Some multidimensional generalizations of the Fokker-Planck equation used by R. Friedrich and J. Peinke for the description of a turbulent cascade as a stochastic process of Markovian type, are considered. The exact solutions of the Cauchy problems for these equations are found with the operator methods.

A. A. Donkov A. D. Donkov E. I. Grancharova

We give explicit formulae for the generators of H^2(Hol(\Sigma_r,{\cal F}_{\lambda}(\Sigma_r)) in terms of affine and projective connections. This is done using the cocycles of V. Ovsienko and C. Roger for the case of the circle and globalizing them to an open Riemann surface \Sigma_r.

Friedrich Wagemann

We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant $J$-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for $J$-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with $L^p$-potentials. Known bounds on the non-real eigenvalues of such operators are improved.

Friedrich Philipp

Comment on "Indispensable Finite Time Correlations for Fokker-Planck Equations from Time Series Data"

R. Friedrich Ch. Renner M. Siefert J. Peinke

Within the framework of metric-affine gravity (MAG, metric and an independent linear connection constitute spacetime), we find, for a specific gravitational Lagrangian and by using {\it prolongation} techniques, a stationary axially symmetric exact solution of the vacuum field equations. This black hole solution embodies a Kerr-deSitter metric and the post-Riemannian structures of torsion and nonmetricity. The solution is characterized by mass, angular momentum, and shear charge, the latter of which is a measure for violating Lorentz invariance.

Peter Baekler Friedrich W. Hehl

We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.

Friedrich Knop

We classify subalgebras of a ring of differential operators which are big in the sense that the extension of associated graded rings is finite. We show that these subalgebras correspond, up to automorphisms, to uniformly ramified finite morphisms. This generalizes a theorem of Levasseur-Stafford on the generators of the invariants of a Weyl algebra under a finite group.

Friedrich Knop

J. Tuma proved an interesting "congruence amalgamation" result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: --A.P. Huhn proved that every distributive algebraic lattice $D$ with at most $\aleph\_1$ compact elements can be represented as the congruence lattice of a lattice $L$. We show that $L$ can be constructed as a locally finite relatively complemented lattice with zero. --We find a large class of lattices, the $\omega$-congruence-finite lattices, that contains all locally finite countable lattices, in which every lattice has a relatively complemented congruence-preserving extension.

George Grätzer Harry Lakser Friedrich Wehrung

In this work we present a formal solution of the extended version of the Friedrichs Model. The Hamiltonian consists of discrete and continuum bosonic states, which are coupled to fermions. The simultaneous treatment of the couplings of the fermions with the discrete and continuous sectors of the bosonic degrees of freedom leads to a system of coupled equations, whose solutions are found by applying standard methods of representation of bound and resonant states.

O. Civitarese M. Gadella G. P. Pronko

A general method is proposed which allows one to estimate drift and diffusion coefficients of a stochastic process governed by a Langevin equation. It extends a previously devised approach [R. Friedrich et al., Physics Letters A 271, 217 (2000)], which requires sufficiently high sampling rates. The analysis is based on an iterative procedure minimizing the Kullback-Leibler distance between measured and estimated two time joint probability distributions of the process.

D. Kleinhans R. Friedrich A. Nawroth J. Peinke