First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory. In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups. Furthermore, we determine the Euler characteristic for these complexes and prove that several of their homology groups are non-zero. In chapter 3, we introduce the notion of a homomorphism complex for partially ordered sets, placing particular emphasis on maps between chain posets and the Boolean algebras. We extend the notion of folding from general graph homomorphism complexes to the poset case, and we define an iterative discrete Morse matching for these Boolean complexes. We provide formulas for enumerating the number of critical cells arising from this matching as well as for the Euler characteristic. We end with a conjecture on the optimality of our matching derived from connections to 3-equal manifolds

Water and methanol are common solvents used in liquid chromatographic (LC) separations. It is highly desirable to model .the interactions of these solvents in order to better understand the nature of analyte solvation and its effect on retention. Therefore, structure and frequencies of complexes of these solvent molecules have been studied from a theoretical perspective as a first step in this direction. Specifically, cluster structures have been optimized at the RHF and MP2 levels in various flexible basis sets and with the counterpoise correction for basis set superposition error, and trends in the structure and binding energies of several clusters are described. Good agreement wasobtained for the water dimer with the experimental value for the binding energy of D20 using MP2 energies from 6-3 11G**/6-3 l+G** basis sets in conjunction with counterpoise optimizations and full counterpoise corrections. In this investigation harmonic frequencies have been calculated and corrected for the effects of anharmonicity by several methods, two of which are original. The first new method fits a Morse potential function to the energy computed along each normal mode. A second new method is based on fitting a quartic polynomial to energies computed along each normal mode. In cases where the quartic potential function is not very different from the harmonic well, a second order perturbation formula provides a reasonable approximation to the anharmonic vibrational frequencies. When the quartic potential is very far from the harmonic potential, a variational treatment of the vibrations is required. We find that the Morse method delivers reasonable estimates of frequencies of anharmonic motions at lower cost than multi-point potential mapping/multiple geometry optimization/Taylor series methods, and is more successful at predicting intermolecular frequencies than the anharmonic VSCF methods found in GAMESS software. Variational calculations using the quartic polynomials produce estimates of frequencies comparable to the more costly VSCF method. Both the Morse method and polynomial method are very fast computationally relative to these and other methods found in the literature.

In [GH09] M. Goresky and N. Hingston described and investigated various properties of a product on the cohomology of the free loop space of a closed, oriented manifold M relative to the constant loops. In this thesis we will give Morse and Floer theoretic descriptions of the product. There is a theorem due to J. Jones in [JJ87] which describes an isomorphism between cohomology of the free loop space and Hochschild homology of the singular cochain algebra of M with rational coefficients. We will use the theorem of J. Jones to find an algebraic model for the Goresky-Hingston product. We then use the algebraic model to explore further properties and applications of the Goresky Hingston product. In particular we use it to compute the ring structure for the n-spheres.

We prove that the non-k-equal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, K_{1,3}, K_{1,4}, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings.

