Research Seminar in Analysis in conjunction with the Colloquium at UMR meets again in G5 (Rolla Building) on Monday, September 27, 1999, 3:30 pm. All interested faculty and students are invited to attend. 

Positivity of Quadratic Functionals and Oscillation. In this talk we consider continuous and discrete quadratic functionals, which occur as second variations and Hessians, respectively, of continuous and discrete optimization problems. The continuous part of the talk deals with corresponding Hamiltonian differential systems, Riccati matrix differential equations, and eigenvalue problems. The main results are characterizations of positivity (Reid Roundabout Theorem), and, in particular, an oscillation theorem, which yields e.g. Rayleigh's principle for the eigenvalue problem. The discrete part is concerned with the corresponding difference equations and analogous results, but without oscillation results, which remain an open problem. 