Satellite communication promises potential methods for providing global communication. In particular, by the development of a Low Earth Orbital (LEO) satellite constellation, both global coverage and broadband communication will be accessible. Problems arise in situations where various traffic types in broadband communication require different levels of quality of service (QoS). Traffic control is required to make sure that each traffic demand may receive the expected QoS. Another problem is that the dynamic topology of a LEO satellite network requires a traffic allocation control, which is able to allocate traffic demand into the Inter Satellite Links (ISLs) between LEO satellites.In this thesis, traffic allocation strategy in a dynamic LEO satellite communication network is studied and analyzed. The delivery of Quality of Service (QoS) is an important objective. Traffic allocation control is performed in the LEO satellite constellation to provide a near optimal utilization of these ISLs. An alternative solution is proposed in this research, in which a combination of two algorithms will be used to allocate traffic in this dynamic satellite network. The first algorithm allocates traffic during small time intervals, based on an assumption that the topology is unchanged during these intervals. The second algorithm allocates traffic after topology updating has been accomplished. Traffic allocation respects some constraints including QoS (due to multiservice requirements), capacity constraints, traffic distribution, and availability constraints. Both theoretical and empirical studies have been undertaken to examine the performance of the proposed algorithm, denoted GALPEDA (Genetic Algorithm Linear Programming and Extended Dijkstra Algorithm). The proposed algorithm provides privileges to a class of high priority traffic, including benefits for traffic allocation of multiclass traffic in LEO satellite communication. It provides a novel traffic allocation mechanism to cope with the dynamic topology of a LEO satellite; moreover this algorithm distributes multiservice traffic evenly over the network. Simulations results are provided.

Doctor of Philosophy / Department of Mathematics / Robert B. Burckel / In this paper a regularized spectral estimation formula and a regularized iterative algorithm for band-limited extrapolation are presented. The ill-posedness of the problem is taken into account. First a Fredholm equation is regularized. Then it is transformed to a differential equation in the case where the time interval is R. A fast algorithm to solve the differential equation by the finite differences is given and a regularized spectral estimation formula is obtained. Then a regularized iterative extrapolation algorithm is introduced and compared with the Papoulis and Gerchberg algorithm. A time-frequency regularized extrapolation algorithm is presented in the two-dimensional case. The Gibbs phenomenon is analyzed. Then the time-frequency regularized extrapolation algorithm is applied to image restoration and compared with other algorithms.

Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main focus of this work is to study the classical Calder\'n-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in $(\rn, d\mu)$, where $\mu$ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

Doctor of Philosophy / Department of Mathematics / Louis Crane / We exhibit a static, cylindrically symmetric, exact solution to the Euler-Heisenberg field equations (EHFE) and prove that its effective geometry contains (optical) black holes. It is conjectured that there are also soliton solutions to the EHFE which contain black hole geometries.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programs (IP) are a commonly researched class of problems used by governments and businesses to improve decision making through optimal resource allocation and scheduling. However, integer programs require an exponential amount of effort to solve and in some instances a feasible solution is unknown even with the most powerful computers. There are several methods commonly used to reduce the solution time for IPs. One such approach is to generate new valid inequalities through lifting. Lifting strengthens a valid inequality by changing the coefficients of the variables in the inequality. Lifting can result in facet defining inequalities, which are the theoretically strongest inequalities. This thesis introduces the Cutting-plane Algorithm in Three Sets (CATS) that can help reduce the solution time of integer programs. CATS uses synchronized simultaneous lifting to generate a new class of previously undiscovered valid inequalities. These inequalities are based upon three sets of indices from a binary knapsack integer program, which is a commonly studied integer program. CATS requires quartic effort times the number of inequalities generated. Some theoretical results describe easily verifiable conditions under which CATS inequalities are facet defining. A small computational study shows CATS obtains about an 8.9% percent runtime improvement over a commercial IP software. CATS preprocessing time is fast and requires an average time of approximately .032 seconds to perform. With the exciting new class of inequalities produced relatively quickly compared to the solution time, CATS is advantageous and should be implemented to reduce solution time of many integer programs.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / Several methods for a stable solution to the equation $F(u)=f$ have been developed. Here $F:H\to H$ is an operator in a Hilbert space $H$, and we assume that noisy data $f_\delta$, $\|f_\delta-f\|\le \delta$, are given in place of the exact data $f$. When $F$ is a linear bounded operator, two versions of the Dynamical Systems Method (DSM) with stopping rules of Discrepancy Principle type are proposed and justified mathematically. When $F$ is a non-linear monotone operator, various versions of the DSM are studied. A Discrepancy Principle for solving the equation is formulated and justified. Several versions of the DSM for solving the equation are formulated. These methods consist of a Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation $F(u)=f$ is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to the equation $F(u)=f$ is proved. This dissertation consists of six chapters which are based on joint papers by the author and his advisor Prof. Alexander G. Ramm. These papers are published in different journals. The first two chapters deal with equations with linear and bounded operators and the last four chapters deal with non-linear equations with monotone operators.

