Mathematics Commons^™

The Dihedral Genus Of A Knot, Patricia Cahn, Alexandra Kjuchukova

Mathematics Sciences: Faculty Publications

Let K ⊂ S³ be a Fox p-colored knot and assume K bounds a locally flat surface S ⊂ B⁴ over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X → S⁴ . Its branching set is a closed surface embedded in S⁴ locally flatly away from one singularity whose link is K. When S is homotopy ribbon and X a definite four-manifold, a condition relating the signature of X and the Murasugi signature of K guarantees that S in fact realizes the four-genus of K. We exhibit an infinite …

Action Of The Symmetric Group On The Free Lanke: A Catalanke Theorem, Tamar Friedmann, Philip Hanlon, Richard P. Stanley, Michelle L. Wachs

Mathematics Sciences: Faculty Publications

We initiate a study of the representation of the symmetric group on the multilinear component of an n-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group S_2n−1 on the multilinear component of the free LAnKe with S_2n−1 generators is given by an irreducible representation whose dimension is the nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator. A decomposition, into irreducibles, of the representation of S_3n−2 on the multilinear component the free LAnKe …

Strong Comparison Principle For P-Harmonic Functions In Carnot-Caratheodory Spaces, Luca Capogna, Xiaodan Zhou

Mathematics Sciences: Faculty Publications

We extend Bony’s propagation of support argument to C1 solutions of the nonhomogeneous subelliptic p-Laplacian associated to a system of smooth vector fields satisfying Hörmander’s finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf.

How Often Does The Best Team Win? A Unified Approach To Understanding Randomness In North American Sport, Michael J. Lopez, Gregory J. Matthews, Benjamin Baumer

Mathematics Sciences: Faculty Publications

Statistical applications in sports have long centered on how to best separate signal (e.g. team talent) from random noise. However, most of this work has concentrated on a single sport, and the development of meaningful cross-sport comparisons has been impeded by the difficulty of translating luck from one sport to another. In this manuscript, we develop Bayesian state-space models using betting market data that can be uniformly applied across sporting organizations to better understand the role of randomness in game outcomes. These models can be used to extract estimates of team strength, the between-season, within-season, and game-to-game variability of team …

Mapping Class Group Orbits Of Curves With Self-Intersections, Patricia Cahn, Federica Fanoni, Bram Petri

Mathematics Sciences: Faculty Publications

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.

We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.

As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. …

Flagellar Swimming In Viscoelastic Fluids: Role Of Fluid Elastic Stress Revealed By Simulations Based On Experimental Data, Chuanbin Li, Boyang Qin, Arvind Gopinath, Paulo E. Arratia, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

Many important biological functions depend on microorganisms' ability to move in viscoelastic fluids such as mucus and wet soil. The effects of fluid elasticity on motility remain poorly understood, partly because the swimmer strokes depend on the properties of the fluid medium, which obfuscates the mechanisms responsible for observed behavioural changes. In this study, we use experimental data on the gaits of Chlamydomonas reinhardtii swimming in Newtonian and viscoelastic fluids as inputs to numerical simulations that decouple the swimmer gait and fluid type in order to isolate the effect of fluid elasticity on swimming. In viscoelastic fluids, cells employing the …

Lessons From Between The White Lines For Isolated Data Scientists, Benjamin Baumer

Mathematics Sciences: Faculty Publications

Many current and future data scientists will be “isolated”—working alone or in small teams within a larger organization. This isolation brings certain challenges as well as freedoms. Drawing on my considerable experience both working in the professional sports industry and teaching in academia, I discuss troubled waters likely to be encountered by newly minted data scientists and offer advice about how to navigate them. Neither the issues raised nor the advice given are particular to sports and should be applicable to a wide range of knowledge domains.

Okcupid Data For Introductory Statistics And Data Science Courses, Albert Y. Kim, Adriana Escobedo-Land

Mathematics Sciences: Faculty Publications

We present a data set consisting of user profile data for 59,946 San Francisco OkCupid users (a free online dating website) from June 2012. The data set includes typical user information, lifestyle variables, and text responses to 10 essay questions. We present four example analyses suitable for use in undergraduate introductory probability and statistics and data science courses that use R. The statistical and data science concepts covered include basic data visualization, exploratory data analysis, multivariate relationships, text analysis, and logistic regression for prediction.

