Physical Sciences and Mathematics Commons^™

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×𝕋 and prove the stability of the rectangular stationary state χ_|x|<L for large enough L.