Doctor of Philosophy / Department of Mathematics / Virginia Naibo / Investigations of pseudodifferential operators are useful in a variety of applications. These include finding solutions or estimates of solutions to certain partial differential equations, studying boundedness properties of commutators and paraproducts, and obtaining fractional Leibniz rules. A pseudodifferential operator is given through integration involving the Fourier transform of the arguments and a function called a symbol. Pseudodifferential operators were first studied in the linear case and results were obtained to advance both the theory and applicability of these operators. More recently, significant progress has been made in the study of bilinear, and more generally multilinear, pseudodifferential operators. Of special interest are boundedness properties of bilinear pseudodifferential operators which have been examined in a variety of function spaces. Since determining factors in the boundedness of these operators are connected to properties of the corresponding symbols, significant effort has been directed at categorizing the symbols according to size and decay conditions as well as at establishing the associated symbolic calculus. One such category, the bilinear Hörmander classes, plays a vital role in results concerning the boundedness of bilinear pseudodifferential operators in the setting of Lebesgue spaces in particular. The new results in this work focus on the study of bilinear pseudodifferential operators with symbols in weighted Besov spaces of product type. Unlike the Hörmander classes, symbols in these Besov spaces are not required to possess in finitely many derivatives satisfying size or decay conditions. Even without this much smoothness, boundedness properties on Lebesgue spaces are obtained for bilinear operators with symbols in certain Besov spaces. Important tools in the proofs of these new results include the demonstration of appropriate estimates and the development of a symbolic calculus for some of the Besov spaces along with duality arguments. In addition to the new boundedness results and as a byproduct of studying operators with symbols in Besov spaces, it is possible to quantify the smoothness of the symbols, in terms of the conditions that define the Hörmander classes, that is sufficient for boundedness of the operators in the context of Lebesgue spaces.

Doctor of Philosophy / Curriculum and Instruction Programs / Andrew G. Bennett / Online assessments afford many advantages for teachers and students. Okolo (2006) stated, “As the power, sophistication, and availability of technology have increased in the classroom, online assessments have become a viable tool for providing the type of frequent and dynamic assessment information that educators need to guide instructional decisions,” (pp 67-68). As post secondary institutes use online learning environments, education has molded into hybrid experiences. Traditional courses now regularly infuse components of online learning and assessments by required student participation both in person and online. Research is needed to analyze online components of assessment and student achievement. Data was gathered from an undergraduate mathematics course designed for students seeking a bachelor’s degree in elementary education. The course was entitled MATH 320: Mathematics for Elementary School Teachers. Synergies of quantitative and qualitative data were evaluated to assess the impact of written and video help tutorials in online quizzes on student achievement. Three forms of data were collected: student interviews, surveys about students’ online quiz experiences and learning style preferences, and student performance and tutorial usage statistics from seven online quizzes. Student interviews were conducted mid-semester by the researcher who also transcribed and analyzed data. Graphical schemes were used to identify and categorize responses to interview questions. Students’ responses were summarized and quantified in frequency tables. Surveys about students’ online quiz experiences and learning style preferences were analyzed through descriptive statistical methods to describe the data with numerical indices and in graphical form. Correlation matrices and linear regression models were used to identify relationships among survey items. Additionally, Analysis of Variance (ANOVA) techniques were used to explore the data for statistical significance. Students were assigned seven online quizzes throughout the semester. Descriptive statistics were calculated to describe the online quiz data. Regression models were used to determine correlations between use of help tutorials and performance on online quizzes. Data analysis revealed students were persistent and motivated to retake similar quizzes multiple times until a high or perfect score was obtained. After missing a problem, students selected written help tutorials more often than video help tutorials to identify mistakes and understand how to solve the particular problem. The proportion of students whose scores improved after using both written and video help tutorials was greater than those who used the written help tutorials alone. Although the number of students who benefited from the video help tutorials was smaller than expected, the increased performance could be appreciated by students and educators alike. The research presented herein should serve as a base for curriculum development in university mathematics programs utilizing or considering implementation of online tutorials coupled with student evaluation.

Master of Science / Department of Statistics / Diego M. Maldonado / Haiyan Wang / This report addresses some mathematical and statistical techniques of image processing and their computational implementation. Fundamental theories have been presented, applied and illustrated with examples. To make the report as self-contained as possible, key terminologies have been defined and some classical results and theorems are stated, in the most part, without proof. Some algorithms and techniques of image processing have been described and substantiated with experimentation using MATLAB. Several ways of estimating original images from noisy image data and their corresponding risks are discussed. Two image processing concepts selected to illustrate computational implementation are: "Bayes classification" and "Wavelet denoising". The discussion of the latter involves introducing a specialized area of mathematics, namely, wavelets. A self-contained theory for wavelets is built by first reviewing basic concepts of Fourier Analysis and then introducing Multi-resolution Analysis and wavelets. For a better understanding of Fourier Analysis techniques in image processing, original solutions to some problems in Fourier Analysis have been worked out. Finally, implementation of the above-mentioned concepts are illustrated with examples and MATLAB codes.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / A representation of a quiver [Gamma] over a commutative ring R assigns an R-module to each vertex and an R-linear map to each arrow. In this dissertation, we consider R = k[t]/(t[superscript]n) and all R-free representations of [Gamma] which assign a free R-module to each vertex. The category, denoted by Rep[superscript]f[subscript] R([Gamma]), containing all such representations is not an abelian category, but rather an exact category. In this dissertation, we firstly study the Hall algebra of the category Rep[superscript]f[subscript] R([Gamma]), denote by [Eta](R[Gamma]), for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of [Eta](R[Gamma]) is given under the framework of Lusztig's geometric setting. Moreover, the canonical basis and a monomial basis of this subalgebra are constructed by using perverse sheaves. This generalizes Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebras is obtained. If [Gamma] is a Jordan quiver, which is a quiver with one vertex and one loop, each representation in Rep[superscript]f[subscript] R([Gamma]), gives a matrix over R when we fix a basis of the free R-module. An interesting case arises when considering invertible matrices. It then turns out that one is dealing with representations of the group GL[subscript]m(k[t]/(t[superscript]n)). Character sheaf theory is a geometric character theory of algebraic groups. In this dissertation, we secondly construct character sheaves on GL[subscript]m(k[t]/(t[superscript]2)). Then we define an induction functor and restriction functor on these perverse sheaves.