Galaxy population properties in the rich clusters MS0839.8+2938, 1224.7+2007 and 1231.3+1542

This paper discusses the galaxy populations of three rich clusters, with redshift 0.19 (0839+29), 0.24 (1231+15), and 0.32 (1224+20), from the database of the CNOC1 consortium. The data consist of spectra of 52 cluster members for 0839+29, 30 members for 1224+15, and 82 members for 1231+15, and there are comparable numbers of field galaxy spectra. 0839+29 is compact with no strong radial gradients, and possibly dusty. 1224+20 is isolated in redshift, has low velocity dispersion around the cD galaxy, and low 4000A break. 1231+15 is asymmetrical and we discuss the possibility that it may be a recent merger of two old clusters. We find few galaxies in 0839+29 and 1231+15 with ongoing or recently truncated star-formation.Comment: 16 pages and 20 diagrams, to appear in A

Virtual moduli cycles and Gromov-Witten invariants of noncompact symplectic manifolds

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).Comment: Final versio