This thesis deals with optimizing extremal eigenvalues of weighted graph Laplacian matrices. In general, the Laplacian matrix of a (weighted) graph is of particular importance in spectral graph theory and combinatorial optimization (e.g., graph partition like max-cut and graph bipartition). Especially the pioneering work of M. Fiedler investigates extremal eigenvalues of weighted graph Laplacians and provides close connections to the node- and edge-connectivity of a graph. Motivated by Fiedler, Göring et al. were interested in further connections between structural properties of the graph and the eigenspace of the second smallest eigenvalue of weighted graph Laplacians using a semidefinite optimization approach. By redistributing the edge weights of a graph, the following three optimization problems are studied in this thesis: maximizing the second smallest eigenvalue (based on the mentioned work of Göring et al.), minimizing the maximum eigenvalue and minimizing the difference of maximum and second smallest eigenvalue of the weighted Laplacian. In all three problems a semidefinite optimization formulation allows to interpret the corresponding semidefinite dual as a graph realization problem. That is, to each node of the graph a vector in the Euclidean space is assigned, fulfilling some constraints depending on the considered problem. Optimal realizations are investigated and connections to the eigenspaces of corresponding optimized eigenvalues are established. Furthermore, optimal realizations are closely linked to the separator structure of the graph. Depending on this structure, on the one hand folding properties of optimal realizations are characterized and on the other hand the existence of optimal realizations of bounded dimension is proven. The general bounds depend on the tree-width of the graph. In the case of minimizing the maximum eigenvalue, an important family of graphs are bipartite graphs, as an optimal one-dimensional realization may be constructed. Taking the symmetry of the graph into account, a particular optimal edge weighting exists. Considering the coupled problem, i.e., minimizing the difference of maximum and second smallest eigenvalue and the single problems, i.e., minimizing the maximum and maximizing the second smallest eigenvalue, connections between the feasible (optimal) sets are established.

Least Squares Kinetic Upwind Method (LSKUM), a grid free method based on kinetic schemes has been gaining popularity over the conventional CFD methods for computation of inviscid and viscous compressible flows past complex configurations. The main reason for the growth of popularity of this method is its ability to work on any point distribution. The grid free methods do not require the grid for flow simulation, which is an essential requirement for all other conventional CFD methods. However, they do require point distribution or a cloud of points. Point generation is relatively simple and less time consuming to generate as compared to grid generation. There are various methods for point generation like an advancing front method, a quadtree based point generation method, a structured grid generator, an unstructured grid generator or a combination of above, etc. One of the easiest ways of point generation around complex geometries is to overlap the simple point distributions generated around individual constituent parts of the complex geometry. The least squares grid free method has been successfully used to solve a large number of flow problems over the years. However, it has been observed that some problems are still encountered while using this method on point distributions around complex configurations. Close analysis of the problems have revealed that bad connectivity of the nodes is the cause and this leads to bad connectivity related code divergence. The least squares (LS) grid free method called LSKUM involves discretization of the spatial derivatives using the least squares approach. The formulae for the spatial derivatives are obtained by minimizing the sum of the squares of the error, leading to a system of linear algebraic equations whose solution gives us the formulae for the spatial derivatives. The least squares matrix A for 1-D and 2-D cases respectively is given by (Refer PDF File for equation) The 1-D LS formula for the spatial derivatives is always well behaved in the sense that ∑∆xi2 can never become zero. In case of 2-D problems can arise. It is observed that the elements of the Ls matrix A are functions of the coordinate differentials of the nodes in the connectivity. The bad connectivity of a node thus can have an adverse effect on the nature of the LS matrices. There are various types of bad connectivities for a node like insufficient number of nodes in the connectivity, highly anisotropic distribution of nodes in the connectivity stencil, the nodes falling nearly on a line (or a plane in 3-D), etc. In case of multidimensions, the case of all nodes in a line will make the matrix A singular thereby making its inversion impossible. Also, an anisotropic distribution of nodes in the connectivity can make the matrix A highly illconditioned thus leading to either loss in accuracy or code divergence. To overcome this problem, the approach followed so far is to modify the connectivity by including more neighbours in the connectivity of the