Pismak and Shukhobodskaia recently proposed a model of a Dirac particle interacting with a material plane. Depending on the interaction constants, the Dirac particle can have bound states moving parallel to the plane and decaying exponentially perpendicular to the plane. Moreover, the waves on the two sides of the plane can move with different velocities. Although states with sharp momenta parallel to the plane do not have currents perpendicular to it, it turns out that the wave packets considered here do have a weak current component in this direction, which keeps the wave packets above and below the plane together.

A short historical account of the curves related to the two-dimensional floating bodies of equilibrium and the bicycle problem is given. Bor, Levi, Perline and Tabachnikov found, quite a number had already been described as Elastica by Bernoulli and Euler and as Elastica under Pressure or Buckled Rings by Levy and Halphen. Auerbach already realized that Zindler had described curves for the floating bodies problem. An even larger class of curves solves the bicycle problem. The subsequent sections deal with some supplemental details: Several derivations of the equations for the elastica and elastica under pressure are given. Properties of Zindler curves and some work on the problem of floating bodies of equilibrium by other mathematicians are considered. Special cases of elastica under pressure reduce to algebraic curves, as shown by Greenhill. Since most of the curves considered here are bicycle curves, a few remarks concerning them are added.

Leo Kadanoff has worked in many fields of statistical mechanics. His contributions had an enormous impact. This holds in particular for critical phenomena, where he explained Widom’s homogeneity laws by means of block-spin transformations and laid the basis for Wilson’s renormalization group equation. I had the pleasure to work in his group for 1 year. A short historical account is given.

This paper rests to a large extend on a paper I wrote some time ago on 'Duality in generalized Ising models and phase transitions without local order parameter'. It deals with Ising models with interactions containing products of more than two spins. In contrast to the old paper I will first give examples before I come to the general statements. Of particular interest is a gauge invariant Ising model in four dimensions. It has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson-loop yielding an interaction increasing proportional to the distance and thus corresponding to quark-confinement. The other phase yields a perimeter law allowing for a quark-gluon plasma.

We investigated the appearance of Chern-Simons terms in electrodynamics at the surface or interface of materials. The requirement of locality, gauge invariance, and renormalizability in this model is imposed. Scattering and reflection of electromagnetic waves in three different homogeneous layers of media is determined. Snell's law is preserved. However, the transmission and reflection coefficient depend on the strength of the Chern-Simons interaction (connected with Hall conductance), and parallel and perpendicular components are mixed.

Kenneth Wilson had an enormous impact on field theory, in particular on the renormalization group and critical phenomena, and on QCD. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and QCD, and flow equations and similarity renormalization.

Kenneth Wilson had an enormous impact on the renormalization group and field theories in general. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and confinement, and flow equations and similarity renormalization.

The density of states of disordered systems in the Wigner–Dyson classes approaches some finite non-zero value at the mobility edge, whereas the density of states in systems of the chiral and Bogolubov-de Gennes classes shows a divergent or vanishing behavior in the band centre. Such types of behavior were classified as homogeneous and inhomogeneous fixed point ensembles within a real-space renormalization group approach. For the latter ensembles, the scaling law µ = dν-1 was derived for the power laws of the density of states ρ ∝ |E|µ and of the localization length ξ ∝ |E|-ν. This prediction from 1976 is checked against explicit results obtained meanwhile.

Bodies of density one half (of the fluid in which they are immersed) that can float in all orientations are investigated. It is shown that expansions starting from and deforming the (hyper)sphere are possible in arbitrary dimensions and allow for a large manifold of solutions: One may either (i) expand r(n)+r(-n) in powers of a given difference r(u)-r(-u), (r(n) denoting the distance from the origin in direction n). Or (ii) the envelope of the water planes (for fixed body and varying direction of gravitation) may be given. Equivalently r(n) can be expanded in powers of the distance h(u) of the water planes from the origin perpendicular to u.

The critical behaviour of the O(n)-symmetric model with two n-vector fields is studied within the field-theoretical renormalization group approach in a D=4-2 epsilon expansion. Depending on the coupling constants the beta-functions, fixed points and critical exponents are calculated up to the one- and two-loop order, resp. (eta in two- and three-loop order). Continuous lines of fixed points and O(n)*O(2) invariant discrete solutions were found. Apart from already known fixed points two new ones were found. One agrees in one-loop order with a known fixed point, but differs from it in two-loop order.

Explicit solutions of the two-dimensional floating body problem (bodies that can float in all positions) for relative density ρ 6= 1 2 and of the tire track problem (tire tracks of a bicycle, which do not allow to determine, which way the bicycle went) are given, which differ from circles. Starting point is the differential equation given by the author in [8, 9].

Explicit solutions of the two-dimensional floating body problem (bodies that can float in all positions) for relative density ρ 6= 1 2 and of the tire track problem (tire tracks of a bicycle, which do not allow to determine, which way the bicycle went) are given, which differ from circles. Starting point is the differential equation given by the author in [12, 8]. The curves are also trajectories of charges in a perpendicular parabolic magnetic field.

Hier werden einige Formeln und Ergebnisse in Ergänzung zu http://www.tphys.uni-heidelberg.de/∼wegner/Fl2mvs/Filme.html gebracht. Genaueres findet man in [1]. Im ersten Abschnitt gebe ich einige grundlegende Formeln an, im zweiten die Formeln für den Grenzfall, in dem sich die Kurve durch ’elementare’ Funktionen wie Exponentialund Winkelfunktionen darstellen lässt, während man im allgemeinen Fall doppelt-periodische Funktionen benötigt.

A long cylindrical body of circular cross-section and homogeneous density may float in all orientations around the cylinder axis. It is shown that there are also bodies of non-circular cross-sections which may float in any direction. Apart from those found by Auerbach for ρ = 1/2 there are one-parameter families of cross-sections for ρ 6= 1/2 which have a p-fold rotation axis. For given p they exist for p− 2 densities ρ. There are strong indications, that for all p− 2 densities one has the same family of cross-sections.

Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density rho=1/2 there are solutions, if holes in the body are allowed. For rho different from 1/2 the body is deformed from a sphere. A set of nonlinear shape-equations determines the shape in lowest order in the deformation. It is shown that a large number of solutions exists. An expansion scheme is given, which allows a formal expansion in the deformation to arbitrary order under the assumption that apart from x=0,+1,-1 there is no x, which obeys P_{p,2}(x)=0 for two different integer ps, where P are Legendre functions.

Explicit solutions of the two-dimensional floating body problem (bodies that can float in all positions) for relative density different from 1/2 and of the tire track problem (tire tracks of a bicycle, which do not allow to determine, which way the bicycle went) are given, which differ from circles. Starting point is the differential equation given by the author in archive physics/0205059 and Studies in Appl. Math. 111 (2003) 167-183. The curves are also trajectories of charges in a perpendicular magnetic field.

First we give an introduction to the method of diagonalizing or block-diagonalizing continuously a Hamiltonian and explain how this procedure can be used to analyse the two-dimensional Hubbard model. Then we give a short survey on applications of this flow equation on other models. Finally we outline, how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering, not of the Hamiltonian.

Explicit solutions of the two-dimensional floating body problem (bodies that can float in all positions) for relative density rho different from 1/2 and of the tire track problem (tire tracks of a bicycle, which do not allow to determine, which way the bicycle went) are given, which differ from circles. Starting point is the differential equation given in archive physics/0205059 and Studies in Appl. Math. 111 (2003) 167-183.