Department of Mathematical Sciences
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The Department of Mathematical Sciences hosts a variety of exciting undergraduate and graduate programs and courses in pure, applied and industrial mathematics. The faculty, staff and students of the Department of Mathematical Sciences focus on strong research across many subdisciplines and educational innovations at all levels.
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Item A 2+1 Dimensional Insoluble Surfactant Model for a Vertical Draining Free Film(Department of Mathematical Sciences, 2002-05-09) Naire, S.; Braun, Richard J.; Snow, S.A.A 2 + 1 dimensional mathematical model is constructed to study the evolution of a vertically-oriented thin, free liquid film draining under gravity when there is an insoluble surfactant, with finite surface viscosity, on its free surface. Lubrication theory for this free film results in four coupled nonlinear partial differential equations (PDEs) describing the free surface shape, the surface velocities and the surfacant transport, at leading order. Numerical experiments are performed to understand the stability of the system to perturbations across the film. In the limit of large surface viscosities, the evolution of the free surface is that of a rigid film. In addition, these large surface viscosities act as stabilizing factors due to their energy dissipating effect. An instability is seen for the mobile case; this is caused by a competition between gravity and the Marangoni effect. The behavior observed from this model qualitatively matches some structures observed in draining film experimentsItem A Closed-Form EVSI Expression for a Multinomial Data-Generating Process(Decision Analysis, 2022-11-23) Fleischhacker, Adam; Fok, Pak-Wing; Madiman, Mokshay; Wu, NanThis paper derives analytic expressions for the expected value of sample information (EVSI), the expected value of distribution information, and the optimal sample size when data consists of independent draws from a bounded sequence of integers. Because of the challenges of creating tractable EVSI expressions, most existing work valuing data does so in one of three ways: (1) analytically through closed-form expressions on the upper bound of the value of data, (2) calculating the expected value of data using numerical comparisons of decisions made using simulated data to optimal decisions for which the underlying data distribution is known, or (3) using variance reduction as proxy for the uncertainty reduction that accompanies more data. For the very flexible case of modeling integer-valued observations using a multinomial data-generating process with Dirichlet prior, this paper develops expressions that (1) generalize existing beta-binomial computations, (2) do not require prior knowledge of some underlying “true” distribution, and (3) can be computed prior to the collection of any sample data.Item A superconvergent CDG finite element for the Poisson equation on polytopal meshes(Zeitschrift für anorganische und allgemeine Chemie | Journal of Inorganic and General Chemistry, 2023-12-08) Ye, Xiu; Zhang, ShangyouA conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous Pk functions on the two sides, but a lifted Pk+2 function from four Pk functions. When the numerical gradient space is the H (div,T) subspace of piecewise Pdk+1 polynomials on subtriangles/subtehrahedra of a polygon/polyhedron T which have a one-piece polynomial divergence on T, this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post-process which lifts a Pk CDG solution to a quasi-optimal Pk+2 solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.Item A1-L1 Phase Boundaries and Anisotropy via Multiple-Order-Parameter Theory for an FCC Alloy(Department of Mathematical Sciences, 2003) Tanoglu, G.B.; Braun, Richard J.; Cahn, J.W.; McFadden, G.B.The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an FCC lattice is studied as a function of their orientation. Using a recently-developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L12 and L10) such as occurs in the Cu-Au system, we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases. We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L10 ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find the interfacial energy, γ, as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman ‘ξ-vector’ formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is “transversely isotropic” with respect to the tetragonal axis of the L10 phase. We verify the model’s consistency with the Gibbs adsorption equation.Item Accurate Discretisation of a Nonlinear Micromagnetic Problem(Department of Mathematical Sciences, 2000) Monk, Peter B.; Vacus, O.In this paper we propose