Mathematics and Statistics Faculty Publications and PresentationsCopyright (c) 2020 Portland State University All rights reserved.
https://pdxscholar.library.pdx.edu/mth_fac
Recent documents in Mathematics and Statistics Faculty Publications and Presentationsen-usFri, 29 May 2020 11:03:31 PDT3600Some New Results on Stochastic Comparisons of Coherent Systems Using Signatures
https://pdxscholar.library.pdx.edu/mth_fac/286
https://pdxscholar.library.pdx.edu/mth_fac/286Fri, 22 May 2020 12:33:37 PDT
We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.
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Ebrahim Amini-Seresht et al.Comparing Hecke Coefficients of Automorphic Representations
https://pdxscholar.library.pdx.edu/mth_fac/285
https://pdxscholar.library.pdx.edu/mth_fac/285Wed, 22 Apr 2020 10:28:13 PDT
We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin-Selberg -functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to . As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes.
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Liubomir Chiriac et al.Predicting Densities and Elastic Moduli of SiO2-based Glasses by Machine Learning
https://pdxscholar.library.pdx.edu/mth_fac/284
https://pdxscholar.library.pdx.edu/mth_fac/284Mon, 20 Apr 2020 11:25:57 PDT
Chemical design of SiO2-based glasses with high elastic moduli and low weight is of great interest. However, it is difficult to find a universal expression to predict the elastic moduli according to the glass composition before synthesis since the elastic moduli are a complex function of interatomic bonds and their ordering at different length scales. Here we show that the densities and elastic moduli of SiO2-based glasses can be efficiently predicted by machine learning (ML) techniques across a complex compositional space with multiple (>10) types of additive oxides besides SiO2. Our machine learning approach relies on a training set generated by high-throughput molecular dynamic (MD) simulations, a set of elaborately constructed descriptors that bridges the empirical statistical modeling with the fundamental physics of interatomic bonding, and a statistical learning/predicting model developed by implementing least absolute shrinkage and selection operator with a gradient boost machine (GBM-LASSO). The predictions of the ML model are comprehensively compared and validated with a large amount of both simulation and experimental data. By just training with a dataset only composed of binary and ternary glass samples, our model shows very promising capabilities to predict the density and elastic moduli for k-nary SiO2-based glasses beyond the training set. As an example of its potential applications, our GBM-LASSO model was used to perform a rapid and low-cost screening of many (~105) compositions of a multicomponent glass system to construct a compositional-property database that allows for a fruitful overview on the glass density and elastic properties.
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Yong-Jie Hu et al.A Bayesian Nonparametric Testing Procedure for Paired Samples
https://pdxscholar.library.pdx.edu/mth_fac/283
https://pdxscholar.library.pdx.edu/mth_fac/283Tue, 31 Mar 2020 10:12:43 PDT
We propose a Bayesian hypothesis testing procedure for comparing the distributions of paired samples. The procedure is based on a flexible model for the joint distribution of both samples. The flexibility is given by a mixture of Dirichlet processes. Our proposal uses a spike‐slab prior specification for the base measure of the Dirichlet process and a particular parametrization for the kernel of the mixture in order to facilitate comparisons and posterior inference. The joint model allows us to derive the marginal distributions and test whether they differ or not. The procedure exploits the correlation between samples, relaxes the parametric assumptions, and detects possible differences throughout the entire distributions. A Monte Carlo simulation study comparing the performance of this strategy to other traditional alternatives is provided. Finally, we apply the proposed approach to spirometry data collected in the United States to investigate changes in pulmonary function in children and adolescents in response to air polluting factors.
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Luz Adriana Pereira et al.Group Theory Students' Perceptions of Binary Operation
https://pdxscholar.library.pdx.edu/mth_fac/282
https://pdxscholar.library.pdx.edu/mth_fac/282Tue, 31 Mar 2020 10:12:41 PDT
Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students’ perceptions of the general binary operation via a two-phase study consisting of task-based surveys and interviews. We document what attributes of binary operation group theory students perceive as critical and what types of metaphors students use to convey these attributes. We found that many students treat superficial features as critical (such as element-operator-element formatting) and do not always perceive critical features as essential (such as the binary attribute). Further, these attributes were communicated across three metaphor categories: arithmetic-related, function-related, and organization-related.
