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We exhibit multiple periodic, collision-based orbits of the Newtonian n-body problem. Many of these orbits feature regularizable collisions between the masses. We demonstrate existence of the periodic orbits after performing the appropriate regularization. Stability, including linear stability, for the orbits is then computed using a technique due to Roberts. We point out other interesting features of the orbits as appropriate. When applicable, the results are extended to a broader family of orbits with similar behavior.

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[pt] Este estudo aborda um tipo de conflito contemporâneo muito comum na cidade do Rio de Janeiro e no Brasil: as moradias informais. O problema será tratado a partir do conflito em torno das moradias informais no Jardim Botânico do Rio de Janeiro, objeto de estudo da dissertação. O caso envolve um conjunto complexo de atores: o Estado, moradores informais, proprietários e uma grande corporação do setor de comunicação. Com o estudo do problema das moradias informais no Jardim Botânico, observou-se a falta de cuidado do poder público em não permitir que invasões ocorram, depredando assim um bem público como o JBRJ e o Horto Florestal no que tange aos seus talhões e reservas ambientais. Mas o conflito guarda também uma peculiaridade: uma autorização do gestor da instituição para a construção de imóveis de seus funcionários. Como a pesquisa demonstra, o conflito em torno da moradia é construído em torno de duas lógicas distintas: ancestralidade e propriedade. Com o tempo e sem mais nenhum vínculo empregatício junto a instituição em questão, emergiram conflitos acerca da legitimidade para a cessão do direito real de uso do imóvel ou sobre quais benfeitorias poderiam ser indenizáveis, sobre quais os parâmetros desta indenização ou ainda sobre qual seria o local digno para reassentamento desta comunidade. Finalmente, o estudo permitiu compreender os limites do direito para a resolução desse tipo de conflito e a necessidade de um processo de mediação que reconheça a legitimidade das posições presentes no conflito e consiga produzir uma solução inclusiva e efetiva. / [en] This study addresses a type of contemporary conflict that is very common in the city of Rio de Janeiro and in Brazil: informal housing. The problem will be addressed from the conflict surrounding informal housing in the Botanical Garden of Rio de Janeiro, the object of study of the dissertation. The case involves a complex set of actors: the State, informal residents, property owners, and a large communication corporation. By studying the problem of informal housing in the Botanical Garden, it was possible to observe the lack of preoccupation by the public authorities in not allowing invasions to occur, thus depredating a public asset such as the Botanical Garden and the Horto Florestal in regards to its plots and environmental reserves. But the conflict also has a peculiarity: an authorization by the institution s manager for the construction of real estate for his employees. As the research shows, the conflict over housing is built around two distinct logics: ancestry and property. With time and without any further employment ties with the institution in question, conflicts have emerged over the legitimacy of the assignment of the right of use of the property, or over which improvements could be compensated, over the parameters of this compensation, or even over what would be a decent place for resettlement for this community. Finally, the study allowed us to understand the limits of the law in resolving this type of conflict and theneed for a mediation process that recognizes the legitimacy of the positions present in the conflict and manages to produce an inclusive and effective solution.

This thesis concerns the approximation of optimally controlled partial differential equations for applications in optimal design and reconstruction. Such optimal control problems are often ill-posed and need to be regularized to obtain good approximations. We here use the theory of the corresponding Hamilton-Jacobi-Bellman equations to construct regularizations and derive error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method where the first, analytical, step is to regularize the Hamiltonian. Next its stationary Hamiltonian system, a nonlinear partial differential equation, is computed efficiently with the Newton method using a sparse Jacobian. An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems. In the thesis we present solutions to applications such as optimal design and reconstruction of conducting materials and elastic structures. / QC 20101110

