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Most distribution free nonparametric methods depend on the ranks or orderings of the individual observations. This dissertation develops methods for the situation when there is only partial information about the ranks available. A random-linear-extension exact test and an empirical version of the random-linear-extension test are proposed as a new way to compare groups of data with partial orders. The basic computation procedure is to generate all possible permutations constrained by the known partial order using a randomization method similar in nature to multiple imputation. This random-linear-extension test can be simply implemented using a Gibbs Sampler to generate a random sample of complete orderings. Given a complete ordering, standard nonparametric methods, such as the Wilcoxon rank-sum test, can be applied, and the corresponding test statistics and rejection regions can be calculated. As a direct result of our new method, a single p-value is replaced by a distribution of p-values. This is related to some recent work on Fuzzy P-values, which was introduced by Geyer and Meeden in Statistical Science in 2005. A special case is to compare two groups when only two objects can be compared at a time. Three matching schemes, random matching, ordered matching and reverse matching are introduced and compared between each other. The results described in this dissertation provide some surprising insights into the statistical information in partial orderings. / Statistics

Price and inventory decisions are key levers of profit for firms. A manager needs to understand the impacts of pricing, ordering and stocking decisions not only on today's operations but also on future demand. In this dissertation we investigate these intertwining decisions by incorporating inter-temporal effects of pricing decisions through reference prices. We introduce three significant extensions to reference price models to provide more meaningful insight into pricing, inventory and ordering decisions. We first present a threshold reference model. The threshold model incorporates zones of insensitivity around expected price that moderate the reference impacts on demand. This provides a rigourous model that is flexible enough to handle different pricing strategies such as single everyday low pricing (EDLP), high-low pricing (HiLo) and other general price cycles. We develop two solution approaches and provide computational results. We next introduce a reference model with stochastic demand. There is considerable previous research supporting the consideration of variability in pricing and inventory decisions and this is especially true in the context of inter-temporal demand interactions based on pricing decisions. We find that the introduction of stochastic elements can actually increase or decrease the length of the price cycle for some consumers in a reference model depending on the parameters of the model. This extends the stochastic demand model and bridges to reference models for improved managerial insight. The final model presented is the dynamic lot sizing model. When prices and production decisions or order quantities are determined simultaneously the interactions need to be considered to optimize profits. The reference model incorporates the inter-temporal price effects to provide a clearer picture of the optimal decision. The inclusion of reference effects does change the optimal decision. / Thesis / Doctor of Philosophy (PhD)

Yes / A quantum system (n) with variables in Z(n), where n = Qpi (with pi prime numbers), is considered. The non-near-linear geometry G(n) of the phase space Z(n) × Z(n), is studied. The lines through the origin are factorized in terms of ‘prime factor lines’ in Z(pi)×Z(pi). Weak mutually unbiased bases (WMUB) which are products of the mutually unbiased bases in the ‘prime factor Hilbert spaces’ H(pi), are also considered. The factorization of both lines and WMUB is analogous to the factorization of integers in terms of prime numbers. The duality between lines and WMUB is discussed. It is shown that there is a partial order in the set of subgeometries of G(n), isomorphic to the partial order in the set of subsystems of (n).

Spontaneous order happens in non-equilibrium systems composed of interacting elements. This phenomenon manifests in both the formation of space-time patterns in reaction-diffusion systems and collective rhythmic behaviors in coupled oscillators. In this thesis, we present the results of two studies: 1) The response of a multi-species predator-prey system to perturbation. 2) The features of a rich attractor space in a system of repulsively coupled Kuramoto oscillators. In the first part, we address this question: how does a complex coarsening system with non-trivial in-domain dynamics respond to perturbations? We choose a cyclic predator-prey model with six species each attacking three others. As a result of this interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game which results in the formation of spirals inside the domains. We perturb the system by changing the interaction scheme which leads to a change of alliances and therefore a different spatial pattern. As expected, perturbing a complex space-time pattern results in a complex response. In the second part, we explore the attractor space of a system of repulsively coupled oscillators with non-homogeneous natural frequencies on a hexagonal lattice. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. Coupled oscillators with frustration show similar features as frustrated magnetic systems. We use the parameters of the system like the coupling constant and the width of the frequency distribution to understand the system's attractor space. Further, we study the effects of external noise on the system. We also identify the breaking of time-translation invariance in the absence of external noise, in our system. / Doctor of Philosophy / Spontaneous ordering is a ubiquitous phenomenon observed in natural systems containing many interacting elements. In some systems the order is observed in the form of spatial patterns. It also can be seen in a population of coupled oscillators in the form of collective rhythmic behaviors. In this thesis, we present the results of two studies. For the first study, we choose a cyclic predator-prey system that shows a non-trivial space-time pattern. The system consists of six species each attacking three others, cyclically. By choosing such an interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game. As a result of the inner competition, spirals form inside the domains. We study the response of the system to a perturbation. To perturb the system, we change the interaction scheme which leads to a change of alliances and therefore, a different spatial pattern. In the second study, we explore the patterns of clustering and synchronization in a system of repulsively coupled oscillators with non-homogeneous natural frequencies. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. We use the parameters of the system such as the coupling constant and the width of the frequency distribution to understand the system’s attractor space. Further, we examine the effect of external noise on the system.