node. In this thesis, we have followed a different approach of using weights to alter the nature of the LS matrix A. (Refer PDF File for equation) The weighted LS formulae for the spatial derivatives in 1-D and 2-D respectively are are all positive. So we ask a question : Can we reduce the multidimensional LS formula for the derivatives to the 1-D type formula and make use of the advantages of 1-D type formula in multidimensions? Taking a closer look at the LS matrices, we observe that these are real and symmetric matrices with real eigenvalues and a real and distinct set of eigenvectors. The eigenvectors of these matrices are orthogonal. Along the eigendirections, the corresponding LS formulae reduce to the 1-D type formulae. But a problem now arises in combining the eigendirections along with upwinding. Upwinding, which in LS is done by stencil splitting, is essential to provide stability to the numerical scheme. It involves choosing a direction for enforcing upwinding. The stencil is split along the chosen direction. But it is not necessary that the chosen direction is along one of the eigendirections of the split stencil. Thus in general we will not be able to use the 1-D type formulae along the chosen direction. This difficulty has been overcome by the use of weights leading to WLSKUM-ED (Weighted Least Squares Kinetic Upwind Method using Eigendirections). In WLSKUM-ED weights are suitably chosen so that a chosen direction becomes an eigendirection of A(w). As a result, the multi-dimensional LS formulae reduce to 1-D type formulae along the eigendirections. All the advantages of the 1-D LS formuale can thus be made use of even in multi-dimensions. A very simple and novel way to calculate the positive weights, utilizing the coordinate differentials of the neighbouring nodes in the connectivity in 2-D and 3-D, has been developed for the purpose. This method is based on the fact that the summations of the coordinate differentials are of different signs (+ or -) in different quadrants or octants of the split stencil. It is shown that choice of suitable weights is equivalent to a suitable decomposition of vector space. The weights chosen either fully diagonalize the least squares matrix ie. decomposing the 3D vector space R3 as R3 = e1 + e2 + e3, where e1, e2and e3are the eigenvectors of A (w) or the weights make the chosen direction the eigendirection ie. decomposing the 3D vector space R3 as R3 = e1 + ( 2-D vector space R2). The positive weights not only prevent the denominator of the 1-D type LS formulae from going to zero, but also preserve the LED property of the least squares method. The WLSKUM-ED has been successfully applied to a large number of 2-D and 3-D test cases in various flow regimes for a variety of point distributions ranging from a simple cloud generated from a structured grid generator (shock reflection problem in 2-D and the supersonic flow past hemisphere in 3-D) to the multiple chimera clouds generated from multiple overlapping meshes (BI-NACA test case in 2-D and FAME cloud for M165 configuration in 3-D) thus demonstrating the robustness of the WLSKUM-ED solver. It must be noted that the second order acccurate computations using this method have been performed without the use of the limiters in all the flow regimes. No spurious oscillations and wiggles in the captured shocks have been observed, indicating the preservation of the LED property of the method even for 2ndorder accurate computations. The convergence acceleration of the WLSKUM-ED code has been achieved by the use of LUSGS method. The use of 1-D type formulae has simplified the application of LUSGS method in the grid-free framework. The advantage of the LUSGS method is that the evaluation and storage of the jacobian matrices can be eliminated by approximating the split flux jacobians in the implicit operator itself. Numerical results reveal the attainment of a speed up of four by using the LUSGS method as compared to the explicit time marching method. The 2-D WLSKUM-ED code has also been used to perform the internal flow computations. The internal flows are the flows which are confined within the boundaries. The inflow and the outflow boundaries have a significant effect on these flows. The accurate treatment of these boundary conditions is essential particularly if the flow condition at the outflow boundary is subsonic or transonic. The Kinetic Periodic Boundary Condition (KPBC) which has been developed to enable the single-passage (SP) flow computations to be performed in place of the multi-passage (MP) flow computations, utilizes the moment method strategy. The state update formula for the points at the periodic boundaries is identical to the state update formula for the interior points and can be easily extended to second order accuracy like the interior points. Numerical results have shown the successful reproduction of the MP flow computation results using the SP flow computations by the use of KPBC. The inflow and the outflow boundary conditions at the respective boundaries have been enforced by the use of Kinetic Outer Boundary Condition (KOBC). These boundary conditions have been validated by performing the flow computations for the 3rdtest case of the 4thstandard blade configuration of the turbine blade. The numerical results show a good comparison with the experimental results.

