Open Access Publications
Permanent URI for this collection
Open access publications by faculty, postdocs, and graduate students in the Department of Mathematical Sciences
Browse
Browsing Open Access Publications by Issue Date
Now showing 1 - 20 of 47
Results Per Page
Sort Options
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 “I’m like the Sherpa guide”: On Learning to Teach Proof in School Mathematics(The International Group for the Psychology of Mathematics Education, 2011) Cirillo, MichelleThis article describes the experiences of a beginning mathematics teacher, Matt, across his first three years of teaching proof in a high school geometry course. Matt’s past experiences with mathematics influenced his beliefs about what he could and could not do to help his students learn how to prove. During his first year of teaching proof, Matt claimed that you cannot teach someone to write a proof. Over time, however, Matt eventually developed some strategies for teaching proof to his students. Within this work is an interest in learning more about how a teacher learns to teach proof to students who are just learning how to construct a formal proof. This case highlights the importance of pedagogical content knowledge.Item Supporting the Introduction to Formal Proof(The International Group for the Psychology of Mathematics Education (PME), 2014) Cirillo, MichelleIn this study, a tool that worked to support teachers with the introduction to formal proof in geometry is discussed. The tool helped teachers navigate the “shallow end” of proof. More specifically, the tool was shown to support teachers with introducing and scaffolding proof. Findings from this study suggest that the tool may be useful for supporting formal reasoning in geometry as well as other areas.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 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 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 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 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 Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes(Public Library Science, 2016-08-18) Soltani,Mohammad; Vargas-Garcia,Cesar A.; Antunes,Duarte; Singh,Abhyudai; Mohammad Soltani, Cesar A. Vargas-Garcia, Duarte Antunes, Abhyudai Singh; Singh, AbhyudaiInside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between two daughter cells are significant. We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise, but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes, noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where transcription rate is increased at a random point in the cell cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells.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 A stabilizer-free pressure-robust finite element method for the Stokes equations(Advances in Computational Mathematics, 2021-04-08) Ye, Xiu; Zhang, ShangyouIn this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div) conforming finite element methods for the Stokes equations require stabilizers to enforce the weak continuity of velocity in tangential direction. Some stabilizers need to tune penalty parameter and some of them do not. Our method is stabilizer free although discontinuous velocity fields are used. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Extensive numerical investigations are conducted to test accuracy and robustness of the method and to confirm the theory. The numerical examples cover low- and high-order approximations up to the degree four, and 2D and 3D cases.Item Spintronics Meets Density Matrix Renormalization Group: Quantum Spin-Torque-Driven Nonclassical Magnetization Reversal and Dynamical Buildup of Long-Range Entanglement(Physical Review X, 2021-06-23) Petrović, Marko D.; Mondal, Priyanka; Feiguin, Adrian E.; Plecháč, Petr; Nikolić, Branislav K.We introduce the time-dependent density matrix renormalization group (tDMRG) as a solution to a long-standing problem in spintronics—how to describe spin-transfer torque (STT) between flowing spins of conduction electrons and localized spins within a magnetic material by treating the dynamics of both spin species fully quantum mechanically. In contrast to conventional Slonczewski-Berger STT, where the localized spins are viewed as classical vectors obeying the Landau-Lifshitz-Gilbert equation and where their STT-driven dynamics is initiated only when the spin polarization of flowing electrons and localized spins are noncollinear, quantum STT can occur when these vectors are collinear but antiparallel. Using tDMRG, we simulate the time evolution of a many-body quantum state of electrons and localized spins, where the former are injected as a spin-polarized current pulse while the latter comprise a quantum Heisenberg ferromagnetic metallic (FM) spin-1/2XXZ chain initially in the ground state with spin polarization antiparallel to that of injected electrons. The quantum STT reverses the direction of localized spins, but without rotation from the initial orientation, when the number of injected electrons exceeds the number of localized spins. Such nonclassical reversal, which is absent from Landau-Lifshitz-Gilbert dynamics, is strikingly inhomogeneous across the FM chain, and it can be accompanied by reduction of the magnetization associated with localized spins, even to zero at specific locations. This feature arises because quantum STT generates a highly entangled nonequilibrium many-body state of all flowing and localized spins, despite starting from the initially unentangled ground state of a mundane FM. Furthermore, the mutual information between localized spins at the FM edges remains nonzero even at infinite separation as the signature of dynamical buildup of long-range entanglement. The growth in time of entanglement entropy differentiates between the quantum and conventional (i.e., noncollinear) setups for STT, reaching a much larger asymptotic value in the former case.Item Use of variables in calculus class: focusing on Teaching Assistants’ discussion of variables(International Journal of Mathematical Education in Science and Technology, 2021-09-22) Park, Jungeun; Rizzolo, DouglasGiven the importance of the ability to use variables flexibly in Calculus and students’ difficulties related to various uses of variables, this study examined how variables are treated in calculus class. Data for this study came from graduate teaching assistants’ (TAs’) classroom teaching, which plays a crucial role in undergraduate students’ learning of entry-level mathematics, but of which we still have a limited understanding. We analyzed TAs’ uses of variables in terms of prior literature examining how students use variables and what uses of variables cause difficulties for students. Our results show that the uses of variables by the TAs in this study typically aligned with students’ dominant conception of variables as symbols to be manipulated and did not give students many opportunities to consider the uses of variables that commonly cause difficulties for students.Item Identifying brain regions supporting amygdalar functionality: Application of a novel graph theory technique(NeuroImage, 2021-09-25) Matyi, Melanie A.; Cioaba, Sebastian M.; Banich, Marie T.; Spielberg, Jeffrey M.Effective amygdalar functionality depends on the concerted activity of a complex network of regions. Thus, the role of the amygdala cannot be fully understood without identifying the set of brain structures that allow the processes performed by the amygdala to emerge. However, this identification has yet to occur, hampering our ability to understand both normative and pathological processes that rely on the amygdala. We developed and applied novel graph theory methods to diffusion-based anatomical networks in a large sample (n = 1,052, 54.28% female, mean age=28.75) to identify nodes that critically support amygdalar interactions with the larger brain network. We examined three graph properties, each indexing a different emergent aspect of amygdalar network communication: current-flow betweenness centrality (amygdalar influence on information flowing between other pairs of nodes), node communicability (clarity of communication between the amygdala and other nodes), and subgraph centrality (amygdalar influence over local network processing). Findings demonstrate that each of these aspects of amygdalar communication is associated with separable sets of regions and, in some cases, these sets map onto previously identified sub-circuits. For example, betweenness and communicability were each associated with different sub-circuits that have been identified in previous work as supporting distinct aspects of memory-guided behavior. Other regions identified span basic (e.g., visual cortex) to higher-order (e.g., insula) sensory processing and executive functions (e.g., dorsolateral prefrontal cortex). Present findings expand our current understanding of amygdalar function by showing that there is no single ‘amygdala network’, but rather multiple networks, each supporting different modes of amygdalar interaction with the larger brain network. Additionally, our novel method allowed for the identification of how such regions support the amygdala, which has not been previously explored.Item 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 On the propagation of acoustic waves in a thermo-electro-magneto-elastic solid(Applicable Analysis, 2021-10-05) Hsiao, George C.; Wendland, Wolfgang L.In this paper, we are concerned with a time-dependent transmission problem for a thermo-electric-magneto-elastic solid immersed in an inviscid and compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided an appropriate scaling factor is employed. Based on estimates of variational solutions in the Laplace-transformed domain, we obtain the properties of corresponding solutions in the time-domain without having to perform explicit inversions of the variational solutions in the Laplace-transformed domain.Item The Potential of Gamma Secretase as a Therapeutic Target for Cardiac Diseases(Journal of Personalized Medicine, 2021-12-04) Sen, Sujoita; Hallee, Logan; Lam, Chi KeungHeart diseases are some of the most common and pressing threats to human health worldwide. The American Heart Association and the National Institute of Health jointly work to annually update data on cardiac diseases. In 2018, 126.9 million Americans were reported as having some form of cardiac disorder, with an estimated direct and indirect total cost of USD 363.4 billion. This necessitates developing therapeutic interventions for heart diseases to improve human life expectancy and economic relief. In this review, we look into gamma-secretase as a potential therapeutic target for cardiac diseases. Gamma-secretase, an aspartyl protease enzyme, is responsible for the cleavage and activation of a number of substrates that are relevant to normal cardiac development and function as found in mutation studies. Some of these substrates are involved in downstream signaling processes and crosstalk with pathways relevant to heart diseases. Most of the substrates and signaling events we explored were found to be potentially beneficial to maintain cardiac function in diseased conditions. This review presents an updated overview of the current knowledge on gamma-secretase processing of cardiac-relevant substrates and seeks to understand if the modulation of gamma-secretase activity would be beneficial to combat cardiac diseases.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 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.Item Promoting conceptual replications in educational research(Educational Research and Evaluation, 2022-01-31) Cai, Jinfa
- «
- 1 (current)
- 2
- 3
- »