The subgroup growth spectrum of virtually free groups

For a finitely generated group $\Gamma$ denote by $\mu(\Gamma)$ the growth coefficient of $\Gamma$, that is, the infimum over all real numbers $d$ such that $s_n(\Gamma)<n!^d$. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. Moreover, we describe an algorithm to compute $\mu$

Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection

We investigate the utility of the convex hull of many Lagrangian tracers to analyze transport properties of turbulent flows with different anisotropy. In direct numerical simulations of statistically homogeneous and stationary Navier-Stokes turbulence, neutral fluid Boussinesq convection, and MHD Boussinesq convection a comparison with Lagrangian pair dispersion shows that convex hull statistics capture the asymptotic dispersive behavior of a large group of passive tracer particles. Moreover, convex hull analysis provides additional information on the sub-ensemble of tracers that on average disperse most efficiently in the form of extreme value statistics and flow anisotropy via the geometric properties of the convex hulls. We use the convex hull surface geometry to examine the anisotropy that occurs in turbulent convection. Applying extreme value theory, we show that the maximal square extensions of convex hull vertices are well described by a classic extreme value distribution, the Gumbel distribution. During turbulent convection, intermittent convective plumes grow and accelerate the dispersion of Lagrangian tracers. Convex hull analysis yields information that supplements standard Lagrangian analysis of coherent turbulent structures and their influence on the global statistics of the flow.Comment: 18 pages, 10 figures, preprin

Testing Lorentz invariance by use of vacuum and matter filled cavity resonators

We consider tests of Lorentz invariance for the photon and fermion sector that use vacuum and matter-filled cavities. Assumptions on the wave-function of the electrons in crystals are eliminated from the underlying theory and accurate sensitivity coefficients (including some exceptionally large ones) are calculated for various materials. We derive the Lorentz-violating shift in the index of refraction n, which leads to additional sensitivity for matter-filled cavities ; and to birefringence in initially isotropic media. Using published experimental data, we obtain improved bounds on Lorentz violation for photons and electrons at levels of 10^-15 and below. We discuss implications for future experiments and propose a new Michelson-Morley type experiment based on birefringence in matter.Comment: 15 pages, 8 table

Lagrangian Statistics of Navier-Stokes- and MHD-Turbulence

We report on a comparison of high-resolution numerical simulations of Lagrangian particles advected by incompressible turbulent hydro- and magnetohydrodynamic (MHD) flows. Numerical simulations were performed with up to $1024^3$ collocation points and 10 million particles in the Navier-Stokes case and $512^3$ collocation points and 1 million particles in the MHD case. In the hydrodynamics case our findings compare with recent experiments from Mordant et al. [1] and Xu et al. [2]. They differ from the simulations of Biferale et al. [3] due to differences of the ranges choosen for evaluating the structure functions. In Navier-Stokes turbulence intermittency is stronger than predicted by a multifractal approach of [3] whereas in MHD turbulence the predictions from the multifractal approach are more intermittent than observed in our simulations. In addition, our simulations reveal that Lagrangian Navier-Stokes turbulence is more intermittent than MHD turbulence, whereas the situation is reversed in the Eulerian case. Those findings can not consistently be described by the multifractal modeling. The crucial point is that the geometry of the dissipative structures have different implications for Lagrangian and Eulerian intermittency. Application of the multifractal approach for the modeling of the acceleration PDFs works well for the Navier-Stokes case but in the MHD case just the tails are well described.Comment: to appear in J. Plasma Phy

Rheology of gelling polymers in the Zimm model

In order to study rheological properties of gelling systems in dilute solution, we investigate the viscosity and the normal stresses in the Zimm model for randomly crosslinked monomers. The distribution of cluster topologies and sizes is assumed to be given either by Erd\H os-R\'enyi random graphs or three-dimensional bond percolation. Within this model the critical behaviour of the viscosity and of the first normal stress coefficient is determined by the power-law scaling of their averages over clusters of a given size $n$ with $n$. We investigate these Mark--Houwink like scaling relations numerically and conclude that the scaling exponents are independent of the hydrodynamic interaction strength. The numerically determined exponents agree well with experimental data for branched polymers. However, we show that this traditional model of polymer physics is not able to yield a critical divergence at the gel point of the viscosity for a polydisperse dilute solution of gelation clusters. A generally accepted scaling relation for the Zimm exponent of the viscosity is thereby disproved.Comment: 9 pages, 2 figure

Collusion in Experimental Bertrand Duopolies with Convex Costs:The Role of Information and Cost Asymmetry

We report the results of a series of experimental Bertrand duopolies where firms have convex costs. Theoretically these duopolies are characterized by a multiplicity of Nash equilibria. Using a 2x2 design, we analyze price choices in symmetric and asymmetric markets under 2 information conditions: complete versus incomplete information about profits. We find that information has no effect in symmetric markets with respect to market prices and the time it takes for markets to stabilize. However, in asymmetric markets, complete information leads to higher average market prices and quicker convergence of price choices.

Invertible Dirac operators and handle attachments on manifolds with boundary

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.Comment: Accepted for publication in Journal of Topology and Analysi

Faraday waves on a viscoelastic liquid

We investigate Faraday waves on a viscoelastic liquid. Onset measurements and a nonlinear phase diagram for the selected patterns are presented. By virtue of the elasticity of the material a surface resonance synchronous to the external drive competes with the usual subharmonic Faraday instability. Close to the bicriticality the nonlinear wave interaction gives rise to a variety of novel surface states: Localised patches of hexagons, hexagonal superlattices, coexistence of hexagons and lines. Theoretical stability calculations and qualitative resonance arguments support the experimental observations.Comment: 4 pages, 4figure