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Open access publications by faculty, postdocs, and graduate students in the Department of Mathematical Sciences
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Item Multi-frequency Linear Sampling Method on Experimental Data Sets(IEEE Transactions on Antennas and Propagation, 2023-08-01) Monk, Peter; Pena, Manuel; Selgas, VirginiaWe investigate the use of the Linear Sampling Method (LSM) for determining the shape of a scatterer from multi-frequency experimental data. We study three multi-frequency indicators for two 2D data sets available online: one is provided by the Institut Fresnel, and another by the Electromagnetic Imaging Laboratory of the University of Manitoba. We show that the multi-frequency LSM works exceptionally well on the 2D Fresnel database, and also acceptably well on the Manitoba one. In particular, a new multi-frequency indicator is tested, and data completion for the Fresnel data set is studied. We also test an adaptive technique to cut down on the number of evaluations of the indicator function for well resolved scatterers.Item Integral representation of hydraulic permeability(Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2022-05-06) Bi, Chuan; Ou, Miao-Jung Yvonne; Zhang, ShangyouIn this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda (Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is zμ1 and the viscosity of the hosting fluid is μ1∈R+ , z∈C . The proof is carried out by the construction of solutions for large |z| and small |z| with an iteration process similar to the one used in Bruno and Leo (Arch. Ration. Mech. Anal. 121 (1993), 303–338) and Golden and Papanicolaou (Commun. Math. Phys. 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter s:=1/(z−1) , as long as s is outside the interval [−2E221+2E22,−11+2E21] , where the positive constants E1 and E2 are the extension constants that depend only on the geometry of the periodic pore space of the material.Item Effects of defect density, minority carrier lifetime, doping density, and absorber-layer thickness in CIGS and CZTSSe thin-film solar cells(Journal of Photonics for Energy, 2023-06-02) Ahmad, Faiz; Civiletti, Benjamin J.; Monk, Peter B.; Lakhtakia, AkhleshDetailed optoelectronic simulations of thin-film photovoltaic solar cells (PVSCs) with a homogeneous photon-absorber layer made of with CIGS or CZTSSe were carried out to determine the effects of defect density, minority carrier lifetime, doping density, composition (i.e., bandgap energy), and absorber-layer thickness on solar-cell performance. The transfer-matrix method was used to calculate the electron-hole-pair (EHP) generation rate, and a one-dimensional drift-diffusion model was used to determine the EHP recombination rate, open-circuit voltage, short-circuit current density, power-conversion efficiency, and fill factor. Through a comparison of limited experimental data and simulation results, we formulated expressions for the defect density in terms of the composition parameter of either CIGS or CZTSSe. All performance parameters of the thin-films PVSCs were thereby shown to be obtainable from the bulk material-response parameters of the semiconductor, with the influence of surface defects being small enough to be ignored. Furthermore, unrealistic values of the defect density (equivalently, minority carrier lifetime) will deliver unreliable predictions of the solar-cell performance. The derived expressions should guide fellow researchers in simulating the graded-bandgap and quantum-well-based PVSCs.Item Transcriptional regulation of Sis1 promotes fitness but not feedback in the heat shock response(eLife, 2023-05-17) Grade, Rania; Singh, Abhyudai; Ali, Asif; Pincus, DavidThe heat shock response (HSR) controls expression of molecular chaperones to maintain protein homeostasis. Previously, we proposed a feedback loop model of the HSR in which heat-denatured proteins sequester the chaperone Hsp70 to activate the HSR, and subsequent induction of Hsp70 deactivates the HSR (Krakowiak et al., 2018; Zheng et al., 2016). However, recent work has implicated newly synthesized proteins (NSPs) – rather than unfolded mature proteins – and the Hsp70 co-chaperone Sis1 in HSR regulation, yet their contributions to HSR dynamics have not been determined. Here, we generate a new mathematical model that incorporates NSPs and Sis1 into the HSR activation mechanism, and we perform genetic decoupling and pulse-labeling experiments to demonstrate that Sis1 induction is dispensable for HSR deactivation. Rather than providing negative feedback to the HSR, transcriptional regulation of Sis1 by Hsf1 promotes fitness by coordinating stress granules and carbon metabolism. These results support an overall model in which NSPs signal the HSR by sequestering Sis1 and Hsp70, while induction of Hsp70 – but not Sis1 – attenuates the response.Item Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes(CSIAM Transactions on Applied Mathematics, 2023-02) Ye, Xiu; Zhang, ShangyouIt is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous Pk elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous Pk element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the L2 norm. A local post-process is defined which lifts a Pk CDG solution to a discontinuous Pk+2 solution. It is proved that the lifted Pk+2 solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the Pk CDG finite element is as good as the Pk+2 continuous Galerkin finite element.Item Self-Assembly, Self-Folding, and Origami: Comparative Design Principles(Biomimetics, 2022-12-27) Jungck, John R.; Brittain, Stephen; Plante, Donald; Flynn, JamesSelf-assembly is usually considered a parallel process while self-folding and origami are usually considered to be