Mathematics Commons^™

A Formula For The Cohomology And K-Class Of A Regular Hessenberg Variety, Erik Insko, Julianna Tymoczko, Alexander Woo

Mathematics Sciences: Faculty Publications

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result …

Circada: Shiny Apps For Exploration Of Experimental And Synthetic Circadian Time Series With An Educational Emphasis, Lisa Cenek, Liubou Klindziuk, Cindy Lopez, Eleanor Mccartney, Blanca Martin Burgos, Selma Tir, Mary E. Harrington, Tanya L. Leise

Psychology: Faculty Publications

Circadian rhythms are daily oscillations in physiology and behavior that can be assessed by recording body temperature, locomotor activity, or bioluminescent reporters, among other measures. These different types of data can vary greatly in waveform, noise characteristics, typical sampling rate, and length of recording. We developed 2 Shiny apps for exploration of these data, enabling visualization and analysis of circadian parameters such as period and phase. Methods include the discrete wavelet transform, sine fitting, the Lomb-Scargle periodogram, autocorrelation, and maximum entropy spectral analysis, giving a sense of how well each method works on each type of data. The apps also …

How Emergent Social Patterns In Allogrooming Combat Parasitic Infections, Shelby N. Wilson, Suzanne S. Sindi, Heather Z. Brooks, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Nakeya D. Williams, Nina H. Fefferman

Mathematics Sciences: Faculty Publications

Members of social groups risk infection through contact with those in their social network. Evidence that social organization may protect populations from pathogens in certain circumstances prompts the question as to how social organization affects the spread of ectoparasites. The same grooming behaviors that establish social bonds also play a role in the progression of ectoparasitic outbreaks. In this paper, we model the interactions between social organization and allogrooming efficiency to consider how ectoparasitic threats may have shaped the evolution of social behaviors. To better understand the impacts of social grooming on organizational structure, we consider several dynamic models of …

A Permutation Test And Spatial Cross-Validation Approach To Assess Models Of Interspecific Competition Between Trees, David Allen, Albert Y. Kim

Statistical and Data Sciences: Faculty Publications

Measuring species-specific competitive interactions is key to understanding plant communities. Repeat censused large forest dynamics plots offer an ideal setting to measure these interactions by estimating the species-specific competitive effect on neighboring tree growth. Estimating these interaction values can be difficult, however, because the number of them grows with the square of the number of species. Furthermore, confidence in the estimates can be overestimated if any spatial structure of model errors is not considered. Here we measured these interactions in a forest dynamics plot in a transitional oak-hickory forest. We analytically fit Bayesian linear regression models of annual tree radial …

Three-Dimensional Viscoelastic Instabilities In A Four-Roll Mill Geometry At The Stokes Limit, Paloma Gutierrez-Castillo, Adam Kagel, Becca Thomases

Mathematics Sciences: Faculty Publications

Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force (extended to the third dimension) were performed. Both the Oldroyd-B model and FENE-P model of viscoelastic fluids were used. Different temporal behaviors were observed depending on the Weissenberg number (non-dimensional relaxation time), model, and initial conditions. Temporal dynamics evolve on long time scales, and simulations were accelerated by using a Graphics Processing Unit (GPU). Previously, parameter explorations and long-time simulations in 3D were prohibitively expensive. For a small Weissenberg number, all the solutions are constant in the third dimension, displaying strictly two-dimensional temporal evolutions. …

Fibrations, Unique Path Lifting, And Continuous Monodromy, Hanspeter Fischer, Jacob D. Garcia

Mathematics Sciences: Faculty Publications

Given a path-connected space X and H≤π1(X,x0), there is essentially only one construction of a map pH:(X˜H,x˜0)→(X,x0) with connected and locally path-connected domain that can possibly have the following two properties: (pH)#π1(X˜H,x˜0)=H and pH has the unique lifting property. X˜H consists of equivalence classes of paths starting at x0, appropriately topologized, and pH is the endpoint projection. For pH to have these two properties, T1 fibers are necessary and unique path lifting is sufficient. However, pH always admits the standard lifts of paths. We show that pH has unique path lifting if it has continuous (standard) monodromies toward a T1 …

Analysis In Metric Spaces, Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, Jeremy Tyson

Mathematics Sciences: Faculty Publications

No abstract provided.

