Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Tetali, Prasad; Committee Member: Duke, Richard; Committee Member: Heitsch, Christine; Committee Member: Randall, Dana; Committee Member: Trotter, William T.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 115-119).

The main results of this dissertation are various algorithms related to partially ordered sets. The dissertation basically consists of two parts. The first part treats algorithms that generate ideals of partially ordered sets. The second part concerns the generation of partially ordered sets themselves. First, we present two algorithms for listing ideals of a forest poset. These algorithms generate ideals in a Gray Code manner, that is, consecutive ideals differ by exactly one element. Both algorithms use storage O(n), where n is the number of elements in the poset. The first algorithm traverses, at each phase, the current ideal being listed and runs in time O(nN), where N is the number of ideals of the poset. The second algorithm mimics the first but eliminates the traversal and runs in time O(N). This algorithm has the property that the amount of computation between successive ideals is O(1). Secondly, we give orderly algorithms for constructing acyclic digraphs, acyclic transitive digraphs, finite topologies and finite topologies and finite lattices. For the first time we show that the number of finite lattices on 11, 12, and 13 elements are 37622, 262775, and 2018442, respectively, and the number of finite topologies on 8 and 9 elements are 35979 and 363083, respectively. We also describe orderly algorithms for generating k-colored graphs. We present, in particular, an algorithm for generating connected bicolorable graphs. We also prove some properties of a canonic matrix which might be generally useful for improving the efficiency of orderly algorithms. / Graduate

Autonomous agents may not have access to complete information about the state of the environment. For example, a robot soccer player may only be able to estimate the locations of other players not in the scope of its sensors. However, even though all the information needed for ideal decision making cannot be sensed, all that is sensed is usually not needed. The noise and motion of spectators, for example, can be ignored in order to focus on the game field. Standard formulations do not consider this situation, assuming that all the can be sensed must be included in any useful abstraction. This dissertation extends the Markov Decision Process Homomorphism framework (Ravindran, 2004) to partially observable domains, focusing specically on reducing Partially Observable Markov Decision Processes (POMDPs) when the model is known. This involves ignoring aspects of the observation function which are irrelevant to a particular task. Abstraction is particularly important in partially observable domains, as it enables the formation of a smaller domain model and thus more efficient use of the observed features.

<p> In this thesis we introduce the concept of a dense extension of a partially ordered set and study some of the properties of the resulting class of extensions. In particular we study the dense distributive extensions, dense Boolean extensions and dense meet continuous extensions of distributive, Boolean and meet continuous lattices respectively.</p> / Thesis / Doctor of Philosophy (PhD)

The major results of this work concern perfect ideals of ordered vector spaces, and a representation theory for ordered vector spaces. Perfect ideals are characterized by the property that their annihilators in the order dual are ideals. We obtain a number of conditions for an ordered vector space which are equivalent to the intersection of the set of perfect maximal ideals being 0. We also obtain conditions which permit an ordered vector space to be represented as a subspace of the sections of a vector bundle. This generalizes the representation theory for odered vector spaces with unit. / Thesis / Doctor of Philosophy (PhD)

Nonsymmetric Macdonald polynomials can be symmetrized in all their variables to obtain the (symmetric) Macdonald polynomials. We generalize this process, symmetrizing the nonsymmetric Macdonald polynomials in only the first k out of n variables. The resulting partially-symmetric Macdonald polynomials interpolate between the symmetric and nonsymmetric types. We begin developing theory for these partially-symmetric polynomials, and prove results including their stability, an integral form, and a Pieri-like formula for their multiplication with certain elementary symmetric functions. / Doctor of Philosophy / There are two well-understood types of polynomials known as the nonsymmetric Macdonald polynomials and symmetric Macdonald polynomials. We define a new form of Macdonald polynomials, which we call partially-symmetric, that are somewhere between the symmetric and nonsymmetric versions. We examine properties of these new partially-symmetric polynomials, including what happens when adding additional symmetric variables, how to multiply them by a constant to clear out denominators in their coefficients, and what happens when multiplying them by another symmetric polynomial.

