In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one e...

“From independence to dependence”: the head of Chapter IX in M. Lokve’s book [ 181 must be a program of any attempt to build a limit theory extending the classical one for sums of independent random variables. Lo&e himself suggested replacing dependent summands by their independent copies, but this could not bring much success. The next step in complication consists in dividing the sum into alm...

Let k be a field, and let R = k[x1, x2, x3]. Given a Hilbert function H for a cyclic module over R, we give an algorithm to produce a stable ideal I such that R/I has Hilbert function H and uniquely minimal graded Betti numbers among all R/J with the same Hilbert function, where J is another stable ideal in R. We also show that such an algorithm is impossible in more variables and disprove a re...