Physical Sciences and Mathematics Commons^™

Localized Eigenvectors On Metric Graphs, Hannah Kravitz, Moysey Brio, J G, Caputo

Mathematics and Statistics Faculty Publications and Presentations

We analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. In accordance with random network theory, localization of an eigenvector is rare and the network should be tuned to observe exactly localized eigenvectors. We derive the resonance conditions to obtain localized eigenvectors for various geometric configurations and their combinations to form more complicated resonant structures. These localized eigenvectors suggest new indicators based on the energy density; in contrast to standard criteria, ours provide the number of active edges. We also suggest practical ways to design resonating systems based on metric graphs. Finally, …

Proof Of The Kresch-Tamvakis Conjecture, John Caughman, Taiyo S. Terada

Mathematics and Statistics Faculty Publications and Presentations

In this paper we resolve a conjecture of Kresch and Tamvakis.

Estimating Ocean Observation Impacts On Coupled Atmosphere-Ocean Models Using Ensemble Forecast Sensitivity To Observation (Efso), Chu-Chun Chang, Tse-Chun Chen, Eugenia Kalnay, Cheng Da, Safa Mote

Mathematics and Statistics Faculty Publications and Presentations

Ensemble Forecast Sensitivity to Observation (EFSO) is a technique that can efficiently identify the beneficial/detrimental impacts of every observation in ensemble-based data assimilation (DA). While EFSO has been successfully employed on atmospheric DA, it has never been applied to ocean or coupled DA due to the lack of a suitable error norm for oceanic variables. This study introduces a new density-based error norm incorporating sea temperature and salinity forecast errors, making EFSO applicable to ocean DA for the first time. We implemented the oceanic EFSO on the CFSv2-LETKF and investigated the impact of ocean observations under a weakly coupled DA …

Dependence Among Order Statistics For Time-Transformed Exponential Models, Subhash C. Kochar, Fabio Spizzichino

Mathematics and Statistics Faculty Publications and Presentations

Let X₁, ..., X_n be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for 1 i n, X_i:n denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of X_i’s with that of the corresponding order statistics. It is in particular proved that for m = 2, ..., n, the dependence of X2:m on X1:m is more than that of X₂ on X …

Parallel Element-Based Algebraic Multigrid For H (Curl) And H (Div) Problems Using The Parelag Library, Delyan Z. Kalchev, Panayot S. Vassilevski, Umberto Villa

Mathematics and Statistics Faculty Publications and Presentations

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers (ParELAG) library, to produce multilevel preconditioners and solvers for H (curl) and H (div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and the Auxiliary-Space Maxwell Solver or the Auxiliary-Space Divergence Solver on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this …

Area, Perimeter, Height, And Width Of Rectangle Visibility Graphs, John Caughman, Charles L. Dunn, Joshua Laison, Nancy Ann Neudauer, Colin L. Starr

Mathematics and Statistics Faculty Publications and Presentations

A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, height, or width (height is assumed not to exceed width).

A Nested Semiparametric Method For Case-Control Study With Missingness, Ge Zhao, Yanyuan Ma, Jill Schnall Hasler, Scott Damrauer, Michael Levin, Jinbo Chen

Mathematics and Statistics Faculty Publications and Presentations

We propose a nested semiparametric model to analyze a case-control study where genuine case status is missing for some individuals. The concept of a noncase is introduced to allow for the imputation of the missing genuine cases. The odds ratio parameter of the genuine cases compared to controls is of interest. The imputation procedure predicts the probability of being a genuine case compared to a noncase semiparametrically in a dimension reduction fashion. This procedure is flexible, and vastly generalizes the existing methods. We establish the root-n asymptotic normality of the odds ratio parameter estimator. Our method yields stable odds ratio …

What Is The Mathematics In Mathematics Education?, Eva Thanheiser

Mathematics and Statistics Faculty Publications and Presentations

In this paper I tackle the question What is the mathematics in mathematics education? By providing three different frames for the word mathematics.

1.

Frame 1: Mathematics as an abstract body of knowledge/ideas, the organization of that into systems and structures, and a set of methods for reaching conclusions.

2.

Frame 2: Mathematics as contextual, ever present, as a lens or language to make sense of the world.

3.

