In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.

General expressions for the anomalies appearing in pure W 3 gravity are found by requiring that they satisfy a modified version of the Wess-Zumino consistency conditions in which the Ward identities are treated as nonvanishing quantities.

A cyclic symmetry among the left-handed doublets of the three families is proposed. This symmetry can naturally result in a realistic hierarchical pattern of the fermion masses within the framework of supersymmetry with nonvanishing sneutrino vacuum expectation values.

The area of the T 2 group orbits takes on all positive values in the maximal Cauchy development of T 2 symmetric initial data for the Einstein-Vlasov equations on T 3 with a nonvanishing number of equal mass particles.

In a seminal paper, Erdős and Rényi identified a sharp threshold for connectivity of the random graph G(n, p). In particular, they showed that if p log n/n then G(n, p) is almost always connected, and if p log n/n then G(n, p) is almost always disconnected, as n→∞. The clique complex X (H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the z...