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Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.

Given a simple graph G with vertex set V(G)={1,2,...,n} define S(G) to be the set of all real symmetric matrices A such that for all i not equal to j, the ijth entry of A is nonzero if and only if ij is in E(G). The range of the ranks of matrices in S(G) is of interest and can be determined by finding the minimum rank. The minimum rank of a graph, denoted mr(G), is the minimum rank achieved by a matrix in S(G). The maximum nullity of a graph, denoted M(G), is the maximum nullity achieved by a matrix in S(G). Note that mr(G)+M(G)=|V(G)| and so in finding the maximum nullity of a graph, the minimum rank of a graph is also determined. The minimum rank problem for a graph G asks us to determine mr(G) which in general is very difficult. A simple graph is planar if there exists a drawing of G in the plane such that any two line segments representing edges of G intersect only at a point which represents a vertex of G. A planar drawing partitions the rest of the plane into open regions called faces. A graph is outerplanar if there exists a planar drawing of G such that every vertex lies on the outer face. We consider the class of outerplanar graphs and summarize some of the recent results concerning the minimum rank problem for this class. The path cover number of a graph, denoted P(G), is the minimum number of vertex-disjoint paths needed to cover all the vertices of G. We show that for all outerplanar graphs G, P(G)is greater than or equal to M(G). We identify a subclass of outerplanar graphs, called partial 2-paths, for which P(G)=M(G). We give a different characterization for another subset of outerplanar graphs, unicyclic graphs, which determines whether M(G)=P(G) or M(G)=P(G)-1. We give an example of a 2-connected outerplanar graph for which P(G) ≥ M(G).A cover of a graph G is a collection of subgraphs of G such that the union of the edge sets of the subgraphs is equal to the E(G). The rank-sum of a cover C of G is denoted as rs(C) and is equal to the sum of the minimum ranks of the subgraphs in C. We show that for an outerplanar graph G, there exists an edge-disjoint cover of G consisting of cliques, stars, cycles, and double cycles such that the rank-sum of the cover is equal to the minimum rank of G. Using the fact that such a cover exists allows us to show that the minimum rank of a weighted outerplanar graph is equal to the minimum rank of its underlying simple graph.

This is a linguistic study that will apply theories as a way of understanding the contexts of aspects of the play King Lear by William Shakespeare, as they relate to the possession, and exercise of power. It focuses on targeting and exploring the language of the play and how it impacts characters’ behaviour to gain or sustain power. To do this, specific theoretical frameworks have been applied, including semantics and pragmatics in the analysis of a passage. Examples from the opening scene of King Lear are displayed in order to answer three research questions. Among the findings are differences in the selection of nouns and pronouns with references to authority such as when females tend to overuse “I”, “love” and “lord” when conversing. This research discovered that semantic approaches therefore can be used to explain how Shakespeare portrays, for example, gender differences between the characters by his selection of words, metaphors, and metonymic expressions. Since Lear does not speak in the same manner as his later self as he would have done when at the heights of his power, his linguistic shift mirrors his shift in status following abdication. The analysis also draws certain conclusions with regard to implicatures that are derived from the use of vagueness and ambiguity as outlined in the field of pragmatics, including Speech Act Theory, Deixis and Grice´s Cooperative Principle. However, this essay argues that Grice´s Theory of Implicature and his Maxims can be insightful when analysing Shakespearean dramas, especially floutings and violations of the Maxim of Manner. By applying approaches from the fields of semantics and pragmatics this study concludes that the findings relate to Shakespearean works in general and other works from that period and genre.

