The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with $R$.Comment: A Remark was added at the end of the paper. To appear in Comm. Algebr

Nonlinear r-Modes in Neutron Stars: Instability of an unstable mode

We study the dynamical evolution of a large amplitude r-mode by numerical simulations. R-modes in neutron stars are unstable growing modes, driven by gravitational radiation reaction. In these simulations, r-modes of amplitude unity or above are destroyed by a catastrophic decay: A large amplitude r-mode gradually leaks energy into other fluid modes, which in turn act nonlinearly with the r-mode, leading to the onset of the rapid decay. As a result the r-mode suddenly breaks down into a differentially rotating configuration. The catastrophic decay does not appear to be related to shock waves at the star's surface. The limit it imposes on the r-mode amplitude is significantly smaller than that suggested by previous fully nonlinear numerical simulations.Comment: Published in Phys. Rev. D Rapid Comm. 66, 041303(R) (2002

On a Penrose Inequality with Charge

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfies the inequality: m - 1/2(R + Q^2/R) < 0, where m is the total mass, R=sqrt(A/4) is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality to charged black holes.Comment: Minor revision: some typos; author's address updated; bibliographical reference added; journal information: to appear in Comm. Math. Phy