Physical Sciences and Mathematics Commons^™

Frameworks With Crystallographic Symmetry, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

Periodic frameworks with crystallographic symmetry are investigated from the perspective of a general deformation theory of periodic bar-and-joint structures in R^d. It is shown that natural parametrizations provide affine section descriptions for families of frameworks with a specified graph and symmetry. A simple geometric setting for diaplacive phase transitions is obtained. Upper bounds are derived for the number of realizations of minimally rigid periodic graphs.

A Subelliptic Analogue Of Aronson-Serrin's Harnack Inequality, Luca Capogna, Giovanna Citti, Garrett Rea

Mathematics Sciences: Faculty Publications

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE,(Formula presented.) in cylinders Ω × (0,T) where Ω ⊂ M is an open subset of a manifold M endowed with control metric d corresponding to a system of Lipschitz continuous vector fields X=(X_1,..., X_m) and a measure dσ. We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space (M,d,dσ). We also show that such hypothesis hold for a class of Riemannian metrics gε collapsing to a sub-Riemannian metric limε → …

A Bayesian Model For Cluster Detection, Jonathan Wakefield, Albert Y. Kim

Statistical and Data Sciences: Faculty Publications

The detection of areas in which the risk of a particular disease is significantly elevated, leading to an excess of cases, is an important enterprise in spatial epidemiology. Various frequentist approaches have been suggested for the detection of “clusters” within a hypothesis testing framework. Unfortunately, these suffer from a number of drawbacks including the difficulty in specifying a p-value threshold at which to call significance, the inherent multiplicity problem, and the possibility of multiple clusters. In this paper, we suggest a Bayesian approach to detecting “areas of clustering” in which the study region is partitioned into, possibly multiple, “zones” …

A 2-Chain Can Interlock With An Open 10-Chain, Bin Lu, Joseph O'Rourke, Jianyuan K. Zhong

Computer Science: Faculty Publications

Abstract. It is an open problem, posed in [3], to determine the minimal k such that an open flexible k-chain can interlock with a flexible 2-chain. It was first established in [5] that there is an open 16-chain in a trapezoid frame that achieves interlocking. This was subsequently improved in [6] to establish interlocking between a 2-chain and an open 11-chain. Here we improve that result once more, establishing interlocking between a 2-chain and a 10-chain. We present arguments that indicate that 10 is likely the minimum.

Differential Graded Contact Geometry And Jacobi Structures, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We study contact structures on nonnegatively graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.

Sub-Riemannian Heat Kernels And Mean Curvature Flow Of Graphs, Luca Capogna, Giovanna Citti, Cosimo Senni Guidotti Magnani

Mathematics Sciences: Faculty Publications

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher algorithm (Merriman et al., 1992 [42]) and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L.C. Evans (1993) [27] in the Euclidean setting and is based on new results on the relation between sub-Riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs.

The String Landscape: On Formulas For Counting Vacua, Tamar Friedmann, Richard P. Stanley

Mathematics Sciences: Faculty Publications

We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G₂ holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U (1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside Lie groups. We also point out certain cases where the …

Repeated Changes In Reported Sexual Orientation Identity Linked To Substance Use Behaviors In Youth, Miles Q. Ott, David Wypij, Heather L. Corliss, Margaret Rosario, Sari L. Reisner, Allegra R. Gordon, S. Bryn Austin

Statistical and Data Sciences: Faculty Publications

Purpose—Previous studies have found that sexual minority (e.g., lesbian, gay, bisexual) adolescents are at higher risk of substance use than heterosexuals, but few have examined how changes in sexual orientation over time may relate to substance use. We examined the associations between change in sexual orientation identity and marijuana use, tobacco use, and binge drinking in U.S. youth.

Methods—Prospective data from 10,515 U.S. youth ages 12-27 years in a longitudinal cohort study were analyzed using sexual orientation identity mobility measure M (frequency of change from 0 [no change] to 1 [change at every wave]) in up to five waves of …

The K-Dominating Graph, Ruth Haas, Karen Seyffarth

Mathematics Sciences: Faculty Publications

Abstract. Given a graph G, the k-dominating graph of G, D_k(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D_k(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D_k(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of …

No Radial Excitations In Low Energy Qcd. Ii. The Shrinking Radius Of Hadrons, Tamar Friedmann

Mathematics Sciences: Faculty Publications

We discuss the implications of our prior results obtained in our companion paper (Eur. Phys. J. C (2013). doi:10.1140/epjc/s10052-013-2298-9). Inescapably, they lead to three laws governing the size of hadrons, including in particular protons and neutrons that make up the bulk of ordinary matter: (a) there are no radial excitations in low-energy QCD; (b) the size of a hadron is largest in its ground state; (c) the hadron’s size shrinks when its orbital excitation increases. The second and third laws follow from the first law. It follows that the path from confinement to asymptotic freedom is a Regge trajectory. It …

No Radial Excitations In Low Energy Qcd. I. Diquarks And Classification Of Mesons, Tamar Friedmann

Mathematics Sciences: Faculty Publications

We propose a new schematic model for mesons in which the building blocks are quarks and flavor-antisymmetric diquarks. The outcome is a new classification of the entire meson spectrum into quark–antiquark and diquark– antidiquark states which does not give rise to a radial quantum number: all mesons which have so far been believed to be radially excited are orbitally excited diquark–antidiquark states; similarly, there are no radially excited baryons. Further, mesons that were previously viewed as “exotic” are no longer exotic as they are now naturally integrated into the classification as diquark–antidiquark states. The classification also leads to the introduction …

Generalizing Tanisaki's Ideal Via Ideals Of Truncated Symmetric Functions, Aba Mbirika, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Abstract. We define a family of ideals I_h in the polynomial ring Z[_{x1, . . . , xn}] that are parametrized by Hessenberg functions h (equivalently Dyck paths or ample partitions). The ideals I_h generalize algebraically a family of ideals called the Tanisaki ideal, which is used in a geometric construction of permutation representations called Springer theory. To define I_h, we use polynomials in a proper subset of the variables {_{x1, . . . , xn}} that are symmetric under the corresponding permutation subgroup. We call these polynomials truncated symmetric functions and …

Billey's Formula In Combinatorics, Geometry, And Topology, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of …

A Generalization Of The Turaev Cobracket And The Minimal Self-Intersection Number Of A Curve On A Surface, Patricia Cahn

Mathematics Sciences: Faculty Publications

Goldman and Turaev constructed a Lie bialgebra structure on the free Zmodule generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket ∆(α) is zero if and only if α is a power of a simple class. Chas constructed examples that show Turaev’s conjecture is, unfortunately, false. We define an operation µ in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through µ, so we can view µ as a generalization of ∆. We show that Turaev’s conjecture holds when ∆ is replaced with µ. We also show that µ(α) …

Intersections Of Loops And The Andersen–Mattes–Reshetikhin Algebra, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

Given two free homotopy classes α₁,α₂ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points # (α₁,α₂) of loops in these two classes.

We show that, for α₁≠α₂, the number of terms in the Andersen–Mattes–Reshetikhin Poisson bracket of α₁ and α₂ is equal to # (α₁,α₂). Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of α₁ and α₂.

The …

Uniform Gaussian Bounds For Subelliptic Heat Kernels And An Application To The Total Variation Flow Of Graphs Over Carnot Groups, Luca Capogna, Giovanna Citti, Maria Manfredini

Mathematics Sciences: Faculty Publications

In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σϵ which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ϵ rarr; 0. The main new contribution are Gaussian-Type bounds on the heat kernel for the σϵ metrics which are stable as ϵ rarr; 0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σϵ ). We establish …