a finite element discretization of the Maxwell-Landau-Lifchitz-Gilbert equations governing the electromagnetic field in a ferromagnetic material. Our point of view is that it is desirable for the discrete problem to possess conservation properties similar to the continuous system. We first prove the existence of a new class of Liapunov functions for the continuous problem, and then for a variational formulation of the continuous problem. We also show a special continuous dependence result. Then we propose a family of mass-lumped finite element schemes for the problem. For the resulting semi-discrete problem we show that magnetization is conserved and that semi-discrete Liapunov functions exist. Finally we show the results of some computations that show the behavior of the fully discrete Liapunov functions.Item Achieving High-Order Convergence Rates with Deforming Basis Functions(Department of Mathematical Sciences, 2003) Rossi, Louis F.This article studies the use of moving, deforming elliptical Gaussian basis functions to compute the evolution of passive scalar quantities in a two-dimensional, incompressible flow field. We compute an evolution equation for the velocity, rotation, extension and deformation of the com- putational elements as a function of flow quantities. We find that if one uses the physical flow velocity data calculated from the basis function centroid, the method has only second order spatial accuracy. However, by computing the residual of the numerical method, we can determine adjustments to the centroid data so that the scheme will achieve fourth-order spatial accuracy. Simulations with nontrivial flow parameters demonstrate that the methods exhibit the properties predicted by theory.Item Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: The CDG Method(East Asian Journal on Applied Mathematics, 2022-04-06) Ye, Xiu; Zhang, ShangyouNovelty of this work is the development of a finite element method using discontinuous Pk element, which has two-order higher convergence rate than the optimal order. The method is used to solve a one-dimensional second order elliptic problem. A totally new approach is developed for error analysis. Superconvergence of order two for the CDG finite element solution is obtained. The Pk solution is lifted to an optimal order Pk+2 solution elementwise. The numerical results confirm the theory.Item Approximations in Canonical Electrostatic MEMS Models(Kluwer Academic Publishers (see: http://www.sherpa.ac.uk/romeo.php for publisher's conditions for archiving in an institutional repository), 2004-09-28) Pelesko, John A.; Driscoll, Tobin A.The mathematical modeling and analysis of electrostatically actuated micro- and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace's equation. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general "drum shaped" electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored.Item Area and Perimeter Geogebra Applet(2015-05-04) Cirillo, MichelleThis applet was designed to be used with the "Mathematics Discourse in Secondary Classrooms" professional development materials. The applet allows the user to explore questions about how the area and perimeter of a triangle interact with one another. The applet works with free Geogrebra open source software which may be downloaded from this site: https://www.geogebra.org/Item Axisymmetric Finite Element Solution of Non-isothermal Parallel-plate Flow(Department of Mathematical Sciences, 2004-11-19) Zhang, Shangyou; Olagunju, David O.Steady non-isothermal parallel-plate flow of a Newtonian fluid with a temperature dependent viscosity is considered. The viscosity is modelled by a Nahme type law. We apply axisymmetric Q k finite elements to the coupled nonlinear system to obtain numerical solutions for a wide range of parameters.Item Basis Properties of Traces and Normal Derivatives of Spherical-Separable Solutions of the Helmholtz Equation(Department of Mathematical Sciences, 2000) Dallas, A.G.The classical solutions of the Heimholtz equation resulting from the separation-of-variables procedure in spherical coördinates are frequently used in one way or another to approximate other solutions. In particular, traces and/or normal derivatives of certain sequences of these spherical-separable solutions are commonly used as trial-and-test-functions in Galerkin procedures for the approximate solution of boundary-operator problems arising from the reformulation of exterior or interior boundary-value problems and set on the boundary Γ of the domain where a solution is wanted. While the completeness properties of these traces and normal derivatives in the usual Hilbert space L2( Γ) are well known, their basis properties are not. We show that such sequences of traces