The expected degree of minimal spanning forests

We give a lower bound on the expected degree of the free minimal spanning forest of a vertex transitive graph in terms of its spectral radius. This result answers a question of Lyons-Peres-Schramm and simplifies the Gaboriau-Lyons proof of the measurable-group-theoretic solution to von Neumann’s problem.In the second part we study a relative version of the free minimal spanning forest. As a consequence of this study we can show that non-torsion unitarizable groups have fixed price one.     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

A convergent overlapping domain decomposition method for total variation minimization

In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.     

Complete Duality for Quasiconvex and Convex Set-Valued Functions

This paper provides a unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set-valued functions.