Teaching Introductory Statistics With Datacamp, Benjamin Baumer, Andrew P. Bray, Mine Çetinkaya-Rundel, Johanna S. Hardin

Statistical and Data Sciences: Faculty Publications

We designed a sequence of courses for the DataCamp online learning platform that approximates the content of a typical introductory statistics course. We discuss the design and implementation of these courses and illustrate how they can be successfully integrated into a brick-and-mortar class. We reflect on the process of creating content for online consumers, ruminate on the pedagogical considerations we faced, and describe an R package for statistical inference that became a by-product of this development process. We discuss the pros and cons of creating the course sequence and express our view that some aspects were particularly problematic. The issues …

Neural Networks For Threshold Dynamics Reconstruction, Elisa Negrini, Almanzo Jiahe Gao, Abigail Bowering, Wei Zhu, Luca Capogna

Mathematics Sciences: Faculty Publications

We introduce two convolutional neural network (CNN) architec- tures, inspired by the Merriman-Bence-Osher (MBO) algorithm and by cellular automatons, to model and learn threshold dynamics for front evolution from video data. The first model, termed the (single-dynamics) MBO network, learns a specific kernel and threshold for each input video without adapting to new dynamics, while the second, a meta-learning MBO network, generalizes across diverse threshold dynamics by adapting its parameters per input. Both models are evaluated on synthetic and real-world videos (ice melting and fire front propagation), with performance metrics indicating effective reconstruction and extrapolation of evolving boundaries, even under …

A Filtration On The Cohomology Rings Of Regular Nilpotent Hessenberg Varieties, Megumi Harada, Tatsuya Horiguchi, Satoshi Murai, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in GL(n, C) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in GL(n- 1 , C) / B, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce …

Graphs Admitting Only Constant Splines, Katie Anders, Alissa S. Crans, Briana Foster-Greenwood, Blake Mellor, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.

Sleep Stage Prediction With Raw Acceleration And Photoplethysmography Heart Rate Data Derived From A Consumer Wearable Device, Olivia Walch, Yitong Huang, Daniel Forger, Cathy Goldstein

Mathematics Sciences: Faculty Publications

Wearable, multisensor, consumer devices that estimate sleep are now commonplace, but the algorithms used by these devices to score sleep are not open source, and the raw sensor data is rarely accessible for external use. As a result, these devices are limited in their usefulness for clinical and research applications, despite holding much promise. We used a mobile application of our own creation to collect raw acceleration data and heart rate from the Apple Watch worn by participants undergoing polysomnography, as well as during the ambulatory period preceding in lab testing. Using this data, we compared the contributions of multiple …

Reduced Bias For Respondent Driven Sampling: Accounting For Non-Uniform Edge Sampling Probabilities In People Who Inject Drugs In Mauritius, Miles Q. Ott, Krista J. Gile, Matthew T. Harrison, Lisa G. Johnston, Joseph W. Hogan

Statistical and Data Sciences: Faculty Publications

People who inject drugs are an important population to study in order to reduce transmission of blood-borne illnesses including HIV and Hepatitis. In this paper we estimate the HIV and Hepatitis C prevalence among people who inject drugs, as well as the proportion of people who inject drugs who are female in Mauritius. Respondent driven sampling (RDS), a widely adopted link-tracing sampling design used to collect samples from hard-to-reach human populations, was used to collect this sample. The random walk approximation underlying many common RDS estimators assumes that each social relation (edge) in the underlying social network has an equal …

Polymer Stress Growth In Viscoelastic Fluids In Oscillating Extensional Flows With Applications To Micro-Organism Locomotion, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

Simulations of undulatory swimming in viscoelastic fluids with large amplitude gaits show concentration of polymer elastic stress at the tips of the swimmers. We use a series of related theoretical investigations to probe the origin of these concentrated stresses. First the polymer stress is computed analytically at a given oscillating extensional stagnation point in a viscoelastic fluid. The analysis identifies a Deborah number (De) dependent Weissenberg number (Wi) transition below which the stress is linear in Wi, and above which the stress grows exponentially in Wi. Next, stress and velocity are found from numerical simulations in an oscillating 4-roll mill …