This thesis is concerned with the development of two generalized set covering models. The first model is formulated for the total set covering problem where cost is minimized subject to the constraint that each customer must be served by at least one facility. The second model is constructed for the partial set covering problem in which customer coverage is maximized subject to a budget constraint. The conventional formulations of both the total set covering and partial set covering problems are shown to be special cases of the two generalized models that arc developed. Appropriate solution strategies arc discussed for each generalized model. A specialized algorithm for a particular case of the partial covering problem is constructed and computational results are presented. / M.S.

Receiver operating characteristic (ROC) curves are widely used in medical decision making. It was recognized in the last decade that only a specific region of the ROC curve is of clinical interest, which can be summarized by the partial area under the ROC curve (partial AUC). Early statistical methods for evaluating partial AUC assume that the data are from a specified underlying distribution. Nonparametric estimators of the partial AUC emerged recently, but there are theoretical issues to be addressed. In this dissertation, we propose two new nonparametric statistics, partially integrated ROC and partially integrated weighted ROC, for estimating partial AUC. We show that our partially integrated ROC statistic is a consistent estimator of the partial AUC, and derive its asymptotic distribution which is distribution free under the null hypothesis. In the partially integrated ROC statistic, when the ROC curve crosses the Uniform distribution function (CDF) and if the partial area evaluated contains the crossing point, or when there are multiple crossing, the partially integrated ROC statistic might not perform well. To address this issue, we propose the partially integrated weighted ROC statistic. This statistic evaluates the partially weighted AUC, where larger weight is given when the ROC curve is above the Uniform CDF and smaller weight is given when the ROC curve is below the Uniform CDF. We show that our partially integrated weighted ROC statistic is a consistent estimator of the partially weighted AUC. We derive its asymptotic distribution which is distribution free under the null hypothesis. We propose to apply our two nonparametric statistics to functional category analysis in microarray experiments. We define the functional category analysis to be the statistical identification of over-represented functional gene categories in a microarray experiment based on differential gene expression. We compare our statistics with existing methods for the functional category analysis both via simulation study and application to a real microarray data, and demonstrate that our two statistics are effective for identifying over-represented functional gene categories. We also emphasize the essential role of the empirical distribution function plots and the ROC curves in the functional category analysis.

Different scenarios for assembling rod-like and spherical colloidal particles using binary mixtures of partially miscible liquids were investigated experimentally. Suitable rod-like colloids were developed first. The subsequent studies of colloids in binary liquids consisted, on one hand, of systems where particles were partially wetted by both phases and, on the other hand, of systems where particles were completely wetted by the minority phase. A simple method to prepare large quantities of micrometer-sized akagan eite-silica core-shell rods was developed. These were proven to be very versatile, with the possibility of modifying their properties on different levels. The aspect ratio is simply controlled by a gradual growth of the silica shells. From them, hollow silica rods and rods with an increased responsiveness to a magnetic field could be obtained in straightforward ways. Bijels were prepared by trapping rod-like particles on a percolating liquid-liquid interface. The familiar bicontinuous organization of liquid domains was observed after structural arrest. At a fixed volume per particle it is demonstrated that for rod-like particles the domain size decreases faster with increasing quantity of particles than in the case of spherical particles. Additionally, the packing of the rods at the interface was elucidated, revealing several characteristic features. In particle-stabilized droplet emulsions rapid evaporation of the continuous phase and eventual full mixing of the liquid phases can leave a cellular network of particles. The formation and eventual stability of these networks were investigated in detail with confocal microscopy. When colloids are completely wetted by the minority component of an asymmetric binary mixture there can be substantial temperature and composition regimes outside the binodal where shear-induced aggregation can take place. This happens as adsorbed layers present at the particle surfaces coalesce and bind particles through a liquid bridge. Depending on particle concentration, percolating networks can form of rods wetted by the minority phase after temperature quenching such a system just across the binodal.