Mathematics Commons^™

Counting Conjugacy Classes Of Elements Of Finite Order In Lie Groups, Tamar Friedmann, Richard P. Stanley

Mathematics Sciences: Faculty Publications

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define N(G,m) to be the number of conjugacy classes of elements of finite order m in a Lie group G, and N(G,m,s) to be the number of such classes whose elements have s distinct eigenvalues or conjugate pairs of eigenvalues.What is N(G,m) for G a unitary, orthogonal, or symplectic group?What is N(G,m,s) for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques.It appears that the second question has not been asked before; here it is inspired by questions related …

New And Improved Spanning Ratios For Yao Graphs, Luis Barba, Prosenjit Bose, Mirela Damian, Rolf Fagerberg, Wah Loon Keng, Joseph O'Rourke, André Van Renssen, Perouz Taslakian, Sander Verdonschot, Ge Xia

Computer Science: Faculty Publications

For a set of points in the plane and a fixed integer k > 0, the Yao graph Y_k partitions the space around each point into k equiangular cones of angle Θ = 2π/k, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of Y₅, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd k ≥ 5, the spanning ratio of Y_k is at most 1/(1−2sin(3Θ/8)), which gives the first constant upper bound for …

L∞-Extremal Mappings In Amle And Teichmüller Theory, Luca Capogna

Mathematics Sciences: Faculty Publications

These lecture focus on two vector-valued extremal problems which have a common feature in that the corresponding energy functionals involve L ∞ norm of an energy density rather than the more familiar L p norms. Specifically, we will address (a) the problem of extremal quasiconformal mappings and (b) the problem of absolutely minimizing Lipschitz extensions.

The Classification Of V -Transverse Knots And Loose Legendrians, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

We classify knots in a 3-manifold M that are transverse to a nowhere zero vector field V up to the corresponding isotopy relation. When V is the coorienting vector field of a contact structure, these knots are the same as pseudo-Legendrian knots, which were introduced by Benedetti and Petronio. We show that two loose Legendrian knots with the same overtwisted disk in their complement are Legendrian isotopic if and only if they are pseudoLegendrian isotopic.

V -transverse knots are naturally framed. We show that each framed isotopy class corresponds to infinitely many V -transverse isotopy classes whose elements are pairwise …

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.

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 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 …

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 …

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 …

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 …

L ∞-Algebra Actions, Rajan Amit Mehta, Marco Zambon

Mathematics Sciences: Faculty Publications

We define the notion of action of an L -algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of g on a second L -algebra, leading to a notion of "semidirect product" of L -algebras more general than those we found in the literature.

Group-Theoretical Derivation Of Angular Momentum Eigenvalues In Spaces Of Arbitrary Dimensions, Tamar Friedmann, C. R. Hagen

Mathematics Sciences: Faculty Publications

The spectrum of the square of the angular momentum in arbitrary dimensions is derived using only group theoretical techniques. This is accomplished by application of the Lie algebra of the noncompact group O(2, 1). Illuminating discussions with Jonathan Pakianathan and his comments on a draft are gratefully acknowledged. This work was supported in part by (U.S.) Department of Energy (DOE), Grant No. DE-FG02-91ER40685.

An Aronsson Type Approach To Extremal Quasiconformal Mappings, Luca Capogna, Andrew Raich

Mathematics Sciences: Faculty Publications

We study C 2 extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a 'localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short-time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

From Pop-Up Cards To Coffee-Cup Caustics: The Knight's Visor, Stephanie Jakus, Joseph O'Rourke

Computer Science: Faculty Publications

As a pedagogical exercise, we derive the shape of a particularly elegant pop-up card design, and show that it connects to a classically studied plane curve that is (among other interpretations) a caustic of a circle

A Simple Bijection Between Standard 3×N Tableaux And Irreducible Webs For ����3, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and ����₂. Building on a result of Kuperberg’s, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3 × n and irreducible webs for ����₃whose boundary vertices are all sources. In this paper, we give a simple and explicit map from standard Young tableaux of shape 3 …

Unfolding Prismatoids As Convex Patches: Counterexamples And Positive Results, Joseph O'Rourke

Computer Science: Faculty Publications

We address the unsolved problem of unfolding prismatoids in a new context, viewing a “topless prismatoid” as a convex patch—a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for “petal unfolding” topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads …

Source Unfoldings Of Convex Polyhedra Via Certain Closed Curves, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

Abstract. We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q “lives on a cone” to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygon

Π/2-Angle Yao Graphs Are Spanners, Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'Rourke, Ben Seamone, Michiel Smid, Stefanie Wuhrer

Computer Science: Faculty Publications

We show that the Yao graph Y₄ in the L₂ metric is a spanner with stretch factor 8(29+23√ 2). Enroute to this, we also show that the Yao graph Y^∞₄ in the L_∞ metric is a planar spanner with stretch factor 8.

Permutation Notations For The Exceptional Weyl Group F4, Patricia Cahn, Ruth Haas, Aloysius G. Helminck, Juan Li, Jeremy Schwartz

Mathematics Sciences: Faculty Publications

This paper describes a permutation notation for the Weyl groups of type F4 and G2. The image in the permutation group is presented as well as an analysis of the structure of the group. This description enables faster computations in these Weyl groups which will prove useful for a variety of applications.

A New Qcd Effect: The Shrinking Radius Of Hadrons, Tamar Friedmann

Mathematics Sciences: Faculty Publications

We propose an extended schematic model for hadrons in which quarks as well as diquarks serve as building blocks. The outcome is a reclassification of the hadron spectrum in which there are no radially excited hadrons: all mesons and baryons previously believed to be radial excitations are orbitally excited states involving diquarks. Also, there are no exotic hadrons: all hadrons previously believed to be exotic are states involving diquarks and are an integral part of the model. We discuss the implications of this result for a new understanding of confinement and its relation to asymptotic freedom, as well as its …