Our domain G = (0,L) is an interval of length L. The boundary ∂G = {0,L} are the two endpoints. We consider here as an example the case (DD) of Dirichlet boundary conditions: Dirichlet conditions at x = 0 and x = L. For other boundary conditions (NN), (DN), (ND) one can proceed similarly. In one dimension the Laplace operator is just the second derivative with respect to x: ∆u(x, t) = uxx(x, t)...

We consider stability properties of discrete-time bilinear lters. Simple su cient conditions are given for bounded-input bounded-output stability (with not necessarily zero initial conditions), lp stability, and three other important types of stability. In particular, conditions are given under which asymptotically periodic inputs produce asymptotically periodic outputs with the same period. Re...

Periodic travelling waves are a fundamental solution form in oscillatory reactiondiffusion equations. Here I discuss the generation of periodic travelling waves in a reaction-diffusion system of the generic λ-ω form. I present numerical results suggesting that when this system is solved on a semi-infinite domain subject to Dirichlet boundary conditions in which the variables are fixed at zero, ...

This paper examines initial conditions dependence and initial conditions uncertainty for climate projections and predictions. The first contribution is to provide a clear conceptual characterisation of predictions and projections. Concerning initial conditions dependence, projections are often described as experiments that do not depend on initial conditions. Although prominent, this claim has ...

The Roman imperial monarchy is generally studied from the vantage point of ancient Roman history: the “Roman emperor” is viewed and analyzed as an element of the Roman world. This conventional approach fails to place this institution in a broader comparative context, that of monarchical rulers across world history. Systematic comparison opens up new perspectives and is indispensable in identify...