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Kathleen Mary Melhuish et al.Developing an Active Approach to Chemistry-Based Group Theory
https://pdxscholar.library.pdx.edu/mth_fac/281
https://pdxscholar.library.pdx.edu/mth_fac/281Thu, 26 Mar 2020 14:52:59 PDT
Chapter 13. Group theory, particularly the concept of symmetry, has applications in many different scientific fields and is an important part of the undergraduate curriculum in mathematics. In upper-level mathematics courses, group theory is only discussed abstractly, and students are rarely given an opportunity to apply these ideas to real-world problems. To better appreciate the applicability of group theory and symmetry, a local instructional theory is being developed, where students reinvent a classification scheme for chemically important point groups. In a pilot study, two mathematics education graduate students with limited knowledge of chemistry were given ball and stick models of water, ammonia, and ethane and were asked to develop and describe a procedure for efficiently and comprehensively finding all the symmetries of any given molecule. Video recordings of the students successfully completing this task and the corresponding inscriptions they made were interpreted using the emergent model heuristic in order to understand the evolution of the students’ model from model-of to model-for. Implications of these results on the development of the local instructional theory and for future experiments are also discussed.
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Anna Marie Bergman et al.An Essay on Proof, Conviction, and Explanation: Multiple Representation Systems in Combinatorics
https://pdxscholar.library.pdx.edu/mth_fac/280
https://pdxscholar.library.pdx.edu/mth_fac/280Wed, 18 Mar 2020 15:42:32 PDT
There is a longstanding conversation in the mathematics education literature about proofs that explain versus proofs that only convince. In this essay, we offer a characterization of explanatory proofs with three goals in mind. We first propose a theory of explanatory proofs for mathematics education in terms of representation systems. Then, we illustrate these ideas in terms of combinatorial proofs, focusing on binomial identities. Finally, we leverage our theory to explain audience-dependent and audience-invariant aspects of explanatory proof. Throughout, we use the context of combinatorics to emphasize points and to offer examples of proofs that can be explanatory or only convincing, depending on how one understands the claim being made.
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Elise Nicole Lockwood et al.An Enhanced Finite Difference Time Domain Method for Two Dimensional Maxwell's Equations
https://pdxscholar.library.pdx.edu/mth_fac/279
https://pdxscholar.library.pdx.edu/mth_fac/279Tue, 17 Mar 2020 11:18:42 PDT
An enhanced finite-difference time-domain (FDTD)algorithm is built to solve the transverse electrictwo-dimensional Maxwell’s equations with inhomogeneousdielectric media where the electric fields are discontinuousacross the dielectric interface. The new algorithm is derivedbased upon the integral version of the Maxwell’s equationsas well as the relationship between the electric fields acrossthe interface. To resolve the instability issue of Yee’s scheme(staircasing) caused by discontinuous permittivity across theinterface, our algorithm revises the permittivities and makessome corrections to the scheme for the cells around the inter-face. It is also an improvement over the contour-path effectivepermittivity algorithm by including some extra terms in theformulas. The scheme is validated in solving the scatteringof a dielectric cylinder with exact solution from Mie theoryand is then compared with the above contour-path method,the usual staircasing and the volume-average method. Thenumerical results demonstrate that the new algorithm hasachieved significant improvement in accuracy over othermethods. Furthermore, the algorithm has a simple structureand can be merged into current FDTD software packageseasily. The C++source code for this paper is provided assupporting information for public access.
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Timothy Meagher et al.Simulation of Optical Fiber Amplifier Gain Using Equivalent Short Fibers
https://pdxscholar.library.pdx.edu/mth_fac/278
https://pdxscholar.library.pdx.edu/mth_fac/278Tue, 17 Mar 2020 11:18:38 PDT
Electromagnetic wave propagation in optical fiber amplifiers obeys Maxwell equations. Using coupled mode theory, the full Maxwell system within an optical fiber amplifier is reduced to a simpler model. The simpler model is made more efficient through a new scale model, referred to as an equivalent short fiber, which captures some of the essential characteristics of a longer fiber. The equivalent short fiber can be viewed as a fiber made using artificial (unphysical) material properties that in some sense compensates for its reduced length. The computations can be accelerated by a factor approximately equal to the ratio of the original length to the reduced length of the equivalent fiber. Computations using models of two commercially available fibers – one doped with ytterbium, and the other with thulium – show the practical utility of the concept. Extensive numerical studies are conducted to assess when the equivalent short fiber model is useful and when it is not.
