Ein geometrisches Überdeckungsproblem

Ohne Zusammenfassung Alexander Ostrowski zum 60. Geburtstag gewidmet     

Atom-surface elastic scattering with additive potential

An additive corrugated potential with linear repulsion and long range attractive well is proposed for atom-surface scattering. The computational procedure yielding the scattering probabilities (essentially linear algebra) proves to be much simpler than with other potentials. For a given shape of the corrugation function and for high values of the steepness parameter one obtains results close to those of the hard corrugated wall model, while an important enhancement of the specular intensity appears, in particular at large angles of incidence, when the steepness parameter is small.     

über angeordnete projektive Ebenen

Ohne Zusammenfassung Reinhold Baer zu seinem sechzigsten Geburtstag in alter unveränderter Freundschaft gewidmet     

Deformations of 2k-Einstein structures

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.     

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E₂. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.     

Networks of polynomial pieces with application to the analysis of point clouds and images

We consider Hölder smoothness classes of surfaces for which we construct piecewise polynomial approximation networks, which are graphs with polynomial pieces as nodes and edges between polynomial pieces that are in ‘good continuation’ of each other. Little known to the community, a similar construction was used by Kolmogorov and Tikhomirov in their proof of their celebrated entropy results for Hölder classes.We show how to use such networks in the context of detecting geometric objects buried in noise to approximate the scan statistic, yielding an optimization problem akin to the Traveling Salesman. In the same context, we describe an alternative approach based on computing the longest path in the network after appropriate thresholding.For the special case of curves, we also formalize the notion of ‘good continuation’ between beamlets in any dimension, obtaining more economical piecewise linear approximation networks for curves.We include some numerical experiments illustrating the use of the beamlet network in characterizing the filamentarity content of 3D data sets, and show that even a rudimentary notion of good continuity may bring substantial improvement.     

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k‐Einstein (in the sense that their 2k‐Ricci tensor is constant) or have constant 2k‐Gauss‐Bonnet curvature. The results hold for a family of manifolds containing all non‐flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.