Digital Commons Network^™

Computational Methods For Asynchronous Basins, Ian H. Dinwoodie

Mathematics and Statistics Faculty Publications and Presentations

For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.

Temporal Order Of Alzheimer's Disease-Related Cognitive Marker Changes In Blsa And Wrap Longitudinal Studies, Murat Bilgel, Rebecca L. Koscik, Yang An, Jerry L. Prince, Susan M. Resnick, Sterling C. Johnson, Bruno Jedynak

Mathematics and Statistics Faculty Publications and Presentations

Investigation of the temporal trajectories of currently used neuropsychological tests is critical to identifying earliest changing measures on the path to dementia due to Alzheimer's disease (AD). We used the Progression Score (PS) method to characterize the temporal trajectories of measures of verbal memory, executive function, attention, processing speed, language, and mental state using data spanning normal cognition, mild cognitive impairment (MCI), and AD from 1661 participants with a total of 7839 visits (age at last visit 77.6 SD 9.2) in the Baltimore Longitudinal Study of Aging and 1542 participants with a total of 4467 visits (age at last visit …

Cycle Structures Of Orthomorphisms Extending Partial Orthomorphisms Of Boolean Groups, Nichole Louise Schimanski, John S. Caughman Iv

Mathematics and Statistics Faculty Publications and Presentations

A partial orthomorphism of a group GG (with additive notation) is an injection π:S→G for some S⊆G such that π(x)−x ≠ π(y) for all distinct x,y∈S. We refer to |S| as the size of π, and if S=G, then π is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, and what cycle types these permutations have.

It is known that conjugation by automorphisms of G forms a group action on the set of orthomorphisms of G. In this paper, we consider the …

Transients In The Synchronization Of Asymmetrically Coupled Oscillator Arrays, Carlos E. Cantos, David K. Hammond, J.J.P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We consider the transient behavior of a large linear array of coupled linear damped harmonic oscillators following perturbation of a single element. Our work is motivated by modeling the behavior of flocks of autonomous vehicles. We first state a number of conjectures that allow us to derive an explicit characterization of the transients, within a certain parameter regime Ω. As corollaries we show that minimizing the transients requires considering non-symmetric coupling, and that within Ω the computed linear growth in N of the transients is independent of (reasonable) boundary conditions.

Breaking Spaces And Forms For The Dpg Method And Applications Including Maxwell Equations, Carsten Carstensen, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. …

An Ensemble Approach To Weak-Constraint Four-Dimensional Variational Data Assimilation, Jeremy A. Shaw, Dacian Daescu

Mathematics and Statistics Faculty Publications and Presentations

This article presents a framework for performing ensemble and hybrid data assimilation in a weak-constraint four-dimensional variational data assimilation system (w4D-Var). A practical approach is considered that relies on an ensemble of w4D-Var systems solved by the incremental algorithm to obtain flow-dependent estimates to the model error statistics. A proof-of-concept is presented in an idealized context using the Lorenz multi-scale model. A comparative analysis is performed between the weak- and strong-constraint ensemble-based methods. The importance of the weight coefficients assigned to the static and ensemble-based components of the error covariances is also investigated. Our preliminary numerical experiments indicate that an …

The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, N. T. An, Daniel J. Giles, Nguyen Mau Nam, R. Blake Rector

Mathematics and Statistics Faculty Publications and Presentations

The Sylvester or smallest enclosing circle problem involves finding the smallest circle enclosing a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method, we present an effective numerical algorithm for solving these problems.

Minimizing Differences Of Convex Functions With Applications To Facility Location And Clustering, Mau Nam Nguyen, R. Blake Rector, Daniel J. Giles

Mathematics and Statistics Faculty Publications and Presentations

In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems. We demonstrate the algorithms with a variety of numerical examples.

Measuring Fidelity Of Implementation In A Large-Scale Professional Development Efficacy Study, Kathleen Mary Melhuish, Eva Thanheiser, Brenda Rosencrans

Mathematics and Statistics Faculty Publications and Presentations

Fidelity of implementation is the essential link between professional development (PD) experiences and changes in outcomes. Yet, as a field we have few measures available to directly look at implementation. In this poster, we present our preliminary work measuring teachers’ implementation of a research-based PD.

Tridiagonal Matrices And Boundary Conditions, J. J. P. Veerman, David K. Hammond

Mathematics and Statistics Faculty Publications and Presentations

We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one-dimensional flock. We apply our results to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition.

Full State Revivals In Linearly Coupled Chains With Commensurate Eigenspectra, J. J. P. Veerman, Jovan Petrovic

Mathematics and Statistics Faculty Publications and Presentations

Coherent state transfer is an important requirement in the construction of quantum computer hardware. The state transfer can be realized by linear next-neighbour-coupled finite chains. Starting from the commensurability of chain eigenvalues as the general condition of periodic dynamics, we find chains that support full periodic state revivals. For short chains, exact solutions are found analytically by solving the inverse eigenvalue problem to obtain the coupling coefficients between chain elements. We apply the solutions to design optical waveguide arrays and perform numerical simulations of light propagation thorough realistic waveguide structures. Applications of the presented method to the realization of a …

A Multivariate Nonlinear Mixed Effects Model For Longitudinal Image Analysis: Application To Amyloid Imaging, Murat Bilgel, Jerry L. Prince, Dean F. Wong, Susan M. Resnick, Bruno M. Jedynak