This dissertation is part of an effort to understand and to facilitate the modeling of the ordering kinetics of block copolymers in solution during the extraction of solvent from a solution-cast film. Central to this work was determining a suitable method for measuring the ordering kinetics during solvent removal and being able to interpret the measurements in terms of structure development. It was also necessary to assess a model for quantifying the ordering kinetics to use in conjunction with a mass transfer model to predict structure formation during solvent extraction. Changes in the dynamic mechanical response (DMR) over time of block copolymer solutions at fixed concentrations following solvent removal were explored as a means to track the growth of ordered domains. It was found that DMR measurements performed following solvent extraction were sensitive to the nucleation and growth process of the phase separation process over a wide range of concentrations, beginning near the order-disorder transition concentration. Based on complimentary small angle X-ray measurements, it was determined that the changes in the DMR are caused by the development of individual microstructures, The SAXS experiments also indicated that the DMR is insensitive to late stages of the growth process. Ultimately, DMR measurements under-predicted the ordering times at several concentrations and did not detect ordering at concentrations above which SAXS data indicated ordering was still occurring. The ability to use the parallel and series rules of mixtures for determining ï ¦(t) in conjunction with the Avrami equation to quantitatively model the ordering kinetics was also determined. These models allowed the ordering kinetics during solvent removal to be qualitatively analyzed. However, using the two different rules of mixtures resulted in a wide range of possible ordering times for a given copolymer concentration, making these approximations unsuitable for modeling a real solvent extraction process. Further, the parameters of the model were insensitive to the type of microstructures developing. As a continuation of this work, a new apparatus to track block copolymer ordering in situ during solvent extraction was designed. Experiments using the apparatus allowed the ordering kinetics and domain dimensions as a function of concentration to be monitored in real-time under several solvent removal conditions. These experiments study the ordering kinetics is a manner more akin to real processing conditions and will allow future assessment of the ability of iso-concentration ordering kinetics to predict phase separation during film processing. / Ph. D.

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A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction £Ux_i=1 on the (k-1)-dimensional probability simplex S^{k-1}. This dissertation discusses optimal designs for linear and quadratic log contrast models for experiments with mixtures suggested by Aitchison and Bacon-Shone (1984), where the experimental domain is restricted further as in Chan (1992). In this study, firstly, an essentially complete class of designs under the Kiefer ordering for linear log contrast models with mixture experiments is presented. Based on the completeness result, £X_p-optimal designs for all p, -¡Û<p≤1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these £X_p-optimal designs behave. Following that, the exact N-point D-optimal designs for linear log contrast models with three and four ingredients are further investigated. The results show that for k=3 and N=3p+q ,1 ≤q≤2, there is an exact N-point D-optimal design supported at the points of S_1 (S_2) with equal weight n/N, 0≤n≤p , and puts the remaining weight (N-3n)/N uniformly on the points of S_2 (S_1) as evenly as possible, where S_1 and S_2 are sets of the supports of the approximate D-optimal designs. When k=4 and N=6p+q , 1 ≤q≤5, an exact N-point design which distributes the weights as evenly as possible among the supports of the approximate D-optimal design is proved to be exact D-optimal. Thirdly, the approximate D_s-optimal designs for discriminating between linear and quadratic log contrast models for experiments with mixtures are derived. It is shown that for a symmetric subspace of the finite dimensional simplex, there is a D_s-optimal design with the nice structure that puts a weight 1/(2^{k-1}) on the centroid of this subspace and the remaining weight is uniformly distributed on the vertices of the experimental domain. Moreover, the D_s-efficiency of the D-optimal design for quadratic model and the design given by Aitchison and Bacon-Shone (1984) are also discussed Finally, we show that an essentially complete class of designs under the Kiefer ordering for the quadratic log contrast model is the set of all designs in the boundary of T or origin of T . Based on the completeness result, numerical £X_p -optimal designs for some p, -¡Û<p≤1 are obtained.

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