Numerical simulations of Mach shock reflections have shown that the Mach stem can bifurcate as a result of the slip line jetting forward. Numerical simulations were conducted in this study which determined that these bifurcations occur when the Mach number is high, the ramp angle is high, and specific heat ratio is low. It was clarified that the bifurcation is a result of a sufficiently large velocity difference across the slip line which drives the jet. This bifurcation phenomenon has also been observed after triple point collisions in detonation simulations. A triple point reflection was modelled as an inert shock reflecting off a wedge, and the accuracy of the model at early times after reflection indicates that bifurcations in detonations are a result of the shock reflection process. Further investigations revealed that bifurcations likely contribute to the irregular structure observed in certain detonations.

Thesis (M.A. (Human Movement Science))--North-West University, Potchefstroom Campus, 2009.

For the microprocessor technology of today and the foreseeable future, multi-core is a key engine that drives performance growth under very tight power dissipation constraints. While previous research has been mostly focused on individual processor cores, there is a compelling need for studying how to efficiently manage shared resources among cores, including physical space, on-chip communication and on-chip storage. In managing physical space, floorplanning is the first and most critical step that largely affects communication efficiency and cost-effectiveness of chip designs. We consider floorplanning with regularity constraints that requires identical processing/memory cores to form an array. Such regularity can greatly facilitate design modularity and therefore shorten design turn-around time. Very little attention has been paid to automatic floorplanning considering regularity constraints because manual floorplanning has difficulty handling the complexity as chip core count increases. In this dissertation work, we investigate the regularity constraints in a simulated-annealing based floorplanner for multi/many core processor designs. A simple and effective technique is proposed to encode the regularity constraints in sequence-pair, which is a classic format of data representation in automatic floorplanning. To the best of our knowledge, this is the first work on regularity-constrained floorplanning in the context of multi/many core processor designs. On-chip communication and shared last level cache (LLC) play a role that is at least as equally important as processor cores in terms of chip performance and power. This dissertation research studies dynamic voltage and frequency scaling for on-chip network and LLC, which forms a single uncore domain of voltage and frequency. This is in contrast to most previous works where the network and LLC are partitioned and associated with processor cores based on physical proximity. The single shared domain can largely avoid the interfacing overhead across domain boundaries and is practical and very useful for industrial products. Our goal is to minimize uncore energy dissipation with little, e.g., 5% or less, performance degradation. The first part of this study is to identify a metric that can reflect the chip performance determined by uncore voltage/frequency. The second part is about how to monitor this metric with low overhead and high fidelity. The last part is the control policy that decides uncore voltage/frequency based on monitoring results. Our approach is validated through full system simulations on public architecture benchmarks.

Teste de propriedades em grafos consiste no estudo de algoritmos aleatórios sublineares que determinam se um grafo $G$ de entrada com $n$ vértices satisfaz uma dada propriedade ou se é necessário adicionar ou remover mais do que $\\epsilon{n \\choose 2}$ arestas para fazer $G$ satisfazê-la, para algum parâmetro $\\epsilon$ de erro fixo. Uma propriedade de grafos $P$ é dita testável se, para todo $\\epsilon > 0$, existe um tal algoritmo para $P$ cujo tempo de execução é independente de $n$. Um dos resultados de maior importância nesta área, provado por Alon e Shapira, afirma que toda propriedade hereditária de grafos é testável. Neste trabalho, apresentamos resultados análogos para torneios --- grafos completos nos quais são dadas orientações para cada aresta. / Graph property testing is the study of randomized sublinear algorithms which decide if an input graph $G$ with $n$ vertices satisfies a given property or if it is necessary to add or remove more than $\\epsilon{n \\choose 2}$ edges to make $G$ satisfy it, for some fixed error parameter $\\epsilon$ . A graph property $P$ is called testable if, for every $\\epsilon > 0$, there is such an algorithm for $P$ whose run time is independent of $n$. One of the most important results in this area is due to Alon and Shapira, who showed that every hereditary graph property is testable. In this work, we show analogous results for tournaments --- complete graphs in which every edge is given an orientation.

