The vanishing of Tor_1^R(R^+,k) implies that R is regular

Let (R,m,k) be an excellent local ring of positive prime characteristic. We show that if Tor_1^R(R^+,k) = 0 then R is regular. This improves a result of Schoutens, in which the additional hypothesis that R was an isolated singularity was required for the proof.Comment: 3 pages, to appear in Proceedings of the AM

Effects of architectural issues on a km3 scale detector

Simulation results showing the comparison between the performance of different km3 detector geometries are reported. Effective neutrino areas and angular resolutions are reported for three different geometries based on NEMO-towers and strings. The results show that the NEMO-tower based detector has the best performance concerning both the effective area and the angular resolution isotropyComment: to be published on VVVNT2 proceedings (Catania, Italy, November 8-11, 2005

Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices

This paper's aim is to present recent combinatorial considerations on r-Dyck paths, r-Parking functions, and the r-Tamari lattices. Giving a better understanding of the combinatorics of these objects has become important in view of their (conjectural) role in the description of the graded character of the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the symmetric group.Comment: 36 pages, 12 figure

The rr-Bell numbers

The notion of generalized Bell numbers has appeared in several works but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers which generalize the similar properties of the Bell numbers. Most of these results seem to be new. It turns out that in a paper of Whitehead these numbers appeared in a very different context. In addition, we introduce the so-called $r$-Bell polynomials