The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).

[pt] O planejamento da expansão da transmissão tem por objetivo identificar o conjunto de novas linhas de transmissão que serão instaladas na rede elétrica, a fim de garantir a segurança e a qualidade do sistema, atendendo ao crescimento da demanda. No Brasil, o primeiro estágio do planejamento consiste em definir e avaliar o corredor para locação das linhas candidatas. Nesta etapa são definidas informações como cabo escolhido, preços médios, tempo de comissionamento, impactos socioambientais e informações sobre a região. No estágio seguinte ocorre o leilão de transmissão em que vários investidores competem pelos lotes de transmissão. No último estágio, o ganhador do leilão é o responsável pela instalação e comissionamento da linha de transmissão. Apesar do levantamento feito no estágio inicial, os projetos de linha de transmissão podem sofrer atrasos e aumentos no custo final, principalmente, por questões de licenciamentos ambientais, desvios do traçado devido a travessias e interferências com áreas protegidas. A presente dissertação propõe metodologia para dimensionamento e precificação da linha de transmissão, ambos de acordo com os critérios e banco de dados da ANEEL, de forma a acelerar e facilitar o processo de planejamento. Com o intuito de evitar atrasos e prever interferências socioambientais, a segunda parte do trabalho apresenta uma metodologia que consiste em localizar e identificar cruzamentos e interferências com áreas protegidas de caráter ambiental e/ou infraestrutural no território brasileiro, a partir das coordenadas do traçado da linha de transmissão em análise. A terceira parte do trabalho verifica, a partir dos resultados anteriores, o impacto no planejamento da expansão da transmissão através de um modelo de programação linear inteira mista considerando fluxo de potência CC, despacho econômico e critério N-1. / [en] The transmission expansion planning aims at identifying a set of new power lines to be installed on the grid, in order to guarantee the reliability and quality of the system, meeting the demand growth. In Brazil, the first stage of planning consists of defining e survey the corridor for the transmission line siting. At this stage it is defined data such as conductor chosen, average prices, commissioning time, socioenvironmental impacts and information about the region. In the following stage, occurs the auction which several investors compete for the transmission lots. In the last stage, the winner of the auction becomes responsible for the installation and commissioning. Despite the survey of the initial stage, the transmission lines undergo delays and additional costs, mainly, due to environmental licensing, changes of the defined route due to crossing and interferences with protected areas. The present dissertation proposes a methodology to design and price the transmission line, both accordingly to the ANEEL database and criteria, in order to accelerate and facilitate the planning process. Aiming to avoid delays and foreseeing socioenvironmental impacts, the second part of the dissertation presents a methodology that consists of locating and identifying crossings and interferences with protected areas of environmental and/or infrastructural aspect on the Brazilian territory, from the coordinates of the transmission line route in analysis. The third part verifies, from the previous results, the impact on the transmission expansion planning through a model based on a mixed-integer linear programming, considering a DC power flow, unit commitment and N-1 criteria

[pt] O Degree-Corrected Stochastic Block Model (DCSBM) é um modelo popular para geração de grafos aleatórios com estrutura de comunidade, dada uma sequência de graus esperados. O princípio básico de algoritmos que utilizam o DCSBM para detecção de comunidades é ajustar os parâmetros do modelo a dados observados, de forma a encontrar a estimativa de máxima verossimilhança, ou maximum likelihood estimate (MLE), dos parâmetros do modelo. O problema de otimização para o MLE é comumente resolvido por meio de heurísticas. Neste trabalho, propomos métodos de programação matemática, para resolver de forma exata o problema de otimização descrito, e comparamos os métodos propostos com heurísticas baseadas no algoritmo de expectation-maximization (EM). Métodos exatos são uma ferramenta fundamental para a avaliação de heurísticas, já que nos permitem identificar se uma solução heurística é sub-ótima e medir seu gap de otimalidade. / [en] The Degree-Corrected Stochastic Block Model (DCSBM) is a popular model to generate random graphs with community structure given an expected degree sequence. The standard approach of community detection algorithms based on the DCSBM is to search for the model parameters which are the most likely to have produced the observed network data, via maximum likelihood estimation (MLE). Current techniques for the MLE problem are heuristics and therefore do not guarantee convergence to the optimum. We present mathematical programming formulations and exact solution methods that can provably find the model parameters and community assignments of maximum likelihood given an observed graph. We compare the proposed exact methods with classical heuristic algorithms based on expectation-maximization (EM). The solutions given by exact methods give us a principled way of recognizing when heuristic solutions are sub-optimal and measuring how far they are from optimality.

