Research and Publications - Department of Mathematics
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Item A proof of some Schützenberger-type results for Eulerian paths and circuits on digraphs(Wiley, 1994-01) Chwe, Byoung-SongThis paper shows that the number of even Eulerian paths equals the number of odd Eulerian paths when the number of arcs is at least twice the number of vertices of a digraph.Item 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 TuscaloosaThe 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.Item 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 UniversityFractional 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.Item Fibonacci sequences in groupoids(Springer, 2012) Han, Jeong Soon; Kim, Hee Sik; Neggers, Joseph; Hanyang University; University of Alabama TuscaloosaIn 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.Item 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 UniversityIn 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.Item Fuzzy Upper Bounds in Groupoids(Hindawi, 2014) Ahn, Sun Shin; Kim, Young Hee; Neggers, J.; Dongguk University; Chungbuk National University; University of Alabama TuscaloosaThe 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..Item Generalized Fibonacci sequences in groupoids(Springer, 2013) Kim, Hee Sik; Neggers, J.; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym UniversityIn 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.Item 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 MexicoRegression 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.Item 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 UniversityWe 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.Item 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 TuscaloosaWe 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.Item Mathematics: Selected Reference Sources(1980-08-04) Sandy, John H.Item Modeling and Analysis of Ensemble Average Solvation Energy and Solute–Solvent Interfacial Fluctuations(De Gruyter, 2024-12-31) Shao, Yuanzhen; Chen, Zhan; Zhao, ShanVariational implicit solvation models (VISMs) have gained extensive popularity in the molecular level solvation analysis of biological systems due to their cost-effectiveness and satisfactory accuracy. Central in the construction of VISM is an interface separating the solute and the solvent. However, traditional sharp interface VISMs fall short in adequately representing the inherent randomness of the solute–solvent interface, a consequence of thermodynamic fluctuations within the solute–solvent system. Given that experimentally observable quantities are ensemble averaged, the computation of the ensemble average solvation energy (EASE)–the averaged solvation energy across all thermodynamic microscopic states–emerges as a key metric for reflecting thermodynamic fluctuations during solvation processes. This study introduces a novel approach to calculating the EASE. We devise two diffuse-interface VISMs: one within the classic Poisson–Boltzmann (PB) framework and another within the framework of size-modified PB theory, accounting for the finite-size effects. The construction of these models relies on a new diffuse interface definition u(x), which represents the probability of a point x found in the solute phase among all microstates. Drawing upon principles of statistical mechanics and geometric measure theory, we rigorously demonstrate that the proposed models effectively capture EASE during the solvation process. Moreover, preliminary analyses indicate that the size-modified EASE functional surpasses its counterpart based on the classic PB theory across various analytic aspects. Our work is the first step toward calculating EASE through the utilization of diffuse-interface VISM.Item (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 TuscaloosaWe 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.Item 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 TuscaloosaWe 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.Item On Fibonacci functions with Fibonacci numbers(Springer, 2012) Han, Jeong Soon; Kim, Hee Sik; Neggers, Joseph; Hanyang University; University of Alabama TuscaloosaIn 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.Item On Fibonacci functions with periodicity(Springer, 2012) Kim, Hee Sik; Neggers, Joseph; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym UniversityIn 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.Item On Integral Operators with Operator-Valued Kernels(Hindawi, 2010) Shahmurov, Rishad; Ministry of National Education - Turkey; Okan University; University of Alabama TuscaloosaHere, 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.Item ON Q-ALGEBRAS(Wiley, 2001) Neggers, Joseph; Ahn, Sun Shin; Kim, Hee SikWe introduce a new notion, called a Q-algebra, which is a generalization of the idea of BCH /BCI /BCK -algebras and we generalize some theorems discussed in BCI -algebras. Moreover, we introduce the notion of “quadratic” Q-algebra, and show that every quadratic Q-algebra (X; ∗, e), e ∈ X, has a product of the form x ∗ y = x − y + e, where x, y ∈ X when X is a field with |X| ≥ 3. 2000 Mathematics Subject Classification. 06F35, 03G25.Item Rational Modules and Higher Order Cauchy Transforms(Hindawi Publishing Corporation, 1981) Wang, James Li-MingWe apply the higher order Cauchy transforms to describe the closures of rational modules with respect to the L^P norms, the uniform norm and different Lipschitz norms on a compact set in the plane.Item Several types of groupoids induced by two-variable functions(Springer, 2016) Allen, P. J.; Kim, Hee Sik; Neggers, J.; University of Alabama Tuscaloosa; Hanyang UniversityIn 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.