Mapping class group representations and generalized Verlinde formula

Unitary representations of centrally extended mapping class groups \tilde M_{g,1}, g\geq 1 are given in terms of a rational Hopf algebra H, and a related generalization of the Verlinde formula is presented. Formulae expressing the traces of \mcg elements in terms of the fusion rules, quantum dimensions and statistics phases are proposed

Modular differential equations for characters of RCFT

We discuss methods, based on the theory of vector-valued modular forms, to determine all modular differential equations satisfied by the conformal characters of RCFT; these modular equations are related to the null vector relations of the operator algebra. Besides describing effective algorithmic procedures, we illustrate our methods on an explicit example.Comment: 13 page

Tracer Dispersion in a Self-Organized Critical System

We have studied experimentally transport properties in a slowly driven granular system which recently was shown to display self-organized criticality [Frette {\em et al., Nature} {\bf 379}, 49 (1996)]. Tracer particles were added to a pile and their transit times measured. The distribution of transit times is a constant with a crossover to a decaying power law. The average transport velocity decreases with system size. This is due to an increase in the active zone depth with system size. The relaxation processes generate coherently moving regions of grains mixed with convection. This picture is supported by considering transport in a $1D$ cellular automaton modeling the experiment.Comment: 4 pages, RevTex, 1 Encapsulated PostScript and 4 PostScript available upon request, Submitted to Phys. Rev. Let

Random walk through fractal environments

We analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e. of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D of the fractal is less than 2, there is though always a finite rate of unaffected escape. Random walks through fractal sets with D less or equal 2 can thus be considered as defective Levy walks. The distribution of jump increments for D > 2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk-increments. It is shown that the particles undergo anomalous, enhanced diffusion for D_F < 2, the diffusion is dominated by the finite escape rate. Diffusion for D_F > 2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality (SOC) models give rise to enhanced diffusion. The analytical results are illustrated by Monte-Carlo simulations.Comment: 22 pages, 16 figures; in press at Phys. Rev. E, 200

Traffic and Related Self-Driven Many-Particle Systems

Since the subject of traffic dynamics has captured the interest of physicists, many astonishing effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by so-called ``phantom traffic jams'', although they all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction of the traffic volume cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize in lanes, while similar systems are ``freezing by heating''? Why do self-organizing systems tend to reach an optimal state? Why do panicking pedestrians produce dangerous deadlocks? All these questions have been answered by applying and extending methods from statistical physics and non-linear dynamics to self-driven many-particle systems. This review article on traffic introduces (i) empirically data, facts, and observations, (ii) the main approaches to pedestrian, highway, and city traffic, (iii) microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts like a general modelling framework for self-driven many-particle systems, including spin systems. Subjects such as the optimization of traffic flows and relations to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are discussed as well.Comment: A shortened version of this article will appear in Reviews of Modern Physics, an extended one as a book. The 63 figures were omitted because of storage capacity. For related work see http://www.helbing.org

Statistical test of throwing events on the rotating Earth

In a recent paper, Mizera and Horváth computed the effects of environmental factors on shot put and hammer throw ranges [J. Biomech. 35, (2002) 785–796]. They found that the geographic location (latitude and altitude) influences throwing distances as strongly as meteorological conditions (wind and air density). Considering the small differences in record-breaking results, they proposed that normalization to a reference stadium should be introduced. Here we attempt to detect possible correlations between geographic location and throwing ranges by using all-time best result lists. Unfortunately the separation of the effects of different environmental factors is not possible, simply because they are not documented. Our tests failed to find the expected correlation. We conclude that the variance of human factors seems to dominate, thus any correction of measured results is probably unnecessary