In large and complex systems, failures can have dramatic consequences, such as black-outs, pandemics or the loss of entire classes of an ecosystem. Nevertheless, it is a centuries-old intuition that by using networks to capture the core of the complexity of such systems, one might understand in which part of a system a phenomenon originates. I investigate this intuition using spectral methods to decouple the dynamics of complex systems near stationary states into independent dynamical modes. In this description, phenomena are tied to a specific part of a system through localized eigenvectors which have large amplitudes only on a few nodes of the system's network. Studying the occurrence of localized eigenvectors, I find that such localization occurs exactly for a few small network structures, and approximately for the dynamical modes associated with the most prominent failures in complex systems. My findings confirm that understanding the functioning of complex systems generally requires to treat them as complex entities, rather than collections of interwoven small parts. Exceptions to this are only few structures carrying exact localization, whose functioning is tied to the meso-scale, between the size of individual elements and the size of the global network. However, while understanding the functioning of a complex system is hampered by the necessary global analysis, the prominent failures, due to their localization, allow an understanding on a manageable local scale. Intriguingly, food webs might exploit this localization of failures to stabilize by causing the break-off of small problematic parts, whereas typical attempts to optimize technological systems for stability lead to delocalization and large-scale failures. Thus, this thesis provides insights into the interplay of complexity and localization, which is paramount to ascertain the functioning of the ever-growing networks on which we humans depend.

Fullerenes are all-carbon molecules with polyhedral structures where each atom is bonded with three other atoms and the faces of the polyhedron are pentagons and hexagons. Fullerene graphs model the fullerene structures and are cubic planar graphs having twelve pentagonal faces and the remaining faces are hexagonal. This work explores two models that seek to determine the maximum number of bulky addends that may bond to the surface of a fullerene. The first model assumes that any two bulky addends are too large to bond to adjacent carbon atoms. This is equivalent to finding a graph-theoretical maximum independent set: a vertex subset of maximum size such that no two vertices are adjacent. The problem of determining the maximum independent set order is NP-hard for general cubic planar graphs and the complexity for the fullerene subclass was previously unknown. By extending the work of Graver, a graph-theoretical foundation is laid then used to derive a linear-time algorithm for solving the maximum independent set problem for fullerenes. A discussion of the relationship between maximum independent sets and some specific families of fullerenes follows. The second model refines the first by adding an additional requirement that the resulting molecule is stable according to Hückel theory: the molecule exhibits a stable distribution of π electrons. The graph-theoretical description of this model is a maximum closed-shell independent set: a vertex subset of maximum size such that no two vertices are adjacent and exactly half of the eigenvalues of the adjacency matrix of the graph that results from the deletion of the vertex subset are positive. Computations for finding a maximum closed-shell independent set rely on determining whether fullerene subgraphs are closed-shell (satisfy the eigenvalue requirement) so a linear-time algorithm for finding the inertia (number of negative, zero, and positive eigenvalues) of unicyclic graphs is given. This algorithm is part of an exponential-time algorithm for finding a maximum closed-shell independent set of a fullerene molecule that is fast enough for practical use. An improved upper bound of 3n/8 + 3/2 for the closed-shell independence number is included.

