Mathematics and Statistics Faculty Publications and PresentationsCopyright (c) 2021 Portland State University All rights reserved.
https://pdxscholar.library.pdx.edu/mth_fac
Recent documents in Mathematics and Statistics Faculty Publications and Presentationsen-usThu, 10 Jun 2021 02:40:36 PDT3600Revealing Students' Stories As They Construct and Use a Statistical Model in Tinkerplots to Conduct a Randomization Test for Comparing Two Groups
https://pdxscholar.library.pdx.edu/mth_fac/315
https://pdxscholar.library.pdx.edu/mth_fac/315Tue, 08 Jun 2021 15:13:24 PDT
Using simulation approaches when conducting randomization tests for comparing two groups in the context of experimental studies has been promoted as a beneficial approach for supporting student learning of statistical inference. Many researchers have suggested that the data production process in simulations for the randomization test intuitively connects to the random assignment used in the original study design, thus supporting students’ understanding of the logic of inference. Yet, there is little empirical research on how students initially think about the concepts and processes underlying the randomization test as they engage in constructing and using probability models to solve a problem. This work makes a contribution by deepening our understanding of students’ reasoning about randomization tests by focusing on a group of three students as they create and use a TinkerPlots model to simulate data and use this data to make a statistical inference. This work adopts a narrative lens through which to view these students’ reasoning and modeling activity. We compare and contrast the narratives we constructed for these students along with a narrative we constructed for a statistician. We discuss possible implications for teaching randomization tests for comparing two groups using a modeling and simulation approach.
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Jennifer Noll et al.Regularized Semi-Nonnegative Matrix Factorization Using L2,1-Norm for Data Compression
https://pdxscholar.library.pdx.edu/mth_fac/314
https://pdxscholar.library.pdx.edu/mth_fac/314Tue, 08 Jun 2021 15:13:23 PDT
We present a robust, parts-based data compression algorithm, L21 Semi-Nonnegative Matrix Factorization (L21 SNF) for mixed-sign data. To resolve the instability issue caused by the Frobenius norm due to the effects of outliers, we utilize the noise-free L2,1 norm and a regularization parameter in our algorithm. We derive a rigorous proof of convergence of our algorithm. Based on experiments on large-scale over-determined matrices and real facial image data, L21 SNF demonstrates a significant improvement in accuracy over other classical methods. Furthermore, L21 SNF has a simple programming structure and can be implemented within data compression software for compression of highly over-determined systems encountered broadly across many general machine learning processes.
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Anthony Rhodes et al.The Cycle Structure of Unicritical Polynomials
https://pdxscholar.library.pdx.edu/mth_fac/313
https://pdxscholar.library.pdx.edu/mth_fac/313Tue, 08 Jun 2021 15:13:22 PDT
A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime p" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">pp, reduce its coefficients mod p" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">pp and consider its action on the field Fp" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">FpFp. The questions of whether and in what sense these families are random have been studied extensively, spurred in part by Pollard’s famous “rho” algorithm for integer factorization (the heuristic justification of which is the conjectural randomness of one such family). However, the cycle structure of these families cannot be random, since in any such family, the number of cycles of a fixed length in any dynamical system in that family is bounded. In this paper, we show that the cycle statistics of many of these families are as random as possible. As a corollary, we show that most members of these families have many cycles, addressing a conjecture of Mans et al.
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Andrew Bridy et al.Networking Frameworks: a Method for Analyzing the Complexities of Classroom Cultures Focusing on Justifying
https://pdxscholar.library.pdx.edu/mth_fac/312
https://pdxscholar.library.pdx.edu/mth_fac/312Thu, 06 May 2021 10:40:09 PDT
In this paper, we network five frameworks (cognitive demand, lesson cohesion, cognitive engagement, collective argumentation, and student contribution) for an analytic approach that allows us to present a more holistic picture of classrooms which engage students in justifying. We network these frameworks around the edges of the instructional triangle as a means to coordinate them to illustrate the observable relationships among teacher, students(s), and content. We illustrate the potential of integrating these frameworks via analysis of two lessons that, while sharing surface level similarities, are profoundly different when considering the complexities of a classroom focused on justifying. We found that this integrated comparison across all dimensions (rather than focusing on just one or two) was a useful way to compare lessons with respect to a classroom culture that is characterized by students engaging in justifying.