Primeiramente consideramos uma família de \'C POT. 2\'-ações de \'R POT. 2\' sobre uma 3-variedade fechada. Uma das condições que esta família satisfaz é que admite apenas um número finito de órbitas singulares, sendo todas estas difeomorfas ao círculo. Para esta família, daremos uma descrição da estrutura das órbitas assim como das 3-variedades que as suportam. Isto generaliza resultados de classificação de ações localmente livres (isto é, sem singularidades) de \'R POT.2\' sobre 3-variedades fechadas obtidos por Chatelet-Rosenberg- Roussarie-Weil em [12], [30] e [31]. Finalmente, consideramos uma ação \\\'phi\' de \'RPOT. 2\' sobre uma 3-variedade fechada N topologicamente transitiva (isto é, tem uma órbita densa em N). Diremos que \\\'phi\' é metricamente transitiva se dado qualquer conjunto compacto e \\\'phi\'-invariante K, então ou K ou seu complementar tem medida nula com respeito a medida de Lebesgue. É conhecido que toda ação \\\'phi\' metricamente transitiva é topologicamente transitiva e que, em geral, a reciproca não é certa. No Entanto, Morse [27] em 1946 propôs a seguinte conjectura: Qualquer sistema dinâmico topologicamente transitivo, com algum grau de regularidade, é metricamente transitivo. A frase \"algum grau de regularidade\", pode significar, por exemplo, que o sistema dinâmico é analítico real, suave, tem um número finito de singularidades, etc. Na segunda parte da tese, mostramos a conjectura de Morse para o sistema dinâmico definido por uma \'R POT. 2\'-ação sobre uma 3-variedade fechada, cujo conjunto singular é uma união finita de órbitas círculo. Isto generaliza um resultado análogo obtido por Ding [18] para fluxos sobre superfícies fechadas / First we consider a family of \'C POT. 2\' actions of \'R POT. 2\' on a closed 3-manifold. One of the conditions of this family is that it admits only a finite number of singular orbits, which are all diffeomorphic to circle. For this family we will give a description of the structure of the orbits as well the 3-manifolds supporting this actions. This generalizes results of classification for locally free actions (i. e. without singularities) of \'R POT. 2\' on closed 3- manifolds obtained by Chatelet-Rosenberg-Roussarie-Weil in [12], [30] and [31]. Finally, we consider an action \\\'phi\' of \'R POT. 2\' on a closed 3-manifold N which is topologically transitive (i.e. has a dense orbit in N). We will say that \\\'phi\' is metrically transitive if, given any \\\'phi\'-invariant compact set K, then, either K or its complement has zero measure with respect to Lebesgue measure. It is known that every action \\\'phi\' topologically transitive is metrically transitive and that, in general, the reciprocal is not true. However, Morse [27] in 1946 proposed the following conjecture: any topologically transitive dynamical system with any degree of regularity is metrically transitive. The phrase \"some degree of regularity\" may mean, for example, that the dynamical system is real analytic, smooth, have a finite number of singularities, etc. In the second part of the thesis, we show the conjecture to the Morse for an dynamical system defined by a \'R POT. 2\'-action on a closed 3-manifold whose singular set is a finite union of orbits circle. This is a generalization of a similar result obtained by Ding in [18] for flows on closed surfaces

Esta tese analisa a obra do historiador norte-americano Richard Morse (1922-2001) sobre a história de São Paulo. Publicado pela primeira vez em 1954 nas comemorações do IV Centenário de São Paulo como De comunidade a metrópole: biografia de São Paulo, o trabalho é fonte importante dos estudos históricos urbanos sobre São Paulo ainda hoje. Editado em inglês em 1958 nos Estados Unidos, foi republicado no Brasil em 1970 - com algumas diferenças importantes - com o título Formação histórica de São Paulo: de comunidade à metrópole. Essa obra é examinada aqui por três perspectivas - como história urbana, como uma história cultural e como parte do debate sobre a urbanização das cidades na América Latina. Inserindo-a no campo dos estudos históricos urbanos norte-americano e brasileiro, no momento de sua constituição, pretende-se contribuir para o campo da história urbana preenchendo certas lacunas da historiografia da cidade de São Paulo. Ao retraçar os vínculos entre cidade, história e literatura que a obra parece conter, a tese contribui para a discussão de uma história cultural urbana de São Paulo. Reconhecendo o autor como personagem chave na constituição do debate sobre a cidade latino-americana entre as décadas de 1940 e 1970 - justamente o período em que se realizam as três edições -, o exame da obra permite ainda lançar luz em décadas fundamentais da urbanização latino-americana, retomando temas e questões sobre a metropolização em curso na cidade de São Paulo e no continente latino-americano. / This thesis analyzes the work of the American historian Richard Morse (1922-2001) on the history of São Paulo (Brazil). First published in 1954 in celebration of the fourth centenary of São Paulo, under the title of De Comunidade a Metrópole: Biografia de São Paulo, until today the work is an impor6 tant source of urban historical studies of São Paulo. Published in English in 1958 in the United States, was republished in Brazil in 1970 as Formação Histórica de São Paulo: de Comunidade à Metrópole, with some important differences. This work is examined here from three different perspectives - urban history, cultural history and as part of the urbanization of Latin American cities discussion. Entering on the urban historical studies in the U.S. and Brazil, at the time of its constitution, the thesis aims at contributing to the urban history field in order to fill certain gaps in the historiography of the city of São Paulo. By retracing the links between city, history and literature that seems to contain the work, the thesis contributes to the discussion of an urban cultural history of São Paulo. Recognizing the author as a key person in the establishment of the debate on the Latin American city between the 1940s and 1970s - precisely the period in which are held the three editions - it also allows the examination of the work and shed light on fundamental decades of urbanization in Latin America, taking up issues and questions about the ongoing metropolisation in the city of São Paulo and in the Latin American continent.