or normal derivatives of the outgoing spherical-separable solutions form bases for L2( Γ) only when Γ is a sphere centered at the pole of the spherical solutions; corresponding results are given for the entire solutions, accounting for the possibility of an interior eigenvalue. We identify other Hilbert spaces, connected with the far-field pattern, for which these functions do provide bases. We apply the results to discuss some aspects of the Waterman schemes for approximate solutions of scattering problems (the so-called “T-matrix method”), including the previous article of KRISTENSSON, RAMM, and STRÖM (J. Math. Phys.24 (1983), 2619-2631) on the convergence of such methods.Item Beyond computation: Assessing in-service mathematics teachers’ conceptual understanding of fraction division through problem posing(Asian Journal for Mathematics Education, 2023-12-15) Yao, Yiling; Jia, Suijun; Cai, JinfaProblem posing has long been recognized as a critically important teaching method and goal in the area of mathematics education. However, few studies have used problem posing to assess in-service teachers’ mathematical understanding. The present study investigated in-service teachers’ mathematical understanding of fraction division, which is often considered challenging content in elementary school, from three angles: computation, drawing, and problem posing. Two studies involving 66 and 193 primary and middle school teachers were conducted to reveal the in-service teachers’ mathematical understanding and whether drawing and problem posing affected each other. Although the in-service teachers rarely had the opportunity to pose mathematical problems in their daily teaching, they were able to pose mathematical problems in this study. In addition, problem-posing tasks were more useful in diagnosing the in-service teachers’ conceptual understanding than were computation or drawing. Thus, problem posing seems to have contributed to their conceptual understanding of fraction division on the drawing task.Item Bifacial flexible CIGS thin-film solar cells with nonlinearly graded-bandgap photon-absorbing layers(JPhys Energy, 2024-03-06) Ahmad, Faiz; Monk, Peter B.; Lakhtakia, AkhleshThe building sector accounts for 36% of energy consumption and 39% of energy-related greenhouse-gas emissions. Integrating bifacial photovoltaic solar cells in buildings could significantly reduce energy consumption and related greenhouse gas emissions. Bifacial solar cells should be flexible, bifacially balanced for electricity production, and perform reasonably well under weak-light conditions. Using rigorous optoelectronic simulation software and the differential evolution algorithm, we optimized symmetric/asymmetric bifacial CIGS solar cells with either (i) homogeneous or (ii) graded-bandgap photon-absorbing layers and a flexible central contact layer of aluminum-doped zinc oxide to harvest light outdoors as well as indoors. Indoor light was modeled as a fraction of the standard sunlight. Also, we computed the weak-light responses of the CIGS solar cells using LED illumination of different light intensities. The optimal bifacial CIGS solar cell with graded-bandgap photon-absorbing layers is predicted to perform with 18%–29% efficiency under 0.01–1.0-Sun illumination; furthermore, efficiencies of 26.08% and 28.30% under weak LED light illumination of 0.0964 mW cm−2 and 0.22 mW cm−2 intensities, respectively, are predicted.Item Boundary Element Methods – An Overview(Department of Mathematical Sciences, 2004) Hsiao, George C.Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. This paper gives an overview of the method from both theoretical and numerical point of view. It summaries the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.Item Boundary Integral Methods in Low Frequency Acoustics(Department of Mathematical Sciences, 2000) Hsiao, George C.; Wendland, W.L.This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue value problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.Item Bounded Film Evolution with Nonlinear Surface Properties(Department of Mathematical Sciences, 2001-09-12) Debisschop, C.A.; Braun, Richard J.; Snow, S.A.We study the evolution of a Newtonian free surface of a thin film above a solid wall. We consider the case in which the horizontal solid is covered by a non-wetting fluid and an insoluble monolayer of surfactant is present on the fluid-air interface. We pose a model that incorporates a variety of interfacial effects: van der Waals forces, variable surface tension and surface viscosity. The surface tension and surface viscosity depend nonlinearly on the surfactant concentration. Using lubrication theory we obtain a leading order description of the shape and velocity of the