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In many natural and technological systems the rule that governs the system’s dynamics changes over time. In such switched systems the system’s switching can be a significant source of instability. Here we give a simple sufficient criterium to determine if an i.i.d. stochastically switched system is stable in expectation. This method extends recent results for linear switched systems to nonlinear switched systems. It also extends results known for general switched systems giving improved results for systems with i.i.d. stochastic switching. The paper also considers the effects of time-delays on the stability of switched systems. Such time delays, which are intrinsic to any real-world system, can also have a destabilising effect on the system’s dynamics. Previously, it has been shown that if a dynamical system is intrinsically stable, which is a stronger form of global stability, then it maintains its stability even when time-delays are introduced into the system. Here we extend this notion to stochastically switched systems. We refer to this type of stability as patient stability and give a simple sufficient criterium under which such systems are patiently stable, i.e. cannot be destabilised by time delays. Both criteria introduced in this paper side step the need to use Lyapunov, linear matrix inequalities, and semi-definite programming-type methods. Our examples in this paper demonstrate the simplicity of these criteria.

We investigate the claim that the Utah congressional districts enacted in 2011 represent an unfair Republican gerrymander, and we evaluate the most common measures of partisan fairness and gerrymandering in the context of Utah's congressional districts. We do this by generating large ensembles of alternative redistricting plans using Markov chain Monte Carlo methods. We also propose a new metric of partisan fairness in Utah, namely, the Republican vote share in the least-Republican district. This metric only makes sense in settings with at most one competitive district, but it is very effective for quantifying gerrymandering and for evaluating other metrics used in such settings.