_I The theory of impulsive di&reg / erential equations has become an important area of research in recent years. Linear equations, meanwhile, are fundamental in most branches of applied mathematics, science, and technology. The theory of higher order linear impulsive equations, however, has not been studied as much as the cor- responding theory of ordinary di&reg / erential equations. In this work, higher order linear impulsive equations at &macr / xed moments of impulses together with certain boundary conditions are investigated by making use of a Green&#039 / s formula, constructed for piecewise di&reg / erentiable functions. Existence and uniqueness of solutions of such boundary value problems are also addressed. Properties of Green&#039 / s functions for higher order impulsive boundary value prob- lems are introduced, showing a striking di&reg / erence when compared to classical bound- ary value problems of ordinary di&reg / erential equations. Necessarily, instead of an or- dinary Green&#039 / s function there corresponds a sequence of Green&#039 / s functions due to impulses. Finally, as a by-product of boundary value problems, eigenvalue problems for higher order linear impulsive di&reg / erential equations are studied. The conditions for the existence of eigenvalues of linear impulsive operators are presented. Basic properties of eigensolutions of self-adjoint operators are also investigated. In particular, a necessary and su&plusmn / cient condition for the self-adjointness of Sturm-Liouville opera- tors is given. The corresponding integral equations for boundary value and eigenvalue problems are also demonstrated in the present work.

With the growth of power system interconnections, the economic drivers encourage the electric companies to load the transmission lines near their limits, therefore it is critical to know those limits well. One important limiting issue is the damping of inter-area oscillation (IAO) between groups of synchronous machines. In this Ph.D. thesis, the contribution of power system components such as load and static var compensators (SVC) that affect the IAO of the power system, are analysed. The original contributions of this thesis are as follows: 1-Identification of eigenvalues and mode shapes of the IAO: In the first contribution of this thesis, the eigenvalues of the IAO are identified using a correlation based method. Then, the mode shape at each identified resonant frequency is determined to show how the synchronous generators swing against each other at the specific resonant frequencies. 2-Load modelling and load contribution to damping: The first part of this contribution lies in identification of the load model using cross-correlation and autocorrelation functions . The second aspect is the quantification of the load contribution to damping and sensitivity of system eigenvalues with respect to the load. 3- SVC contribution to damping: In this contribution the criteria for SVC controller redesign based on complete testing is developed. Then the effect of the SVC reactive power on the measured power is investigated. All of the contributions of this thesis are validated by simulation on test systems. In addition, there are some specific application of the developed methods to real data to find a.) the mode shape of the Australian electricity network, b.) the contribution of the Brisbane feeder load to damping and c.) the effect of the SVC reactive power of the Blackwall substations on the active power supplying Brisbane.

Les travaux présentés dans cette thèse concernent le problème d'identification des systèmes à retards et d'une certaine classe de systèmes hybrides appelés systèmes "impulsifs".Dans la première partie, un algorithme d'identification rapide a été proposé pour les systèmes à entrée retardée. Il est basé sur une méthode d'estimation distributionnelle non asymptotique initiée pour les systèmes sans retard. Une telle technique mène à des schémas de réalisation simples, impliquant des intégrateurs, des multiplicateurs et des fonctions continues par morceaux polynomiales ou exponentielles. Dans le but de généraliser cette approche pour les systèmes à retard, trois exemples d'applications ont été étudiées. La deuxième partie a été consacrée à l'identification des systèmes impulsifs. En se basant sur le formalisme des distributions, une procédure d'identification a été élaborée afin d'annihiler les termes singuliers des équations différentielles représentant ces systèmes. Par conséquent, une estimation en ligne des instants de commutations et des paramètres inconnus est prévue indépendamment des lois de commutations. Des simulations numériques d'un pendule simple soumis à des frottements secs illustrent notre méthodologie / This PhD thesis concerns the problem of identification of the delays systems and the continuous-time systems subject to impulsive terms.Firstly, a fast identification algorithm is proposed for systems with delayed inputs. It is based on a non-asymptotic distributional estimation technique initiated in the framework of systems without delay. Such technique leads to simple realization schemes, involving