Department of Mathematics

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Adaptive pseudo-time methods for the Poisson-Boltzmann equation with eulerian solvent excluded surface
(University of Alabama Libraries, 2021) Jones, Benjamin Daniel; Zhao, Shan; University of Alabama Tuscaloosa
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.
Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients
(MDPI, 2021) Li, Chuan; Long, Guangqing; Li, Yiquan; Zhao, Shan; Nanning Normal University; University of California System; University of California Los Angeles; University of Alabama Tuscaloosa
The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.
Applicability of time fractional derivative models for simulating the dynamics and mitigation scenarios of COVID-19
(Pergamon, 2020) Zhang, Yong; Yu, Xiangnan; Sun, HongGuang; Tick, Geoffrey R.; Wei, Wei; Jin, Bin; University of Alabama Tuscaloosa; Hohai University; Nanjing Normal University; Nanjing Medical University
Fractional calculus provides a promising tool for modeling fractional dynamics in computational biology, and this study tests the applicability of fractional-derivative equations (FDEs) for modeling the dynamics and mitigation scenarios of the novel coronavirus for the first time. The coronavirus disease 2019 (COVID19) pandemic radically impacts our lives, while the evolution dynamics of COVID-19 remain obscure. A time-dependent Susceptible, Exposed, Infectious, and Recovered (SEIR) model was proposed and applied to fit and then predict the time series of COVID-19 evolution observed over the last three months (up to 3/22/2020) in China. The model results revealed that 1) the transmission, infection and recovery dynamics follow the integral-order SEIR model with significant spatiotemporal variations in the recovery rate, likely due to the continuous improvement of screening techniques and public hospital systems, as well as full city lockdowns in China, and 2) the evolution of number of deaths follows the time FDE, likely due to the time memory in the death toll. The validated SEIR model was then applied to predict COVID-19 evolution in the United States, Italy, Japan, and South Korea. In addition, a time FDE model based on the random walk particle tracking scheme, analogous to a mixing-limited bimolecular reaction model, was developed to evaluate non-pharmaceutical strategies to mitigate COVID-19 spread. Preliminary tests using the FDE model showed that self-quarantine may not be as efficient as strict social distancing in slowing COVID-19 spread. Therefore, caution is needed when applying FDEs to model the coronavirus outbreak, since specific COVID-19 kinetics may not exhibit nonlocal behavior. Particularly, the spread of COVID-19 may be affected by the rapid improvement of health care systems which may remove the memory impact in COVID-19 dynamics (resulting in a short-tailed recovery curve), while the death toll and mitigation of COVID-19 can be captured by the time FDEs due to the nonlocal, memory impact in fatality and human activities. (C) 2020 Elsevier Ltd. All rights reserved.
Asymptotic analysis of mass-dominated convection in a nanofluid
(University of Alabama Libraries, 2014) Dar Assi, Mahmoud H.; Hadji, Layachi; University of Alabama Tuscaloosa
The threshold conditions for the onset of convection in colloidal suspensions is investigated using the particulate medium formulation. We consider a dilute liquid suspension of solid spherical particles that is confined between two horizontal plates of infinite extent placed at the vertical coordinates Z=0 and at Z=H. The plates are assumed to be rigid, perfectly conducting and impermeable to mass flow. The suspension is heated from below. A quasi-Boussinesq approximation has been adopted i.e. the density will be assumed constant except in the gravity term where it depends on both temperature and concentration. But both the fluid viscosity and the coefficient of the particle diffusion are allowed to depend on the particle concentration through the Einstein formula for the dilute case and through the general empirical formula of suspension viscosity μ=μ₀(1-C/C_M)-2, where μ₀ is the dynamic viscosity of the base fluid and C_M is the maximum packing volume fraction of hard-sphere particles suspension for the moderately concentrated case. An experimental parameter, β, is introduced to depict the coupled effects of thermophoresis, sedimentation and particle diffusion. For a given experimental setup, β is a function of the particle size. The graph of β as function of the particle radius is an inverted parabola with two zero crossings. The first zero crossing occurs near zero particle radius. The second zero crossing occurs at larger size particle radius, although still in the nanosize range.
