Tangent unit-vector fields: nonabelian homotopy invariants and the Dirichlet energy

Let O be a closed geodesic polygon in S 2 . Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2 , we compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for E(H) involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, π1 (S 2 − {s1 , . . . , sn }, ∗). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for E(H) reduces to a previous result involving the degrees of a set of regular values s1 , . . . , sn in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π1 (S 2 − {s1 , . . . , sn }, ∗). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.\ud \ud This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism

Wilson loops in heavy ion collisions and their calculation in AdS/CFT

Expectation values of Wilson loops define the nonperturbative properties of the hot medium produced in heavy ion collisions that arise in the analysis of both radiative parton energy loss and quarkonium suppression. We use the AdS/CFT correspondence to calculate the expectation values of such Wilson loops in the strongly coupled plasma of N=4 super Yang-Mills (SYM) theory, allowing for the possibility that the plasma may be moving with some collective flow velocity as is the case in heavy ion collisions. We obtain the N=4 SYM values of the jet quenching parameter $\hat q$, which describes the energy loss of a hard parton in QCD, and of the velocity-dependence of the quark-antiquark screening length for a moving dipole as a function of the angle between its velocity and its orientation. We show that if the quark-gluon plasma is flowing with velocity v_f at an angle theta with respect to the trajectory of a hard parton, the jet quenching parameter $\hat q$ is modified by a factor gamma_f(1-v_f cos theta), and show that this result applies in QCD as in N=4 SYM. We discuss the relevance of the lessons we are learning from all these calculations to heavy ion collisions at RHIC and at the LHC. Furthermore, we discuss the relation between our results and those obtained in other theories with gravity duals, showing in particular that the ratio between $\hat q$ in any two conformal theories with gravity duals is the square root of the ratio of their central charges. This leads us to conjecture that in nonconformal theories $\hat q$ defines a quantity that always decreases along renormalization group trajectories and allows us to use our calculation of $\hat q$ in N=4 SYM to make a conjecture for its value in QCD.Comment: 61 pages, 8 figures. Note added discussing relation between our work and that in several papers that have appeared recently. References adde

Spacetime in String Theory

We give a brief overview of the nature of spacetime emerging from string theory. This is radically different from the familiar spacetime of Einstein's relativity. At a perturbative level, the spacetime metric appears as ``coupling constants" in a two dimensional quantum field theory. Nonperturbatively (with certain boundary conditions), spacetime is not fundamental but must be reconstructed from a holographic, dual theory.Comment: 20 pages; references adde

On Dynamics of Cubic Siegel Polynomials

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given irrational number of Brjuno type. Our main goal is to prove that when theta is of bounded type, the boundary of the Siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial. We also prove that the locus of all cubics with both critical points on the boundary of their Siegel disk is a Jordan curve, which is in some sense parametrized by the angle between the two critical points. A main tool in the bounded type case is a related space of degree 5 Blaschke products which serve as models for our cubics. Along the way, we prove several results about the connectedness locus of these cubic polynomials.Comment: 58 pages. 20 PostScript figure

The solution of the Elrod algorithm for a dynamically loaded journal bearing using multigrid techniques

A numerical solution to a theoretical model of vapor cavitation in a dynamically loaded journal bearing is developed utilizing a multigrid iteration technique. The method is compared with a noniterative approach in terms of computational time and accuracy. The computational model is based on the Elrod algorithm, a control volume approach to the Reynolds equation which mimics the Jakobsson-Floberg and Olsson cavitation theory. Besides accounting for a moving cavitation boundary and conservation of mass at the boundary, it also conserves mass within the cavitated region via a smeared mass or striated flow extending to both surfaces in the film gap. The mixed nature of the equations (parabolic in the full film zone and hyperbolic in the cavitated zone) coupled with the dynamic aspects of the problem create interesting difficulties for the present solution approach. Emphasis is placed on the methods found to eliminate solution instabilities. Excellent results are obtained for both accuracy and reduction of computational time