Numerical simulations of Mach shock reflections have shown that the Mach stem can bifurcate as a result of the slip line jetting forward. Numerical simulations were conducted in this study which determined that these bifurcations occur when the Mach number is high, the ramp angle is high, and specific heat ratio is low. It was clarified that the bifurcation is a result of a sufficiently large velocity difference across the slip line which drives the jet. This bifurcation phenomenon has also been observed after triple point collisions in detonation simulations. A triple point reflection was modelled as an inert shock reflecting off a wedge, and the accuracy of the model at early times after reflection indicates that bifurcations in detonations are a result of the shock reflection process. Further investigations revealed that bifurcations likely contribute to the irregular structure observed in certain detonations.

Os principais objetos de estudo neste trabalho são os números de Ramsey para circuitos e o lema da regularidade de Szemerédi. Dados grafos $L_1, \\ldots, L_k$, o número de Ramsey $R(L_1,\\ldots,L_k)$ é o menor inteiro $N$ tal que, para qualquer coloração com $k$ cores das arestas do grafo completo com $N$ vértices, existe uma cor $i$ para a qual a classe de cor correspondente contém $L_i$ como um subgrafo. Estaremos especialmente interessados no caso em que os grafos $L_i$ são circuitos. Obtemos um resultado original solucionando o caso em que $k=3$ e $L_i$ são circuitos pares de mesmo tamanho. / The main objects of interest in this work are the Ramsey numbers for cycles and the Szemerédi regularity lemma. For graphs $L_1, \\ldots, L_k$, the Ramsey number $R(L_1, \\ldots,L_k)$ is the minimum integer $N$ such that for any edge-coloring of the complete graph with~$N$ vertices by $k$ colors there exists a color $i$ for which the corresponding color class contains~$L_i$ as a subgraph. We are specially interested in the case where the graphs $L_i$ are cycles. We obtained an original result solving the case where $k=3$ and $L_i$ are even cycles of the same length.

Dois problemas combinatórios são estudados: (i) determinar a quantidade de cópias de um hipergrafo fixo em um hipergrafo uniforme pseudoaleatório, e (ii) estimar números de Ramsey de ordem dois e três para grafos com largura de banda pequena e grau máximo limitado. Apresentamos um lema de contagem para estimar a quantidade de cópias de um hipergrafo k-uniforme linear livre de conectores (conector é uma generalização de triângulo, para hipergrafos) que estão presentes em um hipergrafo esparso pseudoaleatório G. Considere um hipergrafo k-uniforme linear H que é livre de conectores e um hipergrafo k-uniforme G com n vértices. Seja d_H=\\max\\{\\delta(J): J\\subset H\\} e D_H=\\min\\{k d_H,\\Delta(H)\\}. Estabelecemos que, se os vértices de G não possuem grau grande, famílias pequenas de conjuntos de k-1 elementos de V(G) não possuem vizinhança comum grande, e a maioria dos pares de conjuntos em {V(G)\\choose k-1} possuem a quantidade ``correta\'\' de vizinhos, então a quantidade de imersões de H em G é (1+o(1))n^{|V(H)|}p^{|E(H)|}, desde que p\\gg n^{1/D_H} e |E(G)|={n\\choose k}p. Isso generaliza um resultado de Kohayakawa, Rödl e Sissokho [Embedding graphs with bounded degree in sparse pseudo\\-random graphs, Israel J. Math. 139 (2004), 93--137], que provaram que, para p dado como acima, esse lema de imersão vale para grafos, onde H é um grafo livre de triângulos. Determinamos assintoticamente os números de Ramsey de ordem dois e três para grafos bipartidos com largura de banda pequena e grau máximo limitado. Mais especificamente, determinamos assintoticamente o número de Ramsey de ordem dois para grafos bipartidos com largura de banda pequena e grau máximo limitado, e o número de Ramsey de ordem três para tais grafos, com a suposição adicional de que ambas as classes do grafo bipartido têm aproximadamente o mesmo tamanho. / Two combinatorial problems are studied: (i) determining the number of copies of a fixed hipergraph in uniform pseudorandom hypergraphs, and (ii) estimating the two and three color Ramsey numbers for graphs with small bandwidth and bounded maximum degree. We give a counting lemma for the number of copies of linear k-uniform \\emph hypergraphs (connector is a generalization of triangle for hypergraphs) that are contained in some sparse hypergraphs G. Let H be a linear k-uniform connector-free hypergraph and let G be a k-uniform hypergraph with n vertices. Set d_H=\\max\\{\\delta(J)\\colon J\\subset H\\} and D_H=\\min\\{kd_H,\\Delta(H)\\}. We proved that if the vertices of G do not have large degree, small families of (k-1)-element sets of V(G) do not have large common neighbourhood and most of the pairs of sets in {V(G)\\choose k-1} have the `right\' number of common neighbours, then the number of embeddings of H in G is (1+o(1))n^p^, given that p\\gg n^ and |E(G)|=p. This generalizes a result by Kohayakawa, R\\\"odl and Sissokho [Embedding graphs with bounded degree in sparse pseudo\\-random graphs, Israel J. Math. 139 (2004), 93--137], who proved that, for p as above, this result holds for graphs, where H is a triangle-free graph. We determine asymptotically the two and three Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that both classes of the bipartite graph have almost the same size.

