Nous étudions, dans la première partie de notre thèse, les polysimplexes d’un espace euclidien de dimension quelconque, c’est-à-dire les objets consistant en une juxtaposition de simplexes réguliers (de tétraèdres si la dimension est 3) accolés le long de leurs faces. Nous étudions principalement le groupe des symétries de ces polysimplexes. Nous présentons une façon de représenter un polysimplexe à l’aide d’un diagramme. Ceci fournit une classification complète des polysimplexes à similitude près. De plus, le groupe des symétries se déduit du groupe des automorphismes du diagramme. Il découle en particulier de notre étude qu’en dimension supérieure à 2, une telle structure ne possède jamais deux faces parallèles et ne contient jamais de circuit fermé de simplexes. Dans la seconde partie de notre thèse, nous abordons un problème classique de combinatoire : l’énumération des pavages d’un rectangle mxn à l’aide de dominos. Klarner et Pollack ont montré qu’en fixant m la suite obtenue vérifie une relation de récurrence linéaire à coefficients constants. Nous établissons une nouvelle méthode nous permettant d’obtenir la fonction génératrice correspondante et la calculons pour m <= 16, alors qu’elle n’était connue que pour m <= 10.

Um importante fator que precisa ser considerado para alcançar alto desempenho em aplicações paralelas é a distribuição dos processos nos núcleos do sistema, denominada mapeamento de processos. Mesmo o mapeamento estático de processos é um problema NP-difícil. Por esse motivo, são utilizadas heurísticas que dependem da aplicação e do hardware no qual a aplicação será mapeada. Nas arquiteturas atuais, além da possibilidade de haver mais de um processador por nó do cluster, é possível haver mais de um núcleo de processamento por processador, assim, o mapeamento estático de processos pode considerar pelo menos três níveis de comunicação entre os processos que executam em um cluster multi-core: intra-chip, intra-nó e inter-nó. Este trabalho propõe a heurística MapEME (Mapeamento Estático MPI com Emparelhamento) que emprega o Emparelhamento Perfeito de Custo Máximo (EPCM) no cálculo do mapeamento estático de processos paralelos MPI em processadores multi-core. Os resultados alcançados pelo mapeamento gerado pela MapEME são comparados aos resultados obtidos pelo mapeamento gerado pela aplicação Scotch, que utiliza o Biparticionamento Recursivo Dual (BRD), já utilizado como heurística para mapeamento estático de processos. Ambas as heurísticas são comparadas à Busca Exaustiva (BE) para verificar o quanto estão próximas do ótimo. Os três métodos têm a complexidade e o ganho no tempo de execução em ralação à distribuição padrão da biblioteca MPICH2 comparados entre si. A principal contribuição deste trabalho é mostrar que a heurística EPCM apresenta ganho de até 40% equivalente a já difundida BRD, e possui uma complexidade menor ao ser aplicado em um cluster multi-core que compartilha cache nível 2 a cada dois núcleos. / An important factor that must be considered to achieve high performance on parallel applications is the mapping of processes on cores. However, since this is defined as an NP-Hard problem, it requires different mapping heuristics that depends on the application and the hardware on which it will be mapped. On the current architectures we can have more than one multi-core processors per node, and consequently the process mapping can consider three process communication types: intrachip, intranode and internode. This work propose the MapEME (Static Mapping MPI using Matching) that use the Maximum Weighted Perfect Matching (MWPM) to calculate the static process mapping and analyze its performance. The results provided by MapEME are compared with the results of application Scotch. It uses Dual Recursive Bipartitioning (DRB), an already used heuristics for static mapping. Both heuristics are compared with Exhaustive Search (ES) to verify how much the two heuristics are near the optimum. The three methods have theirs complexities analyzed. Also the mapping gain when compared with the standard MPICH2 distribution was measured. The main contribution of this work is to show that the heuristic, EPCM, provides gain up to 40%, close of DRB gain. Furthermore, EPCM has a lower complexity when applied to a multicore cluster that shares L2 cache every two cores.

