appears to be outside of the procedure. Instrumentalized by the insolvency law, payment guarantees offer many opportunities to escape the influence of the procedure. If the subject requires a technical approach of the situation of posterior creditors, he supposes to make a global study in order to show the many contradictions which irrigate the discipline.

of doctoral thesis Structural graph theory Jan Hladký In the thesis we make progress on the Loebl-Komlós-Sós Conjecture which is a classic problem in the field of Extremal Graph Theory. We prove the following weaker version of the Conjecture: For every α > 0 there exists a number k0 such that for every k > k0 we have that every n-vertex graph G with at least (1 2 +α)n vertices of degrees at least (1+α)k contains each tree T of order k as a subgraph. The proof of our result follows a strategy common to approaches which employ the Szemerédi Regularity Lemma: the graph G is decomposed, a suitable combinatorial structure inside the decomposition is found, and then the tree T is embedded into G using this structure. However the decomposition given by the Regularity Lemma is not of help when G sparse. To surmount this shortcoming we develop a decomposition technique that applies also to sparse graphs: each graph can be decomposed into vertices of huge degrees, regular pairs (in the sense of the Regularity Lemma), and two other components each exhibiting certain expander-like properties. The results were achieved in a joint work with János Komlós, Diana Piguet, Miklós Simonovits, Maya Jakobine Stein and Endre Szemerédi. 1

anglicky In the presented work, we study the existence and uniqueness of solutions to the generalized Stokes problem. We, further, focus on the higher differentiability and the Hölder continuity of solutions to the generalized Stokes and generalized Navier-Stokes system. We reach the full regularity in an arbitrary dimension for a linear case, while in a nonlinear case we work only in dimensions d = 2, 3. In dimension d = 2 we are able to proof the full regularity of solution, in dimension d = 3 we obtain only a partial regularity. All main results are introduced in the first section. 1

Este trabalho trata de pesquisa de soluções para a síntese física de circuitos integrados menos susceptíveis aos efeitos de variabilidade decorrentes do uso de tecnologias de fabricação com dimensões nanométricas. Também apresenta a pesquisa e o desenvolvimento de uma ferramenta para a geração de leiautes regulares denominada R-CAT. A regularidade geométrica é explorada pela repetição de padrões básicos de leiaute ao longo de uma matriz. A regularidade é apontada como uma das melhores alternativas para lidar com os atuais problemas de fabricação em tecnologias submicrônicas. Projetos regulares são menos suscetíveis aos problemas de litografia, aumentam o yield e diminuem o tempo gasto em re-projeto. Além disso, circuitos regulares apresentam maior previsibilidade de resultados de potência, atraso e yield, principalmente pelo fato das células estarem pré-caracterizadas. A ferramenta desenvolvida visa o trabalho com dois tipos de síntese física para leiautes regulares, produzindo circuitos integrados personalizáveis por todas as máscaras ou circuitos personalizáveis por algumas máscaras. O principal objetivo deste gerador é a facilidade de conversão e adaptação dependendo da abordagem de matriz escolhida. Isso facilitará a comparação entre diferentes alternativas de matrizes, a adoção de blocos lógicos diversos e de novas tecnologias. O gerador de leiautes R-CAT identifica células adjacentes com conexões em comum entre elas e realiza a conexão entre essas células em metal 1, reduzindo o número de conexões a ser realizado pelo roteador em até 10%. A ferramenta R-CAT está inserida em um fluxo de projeto e depende do método de síntese lógica adotado. Duas ferramentas de síntese lógica foram utilizadas: SIS e OrBDDs, oferecendo duas linhas de projeto: a primeira priorizando a área e a segunda priorizando timing e interconexões curtas. Ambas respeitando a mesma regularidade geométrica imposta pela matriz. Os resultados obtidos demonstram que as matrizes SIS ocupam 53% menos área do que a estratégia orBDD e reduzem o wire length em 30%. Uma área menor é obtida devido ao fato da ferramenta SIS gerar descrições com a metade de células lógicas e nets. Entretanto, as matrizes R-CAT OrBDD apresentam menor wire length médio, menor fan-out (redução de 15%), menor delay e maior roteabilidade. As sínteses OrBDD apresentam poucas nets não roteadas sem a inserção de trilhas extras. Além disso, as matrizes R-CAT atingiram resultados até 40% menores em wire length e reduções de área de até 46% em relação às matrizes MARTELO. / This work presents a research for physical synthesis of integrated circuits, which are less susceptible to the effects of variability observed in fabrication technologies using nanometers scale. Moreover, it presents a CAD tool developed to generate regular layouts, which is called R-CAT. The geometric regularity is achieved using basic patterns repeated along one matrix structure. Regularity is pointed like one of the best alternatives to deal with submicron technologies issues. Regular designs are less susceptible to lithographic problems, improve the yield and decrease the time to re-spin. Furthermore, regular circuits improve predictability of power consumption, timing and yield results, because the cells are pre-characterized. The developed tool focuses on two types of physical synthesis for regular layouts, producing either integrated circuit customized using all masks or integrated circuits customized using some masks. The main goal is the facility of conversion and adaptation depending on the chosen matrix approach. This will make easier the comparison of different matrix approaches, besides the adoption of several logic blocks and new technologies. R-CAT layout generator identifies adjacent cells that are placed in a same row and have common connections between them. In this case, the generator can make these connections in Metal 1. This technique reduces the number of connections to be done by the router. The experiments showed that this technique is able to reduce about 10% the number of connections to be done. This tool is inserted into a design flow and it is dependent of the logic synthesis methodology adopted. Two logical syntheses tools were used in the flow: SIS and OrBDDs. R-CAT SIS and R-CAT orBDD Matrices were generated for a set of circuits. The use of R-CAT tool with SIS and orBDD logical synthesis offers two design lines: the first one highlights area and the second one emphasize timing and short connections. Both of them respect the same geometric regularity. The results demonstrate that SIS matrices present 53% less area than orBDD approach and reduce the wire length by 30%. The area reduction is achieved because the SIS tool generates descriptions with the half of logic cells and nets. Nevertheless, the R-CAT orBDD matrices decreased the medium wire length, reduced the fan-out in 15%, reduced the delay and improved the routability. orBDD synthesis presents few non-routed nets without extra tracks insertion. Moreover, the R-CAT matrices obtained about 40% better results in wire length and they reduced area in 46% when compared to MARTELO matrices.

The religious freedom introduced to Russia in 1989, due to perestroika, gave new opportunities for Christianity to expand. People accepted Christ and new churches were founded. Nevertheless after 16 years of transformation the evangelical churches in Russia are diminishing in growth. There are probably several reasons for this. One major reason which is suggested here, is that only a minority of Russian Christian men are willing to take up regular voluntary church ministry: like being a home group leader, taking care of the church building, having a part in Sunday school teaching, helping with the youth group or any other kind of service. As a result, not only are pastors often overloaded with administration work and can not find time for people but also ministry opportunities are not started, developed or expanded. Therefore this research aims to find out which factors are important to Russian Christian men in order for them to engage a regular voluntary church ministry. / Christian Spirituality, Church History and Missiology / M. Th. (Missiology)