The Role Of Body Flexibility In Stroke Enhancements For Finite-Length Undulatory Swimmers In Viscoelastic Fluids, Becca Thomases, Robert D. Guy

Mathematics Sciences: Faculty Publications

The role of passive body dynamics on the kinematics of swimming micro-organisms in complex fluids is investigated. Asymptotic analysis of small-amplitude motions of a finite-length undulatory swimmer in a Stokes-Oldroyd-B fluid is used to predict shape changes that result as body elasticity and fluid elasticity are varied. Results from the analysis are compared with numerical simulations and the numerically simulated shape changes agree with the analysis at both small and large amplitudes, even for strongly elastic flows. We compute a stroke-induced swimming speed that accounts for the shape changes, but not additional effects of fluid elasticity. Elasticity-induced shape changes lead …

How Low Can You Go? New Bounds On The Biplanar Crossing Number Of Low-Dimensional Hypercubes, Gregory J. Clark, Gwen Spencer

Mathematics Sciences: Faculty Publications

In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The k-planar crossing number of a graph cr _k ( G) is the number of crossings required when every edge of G must be drawn in one of k distinct planes. It was shown in [2] that cr ₂ ( Q 8 ) ≤ 256 which we improve to cr ₂ ( Q 8 ) ≤ 128. Our approach highlights the relationship between symmetric drawings and the study of k-planar crossing numbers. We conclude with several open questions concerning this …

Measuring The Value Of Accurate Link Prediction For Network Seeding, Yijin Wei, Gwen Spencer

Mathematics Sciences: Faculty Publications

Merging two classic questions

The influence-maximization literature seeks small sets of individuals whose structural placement in the social network can drive large cascades of behavior. Optimization efforts to find the best seed set often assume perfect knowledge of the network topology. Unfortunately, social network links are rarely known in an exact way. When do seeding strategies based on less-than-accurate link prediction provide valuable insight?

Immersed Boundary Smooth Extension (Ibse): A High-Order Method For Solving Incompressible Flows In Arbitrary Smooth Domains, David B. Stein, Robert D. Guy, Becca Thomases

Mathematics Sciences: Faculty Publications

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor. In a previous work we introduced the Immersed Boundary Smooth Extension (IBSE) method, a variation of the IB method that achieves high-order accuracy for elliptic PDE by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations. In this …

Rhombic Tilings And Primordia Fronts Of Phyllotaxis, Pau Atela, Christophe Golé

Mathematics Sciences: Faculty Publications

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically …

Valid Plane Trees: Combinatorial Models For Rna Secondary Structures With Watson-Crick Base Pairs, Francis Black, Elizabeth Drellich, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

. The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the secondary structure, or folding of the RNA sequence). This paper considers an augmented version of the standard model of plane trees, specifically one that incorporates some observed constraints on how the folding can occur. In particular we assume the alphabet consists of complementary pairs, for instance the Watson–Crick pairs A-U and C-G of RNA. Given a word in the alphabet, a valid plane tree …

Vb-Groupoids And Representation Theory Of Lie Groupoids, Alfonso Gracia-Saz, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the “adjoint representation” of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not. We define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomology with values in the corresponding representations …

2d Euler Equation On The Strip: Stability Of A Rectangular Patch, Jennifer Beichman, Sergey Denisov

Mathematics Sciences: Faculty Publications

We consider the 2D Euler equation of incompressible fluids on a strip ℝ×�� and prove the stability of the rectangular stationary state χ_|x|<L for large enough L.

Advance Care Planning As A Shared Endeavor: Completion Of Acp Documents In A Multidisciplinary Cancer Program, Melissa A. Clark, Miles Q. Ott, Michelle L. Rogers, Mary C. Politi, Susan C. Miller, Laura Moynihan, Katina Robison, Ashley Stuckey, Don Dizon

Statistical and Data Sciences: Faculty Publications

Objective—We examined the roles of oncology providers in advance care planning (ACP) delivery in the context of a multidisciplinary cancer program.

Methods—Semi-structured interviews were conducted with 200 women with recurrent and/or metastatic breast or gynecologic cancer. Participants were asked to name providers they deemed important in their cancer care and whether they had discussed and/or completed ACP documentation. Evidence of ACP documentation was obtained from chart reviews.

Results—Fifty percent of participants self-reported completing an advance directive (AD) and 48.5% had named a healthcare power of attorney (HPA), 38.5% had completed both, and 39.0% had completed neither document. Among women who …

Alcohol Perceptions And Behavior In A Residential Peer Social Network, Shannon R. Kenney, Miles Q. Ott, Matthew Meisel, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

Personalized normative feedback is a recommended component of alcohol interventions targeting college students. However, normative data are commonly collected through campus-based surveys, not through actual participant-referent relationships. In the present investigation, we examined how misperceptions of residence hall peers, both overall using a global question and those designated as important peers using person-specific questions, were related to students’ personal drinking behaviors. Participants were 108 students (88% freshman, 54% White, 51% female) residing in a single campus residence hall. Participants completed an online baseline survey in which they reported their own alcohol use and perceptions of peer alcohol use using both …

Curriculum Guidelines For Undergraduate Programs In Data Science, Richard D. De Veaux, Mahesh Agarwal, Maia Averett, Benjamin Baumer, Andrew Bray, Thomas C. Bressoud, Lance Bryant, Lei Z. Cheng, Amanda Francis, Robert Gould, Albert Y. Kim, Matt Kretchmar, Qin Lu, Ann Moskol, Deborah Nolan, Roberto Pelayo, Sean Raleigh, Ricky J. Sethi, Mutiara Sondjaja, Neelesh Tiruviluamala, Paul X. Uhlig, Talitha M. Washington, Curtis L. Wesley, David White, Ping Ye

Statistical and Data Sciences: Faculty Publications

The Park City Math Institute 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in data science. The group consisted of 25 undergraduate faculty from a variety of institutions in the United States, primarily from the disciplines of mathematics, statistics, and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in data science.

