Mathematics electronic theses and dissertations (MU)
Permanent URI for this collection
The items in this collection are the theses and dissertations written by students of the Department of Mathematics. Some items may be viewed only by members of the University of Missouri System and/or University of Missouri-Columbia. Click on one of the browse buttons above for a complete listing of the works.
Browse
Recent Submissions
Item On the theory of Boolean vector spaces(University of Missouri--Columbia, 1969) Stroup, Fred Oliver Jr.; Zemmer, J. L."The concept of a (abstract) Boolean vector space is due to Subrahmanyam [6]1. In introducing this concept, Subrahmanyam was motivated by Foster [1] who demonstrated that each element of a p-ring R (with unity) can be represented as a type of "Boolean vector" over the Boolean algebra of all the idempotent elements of R. In this representation, Foster made use of a "basis" of R consisting of the nonzero elements of the additive subgroup of R generated by its unity element. Shorter proofs of this latter result concerning a "basis" of R have been given by Penning [3] and Zemmer [9]. The notion of a (abstract) Boolean vector space is a rather natural generalization of this idea of considering a p-ring as a kind of "Boolean vector space" over its Boolean algebra of Idempotent elements. The development of the theory of Boolean vector spaces was continued in subsequent publications by Subrahmanyam [7], [8] and by Jagannadham [2]. Since the postulates for a Boolean vector space bear a strong resemblence to those of a vector space over a field, the study of Boolean vector spaces is influenced to a considerable extent by a comparison of the two mathematical systems. There are many very natural questions concerning the structure of Boolean vector spaces and their associated linear homomorphism (transformation) spaces which at present remain unanswered. The purpose of this dissertation will be to continue the development of the theory of Boolean vector spaces while attempting to provide answers to some of these questions. Chapter II is devoted to summarizing the definitions and previously published results concerning Boolean vector spaces which are deemed essential in the development of subsequent chapters. In Chapter III, the concept of a direct sum of Boolean vector spaces is introduced. Such direct sums play an important role in the structure theory of linear homomorphism spaces of Boolean vector spaces. Subspaces and quotient spaces of Boolean vector spaces are defined and studied in Chapter IV. In [2], Jagannadham continued the study of linear homomorphisms (transformations) of a Boolean vector space into itself which was initiated by Subrahmanyam [7]. In Chapter V, the notion of a linear homomorphism is generalized to include mappings of one Boolean vector space into another Boolean vector space. In addition to generalizing several of the results appearing in [2] and [7], some interesting results are obtained concerning direct sum representations of linear homomorphism spaces. As in the study of vector spaces over a field, the idea of a linear functional arises as a special type of linear homomorphism. Chapter VI is concerned with the theory of linear functionals. The concluding Chapter VII introduces the concept of a duality relationship among Boolean vector spaces. The principal theorem of Chapter VII provides necessary and sufficient conditions for two Boolean vector spaces to be dual spaces. Within each chapter, definitions, lemmas, and theorems are assigned numbers according to the order of their appearance within the chapter. Also, certain key statements throughout each chapter are assigned numbers. A typical reference such as (2.4) will refer to statement (2.4) which will occur as the fourth numbered statement in Chapter II. Theorems, lemmas, and definitions will be referred to as such together with their appropriate numbers."--Introduction.Item Problems in high-dimensional probability(University of Missouri--Columbia, 2024) Sharma, Shivam; Valettas, PetrosIn Chapter 1, we focus on lower estimates for suprema of stochastic processes. We investigate the role of super-gaussianity and the Sudakov Minoration Principle for this concept. We derive an extension of celebrated Majorizing Measures Theorem (MMT) for Gaussian mixtures which in turn yields a comparison inequality between the expected suprema of sub-gaussian random vectors and Gaussian mixtures. In Chapter 2, we focus on an alternative method for deriving Sudakov Minoration type estimates via "discrepancy" bounds. More precisely, we obtain such bounds between the expected suprema