This thesis investigates the properties of the L₂-eigenvalues of singular, elliptic, second order operators, primarily the operator L defined by [formula omitted]. Here the "potential function", V(x), is such that [formula omitted] is a norm on [formula omitted] being the usual norm in the Sobolev space W¹̕²(G) and [formula omitted] is the completion of [formula omitted] in the metric from this norm, identified with a subset L₂(G) ; Δ is the Laplacian and G is an arbitrary open domain of E[superscript n] . Several sufficient conditions are given on V and on G in order that L have spectrum satisfying [formula omitted] , for some real number [formula omitted] denote the spectrum and point spectrum of L , respectively). The properties of these lower eigenvalues are investigated by examining the eigenvalues of a coercive bilinear form corresponding to the operator. Such a form B , having domain [symbol omitted] , say, is defined to have eigenvalueλє¢ with corresponding eigenfunction [symbols omitted] if B[u,f] = λ (u,f) for all f є [symbol omitted] . Variational properties are discussed in detail; In particular, a condition is given which ensures that the numbers sup inf B[u,u] (the sup and inf being over appropriate sets involving [symbol omitted] and n ) are eigenvalues of B . These properties are applied to L to generalize the well-known classical property (G bounded) of monotonic dependence of the eigenvalues on the underlying domain G : G [symbol omitted] G* implies [formula omitted] for corresponding eigenvalues, with strict inclusion implying strict inequality. A few miscellaneous properties of the eigenvalues and eigenfunctions then follow from this dependence. / Science, Faculty of / Mathematics, Department of / Graduate

George Zames. / "Submitted to the Department of Electrical Engineering, M.I.T., August 22, 1960, in partial fulfillment of the requirements for the degree of Doctor of Science." / Bibliography: p.76. / Army Signal Corps Contract DA 36-039-sc-78108. Dept. of the Army Task 3-99-20-001 and Project 3-99-00-000.

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In this thesis we study a mathematical model that describes task partitioning in a colony of ants. This process of self-organization is modeled by a nonlinear coupled system of rst order autonomous ordinary dierential equations. We discuss how this system of equations can be derived based on the behavior of ants in a colony. We use GNU Octave (a high-level programming language) to solve the system of equations numerically for dierent sets of parameters and show how the solutions respond to changes in the parameter values. Finally, we prove that the model is well-posed locally in time. We rewrite the system of ordinary dierential equations in terms of a system of coupled Volterra integral equations and look at the right-hand side of the system as a nonlinear operator on a Banach space. By doing so, we have transformed the problem of showing existence and uniqueness of solutions to a system of ordinary dierential equations into a problem of showing existence and uniqueness of a xed point to the corresponding integral operator. Additionally, we use Gronwall's inequality to prove the stability of solutions with respect to data and parameters.

[pt] Seja F uma função definida de um espaço de Banach real X para um espaço de Banach real Y e g um ponto pertencente a Y. Descrevemos um algoritmo para calcular as soluções u da equação F de u igual a g. Inicialmente, o algoritmo parte de uma curva c no domínio, a qual é escolhida de modo a interceptar substancialmente o conjunto crítico de F. Calculamos através de métodos de continuação uma componente da imagem inversa de F de c e definimos essa componente de forma abstrata: grafo completamente espelhado. Claramente, os métodos de continuação padrão têm melhores chances de sucesso em diferentes pontos iniciais. Fornecemos argumentos geométricos para a abundância ocasional de soluções e uma busca estruturada dessas. Três exemplos são considerados detalhadamente. O primeiro é uma função do plano no plano, em que podemos validar os resultados com auxílio de um software. O segundo conjunto de exemplos é obtido a partir da discretização de um problema de Sturm-Liouville não linear com um número inesperado de soluções. Por último, calculamos as seis soluções aproximadas de um problema estudado por Solimini. / [en] Let F be a definite function from a real Banach space X to a real Banach space Y and g a point belonging to Y. We describe an algorithm for calculating the solutions u of the equation F of u equal to g. Initially, the algorithm starts from a curve c in the domain, which is chosen so as to substantially intercept the critical set of F. We calculate through continuation methods a component of the inverse image of F of c and define this component in an abstract way: graph completely mirrored. Clearly, standard continuation methods have better chances of success at different starting points. We provide geometric arguments for the occasional abundance of solutions and a structured search for these. Three examples are considered in detail. The first is a function of the plan in the plan, in which we can validate the results with the help of software. The second set of examples is obtained from the discretization of a non-linear Sturm-Liouville problem with an unexpected number of solutions. Finally, we calculate the six approximate solutions of a problem studied by Solimini.

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Nonlinear four-wave interactions (Snl) are critical for acquiring realistic spectra needed by operational wave models. High computational demand to calculate these interactions led to an approximation method named the Discrete Interaction Approximation (DIA) to be used broadly in the major operational wave models for a long time. However, the accuracy of the DIA has been controversial since it was first introduced and more precise approximations such as the Two Scale Approximation (TSA in short) are now available. The only issue with the initial TSA`s efficiency is performing an order of a magnitude slower than the DIA in speed. This study questions the exactness of the DIA while trying to increase the competence of the TSA by making improvements on its execution time. Particularly, in this thesis, the main effort is on the local scale term of the TSA since it is the part that consumes the most time while running the code. The findings of this work imply that the TSA can improve its operation speed significantly while maintaining its accuracy with making alterations in the code. By decreasing the number of bands in the local scale it is possible to run the TSA up to 7.5 faster than its initial version.