We discuss two facets of the interaction between geometry and algebra in Banach algebras. In class unital algebras, there is essentially one known example which also strictly convex as a space. recall this example, finite-dimensional, consider open question generalising it to infinite dimensions. C*-algebras, we exhibit striking tighter relationship that exists there.

This paper presents sufficient conditions for the design of strictly positive real (SPR), fixed-order dynamic compensators. The primary motivation for designing SPR compensators is for application to positive real (PR) plants. When an SPR compensator is connected to a PR plant in a negative feedback configuration, the closed loop is guaranteed stable for arbitrary plant variations as long as th...

The Strict Positive Real (SPR) property of discrete-time transfer functions plays a fundamental role in the analysis of the behaviour of several recursive schemes employed in identification and adaptive control. In the context of uncertain systems, this leads to the so called robust SPR problem (RSPR), i.e., given a set P of polynomials and a region Λ of the complex plane, determine, provided i...

Recently, Xu and Cheney (1992) have proved that if all the Legendre coeecients of a zonal function deened on a sphere are positive then the function is strictly positive deenite. It will be shown in this paper, that even if nitely many of the Legendre coeecients are zero, the strict positive deeniteness can be assured. The results are based on approximation properties of singular integrals, and...

We give a necessary and sufficient condition for the strict positivedefiniteness of real and continuous functions on spheres of dimension greater than one.

In the Edgeworth non-tatonnement process, trade occurs if there exists some coalition of agents able to make a Paretoimproving trade among themselves at current prices. It is known that the size of such coalitions is bounded by the number of commodities and that, provided all agents always have strictly positive endowments, bilateral trade suffices. These results are generalized: Let there be h...

We prove that the solutions to fast diffusion stochastic porous media equations have finite time extinction with strictly positive probability.