Augmented Lagrangian method for Euler's elastica based variational models
(University of Alabama Libraries, 2016) Chen, Mengpu; Zhu, Wei; University of Alabama Tuscaloosa
Euler's elastica is widely applied in digital image processing. It is very challenging to minimize the Euler's elastica energy functional due to the high-order derivative of the curvature term. The computational cost is high when using traditional time-marching methods. Hence developments of fast methods are necessary. In the literature, the augmented Lagrangian method (ALM) is used to solve the minimization problem of the Euler's elastica functional by Tai, Hahn and Chung and is proven to be more efficient than the gradient descent method. However, several auxiliary variables are introduced as relaxations, which means people need to deal with more penalty parameters and much effort should be made to choose optimal parameters. In this dissertation, we employ a novel technique by Bae, Tai, and Zhu, which treats curvature dependent functionals using ALM with fewer Lagrange multipliers, and apply it for a wide range of imaging tasks, including image denoising, image inpainting, image zooming, and image deblurring. Numerical experiments demonstrate the efficiency of the proposed algorithm. Besides this, numerical experiments also show that our algorithm gives better results with higher SNR/PSNR, and is more convenient for people to choose optimal parameters.
Augmented matched interface and boundary (AMBI) method for solving interface and boundary value problems
(University of Alabama Libraries, 2021) Feng, Hongsong; Zhao, Shan; University of Alabama Tuscaloosa
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.
A bounded and periodic interest rate model
(University of Alabama Libraries, 2013) Cai, Chen; Wang, Pu; University of Alabama Tuscaloosa
In financial market, interest rate is crucially important. Its changes and moves have a great impact on consumer's products, inflation rate, bond and stock market, and almost all the aspects in the financial world. An ideal stochastic model describing the volatility of the short-term interest rate would possess the following nice properties. First it has to have the periodic behavior; this is different from stock price model in which it has an increasing or decreasing trends. Second, it should maintain in a positive range and be bounded. Third, its differential equation should be simple and have an analytical solution so that its density function as well as any moments can be readily derived. In this dissertation, we propose and investigate such a stochastic differential equation. Its solution involves sine/cosine wave functions of Brownian motion that has all these properties. Their statistical properties such as mean, variance and covariance structure of this interest rate at any time are derived; their relation with martingale is established; both analytical and numerical solutions are obtained. From this interest rate model, the term structures and the yield curves will also be demonstrated for various settings.
Bounds on the number of generators of a module
(University of Alabama Libraries, 2019) Morris, Peyton; Evans, Martin; University of Alabama Tuscaloosa
The aim of this work is to present the Forster-Swan bound on the number of generators of a module. It is not our intention to present a novel finding or research discovery. Instead, we will develop the commutative algebra necessary to understand its proof, as well as the significance of the result. As a consequence, we will not present merely a series of prerequisites to the proof of the Forster-Swan bound, but rather everything which is strictly necessary to obtain a genuine understanding of the nature of the theorem. With this goal in mind, we will develop not only many of the fundamental results of commutative algebra, but in addition present results which are in a more profound way related to or even go beyond the Forster-Swan theorem itself. In section one, we describe the basic properties of commutative rings and modules over them. In particular, we will give the standard results on prime and maximal ideals, finitely generated modules, exact sequences, tensor products and flatness. In section two we define the notion of the spectrum of a ring, show that this is a geometric object associated with the ring, and present a few examples of spectra of rings. In section three, the theory of localization, the driving force behind the Forster-Swan theorem, is developed and several local-global principles of commutative algebra are demonstrated. In section four, the theory of Noetherian rings and modules, the dimension of rings, the concept of an algebraic variety, and Hilbert's basis and zero-locus theorems are presented. Section five contains the full proof of the Forster-Swan theorem. We conclude in section 6 with a discussion of topics surrounding Serre's problem on projective modules and the Eisenbud-Evans conjectures which give an improvement on the Forster-Swan bound.
Capping the variance of cash flow of hedging strategy
(University of Alabama Libraries, 2011) Ginting, Maydison; Wu, Zhijian; University of Alabama Tuscaloosa
This dissertation consolidates previous research on an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-dated futures contracts. By introducing a cap function, this dissertation defines scenarios of running risk over the hedging horizon. We introduce a linear cap function and wish to find a hedging strategy G with the smallest constant F such that the variance of the cumulative cash flow is less than or equal the multiplication of a cap function and the constant F. The objective is to seek the best function G(s) to cap the variance of cash flow under a given non-negative cap function. We also implement the result in MATLAB by creating a Graphical User Interface application that enables the user to see the various results of the variance of cash flow of the best hedging scenario.