Frame 3: Mathematics as a verb (not a noun), a human activity, part of one’s identity.

After introducing the frames and examining their distinction and their overlap, I discuss their implication …

End-To-End Gpu Acceleration Of Low-Order-Refined Preconditioning For High-Order Finite Element Discretizations, Will Pazner, Tzanio Kolev, Jean-Sylvain Camier

Mathematics and Statistics Faculty Publications and Presentations

In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H(curl) and H(div) (e.g., for electromagnetic or radiation diffusion problems), …

Divergence-Conforming Velocity And Vorticity Approximations For Incompressible Fluids Obtained With Minimal Facet Coupling, Jay Gopalakrishnan, Lukas Kogler, Philip L. Lederer, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi–Douglas–Marini space for approximating the velocity and the lowest order Raviart–Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. …

Longitudinal Changes In Alzheimer’S-Related Plasma Biomarkers And Brain Amyloid, Murat Bilgel, Abhay Moghekar, Henrik Zetterberg, Bruno Jedynak, Multiple Additional Authors

Mathematics and Statistics Faculty Publications and Presentations

Introduction Understanding longitudinal plasma biomarker trajectories relative to brain amyloid changes can help devise Alzheimer’s progression assessment strategies.

Methods We examined the temporal order of changes in plasma amyloid-β ratio (Aβ42/Aβ40), glial fibrillary acidic protein (GFAP), neurofilament light chain (NfL), and phosphorylated tau ratios (p-tau181/Aβ42, p-tau231/Aβ42) relative to 11C-Pittsburgh compound B (PiB) positron emission tomography (PET) cortical amyloid burden (PiB−/+). Participants (n = 199) were cognitively normal at index visit with a median 6.1-year follow-up.

Results PiB groups exhibited different rates of longitudinal change in Aβ42/Aβ40 (β = 5.41 × 10-4, SE = 1.95 × 10-4, p = 0.0073). …

Teaching Routines And Student-Centered Mathematics Instruction: The Essential Role Of Conferring To Understand Student Thinking And Reasoning, Eva Thanheiser, Kathleen Melhuish

Mathematics and Statistics Faculty Publications and Presentations

We compare two lessons with respect to how a teacher centers student mathematical thinking to move instruction forward through enactment of five mathematically productive teaching routines: Conferring To Understand Student Thinking and Reasoning, Structuring Mathematical Student Talk, Working With Selected and Sequenced Student Math Ideas, Working with Public Records of Students’ Mathematical Thinking, and Orchestrating Mathematical Discussion. Findings show that the lessons differ in the enactment of teaching routines, especially Conferring to Understand Student Thinking and Reasoning which resulted in a difference in student-centeredness of the instruction. This difference highlights whose mathematics was being centralized in the classroom and whether …

Stability Of Structure-Aware Taylor Methods For Tents, Jay Gopalakrishnan, Zheng Sun

Mathematics and Statistics Faculty Publications and Presentations

Structure-aware Taylor (SAT) methods are a class of timestepping schemes designed for propagating linear hyperbolic solutions within a tent-shaped spacetime region. Tents are useful to design explicit time marching schemes on unstructured advancing fronts with built-in locally variable timestepping for arbitrary spatial and temporal discretization orders. The main result of this paper is that an s-stage SAT timestepping within a tent is weakly stable under the time step constraint

∆t ≤ Ch^1+1/s , where ∆t is the time step size and h is the spatial mesh size. Improved stability properties are also presented for high-order SAT time discretizations coupled …

Dependence Among Order Statistics For Time-Transformed Exponential Models, Subhash C. Kochar, Fabio Spizzichino

Mathematics and Statistics Faculty Publications and Presentations

Let (X1, . . . ,Xn) be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions and with identical marginals. Let for 1 ≤ i ≤ n, Xi:n denote the corresponding ith order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is proved that for m = 2, . . . , n, the dependence of X2:m on X1:m is more than that of X2 …

Chemical Reaction Networks In A Laplacian Framework, J.J.P. Veerman, T. Whalen-Wagner, Ewan Kummel