<div>This thesis mainly focuses on the PDE theories that arise from the study of hydrodynamics of nematic liquid crystals. </div><div><br></div><div>In Chapter 1, we give a brief introduction of the Ericksen--Leslie director theory and Beris--Edwards <i>Q</i>-tensor theory to the PDE modeling of dynamic continuum description of nematic liquid crystals. In the isothermal case, we derive the simplified Ericksen--Leslie equations with general targets via the energy variation approach. Following this, we introduce a simplified, non-isothermal Ericksen--Leslie system and justify its thermodynamic consistency. </div><div><br></div><div>In Chapter 2, we study the weak compactness property of solutions to the Ginzburg--Landau approximation of the simplified Ericksen--Leslie system. In 2-D, we apply the Pohozaev type argument to show a kind of concentration cancellation occurs in the weak sequence of Ginzburg--Landau system. Furthermore, we establish the same compactness for non-isothermal equations with approximated director fields staying on the upper semi-sphere in 3-D. These compactness results imply the global existence of weak solutions to the limit equations as the small parameter tends to zero. </div><div><br></div><div>In Chapter 3, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards system for both the Landau–De Gennes and Ball–Majumdar bulk potentials in 3-D, and then study its partial regularity by proving that the 1-D parabolic Hausdorff measure of the singular set is 0.</div><div><br></div><div>In Chapter 4, motivated by the study of un-corotational Beris--Edwards system, we construct a suitable weak solution to the full Ericksen--Leslie system with Ginzburg--Landau potential in 3-D, and we show it enjoys a (slightly weaker) partial regularity, which asserts that it is smooth away from a closed set of parabolic Hausdorff dimension at most 15/7.</div>

The Reproducing Kernel Element Method (RKEM) is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-delta property. To achieve these properties, the underlying mesh must meet certain regularity constraints, unique to RKEM. The aim of this dissertation is to develop a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. This check is a critical step in the use of RKEM in any application. The general checking algorithm is made more specific to apply to two-dimensional triangular meshes with circular supports and to three-dimensional tetrahedral meshes with spherical supports. The checking algorithm features the output of the uncovered regions that are used to develop a mesh-mending technique for fixing offending meshes. The specific check is used in conjunction with standard quality meshing techniques to produce meshes suitable for use with RKEM. The RKEM quasi-uniformity definitions enable the use of RKEM in solving Galerkin weak forms as well as in general interpolation applications, such as the representation of geometries. A procedure for determining a RKEM representation of discrete point sets is presented with results for surfaces in three-dimensions. This capability is important to the analysis of geometries such as patient-specific organs or other biological objects.

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.