integrators, multipliers andContribution to the identification of time delays systems and a class of hybrid systems piecewise polynomial or exponential time functions. Thus, it allows for a real time implementation. In order to introduce a generalization to systems with input delay, three simple examples are presented.The second part deals with on-line identification of continuous-time systems subject to impulsive terms. Using a distribution framework, a scheme is proposed in order to annihilate singular terms in differential equations representing a class of impulsive systems. As a result, an online estimation of unknown parameters is provided, regardless of the switching times or the impulse rules. Numerical simulations of simple pendulum subjected to dry friction are illustrating our methodology

In this work we present the proofs of the Laplacian comparison theorems for Kähler manifolds Mm of complex dimension m with holomorphic bisectional curvature bounded from below by −1, 1, and 0. The manifolds being compared are the complex hyperbolic space CHm, the complex projective space CPm, and the complex Euclidean space Cm, which holomorphic bisectional curvatures are −1, 1, and 0, respectively. Moreover, as applications of the Laplacian comparison theorems, we describe the proof of the Bishop- Gromov comparison theorem for Kähler manifolds and obtain an estimate for the first eigenvalue λ1(M) of the Laplacian operator, that is, λ1(M) ≤ m2 = λ1(CHm), and show that the volume of Kähler manifolds with holomorphic bisectional curvature bounded from below by 1 is bounded by the volume of CPm. The results cited above have been proved in 2005 by Li and Wang, in an article Comparison theorem for Kähler Manifolds and Positivity of Spectrum , published in the Journal of Differential Geometry. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação, apresentamos as demonstrações dos teoremas de comparação do Laplaciano para variedades Kähler completas Mm de dimensão complexa m com curvatura bisseccional holomorfa limitada inferiormente por −1, 1 e 0. As variedades a serem comparadas são o espaço hiperbólico complexo CHm, o espaço projetivo complexo CPm e o espaço Euclidiano complexo Cm, cujas curvaturas bisseccionais holomorfas são −1, 1 e 0, respectivamente. Além disso, como aplicação dos teoremas de comparação do Laplaciano, descrevemos a prova do Teorema de Comparação de Bishop-Gromov para variedades Kähler; obtemos uma estimativa para o primeiro autovalor λ1(M) do Laplaciano, isto é, λ1(M) ≤ m2 = λ1(CHm); e mostramos que o volume de variedades Kähler, com curvatura bisseccional limitada inferiormente por 1, é limitado pelo volume de CPm. Os resultados citados acima foram provados em 2005 por Li e Wang no artigo Comparison Theorem for Kähler Manifolds and Positivity of Spectrum , publicado no Journal of Differential Geometry.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação o método espectronodal SD-SGF-CN, cf. spectral diamond spectral Green's function - constant nodal, é utilizado para a determinação dos fluxos angulares médios nas faces dos nodos homogeneizados em domínio heterogêneo. Utilizando esses resultados, desenvolvemos um algoritmo para a reconstrução intranodal da solução numérica visto que, em cálculos de malha grossa, soluções numéricas mais localizadas não são geradas. Resultados numéricos são apresentados para ilustrar a precisão do algoritmo desenvolvido. / In this dissertation the spectral nodal method SD-SGF-CN, cf. spectral diamond spectral Green's function - constant nodal, is used to determine the angular fluxes averaged along the edges of the homogenized nodes in heterogeneous domains. Using these results, we developed an algorithm for the reconstruction of the node-edge average angular fluxes within the nodes of the spatial grid set up on the domain, since more localized numerical solutions are not generated by coarse-mesh numerical methods. Numerical results are presented to illustrate the accuracy of the algorithm we offer.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação o método espectronodal SD-SGF-CN, cf. spectral diamond spectral Green's function - constant nodal, é utilizado para a determinação dos fluxos angulares médios nas faces dos nodos homogeneizados em domínio heterogêneo. Utilizando esses resultados, desenvolvemos um algoritmo para a reconstrução intranodal da solução numérica visto que, em cálculos de malha grossa, soluções numéricas mais localizadas não são geradas. Resultados numéricos são apresentados para ilustrar a precisão do algoritmo desenvolvido. / In this dissertation the spectral nodal method SD-SGF-CN, cf. spectral diamond spectral Green's function - constant nodal, is used to determine the angular fluxes averaged along the edges of the homogenized nodes in heterogeneous domains. Using these results, we developed an algorithm for the reconstruction of the node-edge average angular fluxes within the nodes of the spatial grid set up on the domain, since more localized numerical solutions are not generated by coarse-mesh numerical methods. Numerical results are presented to illustrate the accuracy of the algorithm we offer.