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Recommender systems are information filtering tools that discover potential matchings between users and items and benefit both parties. This benefit can be considered a social resource that should be equitably allocated across users and items, especially in critical domains such as education and employment. Biases and unfairness in recommendations raise both ethical and legal concerns. In this dissertation, we investigate the concept of unfairness in the context of recommender systems. In particular, we study appropriate unfairness evaluation metrics, examine the relation between bias in recommender models and inequality in the underlying population, as well as propose effective unfairness mitigation approaches. We start with exploring the implication of fairness in recommendation and formulating unfairness evaluation metrics. We focus on the task of rating prediction. We identify the insufficiency of demographic parity for scenarios where the target variable is justifiably dependent on demographic features. Then we propose an alternative set of unfairness metrics that measured based on how much the average predicted ratings deviate from average true ratings. We also reduce these unfairness in matrix factorization (MF) models by explicitly adding them as penalty terms to learning objectives. Next, we target a form of unfairness in matrix factorization models observed as disparate model performance across user groups. We identify four types of biases in the training data that contribute to higher subpopulation error. Then we propose personalized regularization learning (PRL), which learns personalized regularization parameters that directly address the data biases. PRL poses the hyperparameter search problem as a secondary learning task. It enables back-propagation to learn the personalized regularization parameters by leveraging the closed-form solutions of alternating least squares (ALS) to solve MF. Furthermore, the learned parameters are interpretable and provide insights into how fairness is improved. Third, we conduct theoretical analysis on the long-term dynamics of inequality in the underlying population, in terms of the fitting between users and items. We view the task of recommendation as solving a set of classification problems through threshold policies. We mathematically formulate the transition dynamics of user-item fit in one step of recommendation. Then we prove that a system with the formulated dynamics always has at least one equilibrium, and we provide sufficient conditions for the equilibrium to be unique. We also show that, depending on the item category relationships and the recommendation policies, recommendations in one item category can reshape the user-item fit in another item category. To summarize, in this research, we examine different fairness criteria in rating prediction and recommendation, study the dynamic of interactions between recommender systems and users, and propose mitigation methods to promote fairness and equality. / Doctor of Philosophy / Recommender systems are information filtering tools that discover potential matching between users and items. However, a recommender system, if not properly built, may not treat users and items equitably, which raises ethical and legal concerns. In this research, we explore the implication of fairness in the context of recommender systems, study the relation between unfairness in recommender output and inequality in the underlying population, and propose effective unfairness mitigation approaches. We start with finding unfairness metrics appropriate for recommender systems. We focus on the task of rating prediction, which is a crucial step in recommender systems. We propose a set of unfairness metrics measured as the disparity in how much predictions deviate from the ground truth ratings. We also offer a mitigation method to reduce these forms of unfairness in matrix factorization models Next, we look deeper into the factors that contribute to error-based unfairness in matrix factorization models and identify four types of biases that contribute to higher subpopulation error. Then we propose personalized regularization learning (PRL), which is a mitigation strategy that learns personalized regularization parameters to directly addresses data biases. The learned per-user regularization parameters are interpretable and provide insight into how fairness is improved. Third, we conduct a theoretical study on the long-term dynamics of the inequality in the fitting (e.g., interest, qualification, etc.) between users and items. We first mathematically formulate the transition dynamics of user-item fit in one step of recommendation. Then we discuss the existence and uniqueness of system equilibrium as the one-step dynamics repeat. We also show that depending on the relation between item categories and the recommendation policies (unconstrained or fair), recommendations in one item category can reshape the user-item fit in another item category. In summary, we examine different fairness criteria in rating prediction and recommendation, study the dynamics of interactions between recommender systems and users, and propose mitigation methods to promote fairness and equality.

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In many real-world applications, it is common to have uneven number of examples among multiple classes. The data imbalance, however, usually complicates the learning process, especially for the minority classes, and results in deteriorated performance. Boosting methods were proposed to handle the imbalance problem. These methods need elongated training time and require diversity among the classifiers of the ensemble to achieve improved performance. Additionally, extending the boosting method to handle multi-class data sets is not straightforward. Examples of applications that suffer from imbalanced multi-class data can be found in face recognition, where tens of classes exist, and in capsule endoscopy, which suffers massive imbalance between the classes. This dissertation introduces RegBoost, a new boosting framework to address the imbalanced, multi-class problems. This method applies a weighted stratified sampling technique and incorporates a regularization term that accommodates multi-class data sets and automatically determines the error bound of each base classifier. The regularization parameter penalizes the classifier when it misclassifies instances that were correctly classified in the previous iteration. The parameter additionally reduces the bias towards majority classes. Experiments are conducted using 12 diverse data sets with moderate to high imbalance ratios. The results demonstrate superior performance of the proposed method compared to several state-of-the-art algorithms for imbalanced, multi-class classification problems. More importantly, the sensitivity improvement of the minority classes using RegBoost is accompanied with the improvement of the overall accuracy for all classes. With unpredictability regularization, a diverse group of classifiers are created and the maximum accuracy improvement reaches above 24%. Using stratified undersampling, RegBoost exhibits the best efficiency. The reduction in computational cost is significant reaching above 50%. As the volume of training data increase, the gain of efficiency with the proposed method becomes more significant.

Load control strategies for wind turbines are used to reduce the structural wear of the turbine without reducing energy yield. Until now, these control strategies are almost exclusively built up-on linear approaches like PID and model-based controllers due to their robustness. However, advances in turbine size and capabilities create a need for more complex control strategies that can effectively address design challenges in modern turbines. This work presents WINDL, a load control policy based on a neural network, which is trained through model-free Reinforcement Learning (RL) on a simulated wind turbine. While RL has achieved great success in the past on games and simple simulation benchmarks, applications to more complex control problems are starting to emerge just recently. We show that through the usage of regularization techniques and signal transformations, such an application to the field of wind turbine load control is possible. Using a smoothness regularizer, we incentivize the highly non-linear neural network policy to output control actions that are safe to apply to a wind turbine. The Coleman transformation, a common tool for the design of traditional PID-based load control strategies, is used to project signals into a stationary coordinate space, increasing robustness and final policy performance. Trained to control a large offshore turbine in a model-free fashion, WINDL finds a control policy that outperforms a state-of-the-art controller based on the IPC strategy with respect to the prima-ry optimization goal blade loads. Using the DEL metric, we measure 54.1% lower blade loads in the steady wind and 13.45% lower blade loads in the turbulent wind scenario. While such levels of blade reduction come with slightly worse performance on secondary optimi-zation goals like pitch wear and power production, we demonstrate the ability to control the trade-off between different optimization goals on the example of pitch versus blade loads. To comple-ment our findings, we perform a qualitative analysis of the policy behavior and learning process. We believe our work to be the first application of RL to wind turbine load control that exceeds baseline performance in the primary optimization metric, opening up the possibility of including specialized load controllers for targeting critical design driving scenarios of modern large wind turbines.:Problem Method Aim Results Conclusion