Theses and Dissertations - Department of MathematicsNo Descriptionhttps://ir.ua.edu/handle/123456789/1202024-05-28T06:33:15Z2024-05-28T06:33:15Z811Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous TypeCummings, Jeremyhttps://ir.ua.edu/handle/123456789/83802023-08-08T16:26:06Z2020-01-01T00:00:00Zdc.title: Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous Type
dc.contributor.author: Cummings, Jeremy
dc.description.abstract: Given a space of homogeneous type $(X,\mu,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\pp$. We prove that the variable Muckenhoupt condition $\App$ is necessary and sufficient for the strong type inequality if $\pp$ satisfies log-H\"older continuity conditions and $1 < p_- \leq p_+ < \infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14].
dc.description: Electronic Thesis or Dissertation
2020-01-01T00:00:00ZTopological transformation groups with a fixed end pointGray, William Jessehttps://ir.ua.edu/handle/123456789/82582023-08-08T20:13:31Z1965-01-01T00:00:00Zdc.title: Topological transformation groups with a fixed end point
dc.contributor.author: Gray, William Jesse
dc.description: Electronic Thesis or Dissertation
1965-01-01T00:00:00ZA compiler for the Bama-Bell floating point interpretive systemGray, William J.https://ir.ua.edu/handle/123456789/82572023-08-08T16:40:21Z1962-01-01T00:00:00Zdc.title: A compiler for the Bama-Bell floating point interpretive system
dc.contributor.author: Gray, William J.
dc.description: Electronic Thesis or Dissertation
1962-01-01T00:00:00ZA Classifying Family of Spaces for the Cohomology of Profinite GroupsLee, Evan Matthewhttps://ir.ua.edu/handle/123456789/82042023-08-09T10:58:31Z2021-01-01T00:00:00Zdc.title: A Classifying Family of Spaces for the Cohomology of Profinite Groups
dc.contributor.author: Lee, Evan Matthew
dc.description.abstract: In the study of homological algebra, one useful tool for studying the cohomology of a discrete group is that group's classifying space. In some sense, the classifying space captures both the group itself and a description of its cohomology for any action of the group on any coefficient module. While some constructions for a classifying space also apply to topological groups, the relationship of the resulting space to the group's cohomology is unclear. Profinite groups are a special case of topological groups, determined entirely as the limits of inverse systems of finite, discrete groups. The goal of this work is to construct for profinite groups as close an analog as possible to the classifying space of a discrete group. In particular, we are interested in the construction for a finite, discrete group's classifying space achieved by first producing the nerve of the group as a category and then taking its geometric realization to obtain a space with isomorphic cohomology groups. We proceed by extending each step of this process to apply to a profinite group using inverse limits, followed by correcting for a lack of continuity (in the sense of compatibility with inverse limits) in singular cohomology by applying alternative cohomology theories to the resulting sequence of spaces. The end result has a promising isomorphism to the cohomology of the group, with the possibility of a further isomorphism.
dc.description: Electronic Thesis or Dissertation
2021-01-01T00:00:00ZCoordinate Descent Methods for Sparse Optimal Scoring and its ApplicationsFord, Katie Woodhttps://ir.ua.edu/handle/123456789/81122023-08-08T16:38:34Z2021-01-01T00:00:00Zdc.title: Coordinate Descent Methods for Sparse Optimal Scoring and its Applications
dc.contributor.author: Ford, Katie Wood
dc.description.abstract: Linear discriminant analysis (LDA) is a popular tool for performing supervised classification in a high-dimensional setting. It seeks to reduce the dimension by projecting the data to a lower dimensional space using a set of optimal discriminant vectors to separate the classes. One formulation of LDA is optimal scoring which uses a sequence of scores to turn the categorical variables into quantitative variables. In this way, optimal scoring creates a generalized linear regression problem from a classification problem. The sparse optimal scoring formulation of LDA uses an elastic-net penalty on the discriminant vectors to induce sparsity and perform feature selection. We propose coordinate descent algorithms for finding optimal discriminant vectors in the sparse optimal scoring formulation of LDA, along with parallel implementations for large-scale problems. We then present numerical results illustrating the efficacy of these algorithms in classifying real and simulated data. Finally, we use Sparse Optimal Scoring to analyze and classify visual comprehension of Deaf persons based on EEG data.
dc.description: Electronic Thesis or Dissertation
2021-01-01T00:00:00ZAugmented matched interface and boundary (AMBI) method for solving interface and boundary value problemsFeng, Hongsonghttps://ir.ua.edu/handle/123456789/79292023-08-08T16:33:22Z2021-01-01T00:00:00Zdc.title: Augmented matched interface and boundary (AMBI) method for solving interface and boundary value problems
dc.contributor.author: Feng, Hongsong
dc.description.abstract: This dissertation is devoted to the development the augmented matched interface and boundary(AMIB) method and its applications for solving interface and boundary value problems. We start with a second order accurate AMIB introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients, which illustrates the theory of AMIB illustrated in details. AMIB method is different from its ancestor matched interface and boundary (MIB) method in employing fictitious values to restore the accuracy of central differences for interface and boundary value problems by approximating the corrected terms in corrected central differences with these fictitious values. Through the augmented system and Schur complement, the total computational cost of the AMIB is about $O(N \log N)$ for degree of freedom $N$ on a Cartesian grid in 2D when fast Fourier transform(FFT) based Poisson solver is used. The AMIB method achieves $O(N \log N)$ efficiency for solving interface and boundary value problems, which is a significant advance compared to the MIB method. Following the theory of AMIB in chapter 2, chapter 3 to chapter 6 cover the development of AMIB for a high order efficient algorithm in solving Poisson boundary value problems and a fourth order algorithm for elliptic interface problems as well as efficient algorithm for parabolic interface problems. The AMIB adopts a second order FFT-based fast Poisson solver in solving elliptic interface problems. However, high order FFT-based direct Poisson solver is not available in the literature, which imposes a grand challenge in designing a high order efficient algorithm for elliptic interface problems. The AMIB method investigates efficient algorithm of Poisson boundary value problem (BVP) on rectangular and cubic domains by converting Poisson BVP to an immersed boundary problem, based on which a high order FFT algorithm is proposed. This naturally allows for fulfilling a fourth order fast algorithm for solving elliptic interface problems. Besides the FFT algorithm, a multigrid method is also considered to achieve high efficiency in solving parabolic interface problems. Extensive numerical results are included in each chapter of the concerned problem, and are used to show the robustness and efficiency of AMIB method.