A mixed integer linear programming formulation is used to solve the problem of allocating assets from a bank to its counterparties as collateral within securities lending. The aim of the optimisation is to reduce the cost of allocated collateral, which is broken down into the components opportunity cost, counterparty risk cost and triparty cost. A solution consists of transactions to carry out to improve the allocation cost, each transaction consisting of sending a quantity of some asset from a portfolio to the bank or from the bank to some portfolio. The optimisation process is split into subproblems to separate obvious transactions from more complex optimisations. The reduction of each cost component is examined for all the subproblems. Two subproblems transform an initial collateral allocation into a feasible one which is then improved by the optimisation. Decreasing opportunity cost is shown to be an easier task than decreasing counterparty risk and triparty costs since the latter costs require a comparatively large number of transactions. The optimisation is run several times in a row, performing the suggested transactions after each solved iteration. The cost reduction of k optimisation iterations with 10 transactions per iteration is shown to be similar to the cost reduction of 1 optimisation iteration with 10k transactions. The solution time increases heavily with the number of iterations. The suggested transactions may need to be performed in a certain order. A precedence constrained problem takes this into account. The problem is large and the execution time is slow if a limit is imposed on the number of allowed transactions. A strategic selection of portfolios can limit the number of suggested transactions and still reach a solution which comes close to the optimal one. This can also be done by requiring that all suggested transactions must reduce the cost by a minimum amount. The final model is ready to be used in a semi-automatic fashion, where transactions are verified by a human who checks if they are sound. A fully automated process requires further testing on historical and recent data. / Ett blandat-heltal linjärt optimeringsproblem används för att lösa uppgiften att tilldela värdepapper från en bank till dess kunder som pant för värdepapperslån. Målet med optimeringen är att minska kostnaden av den tilldelade panten. Kostnaden bryts ned i komponenterna alternativkostnad, motpartsrisk och tripartykostnad. En lösning består av föreslagna transaktioner som ska genomföras för att förbättra den nuvarande säkerhetstilldelningens kostnad. En transaktion består av att ta hem eller skicka ut en kvantitet av ett visst värdepapper från eller till en av bankens kunders portföljer. Optimeringsproblemet bryts ned i flera delproblem med syfte att särskilja uppenbara föreslagna säkerheter till en godkänd tildelning som sedan blir en startpunkt för optimeringen. Att minska alternativkostnad visar sig vara enklare än att minska motpartsrisk och tripartykostnader på så sätt att de sistnämnda kostnaderna kräver fler transaktioner för att minskas. Optimeringen körs flera gånger i rad, där alla föreslagna transaktioner från en iteration genomförs innan nästa iteration körs. Kostnadsminskningen av k körningar med 10 transaktioner visar sig vara väldigt nära, om än något mindre, än en körning med 10k transaktioner. Exekveringstiden ökar drastiskt med antalet iterationer. De föreslagna transaktionerna kan behöva genomföras i en viss ordning. En problemformulering konstrueras som tar höjd för detta, men exekveringstiden är extremt lång när antalet transaktioner begränsas. Ett strategiskt urval av portföljer kan begränsa antalet föreslagna transaktioner utan att försämra lösningen särskilt mycket. På ett liknande sätt kan antalet föreslagna transaktioner minskas genom att lägga till ett villkor som säger att lönsamheten av en transaktion måste överskrida en given minsta tröskel. Den slutgiltiga modellen är redo att användas om de föreslagna transaktionerna granskas manuellt innan de genomförs. En helt automatisk process ligger längre fram i tiden efter ytterligare tester på historisk och nuvarande data.

