A fundamental concept in mathematical systems theory concerns the notion of state. Roughly speaking, the state of a system at a particular time instant is all one needs to know about the past to characterize the future behavior. In this talk we revisit the following classical question: given a set of linear higher-order differential equations in the external variables (i.e., inputs and outputs)...

Assume we have a definite integral we wish to evaluate, but it looks nasty because the integrand is wildly oscillatory and the usual numerical techniques based on sampling will not work well. The approach we take is to reformulate the indefinite integral so it doesn’t wiggle so much. That is, we compute another form for the anti-derivative. Using the fundamental theorem of integral calculus (wh...

As a first application of Theorem 0.1 we deduce the concentration of measure property of P� that was alluded to in the first chapter. The claim is simply that with very high probability every Lipschitz-continuous function is very close to its mean, regardless of the value of the ambient dimension �. One can obtain a crude version of this assertion by appealing to the Poincaré inequality of Nash...