Symbols of Applied Algebra

Lectures on Applied Algebra

Some motivation: In this age of computers, algebra may have replaced calculus (analysis) as the most important part of mathematics. For example: 1) Error correcting codes are built into the CD player and the computer. Who do you call to correct errors? The algebraist, that’s who! 2) Digital signal processing (such as that involved in medical scanners, weather prediction, the search for oil) is ...

Applied Linear Algebra

Toeplitz Operators and Toeplitz Algebra with Symbols of Vanishing Oscillation

We study the C∗-algebra generated by Teoplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C∗-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other ...

Binary Relational Algebra Applied to Software Architecture

The architecture of a piece of software refers to its structure as given by its components and their interconnections. An architectural style is a set of commonalities among the structures of a number of software systems. We are concerned with architectures that are characterized by directed graphs. The graphs we are interested in contain edges that are "typed" or "colored", which means that th...

Modal Kleene Algebra Applied to Program Correctness

Modal Kleene algebras are relatives of dynamic logics that support program construction and verification by equational reasoning. We describe their application in implementing versatile program correctness components in interactive theorem provers such as Isabelle/HOL. Starting from a weakest precondition based component with a simple relational store model, we show how variants for Hoare logic...