A case-study of using tensors in multi-way electroencephalograme (EEG) data analysis
(University of Alabama Libraries, 2020) Milligan-Williams, Essence; Sidje, Roger B.; University of Alabama Tuscaloosa
Tensors are multi-dimensional arrays that can represent large datasets. Acquiring large data sets has its pros and cons; a pro being the bigger the data set, the more information could potentially be generated. A con would be the amount of labor needed to process this information. To combat this con, researchers in a variety of fields rely on tensor decomposition. Tensor decomposition's goal is to compress the data without losing any signficant information. Tensor decompositon is also known for its ability to extract underlying features that could not have been seen at face value. One such field is electroencephalography, which is the study of electrograms (EEG). An electrogram is a brain imaging tool that measures brain electrical activity. Having the abilitiy to be continously recorded for long periods of time, this could be hours, days, even weeks, EEG tends to have massive multi-dimensional datasets. In order to process the data, tensor decompositon methods such as Parallel Factor Analysis (PARAFAC) and Tucker decomposition can be executed on these large datasets.
A Classifying Family of Spaces for the Cohomology of Profinite Groups
(University of Alabama Libraries, 2021) Lee, Evan Matthew; Corson, Jon; University of Alabama Tuscaloosa
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.
Cofinite graphs and their profinite completions
(University of Alabama Libraries, 2013) Das, Bikash Chandra; Corson, Jon M.; University of Alabama Tuscaloosa
We generalize the work of B. Hartley, to the category of topological graphs. The completion of a topological group in Hartley's work can be thought of as a particular example of the completion of a general uniform space. We consider topologies that are induced by uniformities, in order to topologize our graph structures and talk about their completions. In this dissertation generalize the idea of cofinite groups, due to B. Hartley. First we define cofinite spaces in general. Then, as a special case, we study cofinite graphs and their uniform completions. We are able to show that these completions are also cofinite graphs and being compact Hausdorff and totally disconnected they are rather regarded as profinite graphs. The idea of constructing a cofinite graph starts with defining a uniform topological graph in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over the graph. By taking finitely many equivalence classes, we want to ensure the production of profinite structures over our topological graphs on taking the projective limit of the corresponding quotient graphs. It is established that for any cofinite graph there exists a unique cofinite completion. Generalizing Hartley's idea of cofinite groups and obtaining the structure for cofinite graphs we start establishing a parallel theory of cofinite graphs which in many ways can also be thought of as generalizations of the well-known works on pronite graphs by Pavel Zalesskii and Luis Ribes. Suitably defining the concept of cofinite connectedness of a cofinite graph we find that many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. As an immediate consequence we obtain the following generalized characterization of the connected Cayley graphs of conite groups: G be a cofinite group, X be a cofinite space, then Cayley graph (G, X) is also cofinite graph and it is cofinitely connected if and only if X generates G (topologically). Our immediate next concern is developing group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we are able to characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the afore said action becomes uniformly continuous. We show that the aforesaid actions can actually be extended to the structures' of corresponding cofinite completions, preserving the underlying character of the original group action.
A combinatorial proof of the invariance of tangle floer homology
(University of Alabama Libraries, 2019) Homan, Timothy Adam; Roberts, Lawrence; University of Alabama Tuscaloosa
The aim of this work is to take the combinatorial construction put forward by Petkova and Vértesi for tangle Floer homology and show that many of the arguments that apply to grid diagrams for knots can be applied to grid diagrams for tangles. In particular, we showed that the stabilization and commutation arguments used in combinatorial knot Floer homology can be applied mutatis mutandis to combinatorial tangle Floer homology, giving us an equivalence of chain complexes (either exactly in the case of commutations or up to the size of the grid in stabilizations). We then added a new move, the stretch move, and showed that the same arguments which work for commutations work for this move as well. We then extended these arguments to the context of A-infinity structures. We developed for our stabilization arguments a new type of algebraic notation and used this notation to demonstrate and simplify useful algebraic results. These results were then applied to produce type D and type DA equivalences between grid complexes and their stabilized counterparts. For commutation moves we proceeded more directly, constructing the needed type D homomorphisms and homotopies as needed and then showing that these give us a type D equivalence between tangle grid diagrams and their commuted counterparts. We also showed that these arguments can also be applied to our new stretch move. Finally, we showed that these grid moves are sufficient to accomplish the planar tangle moves required to establish equivalence of the tangles themselves with the exception of one move.