Mathematics and Statistics Faculty Publications and Presentations

The study of the dynamics of chemical reactions, and in particular phenomena such as oscillating reactions, has led to the recognition that many dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain directed graph, called a Chemical Reaction Network (CRN). In this graph, the edges represent the reactions and the vertices the reacting combinations of chemical substances. In contrast with the classical treatment, in this work, we heavily rely on a recently developed theory of directed graph Laplacians to simplify the traditional treatment of the so-called defi- ciency zero systems of CRN theory. …

Canonical Quantile Regression, Stephen Portnoy

Mathematics and Statistics Faculty Publications and Presentations

In using multiple regression methods for prediction, one often considers the linear combination of explanatory variables as an index. Seeking a single such index when here are multiple responses is rather more complicated. One classical approach is to use the coefficients from the leading Canonical Correlation. However, methods based on variances are unable to disaggregate responses by quantile effects, lack robustness, and rely on normal assumptions for inference. An alternative canonical regression quantile (CanRQ) approach seeks to find the linear combination of explanatory variables that best predicts the τ" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: …

Topological Anomaly Detection In Dynamic Multilayer Blockchain Networks, Dorcas Ofori-Boateng, I. Segovia Dominguez, C. Akcora, M. Kantarcioglu, Y. R. Gel

Mathematics and Statistics Faculty Publications and Presentations

Motivated by the recent surge of criminal activities with cross-cryptocurrency trades, we introduce a new topological perspective to structural anomaly detection in dynamic multilayer networks. We postulate that anomalies in the underlying blockchain transaction graph that are composed of multiple layers are likely to also be manifested in anomalous patterns of the network shape properties. As such, we invoke the machinery of clique persistent homology on graphs to systematically and efficiently track evolution of the network shape and, as a result, to detect changes in the underlying network topology and geometry. We develop a new persistence summary for multilayer networks, …

Multi-Method Investigation Of Factors Influencing Amyloid Onset And Impairment In Three Cohorts, Tobey J. Betthauser, Murat Bilgel, Rebecca L. Koscik, Bruno Jedynak, Yang An, Kristina A. Kellett, Abhay Moghekar, Erin M. Jonaitis, Charles K. Stone, Multiple Additional Authors

Mathematics and Statistics Faculty Publications and Presentations

Alzheimer’s disease biomarkers are becoming increasingly important for characterizing the longitudinal course of disease, predicting the timing of clinical and cognitive symptoms, and for recruitment and treatment monitoring in clinical trials. In this work, we develop and evaluate three methods for modelling the longitudinal course of amyloid accumulation in three cohorts using amyloid PET imaging. We then use these novel approaches to investigate factors that influence the timing of amyloid onset and the timing from amyloid onset to impairment onset in the Alzheimer’s disease continuum.

Data were acquired from the Alzheimer’s Disease Neuroimaging Initiative (ADNI), the Baltimore Longitudinal Study of …

Summing Hecke Eigenvalues Over Polynomials, Liubomir Chiriac, Liyang Yang

Mathematics and Statistics Faculty Publications and Presentations

In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x)∈Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg(f)≤4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.

The Efficacy Of Research-Based "Mathematics For All" Professional, Kathleen Melhuish, Eva P. Thanheiser, Alexander White, Brenda Rosencrans, J. Michael Shaughnessy, Linda Foreman, Andrew Riffel, Layla Guyot

Mathematics and Statistics Faculty Publications and Presentations

This article contributes to the larger narrative around what makes a mathematics professional development (PD) successful and in what ways. We share a research-based PD model that was implemented in elementary schools in an urban school district for 3 years. The model uses a pseudo lesson study approach and emphasizes standards-based instruction. We found that teachers made gains in knowledge and instruction quality. However, whereas some students saw gains on standardized assessments, this was the case only for students who were not members of historically minoritized groups (Black/Latino), countering our assumptions that the PD would lead to equitable achievement results. …

Learning Nonparametric Ordinary Differential Equations: Application To Sparse And Noisy Data, Kamel Lahouel, Michael Wells, David Lovitz, Victor Rielly, Ethan Lew, Bruno Jedynak

Mathematics and Statistics Faculty Publications and Presentations

Learning nonparametric systems of Ordinary Differential Equations (ODEs) x˙=f(t,x) from noisy and sparse data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator. Experiments are provided for the FitzHugh Nagumo oscillator …

The Average Number Of Divisors In Certain Arithmetic Sequences, Liubomir Chiriac

Mathematics and Statistics Faculty Publications and Presentations

In this paper we study the sum p≤xτ(np), where τ(n) denotes the number of divisors of n, and {np} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is x as x → ∞. As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form.

