New life in old linear differential equations
Prof. José M. Gracia-Bondía (Universidad Complutense, Madrid, and UCR San José, Costa Rica)

Linear differential equations of the form: d/dt G(t) = G(t)A(t), where \(G,A\) are matrices or operators, not commuting in general, are pervasive in physics. We review, from a modern viewpoint, two of the three main methods of solution, the Dyson series and the Magnus expansion, and their relations. These solutions naturally involve the Riemann integral. Now, it is not usually realized that, for most purposes, only a purely algebraic property of the integral is used, to wit, integration-by-parts. This opens the door to a generalization of the derivative/integral pair, in terms of skewderivations and Baxter operators, which are also starting to play a prominent role in physics.