In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

Kawasaki has calculated the integral homology groups of weighted projective space, and his results imply the existence of a homotopy equivalence between weighted projective space and a CW-complex, with a single cell in each even dimension less than or equal to that of weighted projective space. When the weights satisfy certain divisibility conditions then the associated weighted projective space is actually homeomorphic to such an minimal CW-complex and such decompositions are well-known in these cases. Otherwise this minimal CW-complex is not a weighted projective space. Our aim is to give an explicit CW-structure on any weighted projective space, using an invariant decomposition of complex projective space with respect to the action of a product of finite cyclic groups. The result has many cells, in both odd and even dimensions; nevertheless, we identify it with a subdivision of the minimal decomposition whenever the weights are divisive. We then extend the decomposition to stunted weighted projective space, defined as the quotient of one weighted projective space by another. Finally, we compute the integral homology groups of stunted weighted projective space, identify generators in terms of cellular cycles, and describe cup products in the corresponding cohomology ring.

<p>The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.</p>

Design optimality for treatment comparison experiments has been intensively studied by numerous researchers, employing a variety of statistically sound criteria. Their general formulation is based on the idea that optimality functions of the treatment information matrix are invariant to treatment permutation. This implies equal interest in all treatments. In practice, however, there are many experiments where not all treatments are equally important. When selecting a design for such an experiment, it would be better to weight the information gathered on different treatments according to their relative importance and/or interest. This dissertation develops a general theory of weighted design optimality, with special attention to the block design problem. Among others, this study develops and justifies weighted versions of the popular A, E and MV optimality criteria. These are based on the weighted information matrix, also introduced here. Sufficient conditions are derived for block designs to be weighted A, E and MV-optimal for situations where treatments fall into two groups according to two distinct levels of interest, these being important special cases of the "2-weight optimality" problem. Particularly, optimal designs are developed for experiments where one of the treatments is a control. The concept of efficiency balance is also studied in this dissertation. One view of efficiency balance and its generalizations is that unequal treatment replications are chosen to reflect unequal treatment interest. It is revealed that efficiency balance is closely related to the weighted-E approach to design selection. Functions of the canonical efficiency factors may be interpreted as weighted optimality criteria for comparison of designs with the same replication numbers. / Ph. D.

Design problems involve finding optimal plans that minimize cost and maximize information about the effects of changing experimental variables on some response. Information is typically measured through statistically meaningful functions, or criteria, of a design's corresponding information matrix. The most common criteria implicitly assume equal interest in all effects and certain forms of information matrices tend to optimize them. However, these criteria can be poor assessments of a design when there is unequal interest in the experimental effects. Morgan and Wang (2010) addressed this potential pitfall by developing a concise weighting system based on quadratic forms of a diagonal matrix W that allows a researcher to specify relative importance of information for any effects. They were then able to generate a broad class of weighted optimality criteria that evaluate a design's ability to maximize the weighted information, ultimately targeting those designs that efficiently estimate effects assigned larger weight. This dissertation considers a much broader class of potential weighting systems, and hence weighted criteria, by allowing W to be any symmetric, positive definite matrix. Assuming the response and experimental effects may be expressed as a general linear model, we provide a survey of the standard approach to optimal designs based on real-valued, convex functions of information matrices. Motivated by this approach, we introduce fundamental definitions and preliminary results underlying the theory of general weighted optimality. A class of weight matrices is established that allows an experimenter to directly assign weights to a set of estimable functions and we show how optimality of transformed models may be placed under a weighted optimality context. Straightforward modifications to SAS PROC OPTEX are shown to provide an algorithmic search procedure for weighted optimal designs, including A-optimal incomplete block designs. Finally, a general theory is given for design optimization when only a subset of all estimable functions is assumed to be in the model. We use this to develop a weighted criterion to search for A-optimal completely randomized designs for baseline factorial effects assuming all high-order interactions are negligible. / Ph. D.

No description available.

Computational modeling is of fundamental significance in mapping possible disease spread, and designing strategies for its mitigation. Conventional contact networks implement the simulation of interactions as random occurrences, presenting public health bodies with a difficult trade off between a realistic model granularity and robust design of intervention strategies. Recently, researchers have been investigating the use of agent-based models (ABMs) to embrace the complexity of real world interactions. At the same time, theoretical approaches provide epidemiologists with general optimization models in which demographics are intrinsically simplified. The emerging study of affiliation networks and co-affiliation networks provide an alternative to such trade off. Co-affiliation networks maintain the realism innate to ABMs while reducing the complexity of contact networks into distinctively smaller k-partite graphs, were each partition represent a dimension of the social model. This dissertation studies the optimization of intervention strategies for infectious diseases, mainly distributed in school systems. First, concepts of synthetic populations and affiliation networks are extended to propose a modified algorithm for the synthetic reconstruction of populations. Second, the definition of multi-coaffiliation networks is presented as the main social model in which risk is quantified and evaluated, thereby obtaining vulnerability indications for each school in the system. Finally, maximization of the mitigation coverage and minimization of the overall cost of intervention strategies are proposed and compared, based on centrality measures.

No description available.

Learning without any prior knowledge in environments that contain large or continuous state spaces is a daunting task. For robots that operate in the real world, learning must occur in a reasonable amount of time. Providing a robot with domain knowledge and also with the ability to learn from watching others can greatly increase its learning rate. This research explores learning algorithms that can learn quickly and make the most use of information obtained from observing others. Domain knowledge is encoded in the form of primitives, small parts of a task that are executed many times while a task is being performed. This thesis explores and presents many challenges involved in programming robots to learn and adapt to environments that humans operate in. A "Learning from Observation Using Primitives" framework has been created that provides the means to observe primitives as they are performed by others. This information is used by the robot in a three level process as it performs in the environment. In the first level the robot chooses a primitive to use for the observed state. The second level decides on the manner in which the chosen primitive will be performed. This information is then used in the third level to control the robot as necessary to perform the desired action. The framework also provides a means for the robot to observe and evaluate its own actions as it performs in the environment which allows the robot to increase its performance of selecting and performing the primitives. The framework and algorithms have been evaluated on two testbeds: Air Hockey and Marble Maze. The tasks are done both by actual robots and in simulation. Our robots have the ability to observe humans as they operate in these environments. The software version of Air Hockey allows a human to play against a cyber player and the hardware version allows the human to play against a 30 degree-of-freedom humanoid robot. The implementation of our learning system in these tasks helps to clearly present many issues involved in having robots learn and perform in dynamic environments.

In this paper, we cite two weighted rank distances (Wilcoxon rank and Log rank) to analyze how many times must a deck of 52 cards be shuffled to become sufficiently randomized. Bayer and Diaconis (1992) used the variation distance as a measure of randomness to analyze the card-shuffling. Lin (2006) used the deviation distance to analyze card-shuffling without complicated mathematics formulas. We provide two new ideas to measure the distance for card-shuffling analysis.