Divergence in 3-manifold groups

The divergence of the fundamental group of compact irreducible 3-manifolds satisfying Thurston's geometrization conjecture is calculated. For every closed Haken 3-manifold group, the divergence is either linear, quadratic or exponential, where quadratic divergence occurs precisely for graph manifolds and exponential divergence occurs when a geometric piece has hyperbolic geometry. An example is given of a closed 3-manifoldN with a Riemannian metric of nonpositive curvature such that the divergence is quadratic and such that there are two geodesic rays in the universal coverN whose divergence is precisely quadratic, settling in the negative a question of Gromov's.Partially supported by NSF grant DMS-9200433.     

Filling Length in Finitely Presentable Groups

We study the filling length function for a finite presentation of a group , and interpret this function as an optimal bound on the length of the boundary loop as a van Kampen diagram is collapsed to the basepoint using a combinatorial notion of a null-homotopy. We prove that filling length is well behaved under change of presentation of . We look at 'AD-pairs' (f,g) for a finite presentation : that is, an isoperimetric function f and an isodiametric function g that can be realised simultaneously. We prove that the filling length admits a bound of the form [g+1][log (f+1)+1] whenever (f,g) is an AD-pair for . Further we show that (up to multiplicative constants) if is an isoperimetric function ( ) for a finite presentation then ( ) is an AD-pair. Also we prove that for all finite presentations filling length is bounded by an exponential of an isodiametric function.Partially supported by NSF grant DMS-9800158Supported by EPSRC Award No. 98001683 and Corpus Christi College, Oxford.     

Tensor Lagrangians,Lagrangians Equivalent to the Hamilton-Jacobi Equation and Relativistic Dynamics

We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell’s equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.     

Flow separation in laminar natural convection

Theoretical and experimental studies of flow separation in laminal natural convection are presented. Since classical boundary-layer theory cannot determine separation on curved walls in natural convection, two extensions of the classical boundary-layer theory are discussed: boundary-layer theory of higher order and double-deck theory. Both theories are applied to experiments on a vertical flat plate with humps.     

Nichtisotherme ebene Freistrahlen unter Schwerkrafteinfluß

Higher-order boundary layer theory is used to study the behaviour of nonisothermal laminar and turbulent free jet flows. In addition to the Prandtl boundary layer equations, an equation is used to describe the equilibrium of forces normal to the flow direction. This equilibrium exists between the buoyancy forces caused by gravity and the centrifugal forces resulting from the curvature in the flow. The proper selection of reference values permits the characteristics of the jet flow to be expressed as universal functions in which only the initial jet orientation and the Prandtl number in the case of laminar flow are input parameters. When the volume flow is given in addition to the momentum and thermal energy, the characteristic parameter are the Archimedes number for turbulent flow and the modified Archimedes number for laminar flow. The jet flow is calculated using an integral method in which the eddy viscosity and the turbulent Prandtl number are given as functions of the local Archimedes number. Comparison of experimental data from the literature and from our laboratory on nonisothermal free jets with the theoretical results, show satisfactory agreement. The universal diagrams given in the paper are valid forall plane laminar (Pr=0.7) and turbulent nonisothermal jets.     

Tests of charge symmetry in few-body systems

Tests of fundamental symmetries offer the promise of possibly unique insights into the nature of the strong interaction. We review the current status of selected experimental and theoretical investigations of charge symmetry in fewbody systems.     

Incomplete Delta Functions

By applying projection operators to state vectors of coordinates, we obtain subspaces in which these states are no longer normalized according to Dirac's delta function but normalized according to what we call incomplete delta functions. We show that this class of functions satisfy identities similar to those satisfied by the Dirac delta function. The incomplete delta functions may be employed advantageously in projected subspaces and in the link between functions defined on the whole space and the projected subspace. We apply a similar procedure to finite-dimensional vector spaces for which we define incomplete Kronecker deltas. Dispersion relations for the momenta are obtained and sums over poles are defined and obtained with the aid of differences of incomplete delta functions.     

Quantum Theory of Massless Particles in Stationary Axially Symmetric Spacetimes

Quantum equations for massless particles of any spin are considered in stationary uncharged axially symmetric spacetimes. It is demonstrated that up to a normalization function, the angular wave function does not depend on the metric and practically is the same as in the Minkowskian case. The radial wave functions satisfy second order nonhomogeneous differential equations with three nonhomogeneous terms, which depend in a unique way on time and space curvatures. In agreement with the principle of equivalence, these terms vanish locally, and the radial equations reduce to the same homogeneous equations as in Minkowski spacetime.     

Small cancellation theory and automatic groups: Part II

Oblatum X-1989 &; 6-II-1991     

Turbulent flow around a rotating cylinder in a quiescent fluid

 Hot-wire measurements have been carried out in the turbulent flow around a rotating circular cylinder in still air for Reynolds numbers Re=∣U _w ∣D/ν=1.5×10⁴ to 10⁵. The experimental results confirm the analysis derived by asymptotic theory for high Reynolds numbers. Two different ways of deriving the friction law from the experiments (via shear stress and via velocity distribution) resulted practically in the same law. It is shown, that in spite of the curvature of the streamlines the universal logarithmic velocity distribution is still valid near the wall. Received: 8 August 1996/Accepted: 24 April 1998