Loose Legendrian And Pseudo-Legendrian Knots In 3-Manifolds, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and DingGeiges. Our approach is to classify knots in a 3-manifold M that are transverse to a nowhere-zero vector field V up to the corresponding isotopy relation. Such knots are called V -transverse. A framed isotopy class is simple if any two V -transverse knots in that class which are homotopic through V -transverse immersions are V -transverse isotopic. We show that all knot types in M are simple if any one of the following three conditions hold: 1. M is closed, …

Convergent Solutions Of Stokes Oldroyd-B Boundary Value Problems Using The Immersed Boundary Smooth Extension (Ibse) Method, David B. Stein, Robert D. Guy, Becca Thomases

Mathematics Sciences: Faculty Publications

The Immersed Boundary (IB) method has been widely used to solve fluid-structure interaction problems, including those where the structure interacts with polymeric fluids. In this paper, we examine the convergence of one such scheme for a well known two-dimensional benchmark flow for the Oldroyd-B constitutive model, and we show that the traditional IB-based scheme fails to adequately capture the polymeric stress near to embedded boundaries. We analyze the reason for such failure, and we argue that this feature is not specific to the case study chosen, but a general feature of such methods due to lack of convergence in velocity …

The Transition Matrix Between The Specht And Web Bases Is Unipotent With Additional Vanishing Entries, Heather M. Russell, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of …

Orientation Dependent Elastic Stress Concentration At Tips Of Slender Objects Translating In Viscoelastic Fluids, Chuanbin Li, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

Elastic stress concentration at the tips of long slender objects moving in viscoelastic fluids has been observed in numerical simulations, but despite the prevalence of flagellated motion in complex fluids in many biological functions, the physics of stress accumulation near the tips has not been analyzed. Here, we theoretically investigate elastic stress development at the tips of slender objects by computing the leading-order viscoelastic correction to the equilibrium viscous flow around long cylinders, using the weak-coupling limit. In this limit, nonlinearities in the fluid are retained, allowing us to study the biologically relevant parameter regime of high Weissenberg number. We …

Proper Orthogonal Decomposition (Pod) Of The Flow Dynamics For A Viscoelastic Fluid In A Four-Roll Mill Geometry At The Stokes Limit, Paloma Gutierrez-Castillo, Becca Thomases

Mathematics Sciences: Faculty Publications

Numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force were performed at a range of Weissenberg number (non-dimensional relaxation time). For small Weissenberg number the flow is steady and symmetric but upon increasing the Weissenberg number (corresponding to increased elasticity or flow memory time), the flow becomes unstable leading to a variety of temporal evolutions to different periodic and aperiodic solutions. These dynamics were analyzed using a Proper Orthogonal Decomposition (POD) that extracted elastic modes in terms of their contribution to the energy of the system. The temporal behavior of the system, captured by …

Springer Fibers And Schubert Points, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

Conformality And Q-Harmonicity In Sub-Riemannian Manifolds, Luca Capogna, Giovanna Citti, Enrico Le Donne, Alessandro Ottazzi

Mathematics Sciences: Faculty Publications

We establish regularity of conformal maps between sub-Riemannian manifolds from regularity of Q-harmonic functions, and in particular we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth in all contact sub-Riemannian manifolds. Together with the recent results in [15], our work yields a new proof of the smoothness of boundary extensions of biholomorphims between strictly pseudoconvex smooth domains [29].