dc.description: Electronic Thesis or Dissertation
2021-01-01T00:00:00ZRegularization methods on solving Poisson’s equation and Poisson Boltzmann equation with singular charge sources and diffuse interfacesWang, Siwenhttps://ir.ua.edu/handle/123456789/79252023-08-08T20:15:35Z2021-01-01T00:00:00Zdc.title: Regularization methods on solving Poisson’s equation and Poisson Boltzmann equation with singular charge sources and diffuse interfaces
dc.contributor.author: Wang, Siwen
dc.description.abstract: Numerical treatment of singular charges is a grand challenge in solving Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this research work, regularization formulations are introduced to capture the singularity analytically, which are the first of their kind for diffuse interface Poisson's equation and PB models. The success lies in a dual decomposition -- besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. Some diffuse interface models including a Gaussian convolution surface (GCS) are also introduced. The GCS efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.
dc.description: Electronic Thesis or Dissertation
2021-01-01T00:00:00ZAdaptive pseudo-time methods for the Poisson-Boltzmann equation with eulerian solvent excluded surfaceJones, Benjamin Danielhttps://ir.ua.edu/handle/123456789/78522023-08-09T02:52:39Z2021-01-01T00:00:00Zdc.title: Adaptive pseudo-time methods for the Poisson-Boltzmann equation with eulerian solvent excluded surface
dc.contributor.author: Jones, Benjamin Daniel
dc.description.abstract: This work further improves the pseudo-transient approach for the Poisson Boltzmann equation (PBE) in the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is known to involve many difficulties, such as exponential nonlinear term, strong singularity by the source terms, and complex dielectric interface. Recently, a pseudo-time ghost-fluid method (GFM) has been developed in [S. Ahmed Ullah and S. Zhao, Applied Mathematics and Computation, 380, 125267, (2020)], by analytically handling both nonlinearity and singular sources. The GFM interface treatment not only captures the discontinuity in the regularized potential and its flux across the molecular surface, but also guarantees the stability and efficiency of the time integration. However, the molecular surface definition based on the MSMS package is known to induce instability in some cases, and a nontrivial Lagrangian-to-Eulerian conversion is indispensable for the GFM finite difference discretization. In this paper, an Eulerian Solvent Excluded Surface (ESES) is implemented to replace the MSMS for defining the dielectric interface. The electrostatic analysis shows that the ESES free energy is more accurate than that of the MSMS, while being free of instability issues. Moreover, this work explores, for the first time in the PBE literature, adaptive time integration techniques for the pseudo-transient simulations. A major finding is that the time increment $\Delta t$ should become smaller as the time increases, in order to maintain the temporal accuracy. This is opposite to the common practice for the steady state convergence, and is believed to be due to the PBE nonlinearity and its time splitting treatment. Effective adaptive schemes have been constructed so that the pseudo-time GFM methods become more efficient than the constant $\Delta t$ ones.
dc.description: Electronic Thesis or Dissertation
2021-01-01T00:00:00ZOn the Theory of Structures in SetsHall, Japheth Jr.https://ir.ua.edu/handle/123456789/73952023-08-09T02:52:57Z1970-01-01T00:00:00Zdc.title: On the Theory of Structures in Sets
dc.contributor.author: Hall, Japheth Jr.
dc.description.abstract: In many branches of mathematics the property of being a subspace receives considerable attention. The subspaces of a given space might be regarded as values of a structure in a set, that is, a function P on the subsets of a set V such that P(X) ⊆ V for all X ⊆ V. Thus, structures in sets form a basis for an abstract treatment of the property of being a subspace.
dc.description: Electronic Thesis or Dissertation
1970-01-01T00:00:00ZConjugate operator on variable harmonic Bergman spaceWang, Xuanhttps://ir.ua.edu/handle/123456789/70232023-08-09T02:51:36Z2020-01-01T00:00:00Zdc.title: Conjugate operator on variable harmonic Bergman space
dc.contributor.author: Wang, Xuan
dc.description.abstract: Complex analytic functions have astonishing and amazing properties. Their real parts and imaginary parts are deeply connected by the Cauchy-Riemann equations. It is natural to ask if we obtain some information about the real part, what can we conclude about the imaginary part, which is called the harmonic conjugate of the real part? Treating the relationship as an operation, the question becomes how well behaved is the harmonic conjugate operator? In this paper, by modifying some classical methods in constant exponent Hardy and Bergman spaces and developing new ways for the modern variable exponent spaces, we will study the harmonic conjugate operator on variable exponent Bergman spaces and prove that the operator is bounded when the exponent has positive minimum and finite maximum and satisfies the log-Holder condition.
dc.description: Electronic Thesis or Dissertation
2020-01-01T00:00:00Z