serial processes. We believe that these distinctions do not hold in actual experiments. Based upon our experience with 4D printing, we have developed three additional hybrid classes: (1) templated-assisted (tethered) self-assembly: e.g., when RNA is bound to viral capsomeres, the subunits are constricted in their interactions to have aspects of self-folding as well; (2) self-folding can depend upon interactions with the environment; for example, a protein synthesized on a ribosome will fold as soon as peptides enter the intracellular environment in a serial process whereas if denatured complete proteins are put into solution, parallel folding can occur simultaneously; and, (3) in turbulent environments, chaotic conditions continuously alternate processes. We have examined the 43,380 Dürer nets of dodecahedra and 43,380 Dürer nets of icosahedra and their corresponding duals: Schlegel diagrams. In order to better understand models of self-assembly of viral capsids, we have used both geometric (radius of gyration, convex hulls, angles) and topological (vertex connections, leaves, spanning trees, cutting trees, and degree distributions) perspectives to develop design principles for 4D printing experiments. Which configurations fold most rapidly? Which configurations lead to complete polyhedra most of the time? By using Hamiltonian circuits of the vertices of Dürer nets and Eulerian paths of cutting trees of polyhedra unto Schlegel diagrams, we have been able to develop a systematic sampling procedure to explore the 86,760 configurations, models of a T1 viral capsid with 60 subunits and to test alternatives with 4D printing experiments, use of MagformsTM, and origami models to demonstrate via movies the five processes described above.Item 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 Two-order superconvergence for a weak Galerkin method on rectangular and cuboid grids(Numerical Methods for Partial Differential Equations, 2022-09-22) Wang, Junping; Wang, Xiaoshen; Ye, Xiu; Zhang, Shangyou; Zhu, PengThis article introduces a particular weak Galerkin (WG) element on rectangular/cuboid partitions that uses k $$ k $$ th order polynomial for weak finite element functions and ( k + 1 ) $$ \left(k+1\right) $$ th order polynomials for weak derivatives. This WG element is highly accurate with convergence two orders higher than the optimal order in an energy norm and the L 2 $$ {L}^2 $$ norm. The superconvergence is verified analytically and numerically. Furthermore, the usual stabilizer in the standard weak Galerkin formulation is no longer needed for this element.Item Seeing Algebra in Arithmetic Through Mathematical Problem Posing(The Journal of Educational Research in Mathematics, 2022-08-31) Cai, Jinfa; Hwang, StephenThis paper proceeds from the position that elementary- and middle-school students can learn and should be exposed to algebraic ideas and that a fruitful mechanism for this is to help them to see the algebra in arithmetic. After a brief survey of the literature on helping students see algebra in arithmetic, the main focus of the paper is on the use of mathematical problem posing in the classroom to help students see the algebra in arithmetic. To illustrate this, we present three cases of teaching mathematics through problem posing and discuss the perspectives they offer on developing students’ algebraic thinking. The paper concludes with an examination of how teachers might be supported in using problem posing to help their students see the algebra in arithmetic.Item Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation(Mathematical Methods in the Applied Sciences, 2022-08-16) Xie, Jiangming; Li, Maojun; Ou, Miao-Jung YvonneIn this work, we investigate the poroelastic waves by solving the time-domain Biot-JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson-Koplik-Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time-domain Biot-JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square-root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge-Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.Item Eulerian--Lagrangian Runge--Kutta Discontinuous Galerkin Method for Transport Simulations on Unstructured Meshes(SIAM Journal on Scientific Computing, 2022-07-26) Cai, Xiaofeng; Qiu, Jing-MeiThe semi-Lagrangian (SL) approach is attractive in transport simulations, e.g., in climate modeling and kinetic models, due to its numerical stability in allowing extra-large time-stepping sizes. For practical problems with complex geometry, schemes on the unstructured meshes are preferred. However, accurate and mass conservative SL methods on unstructured meshes are still under development and encounter several challenges. For instance, when tracking characteristics backward in time, high order curves are required to accurately approximate the shape of upstream cells, which brings in extra computational complexity. To avoid such computational complexity, we propose an Eulerian--Lagrangian Runge--Kutta discontinuous Galerkin method (EL RK DG) in [X. Cai, J.-M. Qiu, and Y. Yang, J. Comput. Phys., 439 (2021), 110392] as an extension of the SL discontinuous Galerkin (DG) methods. This work is a further extension of the algorithm to unstructured triangular meshes with discussion on the treatment of the inflow boundary condition. We also discuss the discrete geometric conservation law. The nonlinear weighted essentially nonoscillatory (WENO) limiter is applied to control oscillations. Desired properties of the proposed method are numerically verified by a set of benchmark tests.