Geodesic Bicombings On Some Hyperspaces, Logan S. Fox

Mathematics and Statistics Faculty Publications and Presentations

We show that if (X,d)" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; 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; border: 0px; padding: 0px; margin: 0px; position: relative;">(X,d)(X,d) is a metric space which admits a consistent convex geodesic bicombing, then we can construct a conical bicombing on CB(X)" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; 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; border: 0px; padding: 0px; margin: 0px; position: relative;">CB(X)CB(X), the hyperspace of nonempty, closed, bounded, and …

The Trace Of T2 Takes No Repeated Values, Liubomir Chiriac, Andrei Jorza

Mathematics and Statistics Faculty Publications and Presentations

We prove that the trace of the Hecke operator T₂" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; 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; border: 0px; position: relative;">T2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.

An Aggregation-Based Nonlinear Multigrid Solver For Two-Phase Flow And Transport In Porous Media, Chak Shing Lee, Francois Hamon, Nicola Castelletto, Panayot S. Vassilevski, Joshua A. White

Mathematics and Statistics Faculty Publications and Presentations

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a …

Dependence Comparisons Of Order Statistics In The Proportional Hazards Model, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

Let X₁,…,X_n be mutually independent exponential random variables with distinct hazard rates λ₁,…,λ_n > 0 and let Y₁,…,Y_n be a random sample from the exponential distribution with hazard rate $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. Also let X_1:n < ⋯ < X_n:n and Y_1:n < ⋯ < Y_n:n be their associated order statistics. It is shown that for 1 ≤ i < j ≤ n, the generalized spacing X_j:n - X _i:n is more dispersed than Y_j:n− Y_i:n according to dispersive ordering. This result is used to solve a long standing open problem …

Fenchel-Rockafellar Theorem In Infinite Dimensions Via Generalized Relative Interiors, Dang Van Cuong, Mau Nam Nguyen, B. S. Mordukhovich, G. Sandine

Mathematics and Statistics Faculty Publications and Presentations

In this paper we provide further studies of the Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel-Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.

An Algorithm For Identifying Eigenvectors Exhibiting Strong Spatial Localization, Jeffrey S. Ovall, Robyn Reid

Mathematics and Statistics Faculty Publications and Presentations

We introduce an approach for exploring eigenvector localization phenomena for a class of (unbounded) selfadjoint operators. More specifically, given a target region and a tolerance, the algorithm identifies candidate eigenpairs for which the eigenvector is expected to be localized in the target region to within that tolerance. Theoretical results, together with detailed numerical illustrations of them, are provided that support our algorithm. A partial realization of the algorithm is described and tested, providing a proof of concept for the approach.

Quadrature For Implicitly-Defined Finite Element Functions On Curvilinear Polygons, Jeffrey S. Ovall, Samuel E. Reynolds

Mathematics and Statistics Faculty Publications and Presentations

Abstract

H¹" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; 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; border: 0px; position: relative;">H1-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined as solutions of Poisson problems having polynomial source and boundary data. Recently, such methods have been extended to allow for mesh cells that are curvilinear polygons. Such extensions present new challenges for determining suitable quadratures. We describe an approach …

A One-Dimensional Field Dislocation Mechanics Model Using Discontinuous Galerkin Method, Ja'nya Breeden, Dow Drake, Jay Gopalakrishnan, Saurabh Puri

Mathematics and Statistics Faculty Publications and Presentations

A numerical solution strategy for a one-dimensional field dislocation mechanics (FDM) model using the Discontinuous Galerkin (DG) method is developed. The FDM model is capable of simulating the dynamics of discrete, nonsingular dislocations using a partial differential equation involving a conservation law for the Burgers vector content with constitutive input for nucleation and velocity. Modeling of individual dislocation lines with an equilibrium compact core structure in the context of this continuum elastoplastic framework requires a non-convex stored energy density. Permanent deformation and stress redistribution caused by the dissipative transport of dislocations is modeled using thermodynamics-based constitutive laws. A DG method …