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Eva Thanheiser et al.Convex Analysis of Minimal Time and Signed Minimal Time Functions
https://pdxscholar.library.pdx.edu/mth_fac/311
https://pdxscholar.library.pdx.edu/mth_fac/311Fri, 16 Apr 2021 11:32:43 PDT
In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.
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D. V. Cuong et al.Response to Grannis FW. Current Controversies in Cardiothoracic Imaging Overdiagnosis at Lung Cancer Screening-No So Bad After All-Counterpoint
https://pdxscholar.library.pdx.edu/mth_fac/310
https://pdxscholar.library.pdx.edu/mth_fac/310Wed, 07 Apr 2021 10:35:25 PDT
To invalidate our estimate 1 of the magnitude of computed tomography (CT) overdiagnosis (OD)—operationally defined as the excess of cancers identified in the screened versus unscreened control cohort at long-term follow-up—would require the author of the counterpoint 2 to either advance persuasive evidence that our methodology is unsound, or that others, employing the same methodology, report a far lower rate. He has done neither. His response contains a number of what might be described as discernable textual and …
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Jerome M. Reich et al.Where Calculus and Engineering Converge: an Analysis of Curricular Change in Calculus for Engineers
https://pdxscholar.library.pdx.edu/mth_fac/309
https://pdxscholar.library.pdx.edu/mth_fac/309Wed, 07 Apr 2021 10:35:24 PDT
Calls to increase the number of STEM graduates on a global scale have created pressure on universities to graduate higher numbers of quality engineers. In response, many engineering and mathematics departments have begun to develop variations of calculus courses specifically for engineering majors. Using a mixed methods research design, we investigated similar curricular changes in calculus that were designed to support engineering students at two large research-intensive universities in the United States. The curricular change at one university was sustained over time while the other was not, which focused our study on understanding what accounted for the curricular sustainment or termination. A finding from our study illustrates that stakeholders’ perceptions of the engineering calculus course impacted the success (or failure) of the variation over time.
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Brittney Marie Ellis et al.Stability Conditions for Coupled Autonomous Vehicles Formations
https://pdxscholar.library.pdx.edu/mth_fac/308
https://pdxscholar.library.pdx.edu/mth_fac/308Wed, 07 Apr 2021 10:35:23 PDT
In this paper, we give necessary conditions for stability of coupled autonomous vehicles in R. We focus on linear arrays with decentralized vehicles, where each vehicle interacts with only a few of its neighbors. We obtain explicit expressions for necessary conditions for stability in the cases that a system consists of a periodic arrangement of two or three different types of vehicles, i.e. configurations as follows: 2-1-2-1 or 3-2-1-3-2-1. Previous literature indicated that the (necessary) condition for stability in the case of a single vehicle type (1-1-1) held that the first moment of certain coefficients of the interactions between vehicles has to be zero. Here, we show that that does not generalize. Instead, the (necessary) condition in the cases considered is that the first moment plus a nonlinear correction term must be zero.
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Pablo Baldivieso et al.Estimating Posterior Quantity of Interest Expectations in a Multilevel Scalable Framework
https://pdxscholar.library.pdx.edu/mth_fac/307
https://pdxscholar.library.pdx.edu/mth_fac/307Wed, 07 Apr 2021 10:35:22 PDT
Scalable approaches for uncertainty quantification are necessary for characterizing prediction confidence in large‐scale subsurface flow simulations with uncertain permeability. To this end we explore a multilevel Monte Carlo approach for estimating posterior moments of a particular quantity of interest, where we employ an element‐agglomerated algebraic multigrid (AMG) technique to generate the hierarchy of coarse spaces with guaranteed approximation properties for both the generation of spatially correlated random fields and the forward simulation of Darcy's law to model subsurface flow. In both these components (sampling and forward solves), we exploit solvers that rely on state‐of‐the‐art scalable AMG. To showcase the applicability of this approach, numerical tests are performed on two 3D examples—a unit cube and an egg‐shaped domain with an irregular boundary—where the scalability of each simulation as well as the scalability of the overall algorithm are demonstrated.
