Research and Publications - Department of Mathematics

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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.
Fuzzy Upper Bounds in Groupoids
(Hindawi, 2014) Ahn, Sun Shin; Kim, Young Hee; Neggers, J.; Dongguk University; Chungbuk National University; University of Alabama Tuscaloosa
The notion of a fuzzy upper bound over a groupoid is introduced and some properties of it are investigated. We also define the notions of an either-or subset of a groupoid and a strong either-or subset of a groupoid and study some of their related properties. In particular, we consider fuzzy upper bounds in Bin(X), where Bin(X) is the collection of all groupoids. Finally, we define a fuzzy-d-subset of a groupoid and investigate some of its properties..
The Interaction between Fuzzy Subsets and Groupoids
(Hindawi, 2014) Shin, Seung Joon; Kim, Hee Sik; Neggers, J.; University of Michigan; Hanyang University; University of Alabama Tuscaloosa
We discuss properties of a class of real-valued functions on a set X-2 constructed as finite (real) linear combinations of functions denoted as [(X, *); mu], where (X, *) is a groupoid (binary system) and mu is a fuzzy subset of X and where [(X., *); mu] (x, y) := mu (x * y) - min {mu(x), mu(y)}. Many properties, for example, mu being a fuzzy subgroupoid of (X, *), can be restated as some properties of [(X, *); mu]. Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin (X); square) for example.
(n-1)-Step Derivations on n-Groupoids: The Case n=3
(Hindawi, 2014) Alshehri, N. O.; Kim, Hee Sik; Neggers, J.; King Abdulaziz University; Hanyang University; University of Alabama Tuscaloosa
We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping d is a self-derivation at each step. Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet (D, d) on X, consisting of a two-step derivation d and its square D = d circle d, for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.
On Abelian and Related Fuzzy Subsets of Groupoids
(Hindawi, 2013) Shin, Seung Joon; Kim, Hee Sik; Neggers, J.; University of Michigan; Hanyang University; University of Alabama Tuscaloosa
We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.
The influence of surfactant on the propagation of a semi-infinite bubble through a liquid-filled compliant channel
(Cambridge University Press, 2012) Halpern, David; Gaver, Donald P., III; University of Alabama Tuscaloosa; Tulane University
We investigate the influence of a soluble surfactant on the steady-state motion of a finger of air through a compliant channel. This study provides a basic model from which to understand the fluid-structure interactions and physicochemical hydrodynamics of pulmonary airway reopening. Airway closure occurs in lung diseases such as respiratory distress syndrome and acute respiratory distress syndrome as a result of fluid accumulation and surfactant insufficiency. This results in 'compliant collapse' with the airway walls buckled and held in apposition by a liquid occlusion that blocks the passage of air. Airway reopening is essential to the recovery of adequate ventilation, but has been associated with ventilator-induced lung injury because of the exposure of airway epithelial cells to large interfacial flow-induced pressure gradients. Surfactant replacement is helpful in modulating this deleterious mechanical stimulus, but is limited in its effectiveness owing to slow surfactant adsorption. We investigate the effect of surfactant on micro-scale models of reopening by computationally modelling the steady two-dimensional motion of a semi-infinite bubble propagating through a liquid-filled compliant channel doped with soluble surfactant. Many dimensionless parameters affect reopening, but we primarily investigate how the reopening pressure p(b) depends upon the capillary number Ca (the ratio of viscous to surface tension forces), the adsorption depth parameter lambda (a bulk concentration parameter) and the bulk Peclet number Pe(b) (the ratio of bulk convection to diffusion). These studies demonstrate a dependence of p(b) on lambda, and suggest that a critical bulk concentration must be exceeded to operate as a low-surface-tension system. Normal and tangential stress gradients remain largely unaffected by physicochemical interactions - for this reason, further biological studies are suggested that will clarify the role of wall flexibility and surfactant on the protection of the lung from atelectrauma.
Impact of an equality constraint on the class-specific residual variances in regression mixtures: A Monte Carlo simulation study
(Springer, 2016) Kim, Minjung; Lamont, Andrea E.; Jaki, Thomas; Feaster, Daniel; Howe, George; Van Horn, M. Lee; University of Alabama Tuscaloosa; University of South Carolina Columbia; Lancaster University; University of Miami; George Washington University; University of New Mexico
Regression mixture models are a novel approach to modeling the heterogeneous effects of predictors on an outcome. In the model-building process, often residual variances are disregarded and simplifying assumptions are made without thorough examination of the consequences. In this simulation study, we investigated the impact of an equality constraint on the residual variances across latent classes. We examined the consequences of constraining the residual variances on class enumeration (finding the true number of latent classes) and on the parameter estimates, under a number of different simulation conditions meant to reflect the types of heterogeneity likely to exist in applied analyses. The results showed that bias in class enumeration increased as the difference in residual variances between the classes increased. Also, an inappropriate equality constraint on the residual variances greatly impacted on the estimated class sizes and showed the potential to greatly affect the parameter estimates in each class. These results suggest that it is important to make assumptions about residual variances with care and to carefully report what assumptions are made.