Mathematics and Statistics Faculty Publications and Presentations

It is important to characterize the temporal trajectories of disease-related biomarkers in order to monitor progression and identify potential points of intervention. These are especially important for neurodegenerative diseases, as therapeutic intervention is most likely to be effective in the preclinical disease stages prior to significant neuronal damage. Neuroimaging allows for the measurement of structural, functional, and metabolic integrity of the brain at the level of voxels, whose volumes are on the order of mm³. These voxelwise measurements provide a rich collection of disease indicators. Longitudinal neuroimaging studies enable the analysis of changes in these voxelwise measures. However, …

Associations Between Performance On An Abbreviated Cogstate Battery, Other Measures Of Cognitive Function, And Biomarkers In People At Risk For Alzheimer's Disease, Annie M. Racine, Lindsay R. Clark, Sara C. Berman, Rebecca L. Koscik, Kimberly D. Mueller, Derek Norton, Christopher R. Nicholas, Kaj Blennow, Henrik Zetterberg, Bruno M. Jedynak, Murat Bilgel, Cynthia M. Carlsson, Bradley T. Christian, Sanjay Asthana, Sterling C. Johnson

Mathematics and Statistics Faculty Publications and Presentations

It is not known whether computerized cognitive assessments, like the CogState battery, are sensitive to preclinical cognitive changes or pathology in people at risk for Alzheimer's disease(AD). In 469 late middle-aged participants from the Wisconsin Registry for Alzheimer's Prevention(mean age 63.8±7 years at testing; 67% female; 39% APOE4+), we examined relationships between a CogState abbreviated battery(CAB) of seven tests and demographic characteristics, traditional paper-based neuropsychological tests as well as a composite cognitive impairment index, cognitive impairment status(determined by consensus review), and biomarkers for amyloid and tau(CSF phosphorylated-tau/Aβ42 and global PET-PiB burden) and neural injury(CSF neurofilament light protein). CSF and PET-PiB …

The Language Of Professional Development Leaders, Andrew Riffel, Kathleen Mary Melhuish, Eva Thanheiser

Mathematics and Statistics Faculty Publications and Presentations

Leaders in professional development (PD) initiatives (such as facilitators, principals, coaches) hold a great deal of power in their language, carrying the ability to manage meaning and frame experiences. Rather than working from interview data, this report addresses a gap in leadership research by examining the words used by leaders in their on-the-job interactions. We present an initial framework for capturing leadership language qualities at the macro level (framing the PD’s purpose) and micro level (rhetorical crafting in terms of metaphor usage, pronoun choice, and other language selections). Our data come from a larger project evaluating the efficacy of a …

Teacher Noticing Of Justification: Attending To The Complexity Of Mathematical Content And Practice, Kathleen Mary Melhuish, Eva Thanheiser, Jodi I. Fasteen, Julie Fredericks

Mathematics and Statistics Faculty Publications and Presentations

In this report, we will consider in-service elementary school teachers' noticing of the mathematical practice: justification. This study is part of a larger project evaluating the efficacy of a three year professional development built around attending to student thinking and promoting mathematical habits of justifying, generalizing and making sense. Noticing just¢cation is a complex task requiring attention to both the (1) mathematical content and strategies and (2) the nature of the argument provided by a student. We have found that teachers' struggle to attend to both aspects simultaneously and offer a framework for considering teacher noticing of mathematical practices.

Greater Number Of Larger Pieces: A Strategy To Promote Prospective Teachers’ Fraction Number Sense Development, Dana Olanoff, Ziv Feldman, Rachael M. Welder, Jennifer M. Tobias, Eva Thanheiser, Amy F. Hillen

Mathematics and Statistics Faculty Publications and Presentations

Prospective teachers (PTs) need opportunities to develop fraction number sense, yet little research has explicated how this development occurs. Our research team collaboratively designed a task targeted at helping PTs develop fraction number sense through an exploration of fraction comparison strategies. This paper focuses on developing one particular strategy, which we call Greater Number of Larger Pieces (GLP). We argue that understanding this strategy has the potential to support PTs’ number sense, particularly in regards to the measure interpretation of fractions. Analysis of data from two iterations of this task (implemented by five mathematics teacher educators at five US institutions …

Signal Velocity In Oscillator Arrays, Carlos E. Cantos, David K. Hammond, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We investigate a system of coupled oscillators on the circle, which arises from a simple model for behavior of large numbers of autonomous vehicles. The model considers asymmetric, linear, decentralized dynamics, where the acceleration of each vehicle depends on the relative positions and velocities between itself and a set of local neighbors. We first derive necessary and sufficient conditions for asymptotic stability, then derive expressions for the phase velocity of propagation of disturbances in velocity through this system. We show that the high frequencies exhibit damping, which implies existence of well-defined signal velocities c+>0 and c−f(x−c+t) in the direction …

Voxel Based Morphometry In Optical Coherence Tomography: Validation & Core Findings, Bhavna J. Antony, Min Chen, Aaron Carass, Bruno M. Jedynak, Omar Al-Louzi, Sharon D. Solomon, Shiv Saidha, Peter Calabresi, Jerry L. Prince

Mathematics and Statistics Faculty Publications and Presentations

Optical coherence tomography (OCT) of the human retina is now becoming established as an important modality for the detection and tracking of various ocular diseases. Voxel based morphometry (VBM) is a long standing neuroimaging analysis technique that allows for the exploration of the regional differences in the brain. There has been limited work done in developing registration based methods for OCT, which has hampered the advancement of VBM analyses in OCT based population studies. Following on from our recent development of an OCT registration method, we explore the potential benefits of VBM analysis in cohorts of healthy controls (HCs) and …