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Hillary R. Fairbanks et al.Structured Layer Surface Segmentation for Retina OCT Using Fully Convolutional Regression Networks.
https://pdxscholar.library.pdx.edu/mth_fac/306
https://pdxscholar.library.pdx.edu/mth_fac/306Wed, 07 Apr 2021 10:35:21 PDT
Optical coherence tomography (OCT) is a noninvasive imaging modality with micrometer resolution which has been widely used for scanning the retina. Retinal layers are important biomarkers for many diseases. Accurate automated algorithms for segmenting smooth continuous layer surfaces with correct hierarchy (topology) are important for automated retinal thickness and surface shape analysis. State-of-the-art methods typically use a two step process. Firstly, a trained classifier is used to label each pixel into either background and layers or boundaries and non-boundaries. Secondly, the desired smooth surfaces with the correct topology are extracted by graph methods (e.g., graph cut). Data driven methods like deep networks have shown great ability for the pixel classification step, but to date have not been able to extract structured smooth continuous surfaces with topological constraints in the second step. In this paper, we combine these two steps into a unified deep learning framework by directly modeling the distribution of the surface positions. Smooth, continuous, and topologically correct surfaces are obtained in a single feed forward operation. The proposed method was evaluated on two publicly available data sets of healthy controls and subjects with either multiple sclerosis or diabetic macular edema, and is shown to achieve state-of-the art performance with sub-pixel accuracy.
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Yufan He et al.A Primer on Laplacian Dynamics in Directed Graphs
https://pdxscholar.library.pdx.edu/mth_fac/305
https://pdxscholar.library.pdx.edu/mth_fac/305Thu, 18 Feb 2021 14:12:19 PST
We analyze the asymptotic behavior of general first order Laplacian processes on digraphs. The most important ones of these are diffusion and consensus with both continuous and discrete time. We treat diffusion and consensus as dual processes. This is the first complete exposition of this material in a single work.
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J. J. P. Veerman et al.Probability Axioms and Set Theory Paradoxes
https://pdxscholar.library.pdx.edu/mth_fac/304
https://pdxscholar.library.pdx.edu/mth_fac/304Tue, 16 Feb 2021 09:24:50 PST
In this paper, we show that Zermelo–Fraenkel set theory with Choice (ZFC) conflicts with basic intuitions about randomness. Our background assumptions are the Zermelo–Fraenekel axioms without Choice (ZF) together with a fragment of Kolmogorov’s probability theory. Using these minimal assumptions, we prove that a weak form of Choice contradicts two common sense assumptions about probability—both based on simple notions of symmetry and independence.
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Ari Herman et al.Mathematics to Understand and Critique the World: Reconceiving Mathematics in a Mathematics Content Course for Elementary School Teachers
https://pdxscholar.library.pdx.edu/mth_fac/303
https://pdxscholar.library.pdx.edu/mth_fac/303Mon, 08 Feb 2021 11:07:14 PST
There are long-standing and ongoing calls for making mathematics meaningful, relevant, and applicable outside the classroom. In other words, to help students see mathematics as a tool for understanding, analyzing, and changing the world. However, there are also tensions between a focus on classical mathematics goals and a focus on analyzing and understanding social and political issues, which does not always lend itself to focusing on a specific mathematical concept. In this study, we redesigned a mathematics content course for prospective elementary teachers (PTs) to examine whether we could engage PTs in learning both about the classical mathematics content and about understanding and critiquing the world. We examined their learning in both areas and their evolving views of mathematics teaching throughout the course. We found that PTs learned both the mathematics and about the world and in addition they reconceived of mathematics as a tool to make sense of the world. Thus, mathematics content courses can be designed to allow PTs to develop their knowledge in both areas and experience a classroom where teaching math for social justice is a focus.