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Dow Drake et al.Supporting Novice Mathematics Teacher Educators Teaching Elementary Mathematics Content Courses for the First Time
https://pdxscholar.library.pdx.edu/mth_fac/277
https://pdxscholar.library.pdx.edu/mth_fac/277Tue, 17 Mar 2020 11:18:35 PDT
In order to be effectively prepared by a teacher education program, prospective elementary teachers (PTs) need to experience high quality mathematics instruction in their mathematics content courses. The instructors of these courses typically consist of individuals (mathematicians and mathematics educators) with ranging experiences, from tenured faculty members to first-year assistant professors or graduate students. This paper explores how to support novice mathematics teacher educators (MTEs) who are teaching elementary content coursework for PTs for the first time. We detail and describe how to implement three systems for supporting novice MTEs: working with a mentor, being provided with educative curriculum materials, and working in a collaborative teaching environment. We close by discussing specific challenges associated with these supports, and call for more institutions to share how they have successfully implemented systems to support novice MTEs.
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Siobahn Suppa et al.Eigenvalue Problems for Exponential-Type Kernels
https://pdxscholar.library.pdx.edu/mth_fac/276
https://pdxscholar.library.pdx.edu/mth_fac/276Tue, 17 Mar 2020 11:18:31 PDT
We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate |λ_{k} − λ_{k,h}| ≤ C_{kh}^{2} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λ_{k} and λ_{k,h} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that |λ_{k,h}^{(G)}-λ_{k,h(N)}| ≤ Ch^{2}, where λ_{k,h}^{(G)} and λ_{k,h}^{(N) }denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments
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Difeng Cai et al.On the Equality Case of the Ramanujan Conjecture for Hilbert Modular Forms
https://pdxscholar.library.pdx.edu/mth_fac/275
https://pdxscholar.library.pdx.edu/mth_fac/275Thu, 30 Jan 2020 16:53:20 PST
The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two.
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Liubomir ChiriacA Computational Method for Earthquake Cycles within Anisotropic Media
https://pdxscholar.library.pdx.edu/mth_fac/274
https://pdxscholar.library.pdx.edu/mth_fac/274Mon, 09 Dec 2019 15:43:40 PST
We present a numerical method for the simulation of earthquake cycles on a 1-D fault interface embedded in a 2-D homogeneous, anisotropic elastic solid. The fault is governed by an experimentally motivated friction law known as rate-and-state friction which furnishes a set of ordinary differential equations which couple the interface to the surrounding volume. Time enters the problem through the evolution of the ordinary differential equations along the fault and provides boundary conditions for the volume, which is governed by quasi-static elasticity. We develop a time-stepping method which accounts for the interface/volume coupling and requires solving an elliptic partial differential equation for the volume response at each time step. The 2-D volume is discretized with a second-order accurate finite difference method satisfying the summation-by-parts property, with boundary and fault interface conditions enforced weakly. This framework leads to a provably stable semi-discretization. To mimic slow tectonic loading, the remote side-boundaries are displaced at a slow rate, which eventually leads to earthquake nucleation at the fault. Time stepping is based on an adaptive, fourth-order Runge–Kutta method and captures the highly varying timescales present. The method is verified with convergence tests for both the orthotropic and fully anisotropic cases. An initial parameter study reveals regions of parameter space where the systems experience a bifurcation from period one to period two behaviour. Additionally, we find that anisotropy influences the recurrence interval between earthquakes, as well as the emergence of aseismic transients and the nucleation zone size and depth of earthquakes.
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Maricela Best Mckay et al.Bornological Coderivative and Subdifferential Calculus in Smooth Banach Spaces
https://pdxscholar.library.pdx.edu/mth_fac/273
https://pdxscholar.library.pdx.edu/mth_fac/273Mon, 09 Dec 2019 15:43:37 PST
In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions.
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Mau Nguyen et al.A DC Programming Approach for Solving Multicast Network Design Problems via the Nesterov Smoothing Technique
https://pdxscholar.library.pdx.edu/mth_fac/272
https://pdxscholar.library.pdx.edu/mth_fac/272Thu, 07 Nov 2019 13:37:40 PST
This paper continues our effort initiated in [19] to study Multicast Communication Networks, modeled as bilevel hierarchical clustering problems, by using mathematical optimization techniques. Given a finite number of nodes, we consider two different models of multicast networks by identifying a certain number of nodes as cluster centers, and at the same time, locating a particular node that serves as a total center so as to minimize the total transportation cost through the network. The fact that the cluster centers and the total center have to be among the given nodes makes this problem a discrete optimization problem. Our approach is to reformulate the discrete problem as a continuous one and to apply Nesterov smoothing approximation technique on the Minkowski gauges that are used as distance measures. This approach enables us to propose two implementable DCA-based algorithms for solving the problems. Numerical results and practical applications are provided to illustrate our approach.