Um importante fator que precisa ser considerado para alcançar alto desempenho em aplicações paralelas é a distribuição dos processos nos núcleos do sistema, denominada mapeamento de processos. Mesmo o mapeamento estático de processos é um problema NP-difícil. Por esse motivo, são utilizadas heurísticas que dependem da aplicação e do hardware no qual a aplicação será mapeada. Nas arquiteturas atuais, além da possibilidade de haver mais de um processador por nó do cluster, é possível haver mais de um núcleo de processamento por processador, assim, o mapeamento estático de processos pode considerar pelo menos três níveis de comunicação entre os processos que executam em um cluster multi-core: intra-chip, intra-nó e inter-nó. Este trabalho propõe a heurística MapEME (Mapeamento Estático MPI com Emparelhamento) que emprega o Emparelhamento Perfeito de Custo Máximo (EPCM) no cálculo do mapeamento estático de processos paralelos MPI em processadores multi-core. Os resultados alcançados pelo mapeamento gerado pela MapEME são comparados aos resultados obtidos pelo mapeamento gerado pela aplicação Scotch, que utiliza o Biparticionamento Recursivo Dual (BRD), já utilizado como heurística para mapeamento estático de processos. Ambas as heurísticas são comparadas à Busca Exaustiva (BE) para verificar o quanto estão próximas do ótimo. Os três métodos têm a complexidade e o ganho no tempo de execução em ralação à distribuição padrão da biblioteca MPICH2 comparados entre si. A principal contribuição deste trabalho é mostrar que a heurística EPCM apresenta ganho de até 40% equivalente a já difundida BRD, e possui uma complexidade menor ao ser aplicado em um cluster multi-core que compartilha cache nível 2 a cada dois núcleos. / An important factor that must be considered to achieve high performance on parallel applications is the mapping of processes on cores. However, since this is defined as an NP-Hard problem, it requires different mapping heuristics that depends on the application and the hardware on which it will be mapped. On the current architectures we can have more than one multi-core processors per node, and consequently the process mapping can consider three process communication types: intrachip, intranode and internode. This work propose the MapEME (Static Mapping MPI using Matching) that use the Maximum Weighted Perfect Matching (MWPM) to calculate the static process mapping and analyze its performance. The results provided by MapEME are compared with the results of application Scotch. It uses Dual Recursive Bipartitioning (DRB), an already used heuristics for static mapping. Both heuristics are compared with Exhaustive Search (ES) to verify how much the two heuristics are near the optimum. The three methods have theirs complexities analyzed. Also the mapping gain when compared with the standard MPICH2 distribution was measured. The main contribution of this work is to show that the heuristic, EPCM, provides gain up to 40%, close of DRB gain. Furthermore, EPCM has a lower complexity when applied to a multicore cluster that shares L2 cache every two cores.