of canonical processes generated by random vectors with independent coordinates and the expected suprema of Gaussian processes. In particular, we obtain a refined proximity estimate for Rademacher and Gaussian complexities. Our estimates are dimension-free and depend only on the geometric parameters and the numerical complexity of the underlying index set. In Chapter 3, we focus on concentration of measure phenomenon and study variancesensitive deviation inequalities. In particular, we derive generalizations of variancesensitive concentration inequalities in [PV18] and in [PV19] to include functions that are "nearly" convex. We also derive an important application in the form of Hanson- Wright type inequality.Item The Neumann problem in rough domains(University of Missouri--Columbia, 2024) Sparrius, Derek; Hofmann, StevenThe purpose of this thesis is to study the solvability of the Neumann problem for divergence form elliptic operators on chord arc domains. We begin, in the setting of a 1 sided chord arc domain, by constructing the Neumann function associated with the differential operator and prove that the Neumann function is Holder continuous up to the boundary. Then, in the setting of a 2 sided chord arc domain, we establish the solvability of the Neumann problem with data in L^q, 1 [less than] q [less than] p as well as in the Hardy space H^1 under the assumptions we have solvability of the Neumann problem and Regularity problem with data in L^p and the Dirichlet problem for the transpose operator with data in L^p.Item Topics in combinatorial phase retrieval(University of Missouri--Columbia, 2024) Gonzalez, Ivan Eduardo; Edidin, DanIn this dissertation, we study and prove results concerning the finite alphabet phase retrieval problem, and the problem of sampling real frames that preserves the phase retrieval property. The finite alphabet phase retrieval problem considers the following: under what conditions can one recover a signal whose entries lie in a small alphabet of possible values from its Fourier magnitudes. We extend the definition of homometric sets to homometric partitions of sets and prove that for generic values of the alphabet, two signals have the same Fourier magnitude if and only if their support partitions are homometric. The real frame sampling phase retrieval problem studies when a subset of a phaseretrievable real frame is phase retrievable. We call a frame for which no subset is phase-retrievable a vital frame. We construct a family of real vital frames of size 2M in dimension M and show that in dimension 3, there are no real vital frames of size 7.Item The capacity of tuples of point clouds and generalized radial isotropy(University of Missouri--Columbia, 2024) Duran, Edward O; Chindris, CalinIn this thesis, we provide both a qualitative and quantitative study of the capacity of tuples of point clouds through the notion of generalized radial isotropy. This is in part motivated by the celebrated Brascamp-Lieb ("BL") constants, and more generally, the Anantharam-Jog-Nair ("AJN") constants that are associated with a tuple of point clouds. Let m, n, and d1, . . . , dn be positive integers and let X = (X^1, . . . X^n) be an n-tuple of point clouds, with each X^i [subset] Rdi being a point cloud of m vectors in R^di. This tuple of point clouds can be encoded in a representation Vx of the complete bipartite quiver with n source vertices and m sink vertices. Let c = (c1, . . . , cm) [element] Qm [greater than] 0 be a vector of positive rational exponents that sum to the total of the dimensions di. The main object of study in this thesis is the capacity, denoted by Cap(X, c), of the quiver datum (Vx, c). When n = 1, the constants of the form Cap(X, c)^-12 are precisely the best constants that occur in the classical BL inequalities associated with single point cloud of m vectors. For arbitrary n, taking the natural logarithm of Cap(X, c)^-12 yields the best constants in the AJN inequalities in Information Theory (in the rank one case). Our first result gives necessary and sufficient conditions for a tuple of point clouds X to be put in generalized c-radial isotropic position. This result, combined with a general character formula for the capacity, allows us to obtain closed formulas for the capacity in several examples. We also describe the capacity Cap(X, c) in a way that enables us to use gradient descent to compute approximations of the capacity. The results in this thesis generalize the work of Barthe and Hardt-Moitra on radial isotropy from a single point cloud (the case of classical BL constants) to tuples of point clouds (the case of AJN constants).