fluid-air interface, and the surfactant concentration, in the form of coupled nonlinear partial differential equations. A linear stability analysis reveals that the wavenumber that characterizes the marginal state is independent of the presence of the surfactant and the nonlinearity of the surface properties. We solve the 1+1-dimensional system numerically to obtain the spatio-temporal evolution of the free surface in the nonlinear regime, and observe the progression to rupture.Item Buffer layer between a planar optical concentrator and a solar cell(American Institute of Physics, 2015-09-15) Solano, Manuel E.; Barber, Greg D.; Lakhtakia, Akhlesh; Faryad, Muhammad; Monk, Peter B.; Mallouk, Thomas E.; Manuel E. Solano, Greg D. Barber, Akhlesh Lakhtakia, Muhammad Faryad, Peter B. Monk and Thomas E. Mallouk; Monk, Peter B.The effect of inserting a buffer layer between a periodically multilayered isotropic dielectric (PMLID) material acting as a planar optical concentrator and a photovoltaic solar cell was theoretically investigated. The substitution of the photovoltaic material by a cheaper dielectric material in a large area of the structure could reduce the fabrication costs without significantly reducing the efficiency of the solar cell. Both crystalline silicon (c-Si) and gallium arsenide (GaAs) were considered as the photovoltaic material. We found that the buffer layer can act as an antireflection coating at the interface of the PMLID and the photovoltaic materials, and the structure increases the spectrally averaged electron-hole pair density by 36% for c-Si and 38% for GaAs compared to the structure without buffer layer. Numerical evidence indicates that the optimal structure is robust with respect to small changes in the grating profile.Item Challenges to Teaching Authentic Mathematical Proof in School Mathematics(The Department of Mathematics, National Taiwan Normal University Taipei, Taiwan, 2009) Cirillo, MichelleAs pointed out by Stylianides (2007), a major reason that proof and proving have been given increased attention in recent years is because they are fundamental to doing and knowing mathematics and communicating mathematical knowledge. Thus, there has been a call over the last two decades to bring the experiences of students in school mathematics closer to the work of practicing mathematicians. In this paper, I discuss the challenges that a beginning teacher faced as he attempted to teach authentic mathematical proof. More specifically, I argue that his past experiences with proof and the curriculum materials made available to him were obstacles to enacting a practice that was more like what he called “real math.”Item Chorded pancyclic properties in claw-free graphs(The Australasian Journal of Combinatorics, 2022) Beck, Kathryn; Cenek, Lisa; Cream, Megan; Gelb, BrittanyA graph G is (doubly) chorded pancyclic if G contains a (doubly) chorded cycle of every possible length m for 4 ≤ m ≤ |V (G)|. In 2018, Cream, Gould, and Larsen completely characterized the pairs of forbidden subgraphs that guarantee chorded pancyclicity in 2-connected graphs. In this paper, we show that the same pairs also imply doubly chorded pancyclicity. We further characterize conditions for the stronger property of doubly chorded (k, m)-pancyclicity where, for k ≤ m ≤ |V (G)|, every set of k vertices in G is contained in a doubly chorded i-cycle for all m ≤ i ≤ |V (G)|. In particular, we examine forbidden pairs and degree sum conditions that guarantee this recently defined cycle property.Item A Comparative Study of Lagrangian Methods Using Axisymmetric and Deforming Blobs(Department of Mathematics, 2003) Rossi, Louis F.This paper presents results from a head-to-head comparison of two Lagrangian methods for solutions to the two-dimensional, incompressible convection-diffusion equations. The first Lagrangian method is an axisymmetric core spreading method using Gaussian basis functions. The second method uses deforming elliptical Gaussian basis functions. Previous results show that the first method has second-order spatial accuracy and the second method has fourth-order spatial accuracy. However, the deforming basis functions require more computational effort per element, so this paper examines computational performance as well as overall accuracy. The test problem is the deformation and diffusion of ellipsoidal distribution of scalar with an underlying flow field that has closed circular streamlines. The test suite includes moderate, high and infinite Peclet number problems. The results indicate that the performance tradeoff for the sample flow calculation occur at modest problem sizes, and that the fourth-order method offers distinct advantages as a general approach for challenging problems.