Determining Gravitational Masses of Galaxy Clusters With (1)Virial Equilibrium And (2)Arc-like Images

The mass derived from gravitational lensing reflects the total mass contained in the lensing system, independent of the specific matter contents and states. A comparison of the dynamical masses from hydrostatic equilibrium with the gravitational masses from arc-like images of background galaxies is made for four clusters of galaxies at intermediate redshits. It is found that virial analysis has underestimated the total cluster masses (from lensing) by a factor of $3\sim6$ within a radius of $\sim0.3$ Mpc $h_{50}^{-1}$ around the cluster centers, indicating that clusters of galaxies might not be regarded as the well relaxed virialized systems. The increase of the total cluster masses obtained from lensing leads to the decrease of the baryon fractions of clusters of galaxies, which provides a crue for solving the ``$\Omega_0$ disprepancy puzzle" in cosmology.Comment: 11 pages plus 1 Table. LATEX style, submitted to ApJ, BAO-BGGC-940

Gravitational Microlensing by the MACHOs of the LMC

The expected microlensing events of the LMC by the MACHOs of the LMC itself are calculated and compared with analogue events by objects in the Galactic halo. The LMC matter distribution is modelled by a spherical halo and an exponential disk while a face-on exponential disk is used for the stellar distribution of the LMC. Among the microlensing events discovered by the MACHOs and EROS projects, a fraction of $22\%$ could be caused by the lenses near the center of the LMC or $13\%$ from lenses at $5^o$ from the LMC center. Therefore, any statistical study of these microlensing events must take the LMC lenses into account.Comment: 12 pages, 6 figures (not included) by fax, ApJ submitted, DAEC-OPM-9

Optimal Design of Multiple Description Lattice Vector Quantizers

In the design of multiple description lattice vector quantizers (MDLVQ), index assignment plays a critical role. In addition, one also needs to choose the Voronoi cell size of the central lattice v, the sublattice index N, and the number of side descriptions K to minimize the expected MDLVQ distortion, given the total entropy rate of all side descriptions Rt and description loss probability p. In this paper we propose a linear-time MDLVQ index assignment algorithm for any K >= 2 balanced descriptions in any dimensions, based on a new construction of so-called K-fraction lattice. The algorithm is greedy in nature but is proven to be asymptotically (N -> infinity) optimal for any K >= 2 balanced descriptions in any dimensions, given Rt and p. The result is stronger when K = 2: the optimality holds for finite N as well, under some mild conditions. For K > 2, a local adjustment algorithm is developed to augment the greedy index assignment, and conjectured to be optimal for finite N. Our algorithmic study also leads to better understanding of v, N and K in optimal MDLVQ design. For K = 2 we derive, for the first time, a non-asymptotical closed form expression of the expected distortion of optimal MDLVQ in p, Rt, N. For K > 2, we tighten the current asymptotic formula of the expected distortion, relating the optimal values of N and K to p and Rt more precisely.Comment: Submitted to IEEE Trans. on Information Theory, Sep 2006 (30 pages, 7 figures