In this dissertation, we investigate methods of modifying a tight frame sequence on a finite subset of the frame so that the result is a tight frame with better properties. We call this a surgery on the frame. There are basically three types of surgeries: transplants, expansions, and contractions. In this dissertation, it will be necessary to consider surgeries on not-necessarily-tight frames because the subsets of frames that are excised and replaced are usually not themselves tight frames on their spans, even if the initial frame and the final frame are tight. This makes the theory necessarily complicated, and richer than one might expect. Chapter I is devoted to an introduction to frame theory. In Chapter II, we investigate conditions under which expansion, contraction, and transplant problems have a solution. In particular, we consider the equiangular replacement problem. We show that we can always replace a set of three unit vectors with a set of three complex unit equiangular vectors which has the same Bessel operator as the Bessel operator of the original set. We show that this can not always be done if we require the replacement vectors to be real, even if the original vectors are real. We also prove that the minimum angle between pairs of vectors in the replacement set becomes largest when the replacement set is equiangular. Iterating this procedure can yield a frame with smaller maximal frame correlation than the original. Frames with optimal maximal frame correlation are called Grassmannian frames and no general method is known at the present time for constructing them. Addressing this, in Chapter III we introduce a spreading algorithm for finite unit tight frames by replacing vectors three-at-a-time to produce a unit tight frame with better maximal frame correlation than the original frame. This algorithm also provides a “good” orientation for the replacement sets. The orientation part ensures stability in the sense that if a selected set of three unit vectors happens to already be equiangular, then the algorithm gives back the same three vectors in the original order. In chapter IV and chapter V, we investigate two special classes of frames called push-out frames and group frames. Chapter VI is devoted to some mathematical problems related to the ”cocktail party problem ”.

[pt] A gestão de ativos é uma abordagem crucial para o desempenho das organizações uma vez que buscam alinhar aspectos técnicos de engenharia com conceitos financeiros para otimizar o ciclo de vida de uma máquina. O Problema de Substituição de Equipamentos é uma das questões tratadas dentro dos estudos de gestão de ativos que visa decidir a melhor opção entre manter ou substituir o equipamento em um determinado intervalo de tempo. Uma das metodologias que vêm sendo utilizadas na literatura para solucionar este problema é a Programação Dinâmica, que se baseia em encontrar soluções parciais em uma série de estágios do problema até alcançar a ótima global. Este trabalho teve como objetivo determinar uma curva de substituição para um conjunto de locomotivas de uma empresa do setor ferroviário, considerando um limite de idade para poderem circular e seus históricos de receitas e custos ao longo dos anos. Os resultados alcançados permitiram que a empresa conhecesse a melhor forma de otimizar seu capital, levando em consideração os impactos financeiros caso opte por antecipar ou postergar o momento ótimo para substituição dos ativos. / [en] Asset management is a crucial approach for the performance of organizations as they seek to align technical aspects of engineering with financial concepts to optimize the life cycle of a machine. The Equipment Replacement Problem is one of the issues addressed within asset management studies that aims to decide the best option between maintaining or replacing equipment in a given time interval. One of the methodologies that have been used in the literature to solve this problem is Dynamic Programming, which is based on finding partial solutions in a series of stages of the problem until reaching the global optimum. This work aimed to determine a substitution curve for a set of locomotives of a company in the railway sector, considering an age limit for them to circulate and their revenue and cost history over the years. The results achieved allowed the company to know the best way to optimize its capital, taking into account the financial impacts if it chooses to anticipate or postpone the optimal moment for the replacement of assets.