Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.Comment: Asian Journal of Mathematics (to appear), Special issue on the occasion of Prof N. Mok's 60th birthda

Trans-Relativistic Supernovae, Circumstellar Gamma-Ray Bursts, and Supernova 1998bw

Supernova (SN) 1998bw and gamma-ray burst (GRB) 980425 offer the first direct evidence that supernovae are the progenitors of some GRBs. However, this burst was unusually dim, smooth and soft compared to other bursts with known afterglows. Whether it should be considered a prototype for cosmological GRBs depends largely on whether the supernova explosion and burst were asymmetrical or can be modeled as spherical. We address this question by treating the acceleration of the supernova shock in the outermost layers of the stellar envelope, the transition to relativistic flow, and the subsequent expansion (and further acceleration) of the ejecta into the surrounding medium. We find that GRB 980425 could plausibly have been produced by a collision between the relativistic ejecta from SN 1998bw and the star's pre-supernova wind; the model requires no significant asymmetry. This event therefore belongs to a dim subclass of GRBs and is not a prototype for jet-like cosmological GRBs.Comment: 5 pages, 2 figures, to appear in Gamma 2001, eds. S. Ritz, N. Gehrels, and C. Shrade

Astrochemical confirmation of the rapid evolution of massive YSOs and explanation for the inferred ages of hot cores

Aims. To understand the roles of infall and protostellar evolution on the envelopes of massive young stellar objects (YSOs). Methods. The chemical evolution of gas and dust is traced, including infall and realistic source evolution. The temperatures are determined self-consistently. Both ad/desorption of ices using recent laboratory temperature-programmed-desorption measurements are included. Results. The observed water abundance jump near 100 K is reproduced by an evaporation front which moves outward as the luminosity increases. Ion-molecule reactions produce water below 100 K. The age of the source is constrained to t \~ 8 +/- 4 x 10^4 yrs since YSO formation. It is shown that the chemical age-dating of hot cores at ~ few x 10^3 - 10^4 yr and the disappearance of hot cores on a timescale of ~ 10^5 yr is a natural consequence of infall in a dynamic envelope and protostellar evolution. Dynamical structures of ~ 350AU such as disks should contain most of the complex second generation species. The assumed order of desorption kinetics does not affect these results.Comment: Accepted by A&A Letters; 4 pages, 5 figure

"Almost stable" matchings in the Roommates problem

An instance of the classical Stable Roommates problem (SR) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an SR instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n^{\frac{1}{2}-\varepsilon}, for any \varepsilon>0, unless P=NP. If the preference lists contain ties, we improve this result to n^{1-\varepsilon}. Also, we show that, given an integer K and an SR instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an SR instance I