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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 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, ShangyouA 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.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 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 An Entropy Power Inequality for Dependent Variables(IEEE Transactions on Information Theory, 2024-04-05) Madiman, Mokshay; Melbourne, James; Roberto, CyrilThe 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.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 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 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 Conceptions and Consequences of What We Call Argumentation, Justification, and Proof(East Lansing, MI: Michigan State University, 2015) Cirillo, Michelle; Kosko, Karl W.; Newton, Jill; Staples, Megan; Weber, Keith; Cirillo, MichelleArgumentation, justification, and proof are conceptualized in many ways in extant mathematics education literature. At times, the descriptions of these objects and processes are compatible or complementary; at other times, they are inconsistent and even contradictory. The inconsistencies in definitions and use of the terms argumentation, justification, and proof highlight the need for scholarly conversations addressing these (and other related) constructs. Collaboration is needed to move toward, not one-size-fits-all definitions, but rather a framework that highlights connections among them and exploits ways in which they may be used in tandem to address overarching research questions. Working group leaders aim to facilitate discussions and collaborations among researchers and to advance our collective understanding of argumentation, justification and proof, particularly the relationships among these important mathematical constructs. Working group sessions will provide opportunities to engage with a panel of researchers and other participants who approach these aspects of reasoning from different perspectives, as well as to: hear findings from a recent analysis of these constructs in research; reflect on one’s own work and position it with respect to the field; and contribute to moving the field forward in this area.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 Discovery of Power-Law Growth in the Self- Renewal of Heterogeneous Glioma Stem Cell Populations(PLOS (Public Library of Science), 2015-08-18) Sugimori, Michiya; Hayakawa, Yumiko; Boman, Bruce M.; Fields, Jeremy Z.; Awaji, Miharu; Kozano, Hiroko; Tamura, Ryoi; Yamamoto, Seiji; Ogata, Toru; Yamada, Mitsuhiko; Endo, Shunro; Kurimoto, Masanori; Kuroda, Satoshi; Michiya Sugimori, Yumiko Hayakawa, Bruce M. Boman, Jeremy Z. Fields, Miharu Awaji, Hiroko Kozano, Ryoi Tamura, Seiji Yamamoto, Toru Ogata, Mitsuhiko Yamada, Shunro Endo, Masanori Kurimoto, Satoshi Kuroda; Boman, Bruce M.BACKGROUND Accumulating evidence indicates that cancer stem cells (CSCs) drive tumorigenesis. This suggests that CSCs should make ideal therapeutic targets. However, because CSC populations in tumors appear heterogeneous, it remains unclear how CSCs might be effectively targeted. To investigate the mechanisms by which CSC populations maintain heterogeneity during self-renewal, we established a glioma sphere (GS) forming model, to generate a population in which glioma stem cells (GSCs) become enriched. We hypothesized, based on the clonal evolution concept, that with each passage in culture, heterogeneous clonal sublines of GSs are generated that progressively show increased proliferative ability. METHODOLOGY/PRINCIPAL FINDINGS To test this hypothesis, we determined whether, with each passage, glioma neurosphere culture generated from four different glioma cell lines become progressively proliferative (i.e., enriched in large spheres). Rather than monitoring self-renewal, we measured heterogeneity based on neurosphere clone sizes (#cells/clone). Log-log plots of distributions of clone sizes yielded a good fit (r>0.90) to a straight line (log(% total clones) = k*log(#cells/ clone)) indicating that the system follows a power-law (y = xk) with a specific degree exponent (k = −1.42). Repeated passaging of the total GS population showed that the same power-law was maintained over six passages (CV = −1.01 to −1.17). Surprisingly, passage of either isolated small or large subclones generated fully heterogeneous populations that retained the original power-law-dependent heterogeneity. The anti-GSC agent Temozolomide, which is well known as a standard therapy for glioblastoma multiforme (GBM), suppressed the self-renewal of clones, but it never disrupted the power-law behavior of a GS population. CONCLUSIONS/SIGNIFICANCE Although the data above did not support the stated hypothesis, they did strongly suggest a novel mechanism that underlies CSC heterogeneity. They indicate that power-law growth governs the self-renewal of heterogeneous glioma stem cell populations. That the data always fit a power-law suggests that: (i) clone sizes follow continuous, non-random, and scale-free hierarchy; (ii) precise biologic rules that reflect self-organizing emergent behaviors govern the generation of neurospheres. That the power-law behavior and the original GS heterogeneity are maintained over multiple passages indicates that these rules are invariant. These self-organizing mechanisms very likely underlie tumor heterogeneity during tumor growth. Discovery of this power-law behavior provides a mechanism that could be targeted in the development of new, more effective, anti-cancer agents. IntroductionItem An effective model for cancellous bone with a viscous interstitial fluid(Applicable Analysis, 2021-09-29) Blaszczyk, Mischa; Gilbert, Robert Pertsch; Hackl, KlausWe outline the mathematical model of the ultrasonic response of cancellous bone and its time harmonic formulation. In contrast to the Biot model, the fluid is not inviscid. Our fluid is viscous, but does not interact with the solid components.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 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 Exploring Side-Side-Angle Triangle Congruence Criterion(2015-05-04) Cirillo, Michelle; Todd, Rachael; Obrycki, JoeWe describe an exploratory task intended to support students’ conceptual understandings of triangle congruence with particular emphasis on the Side-Side-Angle (SSA) case. We reveal how SSA, often dismissed, is actually a challenging an interesting case for exploration.Item 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.Item Fundamental limits of parasitoid-driven host population suppression: Implications for biological control(PLoS ONE, 2023-12-22) Singh, AbhyudaiParasitoid 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.
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