Metric entropy of convex hulls in type spaces--The critical case

Given a precompact subset of a type Banach space , where , we prove that for every and all

holds, where is the absolutely convex hull of and denotes the dyadic entropy number. With this inequality we show in particular that for given and with for all the inequality holds true for all . We also prove that this estimate is asymptotically optimal whenever has no better type than . For this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.

     

A pair of stream functions for three-dimensional vortex flows

Introducing a vector potential, that is based on a pair of stream functions, and a velocity potential, antisymmetric equations for the stream functions are derived with the help of a variational principle. It is found that the equations are in a suitable form to investigate flows with helical symmetry, and, for example, to connect upstream axisymmetric flows with downstream helical flows. The special case of a transition from an upstream solid-body vortex to a downstream helical flow is investigated in detail. Furthermore, the stream-function equations are particularly useful to investigate general small-amplitude inertia waves on vortex flows. Time-dependent helical flows that are time-independent in a suitably rotating frame of reference can also be discussed with the proposed method.     

Constraints on the asymptotic behaviour of total cross sections

We derive constraints on the asymptotic behaviour of total cross sections which follow from dispersion relations and measured real parts of the forward scattering amplitudes. For πN and pp scattering, these constraints are calculated using recent results from FNAL and Serpukhov. The relation to other methods is discussed.     

On the algebra of test functions for field operators

It is shown that every continuous linear functional on the field algebra can be defined by a vector in the Hilbert space of some representation of the algebra. The functionals which can be written as a difference of positive ones are characterized. By an example it is shown that a positive functional on a subalgebra does not always have an extension to a positive functional on the whole algebra.     

Generalized parton distributions from nucleon form factor data

We present a simple empirical parameterization of the x- and t-dependence of generalized parton distributions at zero skewness, using forward parton distributions as input. A fit to experimental data for the Dirac, Pauli and axial form factors of the nucleon allows us to discuss quantitatively the interplay between longitudinal and transverse partonic degrees of freedom in the nucleon (nucleon tomography). In particular we obtain the transverse distribution of valence quarks at given momentum fraction x. We calculate various moments of the distributions, including the form factors that appear in the handbag approximation to wide-angle Compton scattering. This allows us to estimate the minimal momentum transfer required for reliable predictions in that approach to be around . We also evaluate the valence contributions to the energy-momentum form factors entering Jis sum rule.Received: 16 September 2004, Revised: 15 October 2004, Published online: 17 December 2004     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

Unstable extremal surfaces of the “Shiffman-functional”

We establish a new way to Shiffmans result [16], which yields a sufficient condition for the existence of unstable extremal surfaces of a parametric functional with a dominant area term via the mountain-pass-principle. In this paper we consider H ^1,2-surfaces spanned by a piecewise linear closed Jordan-curve in ( ). This enables us to generalize Courants ideas for the area functional to the more complex functional Shiffman was considering.Received: 3 October 2003, Accepted: 18 November 2003, Published online: 2 April 2004Mathematics Subject Classification (2000): 01X23R. Jakob: It is a pleasure for the author to thank Prof. Dr. S. Hildebrandt and Priv.-Doz. Dr. H. von der Mosel for their support during the preparation of this paper.