Um importante fator que precisa ser considerado para alcançar alto desempenho em aplicações paralelas é a distribuição dos processos nos núcleos do sistema, denominada mapeamento de processos. Mesmo o mapeamento estático de processos é um problema NP-difícil. Por esse motivo, são utilizadas heurísticas que dependem da aplicação e do hardware no qual a aplicação será mapeada. Nas arquiteturas atuais, além da possibilidade de haver mais de um processador por nó do cluster, é possível haver mais de um núcleo de processamento por processador, assim, o mapeamento estático de processos pode considerar pelo menos três níveis de comunicação entre os processos que executam em um cluster multi-core: intra-chip, intra-nó e inter-nó. Este trabalho propõe a heurística MapEME (Mapeamento Estático MPI com Emparelhamento) que emprega o Emparelhamento Perfeito de Custo Máximo (EPCM) no cálculo do mapeamento estático de processos paralelos MPI em processadores multi-core. Os resultados alcançados pelo mapeamento gerado pela MapEME são comparados aos resultados obtidos pelo mapeamento gerado pela aplicação Scotch, que utiliza o Biparticionamento Recursivo Dual (BRD), já utilizado como heurística para mapeamento estático de processos. Ambas as heurísticas são comparadas à Busca Exaustiva (BE) para verificar o quanto estão próximas do ótimo. Os três métodos têm a complexidade e o ganho no tempo de execução em ralação à distribuição padrão da biblioteca MPICH2 comparados entre si. A principal contribuição deste trabalho é mostrar que a heurística EPCM apresenta ganho de até 40% equivalente a já difundida BRD, e possui uma complexidade menor ao ser aplicado em um cluster multi-core que compartilha cache nível 2 a cada dois núcleos. / An important factor that must be considered to achieve high performance on parallel applications is the mapping of processes on cores. However, since this is defined as an NP-Hard problem, it requires different mapping heuristics that depends on the application and the hardware on which it will be mapped. On the current architectures we can have more than one multi-core processors per node, and consequently the process mapping can consider three process communication types: intrachip, intranode and internode. This work propose the MapEME (Static Mapping MPI using Matching) that use the Maximum Weighted Perfect Matching (MWPM) to calculate the static process mapping and analyze its performance. The results provided by MapEME are compared with the results of application Scotch. It uses Dual Recursive Bipartitioning (DRB), an already used heuristics for static mapping. Both heuristics are compared with Exhaustive Search (ES) to verify how much the two heuristics are near the optimum. The three methods have theirs complexities analyzed. Also the mapping gain when compared with the standard MPICH2 distribution was measured. The main contribution of this work is to show that the heuristic, EPCM, provides gain up to 40%, close of DRB gain. Furthermore, EPCM has a lower complexity when applied to a multicore cluster that shares L2 cache every two cores.

Berge and Fulkerson conjectured that for each cubic bridgeless graph there are six perfect matchings such that each edge is contained in exactly two of them. Another conjecture due to Berge says that we can cover cubic bridgeless graphs by five perfect matchings. Both conjectures are studied for over forty years. Abreu et al. [2016] introduce a new class of graphs (called treelike snarks) which cannot be covered by less then five perfect matchings. We show that their lower bound on number of perfect matchings is tight. Moreover we prove that a bigger class of cubic bridgeless graphs admits Berge conjecture. Finally, we also show that Berge-Fulkerson conjecture holds for treelike snarks.

There is an increasing need for accurate models describing the electrical behaviour of individual biological cells exposed to electromagnetic fields. In this area of solving linear problem, the most frequently used technique for computing the EM field is the Finite-Difference Time-Domain (FDTD) method. When modelling objects that are small compared with the wavelength, for example biological cells at radio frequencies, the standard Finite-Difference Time-Domain (FDTD) method requires extremely small time-step sizes, which may lead to excessive computation times. The problem can be overcome by implementing a quasi-static approximate version of FDTD, based on transferring the working frequency to a higher frequency and scaling back to the frequency of interest after the field has been computed. An approach to modeling and analysis of biological cells, incorporating the Hodgkin and Huxley membrane model, is presented here. Since the external medium of the biological cell is lossy material, a modified Berenger absorbing boundary condition is used to truncate the computation grid. Linear assemblages of cells are investigated and then Floquet periodic boundary conditions are imposed to imitate the effect of periodic replication of the assemblages. Thus, the analysis of a large structure of cells is made more computationally efficient than the modeling of the entire structure. The total fields of the simulated structures are shown to give reasonable and stable results at 900MHz, 1800MHz and 2450MHz. This method will facilitate deeper investigation of the phenomena in the interaction between EM fields and biological systems. Moreover, the nonlinear response of biological cell exposed to a 0.9GHz signal was discussed on observing the second harmonic at 1.8GHz. In this, an electrical circuit model has been proposed to calibrate the performance of nonlinear RF energy conversion inside a high quality factor resonant cavity with known nonlinear device. Meanwhile, the first and second harmonic responses of the cavity due to the loading of the cavity with the lossy material will also be demonstrated. The results from proposed mathematical model, give good indication of the input power required to detect the weakly effects of the second harmonic signal prior to perform the measurement. Hence, this proposed mathematical model will assist to determine how sensitivity of the second harmonic signal can be detected by placing the required specific input power.