Swarms are groups of agents, which we model as point particles, whose collective behavior emerges from individual interactions. We study a first-order swarming model in a periodic coordinate system with pairwise social forces, investigating its stable configurations for differing numbers of agents relative to the periodic width. Two states emerge from numerical simulations in one dimension: even spacing throughout the period, or clumping within a certain portion of the period. A mathematical analysis of the energy of the system allows us to determine stability of these configurations. We also perform numerical simulations for evolution to equilibrium over time, and find results in agreement with our mathematical analysis. For certain values of the periodic width relative to the number of agents, our numerical simulations show that either clumping or even spacing can be stable equilibria, and which equilibrium is reached depends on on starting conditions, indicating hysteresis.

Variational structure plays an important role in critical point theory and methods. However many differential problems are non-variational i.e. they are not the Euler- Lagrange equations of any variational functionals, which makes traditional critical point approach not applicable. In this thesis, two types of non-variational problems, a nonlinear eigen solution problem and a non-variational semi-linear elliptic system, are studied. By considering nonlinear eigen problems on their variational energy profiles and using the implicit function theorem, an implicit minimax method is developed for numerically finding eigen solutions of focusing nonlinear Schrodinger equations subject to zero Dirichlet/Neumann boundary condition in the order of their eigenvalues. Its mathematical justification and some related properties, such as solution intensity preserving, bifurcation identification, etc., are established, which show some significant advantages of the new method over the usual ones in the literature. A new orthogonal subspace minimization method is also developed for finding multiple (eigen) solutions to defocusing nonlinear Schrodinger equations with certain symmetries. Numerical results are presented to illustrate these methods. A new joint local min orthogonal method is developed for finding multiple solutions of a non-variational semi-linear elliptic system. Mathematical justification and convergence of the method are discussed. A modified non-variational Gross-Pitaevskii system is used in numerical experiment to test the method.

In this master thesis the connection between the two major parts of an osseointegrated prosthesis system for lower limb amputees has been investigated by finite element (FE) analysis. The prosthesis system is developed by Integrum and the current design consists of a fixture, which is integrated in the residual bone, an abutment that penetrates the skin and an abutment screw that holds the parts together. The connection between the fixture and the abutment has a hexagonal section and a press-fit section that together form the connection. Due to wear and fracture problems it is desired to improve the connection. A tapered connection could be an alternative and three different taper angles, the effect of the length of the taper and the smoothness of the outer edge of a tapered fixture have been investigated. The results show that the taper has potential to function well and that a longer connection will give lower stresses in the system, but further investigations are needed.

In this thesis, We study the driven Morse system in a strong time-periodic field. We are interested in the quasi-bound states, which live in the driven system with limit life-times, with an increasing field strength in a low frequency region. We found those states by using Floquet theory, and the exterior complex scaling method (ECCS), which widely use in the resonance system. Choosing the Morse potential with supports 3 bound states, we found that as we increase the time-periodic external field, the number of the quasi-bound states decrease to 2. The distributions of the quasi-bound states which represented by the Husimi distribution were also studied, and compared with the Poincaré surface of section plots of the system. / text