The two volumes of Signal Processing are based on lectures delivered during a six week program held at the IMA from June 27 to August 5, 1988. The first two weeks of the program dealt with general areas and methods of Signal Pro cessing. The problem areas included imaging and analysis of recognition, x-ray crystallography, radar and sonar, signal analysis and 1-D signal processing, speech, vision, and VLSI implementation. The methods discussed included harmonic anal ysis and wavelets, operator theory, algorithm complexity, filtering and estimation, and inverse scattering. The topics of weeks three and four were digital filter, VLSI implementation, and integrable circuit modelling. In week five the concentration was on robust and nonlinear control with aerospace applications, and in week six the emphasis was on problems in radar, sonar and medical imaging. Because of the large overlap between the various one-week and two-week seg ments of the program, we found it more convenient to divide the material somewhat differently. Part I deals with general signal process theory and Part II deals with (i) application of signal processing, (ii) control theory related themes. We are grateful to the scientific organizers: Tom Kailath (Chairman), Louis Aus lander, F. Alberto Grunbaum, J. William Helton, Pramod P. Khargonekar and Sanjoy K. Mitter. We are also grateful for the generous support given to the IMA program by the Office of Naval Research, the Air Force Office of Scientific Research, the Army Research Office and the National Security Agency."

In classical dynamics the action of a one-parameter group-time- on the phase space is one of the principal objects of study. Ergodic theory concerns itself with the case when the phase space is a measure spaces. The purpose of this chapter is to give an outline of the basic theorems in the theory of solvmanifolds.