Solution of the symmetric eigenproblem AX=lambda BX by delayed division

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity

Geometry and observables in (2+1)-gravity

We review the geometrical properties of vacuum spacetimes in (2+1)-gravity with vanishing cosmological constant. We explain how these spacetimes are characterised as quotients of their universal cover by holonomies. We explain how this description can be used to clarify the geometrical interpretation of the fundamental physical variables of the theory, holonomies and Wilson loops. In particular, we discuss the role of Wilson loop observables as the generators of the two fundamental transformations that change the geometry of (2+1)-spacetimes, grafting and earthquake. We explain how these variables can be determined from realistic measurements by an observer in the spacetime.Comment: Talk given at 2nd School and Workshop on Quantum Gravity and Quantum Geometry (Corfu, September 13-20 2009); 10 pages, 13 eps figure

Reconstructing the global topology of the universe from the cosmic microwave background

If the universe is multiply-connected and sufficiently small, then the last scattering surface wraps around the universe and intersects itself. Each circle of intersection appears as two distinct circles on the microwave sky. The present article shows how to use the matched circles to explicitly reconstruct the global topology of space.Comment: 6 pages, 2 figures, IOP format. To be published in the proceedings of the Cleveland Cosmology and Topology Workshop 17-19 Oct 1997. Submitted to Class. Quant. Gra

On the Nature of Andromeda IV

Lying at a projected distance of 40' or 9 kpc from the centre of M31, Andromeda IV is an enigmatic object first discovered during van den Bergh's search for dwarf spheroidal companions to M31. Being bluer, more compact and higher surface brightness than other known dwarf spheroidals, it has been suggested that And IV is either a relatively old `star cloud' in the outer disk of M31 or a background dwarf galaxy. We present deep HST WFPC2 observations of And IV and the surrounding field which, along with ground-based long-slit spectroscopy and Halpha imagery, are used to decipher the true nature of this puzzling object. We find compelling evidence that And IV is a background galaxy seen through the disk of M31. The moderate surface brightness (SB(V)~24), very blue colour (V-I<~0.6), low current star formation rate (~0.001 solar mass/yr) and low metallicity (~10% solar) reported here are consistent with And IV being a small dwarf irregular galaxy, perhaps similar to Local Group dwarfs such as IC 1613 and Sextans A. Although the distance to And IV is not tightly constrained with the current dataset, various arguments suggest it lies in the range 5<~D<~8 Mpc, placing it well outside the confines of the Local Group. It may be associated with a loose group of galaxies, containing major members UGC 64, IC 1727 and NGC 784. We report an updated position and radial velocity for And IV.Comment: 26 pages, LaTex with 9 figures (including 6 jpg plates). Accepted for publication in A

Cosmology, cohomology, and compactification

Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m.Comment: 6 pages, LaTe