A unified discontinuous Galerkin framework for time integration
(Wiley, 2014) Zhao, Shan; Wei, G. W.; University of Alabama Tuscaloosa; Michigan State University
We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time-stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time-stepping schemes, such as low-order one-step methods, Runge-Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge-Kutta methods of different orders are constructed. The derivation of symplectic Runge-Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant-Friedrichs-Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time-stepping schemes. Copyright (c) 2013 John Wiley & Sons, Ltd.
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.
On Integral Operators with Operator-Valued Kernels
(Hindawi, 2010) Shahmurov, Rishad; Ministry of National Education - Turkey; Okan University; University of Alabama Tuscaloosa
Here, we study the continuity of integral operators with operator-valued kernels. Particularly we get L(q) (S;X) -> L(p) (T;Y) estimates under some natural conditions on the kernel k : T x S -> B (X, Y), where X and Y are Banach spaces, and (T, Sigma(T), mu) and (S, Sigma(S), nu) are positive measure spaces: Then, we apply these results to extend the well- known Fourier Multiplier theorems on Besov spaces.
On Fibonacci functions with periodicity
(Springer, 2012) Kim, Hee Sik; Neggers, Joseph; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym University
In this paper we discuss Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions.
On Fibonacci functions with Fibonacci numbers
(Springer, 2012) Han, Jeong Soon; Kim, Hee Sik; Neggers, Joseph; Hanyang University; University of Alabama Tuscaloosa
In this paper we consider Fibonacci functions on the real numbers R, i.e., functions f : R -> R such that for all x is an element of R, f(x + 2) = f(x + 1) + f(x). We develop the notion of Fibonacci functions using the concept of f f-even and f-odd functions. Moreover, we show that if f is a Fibonacci function then lim(x ->infinity) f(x+1)/f(x) = 1+root 5/2.
Fibonacci sequences in groupoids
(Springer, 2012) Han, Jeong Soon; Kim, Hee Sik; Neggers, Joseph; Hanyang University; University of Alabama Tuscaloosa
In this article, we consider several properties of Fibonacci sequences in arbitrary groupoids (i.e., binary systems). Such sequences can be defined in a left-hand way and a right-hand way. Thus, it becomes a question of interest to decide when these two ways are equivalent, i.e., when they produce the same sequence for the same inputs. The problem has a simple solution when the groupoid is flexible. The Fibonacci sequences for several groupoids and for the class of groups as special cases are also discussed. 2000 Mathematics Subject Classification: 20N02; 11B39.
THUNDER: Helping Underfunded NPO’s Distribute Electronic Resources
(2013) Loewen, Gabriel; Galloway, Jeffrey; Robinson, Jeffrey; Hong, Xiaoyan; Vrbsky, Susan; University of Alabama Tuscaloosa
As federal funding in many public non-profit organizations (NPO’s) seems to be dwindling, it is of the utmost importance that efforts are focused on reducing operating costs of needy organizations, such as public schools. Our approach for reducing organizational costs is through the combined benefits of a high performance cloud architecture and low-power, thin-client devices. However, general-purpose private cloud architectures are not easily deployable by average users, or even those with some computing knowledge. For this reason, we propose a new vertical cloud architecture, which is focused on ease of deployment and management, as well as providing organizations with cost-efficient virtualization and storage, and other organization-specific utilities. We postulate that if organizations are provided with on-demand access to electronic resources in a way that is cost-efficient, then the operating costs may be reduced, such that the user experience and organizational efficiency may be increased. In this paper we discuss our private vertical cloud architecture called THUNDER. Additionally, we introduce a number of methodologies that could enable needy non-profit organizations to decrease costs and also provide many additional benefits for the users. Specifically, this paper introduces our current implementation of THUNDER, details about the architecture, and the software system that we have designed to specifically target the needs of underfunded organizations.
Generalized Fibonacci sequences in groupoids
(Springer, 2013) Kim, Hee Sik; Neggers, J.; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym University
In this paper, we introduce the notion of generalized Fibonacci sequences over a groupoid and discuss it in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P-algebra, we obtain several relations on groupoids which are derived from generalized Fibonacci sequences.
Fuzzy rank functions in the set of all binary systems
(Springer, 2016) Kim, Hee Sik; Neggers, J.; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym University
In this paper, we introduce fuzzy rank functions for groupoids, and we investigate their roles in the semigroup of binary systems by using the notions of right parallelisms and rho-shrinking groupoids.
Several types of groupoids induced by two-variable functions
(Springer, 2016) Allen, P. J.; Kim, Hee Sik; Neggers, J.; University of Alabama Tuscaloosa; Hanyang University
In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if (X, *) is a groupoid and if phi : X-2 -> X-2 is a function phi (a, b) = (u, v), then (X, *) is a left-twisted semigroup with respect to phi if for all a, b, c is an element of X, a * (b * c) = (u * v) * c. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule (a * b) * c = u * (v * c) for all a, b, c is an element of X. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field (X,+, .) via a formula x * y = lambda x + mu y, with lambda, mu is an element of X, fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of (X, *) implies the fact that (X, *) is a semigroup.

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