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Eva Thanheiser et al.Boolean Network Control with Ideals
https://pdxscholar.library.pdx.edu/mth_fac/302
https://pdxscholar.library.pdx.edu/mth_fac/302Thu, 28 Jan 2021 11:16:47 PST
A method is given for finding controls to transition an initial state x0 to a target set in deterministic or stochastic Boolean network control models. The algorithms use multivariate polynomial algebra. Examples illustrate the application.
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Ian H. DinwoodieNumerical Results for Adaptive (negative norm) Constrained First Order System Least Squares Formulations
https://pdxscholar.library.pdx.edu/mth_fac/301
https://pdxscholar.library.pdx.edu/mth_fac/301Thu, 28 Jan 2021 10:24:52 PST
We perform a follow-up computational study of the recently proposed space–time first order system least squares ( FOSLS ) method subject to constraints referred to as CFOSLS where we now combine it with the new capability we have developed, namely, parallel adaptive mesh refinement (AMR) in 4D. The AMR is needed to alleviate the high memory demand in the combined space time domain and also allows general (4D) meshes that better follow the physics in space–time. With an extensive set of computational experiments, performed in parallel, we demonstrate the feasibility of the combined space–time AMR approach in both two space plus time and three space plus time dimensions.
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Andreas Schafelner et al.Nonlinear Multigrid Based on Local Spectral Coarsening for Heterogeneous Diffusion Problems
https://pdxscholar.library.pdx.edu/mth_fac/300
https://pdxscholar.library.pdx.edu/mth_fac/300Tue, 29 Dec 2020 13:57:28 PST
This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches—Picard’s and Newton’s methods—are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.
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Chak Shing Lee et al.Newton Polygons of Hecke Operators
https://pdxscholar.library.pdx.edu/mth_fac/299
https://pdxscholar.library.pdx.edu/mth_fac/299Mon, 14 Dec 2020 09:22:42 PST
In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator T2 for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at n≤15, then it agrees with the Newton polygon of T2 up to n.
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Liubomir Chiriac et al.Evolving Order and Chaos: Comparing Particle Swarm Optimization and Genetic Algorithms for Global Coordination of Cellular Automata
https://pdxscholar.library.pdx.edu/mth_fac/298
https://pdxscholar.library.pdx.edu/mth_fac/298Mon, 14 Dec 2020 09:22:40 PST
We apply two evolutionary search algorithms: Particle Swarm Optimization (PSO) and Genetic Algorithms (GAs) to the design of Cellular Automata (CA) that can perform computational tasks requiring global coordination. In particular, we compare search efficiency for PSO and GAs applied to both the density classification problem and to the novel generation of ”chaotic” CA. Our work furthermore introduces a new variant of PSO, the Binary Global-Local PSO (BGL-PSO).
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Anthony D. RhodesA Posteriori Error Estimates for Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques
https://pdxscholar.library.pdx.edu/mth_fac/297
https://pdxscholar.library.pdx.edu/mth_fac/297Tue, 17 Nov 2020 13:42:58 PST
We propose an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate invariant subspaces. Numerical experiments demonstrate the practical effectivity of the approach.
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Stefano Giani et al.Multilevel Graph Embedding
https://pdxscholar.library.pdx.edu/mth_fac/296
https://pdxscholar.library.pdx.edu/mth_fac/296Wed, 04 Nov 2020 10:09:27 PST
The goal of the present paper is the design of embeddings of a general sparse graph into a set of points in "ℝ^{𝑑} " for appropriate d ≥ 2. The embeddings that we are looking at aim to keep vertices that are grouped in communities together and keep the rest apart. To achieve this property, we utilize coarsening that respects possible community structures of the given graph. We employ a hierarchical multilevel coarsening approach that identifies communities (strongly connected groups of vertices) at every level. The multilevel strategy allows any given (presumably expensive) graph embedding algorithm to be made into a more scalable (and faster) algorithm. We demonstrate the presented approach on a number of given embedding algorithms and large‐scale graphs and achieve speed‐up over the methods in a recent paper.
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Benjamin Quiring et al.