Fibonacci Or Quasi-Symmetric Phyllotaxis. Part Ii: Botanical Observations, Stéphane Douady, Christophe Golé

Mathematics Sciences: Faculty Publications

Historically, the study of phyllotaxis was greatly helped by the simple description of botanical patterns by only two integer numbers, namely the number of helices (parastichies) in each direction tiling the plant stem. The use of parastichy num- bers reduced the complexity of the study and created a proliferation of generaliza- tions, among others the simple geometric model of lattices. Unfortunately, these simple descriptive method runs into difficulties when dealing with patterns pre- senting transitions or irregularities. Here, we propose several ways of addressing the imperfections of botanical reality. Using ontogenetic analysis, which follows the step-by-step genesis of the pattern, …

Fibonacci Or Quasi-Symmetric Phyllotaxis. Part I: Why?, Christophe Golé, Jacques Dumais, Stéphane Douady

Mathematics Sciences: Faculty Publications

The study of phyllotaxis has focused on seeking explanations for the occurrence of consecutive Fibonacci numbers in the number of helices paving the stems of plants in the two opposite directions. Using the disk-accretion model, first introduced by Schwendener and justified by modern biological studies, we observe two dis- tinct types of solutions: the classical Fibonacci-like ones, and also more irregular configurations exhibiting nearly equal number of helices in a quasi-square pack- ing, the quasi-symmetric ones, which are a generalization of the whorled patterns. Defining new geometric tools allowing to work with irregular patterns and local transitions, we provide simple …

Unfolding Genus-2 Orthogonal Polyhedra With Linear Refinement, Mirela Damian, Erik Demaine, Robin Flatland, Joseph O'Rourke

Mathematics Sciences: Faculty Publications

We show that every orthogonal polyhedron of genus g ≤ 2 can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus- 0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the “unfolding tree” (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques.

Smoothness Of Subriemannian Isometries, Luca Capogna, Enrico Le Donne

Mathematics Sciences: Faculty Publications

We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of harmonic coordinates, establishing that in an arbitrary subRiemannian manifold there exists an open dense subset where all isometries are smooth.

Bayesian Peer Calibration With Application To Alcohol Use, Miles Q. Ott, Joseph W. Hogan, Krista J. Gile, Crystal Linkletter, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

Peers are often able to provide important additional information to supplement self-reported behavioral measures. The study motivating this work collected data on alcohol in a social network formed by college students living in a freshman dormitory. By using two imperfect sources of information (self-reported and peer-reported alcohol consumption), rather than solely self-reports or peer-reports, we are able to gain insight into alcohol consumption on both the population and the individual level, as well as information on the discrepancy of individual peer-reports. We develop a novel Bayesian comparative calibration model for continuous, count and binary outcomes that uses covariate information to …

Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, Joseph O'Rourke

Computer Science: Faculty Publications

A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does …

Regularity For Subelliptic Pde Through Uniform Estimates In Multi-Scale Geometries, Luca Capogna, Giovanna Citti

Mathematics Sciences: Faculty Publications

We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea (Math Ann 357(3):1175–1198, 2013) and Manfredini (Anal Geom Metric Spaces 1:255–275, 2013) concerning stability of doubling properties, Poincare’ inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of Hörmander vector fields.

Splines In Geometry And Topology, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.

A Bayesian Method For Cluster Detection With Application To Five Cancer Sites In Puget Sound, Albert Y. Kim, Jon Wakefield

Mathematics Sciences: Faculty Publications

Cluster detection is an important public health endeavor and in this paper we describe and apply a recently developed Bayesian method. Commonly-used approaches are based on so-called scan statistics and suffer from a number of difficulties including how to choose a level of significance and how to deal with the possibility of multiple clusters. The basis of our model is to partition the study region into a set of areas which are either “null” or “non-null”, the latter corresponding to clusters (excess risk) or anti-clusters (reduced risk). We demonstrate the Bayesian method and compare with a popular existing approach, using …

Generalized Splines On Arbitrary Graphs, Simcha Gilbert, Julianna Tymoczko, Shira Viel

Mathematics Sciences: Faculty Publications

Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky, Kottwitz, and MacPherson, among others). The central question of this paper asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is “always”, and we …

Sticky Seeding In Discrete-Time Reversible-Threshold Networks, Gwen Spencer

Mathematics Sciences: Faculty Publications

When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of Sticky Interventions: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is Ω(ln(OP T ))-hard to approximate and that maximizing conversion subject to a budget is …