This study examined possibilities of preforming gerrymandering within Swedish municipal elections before 2015, when new regulations contributing to guaranteeing a proportional seat allocation and thereby eliminating the risk of gerrymandering altogether were introduced. As the configuration of the districting problems in Swedish elections follows that of spatial unit allocation, these problems could be modelled as integer-programming problems including previously formulated constraints guaranteeing necessary geometrical and topological properties of districts as well as additional objectives and constraints promoting desired seat allocations. For several of the examined municipalities districts solutions with none-suspicious shapes and a more desirable seat allocation for the considered parties given voting outcomes from the 2014 elections were found. The implication of these results for Swedish municipal elections prior to 2015 as well as other elections using similar seat allocation procedures would however be highly dependent on the ability to make detailed and accurate predictions of voting outcomes prior to actual elections.

Le besoin grandissant pour une planification urbaine plus durable et pour des interventions publiques visant à l'amélioration du bien-être collectif, ont grandement contribué à un engouement pour les espaces verts. Les parcs sont reconnus pour leur impact positif en zone urbaine dense, et nous sommes intéressés par l'application des concepts théoriques du domaine de la recherche opérationnelle pour assister les décideurs publics afin d'améliorer l'accessibilité, la distribution et la conception des parcs. Étant donné le contexte, nous sommes particulièrement motivés par le concept d'équité, et étudions le comportement des usagers des parcs à l'aide d'un modèle d'interaction spatiale, tel qu'appliqué dans les problèmes d'emplacement d'installations dans un marché compétitif. Dans cette recherche, nous présentons un modèle d'emplacement d'installations à deux étapes pouvant être adapté pour assister les décideurs publics à l'échelle de la ville. Nous étudions spécifiquement l'application aux espaces verts urbains, mais soulignons que des extensions du modèle peuvent permettre d'aborder d'autres problèmes d'emplacements d'installations sujets à des enjeux d'équité. La première étape de notre problème d'optimisation a pour but d'évaluer l'allocation la plus équitable du budget de la ville aux arrondissements, basé sur une somme du budget pondérée par des facteurs d'équité. Dans la deuxième étape du modèle, nous cherchons l'emplacement et la conception optimale des parcs, et l'objectif consiste à maximiser la probabilité totale que les individus visitent les parcs. Étant donné la non-linéarité de la fonction objective, nous appliquons une méthode de linéarisation et obtenons un modèle de programmation linéaire mixte en nombres entiers, pouvant être résolu avec des solveurs standards. Nous introduisons aussi une méthode de regroupement pour réduire la taille du problème, et ainsi trouver des solutions quasi optimales dans un délai raisonnable. Le modèle est testé à l'aide de l'étude de cas de la ville de Montréal, Canada, et nous présentons une analyse comparative des résultats afin de justifier la performance de notre modèle. / The recent promotion of sustainable urban planning combined with a growing need for public interventions to improve well-being and health in dense urban areas have led to an increased collective interest for green spaces. Parks have proven a wide range of benefits in urban areas, and we are interested in the application of theoretical concepts from the field of Operations Research to assist decision-makers to improve parks' accessibility, distribution and design. Given the context of public decision-making, we are particularly concerned with the concept of fairness, and are focused on an advanced assessment of users' behavior using a spatial interaction model (SIM) as in competitive facility locations' frameworks. In this research, we present a two-stage fair facility location and design (2SFFLD) model, which serves as a template model to assist public decision-makers at the city-level for the urban green spaces (UGSs) planning. We study the application of the 2SFFLD model to UGSs, but emphasize the potential extension to other applications to location problems concerned with fairness and equity. The first-stage of the optimization problem is about the optimal budget allocation based on a total fair-weighted budget formula. The second-stage seeks the optimal location and design of parks, and the objective consists of maximizing the total expected probability of individuals visiting parks. Given the non-linearity of the objective function, we apply a ``Method-based Linearization'' and obtain a mixed-integer linear program that can be solved with standard solvers. We further introduce a clustering method to reduce the size of the problem and determine a close to optimal solution within reasonable time constraints. The model is tested using the case study of the city of Montreal, Canada, and comparative results are discussed in detail to justify the performance of the model.

