Open Access Publications

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Open access publications by faculty, postdocs, and graduate students in the Department of Mathematical Sciences

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A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids
(Electronic Research Archive, 2024-05-27) Yang, Yan; Ye, Xiu; Zhang, Shangyou
A pressure-robust stabilizer-free weak Galerkin (WG) finite element method has been defined for the Stokes equations on triangular and tetrahedral meshes. We have obtained pressure-independent error estimates for the velocity without any velocity reconstruction. The optimal-order convergence for the velocity of the WG approximation has been proved for the L2 norm and the H1 norm. The optimal-order error convergence has been proved for the pressure in the L2 norm. The theory has been validated by performing some numerical tests on triangular and tetrahedral meshes.
Graduate Teaching Assistants’ Perception of Student Difficulties and Use in Teaching
(International Journal of Research in Undergraduate Mathematics Education, 2024-06-01) Park, Jungeun; Rizzolo, Douglas
Given the important role graduate teaching assistants (TAs) play in undergraduate students’ learning, we investigated what TAs identified as students’ difficulties from students’ written work, their plans to address them, and implementation of their plans in class. Since the difficulties that TAs identified in general matched errors that students made, we analyzed what TAs identified in terms of literature on error handling. We examined levels of specific details of students’ work involved in TAs’ identifying, planning, and teaching. Our results show that (a) TAs often did not identify the most frequent errors students made, which reflected well-documented difficulties from the literature, (b) the errors TAs identified were mainly procedural in nature, (c) specific details of students’ work were mainly included in procedural errors, and (d) the level of specificity of students’ work was generally consistent but showed some drops when going from identifying to planning, then to teaching. Our results highlight interesting questions for future research and could be used as resources to design professional development that helps TAs use students’ errors in teaching to promote students’ learning.
On the analysis of modes in a closed electromagnetic waveguide
(Comptes Rendus Mathématique, 2024-05-31) Halla, Martin; Monk, Peter
Modal expansions are useful to understand wave propagation in an infinite closed electromagnetic waveguide. They can also be used to construct generalized Dirichlet-to-Neumann maps that provide artificial boundary conditions for truncating a computational domain when discretizing the field by finite elements. The modes of a waveguide arise as eigenfunctions of a non-symmetric eigenvalue problem, and the eigenvalues determine the propagation (or decay) of the modes along the waveguide. For the successful use of waveguide modes, it is necessary to know that the modes exist and form a dense set in a suitable function space containing the trace of the electric field in the waveguide. This paper is devoted to proving such a density result using the methods of Keldysh. We also show that the modes satisfy a useful orthogonality property, and show how the Dirichlet-to-Neumann map can be calculated. Our existence and density results are proved under realistic regularity assumptions on the cross section of the waveguide, and the electromagnetic properties of the materials in the waveguide, so generalizing existing results. Résumé Les expansions modales sont utiles pour comprendre la propagation des ondes dans un guide d’ondes électromagnétiques fermé et infini. Elles peuvent également être utilisées pour construire des cartes de Dirichlet à Neumann généralisées qui peuvent être utilisées pour fournir des conditions limites artificielles afin de tronquer un domaine de calcul lors de la discrétisation du champ par des éléments finis. Les modes d’un guide d’ondes apparaissent comme des fonctions propres d’un problème de valeurs propres non symétrique, et les valeurs propres déterminent la propagation (ou la décroissance) des modes le long du guide d’ondes. Pour une utilisation réussie des modes de guide d’ondes, il est nécessaire de savoir que les modes existent et forment un ensemble dense dans un espace de fonctions approprié contenant la trace du champ électrique dans le guide d’ondes. Cet article est consacré à la démonstration d’un tel résultat de densité en utilisant les méthodes de Keldysh. Nous montrons également que les modes satisfont une propriété d’orthogonalité utile, et nous montrons comment la carte de Dirichlet à Neumann peut être calculée. Nos résultats d’existence et de densité sont prouvés sous des hypothèses de régularité réalistes sur la section transversale du guide d’ondes et les propriétés électromagnétiques des matériaux dans le guide d’ondes, généralisant ainsi les résultats existants.
An Entropy Power Inequality for Dependent Variables
(IEEE Transactions on Information Theory, 2024-04-05) Madiman, Mokshay; Melbourne, James; Roberto, Cyril
The entropy power inequality for independent random variables is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Very few extensions of the entropy power inequality have been developed for settings with dependence. We address this gap in the literature by developing entropy power inequalities for dependent random variables. In particular, we highlight the role of log-supermodularity in delivering sufficient conditions for an entropy power inequality stated using conditional entropies.
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, Akhlesh
The 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.
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, Jinfa