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 Introduction to the Andro Mikelic memorial volume(Applicable Analysis, 2022-06-27) Allaire, Grégoire; Amaziane, Brahim; Gilbert, Robert; Piatnitski, Andrey; Vernescu, BogdanAndro Mikelic passed away on 28th November 2020, after a sudden and short illness. He was a professor of Mathematics at the Institut Camille Jordan, the Mathematics Department of the University of Lyon (France), a scientist of great talent and broad range of interests and a person with brilliant individuality. Andro Mikelic was born on 2 October 1956 in Split, Croatia, a country that has always been very dear to his heart. He studied at the University of Zagreb where he obtained his PhD in Mathematics in 1983. After a few years as a researcher at the Ruder Boskovic Institute, he was a visiting professor at the University of Saint-Etienne during the academic year 1990–91, and then joined the University of Lyon as a professor in 1992.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 Quasisteady models for weld temperatures in fused filament fabrication(IMA Journal of Applied Mathematics, 2022-04-03) Edwards, D ADuring fused filament fabrication (FFF), strands of hot extruded polymer are layered onto a cooler substrate. The bond strength between layers is related to the weld temperature at the polymer/substrate interface, and hence understanding temperature evolution is of keen interest. A series of increasingly sophisticated models is presented: a standard heat equation, an unsteady fin equation and a fin equation with a heat-loss jump condition. Each is analytically tractable and uses a quasisteady approximation for the temperature in the growing substrate. The jump condition introduces the complication of a non-self-adjoint problem, but fits the experimental data very well.Item Promoting conceptual replications in educational research(Educational Research and Evaluation, 2022-01-31) Cai, JinfaItem Hamiltonian Dysthe Equation for Three-Dimensional Deep-Water Gravity Waves(Multiscale Modeling and Simulation, 2022-03-17) Guyenne, Philippe; Kairzhan, Adilbek; Sulem, CatherineThis article concerns the water wave problem in a three-dimensional domain of infinite depth and examines the modulational regime for weakly nonlinear wavetrains. We use the method of normal form transformations near the equilibrium state to provide a new derivation of the Hamiltonian Dysthe equation describing the slow evolution of the wave envelope. A precise calculation of the third-order normal form allows for a refined reconstruction of the free surface. We test our approximation against direct numerical simulations of the three-dimensional Euler system and against predictions from the classical Dysthe equation, and find very good agreement.Item Decomposition of transmitted ultrasound wave through a 2-D muscle–bone system(Mathematical Methods in the Applied Sciences, 2022-03-18) Gilbert, Robert P.; Shoushani, MichaelIn this paper, we investigate ultrasound waves passing through a skeletal muscle segment of a specimen. The model is solved using an extension of a method due to Ilya Vekua. An ansatz is made that a solution to a partial differential equation can be found in a form resembling the ansatz of Vekua. It is shown that this is indeed possible. This method is used continuously throughout the paper to find simple representations of the acoustic equations in the different muscle and bone regions.Item Scattering-induced and highly tunable by gate damping-like spin-orbit torque in graphene doubly proximitized by two-dimensional magnet Cr2Ge2Te6 and monolayer WS2(Physical Review Research, 2020-10-09) Zollner, Klaus; Petrović, Marko D.; Dolui, Kapildeb; Plecháč, Petr; Nikolić, Branislav K.; Fabian, JaroslavGraphene sandwiched between semiconducting monolayers of ferromagnet Cr2Ge2Te6 and transition-metal dichalcogenide WS2 acquires both spin-orbit (SO) coupling, of valley-Zeeman and Rashba types, and exchange coupling. Using first principles combined with quantum transport calculations, we predict that such doubly proximitized graphene within van der Waals heterostructure will exhibit SO torque driven by unpolarized charge current. This system lacks spin Hall current which is putatively considered as necessary for the efficient damping-like (DL) SO torque that plays a key role in magnetization switching. Instead, it demonstrates how a DL SO torque component can be generated solely by skew scattering off spin-independent potential barrier or impurities in purely two-dimensional electronic transport due to the presence of proximity SO coupling and its spin texture tilted out of plane. This leads to current-driven nonequilibrium spin density emerging in all spatial directions, whose cross product with proximity magnetization yields DL SO torque, unlike the ballistic regime with no scatterers in which only field-like (FL) SO torque appears. In contrast to SO torque on conventional metallic ferromagnets in contact with three-dimensional SO-coupled materials, the ratio of FL and DL components of SO torque can be tuned by more than an order of magnitude via combined top and back gates.Item Well-posedness of a random coefficient damage mechanics model(Applicable Analysis, 2022-01-07) Plecháč, Petr; Simpson, Gideon; Troy, Jerome R.We study a one-dimensional damage mechanics model in the presence of random materials properties. The model is formulated as a quasilinear partial differential equation of visco-elastic dynamics with a random field coefficient. We prove that in a transformed coordinate system the problem is well-posed as an abstract evolution equation in Banach spaces, and on the probability space it has a strongly measurable and Bochner integrable solution. We also establish the existence of weak solutions in the underlying physical coordinate system. We present numerical examples that demonstrate propagation of uncertainty in the stress–strain relation based on properties of the random damage field.