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Wondi Geremew et al.Racial/Ethnic Differences in the Impact of Adverse Childhood Experiences on Posttraumatic Stress Disorder in a Nationally Representative Sample of Adolescents
https://pdxscholar.library.pdx.edu/mth_fac/270
https://pdxscholar.library.pdx.edu/mth_fac/270Tue, 05 Nov 2019 12:50:56 PST
Adverse childhood experiences (ACE) are associated with increased risk for psychopathology over the life course. However, few studies have examined the differential impact of ACE and posttraumatic stress disorder (PTSD) for racially and ethnically diverse adolescent populations. The findings from the few studies that examine differential effects are contradictory. This study uses data from the National Comorbidity Survey-Adolescent Supplement (NCS-A), a nationally representative adolescent sample (N = 10,123) to examine the impact of race/ethnicity on the association between ACE and PTSD among youth in a nationally representative sample in the United States. A multivariate logistic regression model was used to examine the main effects of ACE and race/ethnicity on PTSD, as well as the moderating role of race/ethnicity adjusting for socio-demographic variables. Findings suggest that race/ethnicity moderates the association between ACE and PTSD. Higher ACE score increased probability of lifetime PTSD for White non-Hispanic, Black, and Hispanic adolescents with White non-Hispanic adolescents presenting with a much higher probability of lifetime PTSD compared to their Hispanic and Black peers. Implications for future research and practice are discussed, specifically trauma informed strategies and culturally inclusive/specific practices that address the structural, interpersonal, and intrapersonal influences adolescent outcomes.
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Jennifer Elkins et al.A Re-emergent Analysis of Early Algebraic Learning
https://pdxscholar.library.pdx.edu/mth_fac/268
https://pdxscholar.library.pdx.edu/mth_fac/268Tue, 05 Nov 2019 12:50:40 PST
In this paper, we discuss a novel approach for collaborative retrospective analysis. One researcher was directly involved in a classroom teaching experiment, adopting an emergent perspective as an interpreter-witness of classroom interactions during a four-week algebra instructional unit with sixth-grade students. The other researcher experienced and analyzed the data in reverse chronological order. We describe how this re-emergent perspective revealed aspects of students’ early algebraic reasoning.
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Steven Boyce et al.Maddie's Units Coordinating Across Contexts
https://pdxscholar.library.pdx.edu/mth_fac/266
https://pdxscholar.library.pdx.edu/mth_fac/266Tue, 05 Nov 2019 12:50:32 PST
Units coordinating has emerged as an important construct for understanding students’ mathematical thinking, particularly their concepts of multiplication and fractions. We conducted an eleven-session constructivist teaching experiment with a pair of sixth-grade students to investigate how children coordinate whole number and fractional units across quantitative contexts. In this paper, we focus on one student, Maddie, who was assessed as reasoning with three levels of whole number units at the onset of the teaching experiment. We illustrate Maddie's units coordinating with fractional units over the course of the teaching experiment: it began with incorrect responses that indicated she had yet to coordinate three levels of fractional units; it progressed through her coordination of three levels of fractional units in situations for which she had a tangible referent for partitioning. We describe how students’ ways of experiencing quantitative situations are important considerations in their development of units coordinating structures.
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Steven Boyce et al.Navigating Around Convex Sets
https://pdxscholar.library.pdx.edu/mth_fac/265
https://pdxscholar.library.pdx.edu/mth_fac/265Tue, 22 Oct 2019 13:45:03 PDT
We review some basic results of convex analysis and geometry in Rn in the context of formulating a differential equation to track the distance between an observer flying outside a convex set K and K itself.
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J. J. P. VeermanEquators Have at Most Countable Many Singularities with Bounded Total Angle
https://pdxscholar.library.pdx.edu/mth_fac/264
https://pdxscholar.library.pdx.edu/mth_fac/264Thu, 17 Oct 2019 09:36:18 PDT
For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. In the case of a topological sphere, mediatrices are called equators and it can benoticed that there are no branching points, thus an equator is a topological circle with possibly many Lipschitz singularities. This paper establishes that mediatrices have the radial …
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Pilar Herreros et al.