Problem 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.
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, Shangyou
A 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.
Saddle point least squares discretization for convection-diffusion
(Applicable Analysis, 2023-12-11) Bacuta, Constantin; Hayes, Daniel; O'Grady, Tyler
We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models.
Fascination with Fluctuation: Luria and Delbrück’s Legacy
(Axioms, 2023-03-07) Robeva, Raina S.; Jungck, John R.
While Luria and Delbrück’s seminal work has found its way to some college biology textbooks, it is now largely absent from those in mathematics. This is a significant omission, and we consider it a missed opportunity to present a celebrated conceptual model that provides an authentic and, in many ways, intuitive example of the quantifiable nature of stochasticity. We argue that it is an important topic that could enrich the educational literature in mathematics, from the introductory to advanced levels, opening many doors to undergraduate research. The paper has two main parts. First, we present in detail the mathematical theory behind the Luria–Delbrück model and make suggestions for further readings from the literature. We also give ideas for inclusion in various mathematics courses and for projects that can be used in regular courses, independent projects, or as starting points for student research. Second, we briefly review available hands-on activities as pedagogical ways to facilitate problem posing, problem-based learning, and investigative case-based learning and to expose students to experiments leading to Poisson distributions. These help students with even limited mathematics backgrounds understand the significance of Luria–Delbrück’s work for determining mutation rates and its impact on many fields, including cancer chemotherapy, antibiotic resistance, radiation, and environmental screening for mutagens and teratogens.
Fundamental limits of parasitoid-driven host population suppression: Implications for biological control
(PLoS ONE, 2023-12-22) Singh, Abhyudai
Parasitoid wasps are increasingly being used to control insect pest populations, where the pest is the host species parasitized by the wasp. Here we use the discrete-time formalism of the Nicholson-Bailey model to investigate a fundamental question—are there limits to parasitoid-driven suppression of the host population density while still ensuring a stable coexistence of both species? Our model formulation imposes an intrinsic self-limitation in the host’s growth resulting in a carrying capacity in the absence of the parasitoid. Different versions of the model are considered with parasitism occurring at a developmental stage that is before, during, or after the growth-limiting stage. For example, the host’s growth limitation may occur at its larval stage due to intraspecific competition, while the wasps attack either the host egg, larval or pupal stage. For slow-growing hosts, models with parasitism occurring at different life stages are identical in terms of their host suppression dynamics but have contrasting differences for fast-growing hosts. In the latter case, our analysis reveals that wasp parasitism occurring after host growth limitation yields the lowest pest population density conditioned on stable host-parasitoid coexistence. For ecologically relevant parameter regimes we estimate this host suppression to be roughly 10-20% of the parasitoid-free carrying capacity. We further expand the models to consider a fraction of hosts protected from parasitism (i.e., a host refuge). Our results show that for a given host reproduction rate there exists a critical value of protected host fraction beyond which, the system dynamics are stable even for high levels of parasitism that drive the host to arbitrary low population densities. In summary, our systematic analysis sheds key insights into the combined effects of density-dependence in host growth and parasitism refuge in stabilizing the host-parasitoid population dynamics with important implications for biological control.
Multi-frequency Linear Sampling Method on Experimental Data Sets
(IEEE Transactions on Antennas and Propagation, 2023-08-01) Monk, Peter; Pena, Manuel; Selgas, Virginia
We 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.
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, Shangyou
In 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.
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, Akhlesh
Detailed 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.
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, David
The 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.
Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes
(CSIAM Transactions on Applied Mathematics, 2023-02) Ye, Xiu; Zhang, Shangyou
It 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.
Self-Assembly, Self-Folding, and Origami: Comparative Design Principles
(Biomimetics, 2022-12-27) Jungck, John R.; Brittain, Stephen; Plante, Donald; Flynn, James
Self-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.
A Closed-Form EVSI Expression for a Multinomial Data-Generating Process
(Decision Analysis, 2022-11-23) Fleischhacker, Adam; Fok, Pak-Wing; Madiman, Mokshay; Wu, Nan
This 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.
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, Peng
This 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.
Seeing Algebra in Arithmetic Through Mathematical Problem Posing
(The Journal of Educational Research in Mathematics, 2022-08-31) Cai, Jinfa; Hwang, Stephen
This 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.
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 Yvonne
In 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.

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