Fixed Choice Design And Augmented Fixed Choice Design For Network Data With Missing Observations, Miles Q. Ott, Matthew T. Harrison, Krista J. Gile, Nancy P. Barnett, Joseph W. Hogan

Statistical and Data Sciences: Faculty Publications

The statistical analysis of social networks is increasingly used to understand social processes and patterns. The association between social relationships and individual behaviors is of particular interest to sociologists, psychologists, and public health researchers. Several recent network studies make use of the fixed choice design (FCD), which induces missing edges in the network data. Because of the complex dependence structure inherent in networks, missing data can pose very difficult problems for valid statistical inference. In this article, we introduce novel methods for accounting for the FCD censoring and introduce a new survey design, which we call the augmented fixed choice …

The Impact Of College Athletic Success On Donations And Applicant Quality, Benjamin Baumer, Andrew Zimbalist

Mathematics Sciences: Faculty Publications

For the 65 colleges and universities that participate in the Power Five athletic conferences (Pac 12, Big 10, SEC, ACC, and Big 12), the football and men’s basketball teams are highly visible. While these programs generate tens of millions of dollars in revenue annually, very few of them turn an operating “profit.” Their existence is thus justified by the claim that athletic success leads to ancillary benefits for the academic institution, in terms of both quantity (e.g., more applications, donations, and state funding) and quality (e.g., stronger applicants, lower acceptance rates, higher yields). Previous studies provide only weak support for …

Convergence In A Disk Stacking Model On The Cylinder, Christophe Golé, Stéphane Douady

Mathematics Sciences: Faculty Publications

We study an iterative process modeling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either …

U.S. College Students’ Social Network Characteristics And Perceived Social Exclusion: A Comparison Between Drinkers And Nondrinkers Based On Pastmonth Alcohol Use, Sara G. Balestrieri, Graham T. Diguiseppi, Matthew Meisel, Melissa A. Clark, Miles Q. Ott, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

There is a general perception on college campuses that alcohol use is normative. However, nondrinking students account for 40% of the U.S. college population. With much of the literature focusing on intervening among drinkers, there has been less of a focus on understanding the nondrinker college experience. The current study has two aims: to describe the social network differences between nondrinkers and drinkers in a college setting, and to assess perceived social exclusion among nondrinkers. METHOD:First-year U.S. college students (n = 1,342; 55.3% female; 47.7% non-Hispanic White) were participants in a larger study examining a social network of one college …

Double Lie Algebroids And Representations Up To Homotopy, A. Gracia-Saz, M. Jotz Lean, K. C. H. Mackenzie, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.

Resistance To Peer Influence Moderates The Relationship Between Perceived (But Not Actual) Peer Norms And Binge Drinking In A College Student Social Network, Graham T. Diguiseppi, Matthew K. Meisel, Sara G. Balestrieri, Miles Q. Ott, Melissa J. Cox, Melissa A. Clark, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

Introduction: Adolescent and young adult binge drinking is strongly associated with perceived social norms and the drinking behavior that occurs within peer networks. The extent to which an individual is influenced by the behavior of others may depend upon that individual’s resistance to peer influence (RPI).

Methods: Students in their first semester of college (N = 1323; 54.7% female, 57% White, 15.1% Hispanic) reported on their own binge drinking, and the perceived binge drinking of up to 10 important peers in the first-year class. Using network autocorrelation models, we investigated cross-sectional relationships between participant’s binge drinking frequency and the perceived …

Constant Symplectic 2-Groupoids, Rajan Amit Mehta, Xiang Tang

Mathematics Sciences: Faculty Publications

We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.

Germs Of Fibrations Of Spheres By Great Circles Always Extend To The Whole Sphere, Patricia Cahn, Herman Gluck, Haggai Nuchi

Mathematics Sciences: Faculty Publications

We prove that every germ of a smooth fibration of an odd-dimensional round sphere by great circles extends to such a fibration of the entire sphere, a result previously known only in dimension three.

An Event- And Network-Level Analysis Of College Students’ Maximum Drinking Day, Matthew K. Meisel, Angelo M. Dibello, Sara G. Balestrieri, Miles Q. Ott, Graham T. Diguiseppi, Melissa A. Clark, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

Background—Heavy episodic drinking is common among college students and remains a serious public health issue. Previous event-level research among college students has examined behaviors and individual-level characteristics that drive consumption and related consequences but often ignores the social network of people with whom these heavy drinking episodes occur. The main aim of the current study was to investigate the network of social connections between drinkers on their heaviest drinking occasions.

Methods—Sociocentric network methods were used to collect information from individuals in the first-year class (N=1342) at one university. Past-month drinkers (N=972) reported on the characteristics of their heaviest drinking occasion …