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This dissertation is organized into three independent chapters. In Chapter 1, I consider the selection of weights for averaging a set of threshold models. Existing model averaging literature primarily focuses on averaging linear models, I consider threshold regression models. The theory I developed in that chapter demonstrates that the proposed jackknife model averaging estimator achieves asymptotic optimality when the set of candidate models are all misspecified threshold models. The simulations study demonstrates that the jackknife model averaging estimator achieves the lowest mean squared error when contrasted against other model selection and model averaging methods. In Chapter 2, I propose a model averaging framework for the synthetic control method of Abadie and Gardeazabal (2003) and Abadie et al. (2010). The proposed estimator serves a twofold purpose. First, it reduces the bias in estimating the weights each member of the donor pool receives. Secondly, it accounts for model uncertainty for the program evaluation estimation. I study two variations of the model, one where model weights are derived by solving a cross-validation quadratic program and another where each candidate model receives equal weights. Next, I show how to apply the placebo study and the conformal inference procedure for both versions of my estimator. With a simulation study, I reveal that the superior performance of the proposed procedure. In Chapter 3, which is co-authored with my advisor Professor Youngki Shin, we provide an exact computation algorithm for the maximum rank correlation estimator using the mixed integer programming (MIP) approach. We construct a new constrained optimization problem by transforming all indicator functions into binary parameters to be estimated and show that the transformation is equivalent to the original problem. Using a modern MIP solver, we apply the proposed method to an empirical example and Monte Carlo simulations. The results show that the proposed algorithm performs better than the existing alternatives. / Dissertation / Doctor of Philosophy (PhD)

Starting with the work of Volker Strassen, algorithms for matrix multiplication have been developed which are time complexity-wise more efficient than the standard algorithm from the definition of multiplication. The general method of the developments has been viewing the bilinear mapping that matrix multiplication is as a three-dimensional tensor, where there is an exact correspondence between time complexity of the multiplication algorithm and tensor rank. The latter can be seen as a generalisation of matrix rank, being the minimum number of terms a tensor can be decomposed as. However, in contrast to matrix rank there is no general method of computing tensor ranks, with many values being unknown for important three-dimensional tensors. To further improve the theoretical bounds of the time complexity of matrix multiplication, support rank of tensors has been introduced, which is the lowest rank of tensors with the same support in some basis. The goal of this master's thesis has been to go through the history of faster matrix multiplication, as well as specifically examining the properties of support rank for general tensors. In regards to the latter, a complete classification of rank structures of support classes is made for the smallest non-degenerate tensor product space in three dimensions. From this, the size of a support can be seen affecting the pool of possible ranks within a support class. At the same time, there is in general no symmetry with regards to support size occurring in the rank structures of the support classes, despite there existing a symmetry and bijection between mirrored supports. Discussions about how to classify support rank structures for larger tensor product spaces are also included. / Från och med forskning gjord av Volker Strassen har flera algoritmer för matrismultiplikation utvecklats som är effektivare visavi tidskomplexitet än standardalgoritmen som utgår från defintionen av multiplikation. Generellt sett har metoden varit att se den bilinjära avbildningen som matrismultiplikation är som en tredimensionell tensor. Där används att det finns en exakt korrespondens mellan multiplikationsalgoritmens tidskomplexitet och tensorrang. Det sistnämnda är ett slags generalisering av matrisrang, och är minsta antalet termer en tensor kan skrivas som. Till skillnad frpn matrisrang finns ingen allmän metod för att beräkna tensorrang, och många värden är okända även för välstuderade och viktiga tensorer. För att hitta fler övre begränsningar på matrismultiplikations tidskomplexitet har stödrang av tensorer införts, som är den lägsta rangen bland tensor med samma stöd i en viss bas. Målet med detta examensarbete har varit att göra en genomgång av historien om snabbare matrismultiplikation, samt att specifikt undersöka egenskaper av stödrang för allmänna tredimensionella tensorer. För det sistnämnda görs en fullständig klassificering av rangstrukturer bland stödklasser för den minsta icke-degenererade tensorprodukten av tre vektorrum. Slutsatser är bl.a. att storleken av ett stöd kan ses påverka antalet möjliga ranger inom en stödklass. Samtidigt finns i allmänhet ingen symmetri med avseende på stödstorlek i stödklassernas rangstrukturer. Detta trots att det finns en symmetri och bijektion mellan speglade stöd. I arbetet ingår även en diskussion om hur stödrangstrukturer skulle kunna klassificeras för större tensorprodukter.