For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson's conjecture and Tutte's 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll's poem, a \emph{snark} is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis.   The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order. In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph $G$ has a cycle of length at least $|V(G)|-9$ then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24. In Paper II we present a new algorithm for generating all non-isomorphic snarks with a given number of vertices. We generate all snarks on 36 vertices and less and study these with respect to various properties. We find that a number of conjectures on cycle covers and colourings holds for all graphs of these orders. Furthermore we present counterexamples to no less than eight published conjectures on cycle coverings, cycle decompositions and the general structure of regular graphs.     In Paper III we show that Jaeger's Petersen colouring conjecture holds for three infinite families of snarks and that a minimum counterexample to this conjecture cannot contain a certain subdivision of $K_{3,3}$ as a subgraph. Furthermore, it is shown that one infinite family of snarks have strong Petersen colourings while another does not have any such colourings. Two simple constructions for snarks with arbitrary high oddness and resistance is given in Paper IV. It is observed that some snarks obtained from this construction have the property that they require at least five perfect matchings to cover the edges. This disproves a suggested strengthening of Fulkerson's conjecture.

Nous étudions, dans la première partie de notre thèse, les polysimplexes d’un espace euclidien de dimension quelconque, c’est-à-dire les objets consistant en une juxtaposition de simplexes réguliers (de tétraèdres si la dimension est 3) accolés le long de leurs faces. Nous étudions principalement le groupe des symétries de ces polysimplexes. Nous présentons une façon de représenter un polysimplexe à l’aide d’un diagramme. Ceci fournit une classification complète des polysimplexes à similitude près. De plus, le groupe des symétries se déduit du groupe des automorphismes du diagramme. Il découle en particulier de notre étude qu’en dimension supérieure à 2, une telle structure ne possède jamais deux faces parallèles et ne contient jamais de circuit fermé de simplexes.<p><p>Dans la seconde partie de notre thèse, nous abordons un problème classique de combinatoire :l’énumération des pavages d’un rectangle mxn à l’aide de dominos. Klarner et Pollack ont montré qu’en fixant m la suite obtenue vérifie une relation de récurrence linéaire à coefficients constants. Nous établissons une nouvelle méthode nous permettant d’obtenir la fonction génératrice correspondante et la calculons pour m <= 16, alors qu’elle n’était connue que pour m <= 10.<p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

yes / There is an increasing need for accurate models describing the electrical behaviour of individual biological cells exposed to electromagnetic fields. In this area of solving linear problem, the most frequently used technique for computing the EM field is the Finite-Difference Time-Domain (FDTD) method. When modelling objects that are small compared with the wavelength, for example biological cells at radio frequencies, the standard Finite-Difference Time-Domain (FDTD) method requires extremely small time-step sizes, which may lead to excessive computation times. The problem can be overcome by implementing a quasi-static approximate version of FDTD, based on transferring the working frequency to a higher frequency and scaling back to the frequency of interest after the field has been computed. An approach to modeling and analysis of biological cells, incorporating the Hodgkin and Huxley membrane model, is presented here. Since the external medium of the biological cell is lossy material, a modified Berenger absorbing boundary condition is used to truncate the computation grid. Linear assemblages of cells are investigated and then Floquet periodic boundary conditions are imposed to imitate the effect of periodic replication of the assemblages. Thus, the analysis of a large structure of cells is made more computationally efficient than the modeling of the entire structure. The total fields of the simulated structures are shown to give reasonable and stable results at 900MHz, 1800MHz and 2450MHz. This method will facilitate deeper investigation of the phenomena in the interaction between EM fields and biological systems. Moreover, the nonlinear response of biological cell exposed to a 0.9GHz signal was discussed on observing the second harmonic at 1.8GHz. In this, an electrical circuit model has been proposed to calibrate the performance of nonlinear RF energy conversion inside a high quality factor resonant cavity with known nonlinear device. Meanwhile, the first and second harmonic responses of the cavity due to the loading of the cavity with the lossy material will also be demonstrated. The results from proposed mathematical model, give good indication of the input power required to detect the weakly effects of the second harmonic signal prior to perform the measurement. Hence, this proposed mathematical model will assist to determine how sensitivity of the second harmonic signal can be detected by placing the required specific input power.