Plusieurs pays possèdent des programmes de don croisé de rein (PDCR). Le but de ces programmes est d’aider les patients ayant un donneur incompatible à obtenir une greffe, en échangeant les donneurs incompatibles entre les patients. Pour pouvoir obtenir des bassins de paires incompatibles de plus grande taille, il est possible d’élargir les PDCR pour y inclure plusieurs pays ou hôpitaux. Par contre, on doit s’attendre à ce que ces derniers agissent de façon stratégique pour maximiser le nombre de leurs patients obtenant une greffe. Avec ce cadre, on peut définir le problème de don croisé de rein à plusieurs agents. Dans ce mémoire, nous modélisons ce problème comme un jeu coopératif à utilité non- transférable et nous présentons le noyau faible comme solution à ce jeu. Nous étudions empiriquement notre solution sur des exemples basés sur des données réelles et montrons qu’elle est atteignable en pratique. Nous comparons aussi le noyau faible à une autre solution présente dans la littérature: les couplages résistants aux rejets. / In various countries, kidney paired donation programs (KPDs) are implemented. These programs aim to help patients with an incompatible donor to obtain a transplant by swapping the donors between the patients. In order to increase the size of the pool of incompatible patient-donor pairs and potentially enhance patient benefits, KPDs can be extended to include multiple countries or hospitals. However, unlike existing nationwide KPDs, strategic behaviour from these entities (agents) is to be expected. This gives rise to the multi-agent kidney exchange problem. In this work, we model for the first time this problem as a non-transferable utility game. We also propose and argue in favour of the use of the weak core as a solution concept for the game. Using integer programming tools, we empirically study our solution concept on instances from the literature, which are derived from real-world data, and show that it is attainable in practice. We also compare the weak core to another recently presented solution concept from the literature, the rejection-proof matching.

[pt] Problemas de emissão de gases de efeito estufa vem sido amplamente discutidos nos últimos anos, uma vez que mais de 70 países já se comprometeram a uma economia neutra em carbono até 2050. A eletrificação dos modais de transporte tem sido ampliadas seguindo essas metas, onde os Veículos Elétricos (VEs) começam a ganhar participação sobre o mercado de Veículos com Motor de Combustão Interna (VMCI) por todo o mundo. Além da particular complexidade na comparação entre VEs e VMCIs, desafios envolvendo a natureza dos VEs e sua integração com as cidades, como a falta de locais públicos para recarga, também são críticos e interferem no seu desenvolvimento. Nesse contexto, este trabalho visa estudar o problema de uma Estação de Troca de Baterias (ETB), uma estrutura onde os usuários de VEs trocam suas baterias descarregadas por outras totalmente ou parcialmente carregadas. No intuito de simular as operações diárias do ETB e o cronograma de carregamento das baterias, um novo modelo de Programação Linear Inteira Mista (PLIM) é proposto, levando em consideração a heterogeneidade da bateria, o uso de geração fotovoltaica (PV) local e a degradação da bateria com base no perfil de carregamento. Uma coleção de métricas de operação do ETB é projetada para avaliar a qualidade da solução do modelo de cronograma proposto. É apresentado um experimento numérico que compreende quatro estudos de caso baseados em dados reais dos sistemas de energia e transporte dos EUA, contendo insights e análises sobre o uso da energia fotovoltaica e da rede, bem como uma comparação financeira do ETB com abordagens de cronograma de benchmarks relacionados, juntamente de sensibilidades no plano de dimensionamento do ETB e atendimento a clientes. / [en] Greenhouse gas emissions-related issues have been extensively discussed in the past years, with over 70 countries already committed to a carbon-neutral economy by 2050. The electrification of transportation modals has increased following these goals, where Electric Vehicles (EVs) are starting to take Internal Combustion Engine Vehicles (ICEV) market share all over the globe. Besides the particular complexity in comparing EVs and ICEVs, challenges involving the nature of EVs and their integration with cities, such as the lack of public locals for charging, are also critical and interfere with their development. In this context, this work aims at studying the problem of a Battery Swapping Station (BSS), a structure where the EVs users swap their depleted batteries for fully or partially charged ones. In order to simulate the BSS daily operations and batteries charging schedule, a novel Mixed Integer Linear Programming (MILP) model is proposed, taking into account battery heterogeneity, the use of local photovoltaic (PV) production and battery degradation based on charging profile. A collection of BSS operation metrics are designed to evaluate the solution quality of the proposed scheduling model. A numerical experiment comprising four case studies based on real data from the US power and transportation systems is presented, with insights and analyses on the PV and grid power use, as well as a BSS financial comparison against close-related benchmark scheduling approaches, together with sensitivities on BSS sizing plan and costumers attendance.

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