Implementation of the Clean Water Act requires agencies to monitor aquatic sites on a regular basis and evaluate the quality of these sites. Sites are evaluated individually even though there may be numerous sites within a watershed. In some cases, sampling frequency is inadequate and the evaluation of site quality may have low reliability. This dissertation evaluates testing procedures for determination of site quality based on modelbased procedures that allow for other sites to contribute information to the data from the test site. Test procedures are described for situations that involve multiple measurements from sites within a region and single measurements when stressor information is available or when covariates are used to account for individual site differences. Tests based on analysis of variance methods are described for fixed effects and random effects models. The proposed model-based tests compare limits (tolerance limits or prediction limits) for the data with the known standard. When the sample size for the test site is small, using model-based tests improves the detection of impaired sites. The effects of sample size, heterogeneity of variance, and similarity between sites are discussed. Reference-based standards and corresponding evaluation of site quality are also considered. Regression-based tests provide methods for incorporating information from other sites when there is information on stressors or covariates. Extension of some of the methods to multivariate biological observations and stressors is also discussed. Redundancy analysis is used as a graphical method for describing the relationship between biological metrics and stressors. A clustering method for finding stressor-response relationships is presented and illustrated using data from the Mid-Atlantic Highlands. Multivariate elliptical and univariate regions for assessment of site quality are discussed. / Ph. D.

The National Council for Accrediting of Teacher Education (NCATE) revised its standards in 1986. Included in this revision was a new entrance criterion for teacher education units: a 2.5 grade point average (GPA). Research indicated that GPA was not a good measure of aptitude or achievement when it was used to compare students. The large error variance involved in using GPA as a measure of aptitude could eliminate many capable teacher candidates. The researcher determined to create a system which would identify students who would not be likely to achieve the 2.5 GPA and which would also suggest methods for motivated students to increase their chances of achieving the 2.5 GPA. A sample was identified: industry and technology students at Millersville University who were sophomores from the fall of 1981 to the fall of 1986. This sample was randomly divided into two groups for the purpose of cross-validation. Multiple regression was used for both the overall group and the two subgroups to create equations which predicted sophomore GPA, using the following independent variables: SAT scores, high school rank, age, sex, race and human social capital. Students who were over 23 years old when they entered the program were eliminated from the study because SAT scores or high school ranks were not available for most of them. Predictors with a significance level of 0.05 had the following squared correlations to sophomore GPA: 1) high school rank: 0.2098, 2) SAT-math: 0.1960, 3) SAT-verbal: 0.1385, 4) special entrance: 0.0566, 5) admission age: 0.0298. Predictors which remained significant when loaded into a multiple prediction equation are listed in order of predictive power with their incremental squared correlation coefficients: 1) high school class rank: 0.2098, 2) SAT-math: 0.0969, 3)admission age: 0.0421, 4) SAT- verbal: 0.0188. The total squared multiple correlation coefficient for the prediction equation was 0.3676. The equation correctly predicted 71.4% of the admission decisions (based on a 2.5 sophomore GPA). Double cross-validation resulted in an average acceptance prediction accuracy of 72.2%. The prediction equation reduced the error of prediction and was recommended for use. / Ph. D.

4D-Flow MRI is a non-invasive imaging technique that can measure temporally resolved 3D images, capturing the flow/velocity in each pixel. The quality of the images and the temporal resolution largely depend on two factors. The acquisition protocol the MRI scanner uses and the reconstruction method used to go from signal to images. In MRI, the signal samples measured are the Fourier coefficients of the sought-after image, and reconstruction is an inverse problem, classically requiring sampling on at least Nyquist rate. Compressed sensing is a framework that allows for reconstruction from fewer samples than the Nyquist rate by incorporating other known information about the images. In this thesis, we evaluate the efficiency of Compressed Sensing for 4D-Flow MRI reconstruction for undersampled signals on synthetic data and compare it to classical reconstruction methods (Gridding and Viewshared Gridding). We specifically focus on the Locally Low Rank (LLR) regularization. The importance of initial-guess, or if it can be beneficial to estimate the temporal images by solving from the difference to the mean, is investigated. After calculating velocity profiles in vessels, we compare the reconstructed velocity profiles to the actual velocity profiles. We look at relative errors and pixel-wise maximum errors, as well as visual inspection. We introduce a velocity error metric aiming at capturing how accurate the reconstructed velocity profile is compared to our synthetic truth. We show that for good choices of regularization strength, the relative, maximum and velocity errors are significantly lower for the Compressed Sensing LLR method compared to the classical methods. We conclude that Compressed sensing with LLR regularization can significantly improve the reconstruction quality of 4D-Flow MRI data.