A compiler for the Bama-Bell floating point interpretive system
(University of Alabama Libraries, 1962) Gray, William J.; University of Alabama Tuscaloosa
Conjugate operator on variable harmonic Bergman space
(University of Alabama Libraries, 2020) Wang, Xuan; Ferguson, Timothy; University of Alabama Tuscaloosa
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.
A constructive nullstellensatz for univariate polynomials
(University of Alabama Libraries, 2014) Netyanun, Anupan; Trent, Tavan T.; University of Alabama Tuscaloosa
In this dissertation, we will take an effective approach to prove the Hilbert's Nullstellensatz in a special case where we have univariate polynomials $f_{i}(z)'s$ for $i\in\{1,2,...,m\}$. This approach will explicitly construct polynomials $p_{i}(z)'s$ for $i\in\{1,2,...,m\}.$ Moreover, we will get the best result on the bounds for the degrees of polynomials $p_{i}(z)'s.$ We then use a similar technique to solve the problems in a matrix case. Previous work motivated by algebraic techniques are from {[}2{]} W.D.Brownawell, {[}5{]} J.Kollar. They made a big improvement on the bounded degree of $p_{i}(z)'s$ in solutions. We are also motivated by works done in analysis from L. Carleson (1962), T. Wolff (1979). These are used to get the best result on the bounds on the degrees of $p_{i}'s$ in the solutions obtained in this dissertation. For the matrix case, we are motivated by {[}11{]} T.T. Trent, X. Zhang. This will enable us to derive the results in the matrix case.
Coordinate Descent Methods for Sparse Optimal Scoring and its Applications
(University of Alabama Libraries, 2021) Ford, Katie Wood; Ames, Brendan P; University of Alabama Tuscaloosa
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.
A corona theorem for certain subalgebras of H∞(D)
(University of Alabama Libraries, 2009) Ryle, Julie; Trent, Tavan T.; University of Alabama Tuscaloosa
[NOTE: Text or symbols not renderable in plain text are indicated by [...]. See PDF document for full abstract.] The corona theorem for the space of bounded analytic functions on the unit disk, [...], which was proven by Carleson in 1962, states that D is dense in the maximal ideal space of [...]. This theorem can be reduced to the following result: [...]. Furthermore, if we have the additional condition that [...]. In this dissertation, we prove that the corona theorem holds for certain subalgebras of [...], and we provide estimates for the sizes of the given solutions. Among the algebras we consider are those which contain bounded analytic functions whose kth derivatives vanish at 0 for all k in K, a subset of the natural numbers, which we call [...]. We give several properties the set K must have in order for [...] to be an algebra. We then prove the corona theorem in both the vector and matrix cases for these algebras. In fact, in the vector case, we prove the corona theorem using two different techniques. Each gives a unique estimate, and one extends our findings to more general algebras. We also settle a conjecture of Mortini, Sasane, and Wick involving the algebra C+BH∞(D), where B is a Blaschke product. We prove the corona theorem in C+BH∞(D) holds for an infinite number of functions. We end with a few suggestions for future research.
The Corona Theorem for the multiplier algebras on weighted Dirichlet spaces
(University of Alabama Libraries, 2009) Kidane, Berhanu Tekle; Trent, Tavan T.; University of Alabama Tuscaloosa
In this dissertation we give a proof of "The Corona Theorem for Infinitely Many Functions for the Multiplier Algebras on Weighted Dirichlet Spaces", and we obtain explicit estimates on the size of the solution. We denote the open unit disc of the complex plane by D, and for α in (0, 1) we denote by Dα the Weighted Dirichlet Spaces of all holomorphic functions on D, and by
Coverings of profinite graphs
(University of Alabama Libraries, 2013) Acharyya, Amrita; Corson, Jon M.; University of Alabama Tuscaloosa
We define a covering of a profinite graph to be a projective limit of a system of covering maps of finite graphs. With this notion of covering, we develop a covering theory for profinite graphs which is in many ways analogous to the classical theory of coverings of abstract graphs. For example, it makes sense to talk about the universal cover of a profinite graph and we show that it always exists and is unique. We define the profinite fundamental group of a profinite graph and show that a connected cover of a connected profinite graph is the universal cover if and only if its profinite fundamental group is trivial.

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