Groupoid Extensions of Mapping Class Representations for Bordered Surfaces

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichm\"uller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representations on the moduli spaces of flat connections.Comment: 24 pages, 4 figures Theorem 3.6 has been strengthened, and Theorems 8.1 and 8.2 have been adde

Thermal roughening of {001} surfaces

Within the framework of a solid-on-solid model that incorporates nearest- (epsilon) and next-nearest-neighbor (delta) interactions we have determined the free energy of the high-symmetry steps on a (001) surface of a cubic crystal. We have found a simple expression that allows one to determine the thermal roughening temperature TR of a (001) surface (2e¿(epsilon/2+delta)/kbTR¿e¿(epsilon+2delta)/kbTR+2e¿(epsilon+delta)/kbTR=1). In a more refined analysis we have explicitly included step-edge overhangs. This results in a slightly lower thermal roughening temperature. Our results are also applicable to the two-dimensional Ising spin system

Finite type invariants and fatgraphs

We define an invariant $\nabla_G(M)$ of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder $S\times I$, S is a connected surface with at least one boundary component, and G is a fatgraph spine of S. In effect, $\nabla_G$ is the composition with the $\iota_n$ maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e., $\nabla_G$ establishes an isomorphism from an appropriate vector space $\bar{H}$ of homology cylinders to a certain algebra of Jacobi diagrams. Via composition $\nabla_{G'}\circ\nabla_G^{-1}$ for any pair of fatgraph spines G,G' of S, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmuller space, as a group of automorphisms of this algebra. The space $\bar{H}$ comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody. We compute how $\nabla_G$ interacts with all three operations explicitly in terms of natural products on Jacobi diagrams and certain diagrammatic constants. Our main result gives an explicit extension of the LMO invariant of 3-manifolds to the Ptolemy groupoid in terms of these operations, and this groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the Morita-Penner cocycle representing the first Johnson homomorphism using a variant/generalization of $\nabla_G$.Comment: 39 page

Comparison of performance of concussed and non-concussed individuals on Subtest VIII of the C-RTT

The purpose of this study was to investigate auditory comprehension in concussed and non-concussed individuals using Subtest VIII of the C-RTT. Thirty non-concussed individuals were matched as closely as possible on gender, age, education and history of concussion to a group of concussed individuals. A Mann-Whitney found that the two samples were significant different at p = .020, and a t-test found a significant difference at p = .008 on the Efficiency Score (ES) of Subtest VIII. The non-concussed individuals performed better than the concussed individuals. Efficiency Score may be useful in assessing comprehension in concussed individuals

Surface bubble nucleation phase space

Recent research has revealed several different techniques for nanoscopic gas nucleation on submerged surfaces, with findings seemingly in contradiction with each other. In response to this, we have systematically investigated the occurrence of surface nanobubbles on a hydrophobised silicon substrate for various different liquid temperatures and gas concentrations, which we controlled independently. We found that nanobubbles occupy a distinct region of this phase space, occurring for gas concentrations of approximately 100-110%. Below the nanobubble phase we did not detect any gaseous formations on the substrate, whereas micropancakes (micron wide, nanometer high gaseous domains) were found at higher temperatures and gas concentrations. We moreover find that supersaturation of dissolved gases is not a requirement for nucleation of bubbles.Comment: 4 pages, 4 figure

Phase diagram of the random field Ising model on the Bethe lattice

The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact upper bound for the existence of a unique paramagnetic phase is found which considerably improves the earlier results. Several numerical estimates of transition lines between a ferromagnetic and a paramagnetic regime are presented. The obtained results do not coincide with a lower bound for the onset of ferromagnetism proposed by Bruinsma [Phys. Rev. B 30, 289 (1984)]. If the latter one proves correct this would hint to a region of coexistence of stable ferromagnetic phases and a stable paramagnetic phase.Comment: Article has been condensed and reorganized; Figs 3,5,6 merged; Fig 4 omitted; Some discussion added at end of Sec. III; 9 pages, 5 figs, RevTeX4, AMSTe

A two-qubit Bell inequality for which POVM measurements are relevant

A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin.Comment: 7 pages, 1 figur

Stationary Properties of a Randomly Driven Ising Ferromagnet

We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. Analytic results for the stationary state are presented in mean-field approximation, exhibiting a novel type of first order phase transition related to dynamic freezing. Monte Carlo simulations performed on a quadratic lattice indicate that many features of the mean field theory may survive the presence of fluctuations.Comment: 5 pages in RevTex format, 7 eps/ps figures, send comments to "mailto:[email protected]", submitted to PR

Vacancy-mediated diffusion of Co atoms embedded in Cu(001)

The diffusion of Co atoms in the Cu(001) surface has been studied using Scanning Tunneling Microscopy (STM). Like other impurities in the Cu(001) surface, the diffusion of Co is mediated by single surface vacancies. STM images reveal that diffusion of the embedded atoms takes place through multi-atom jumps separated by long time intervals, which is characteristic for this type of diffusion. The jump length and frequency are measured to establish the nature of the interaction between surface vacancies and the embedded Co atoms and to extract the relevant formation and diffusion energies