Parry showed that every continuous transitive piecewise monotonic map τ of the interval is conjugate (by an order preserving homeomorphism) to a uniformly piecewise linear map T (i.e., one with slopes ±s). In the current article, it is shown that the map T is unique. This is proven by showing that any order-preserving conjugacy between two continuous transitive uniformly piecewise linear maps i...

Atsuji has internally characterized those metric spaces X for which each real-valued continuous function on X is uniformly continuous as follows: (1) the set X' of limit points of X is compact, and (2) for each £ > 0, the set of points in X whose distance from X' exceeds e is uniformly discrete. We obtain these new characterizations: (a) for each metric space V, the Hausdorff metric on C(X, Y),...

Continuous homogeneous controllers are utilized in a full state feedback setting for the uniform finite time stabilization of a perturbed double integrator in the presence of uniformly decaying piecewise continuous disturbances. Semiglobal strong C1 Lyapunov functions are identified to establish uniform asymptotic stability of the closed-loop planar system. Uniform finite time stability is then...

‎In this paper‎, ‎we define and study the notion of zero elements in topoframes; a topoframe is a pair‎ ‎$(L‎, ‎tau)$‎, ‎abbreviated $L_{ tau}$‎, ‎consisting of a frame $L$ and a‎ ‎subframe $ tau $ all of whose elements are complemented elements in‎ ‎$L$‎. ‎We show that‎ ‎the $f$-ring $ mathcal{R}(L_tau)$‎, ‎the set of $tau$-real continuous functions on $L$‎, ‎is uniformly complete‎. ‎Also‎, ‎t...

We use the simple, but little-known, result that a uniformly continuous function on a convex set is -Lipschitz (as defined below) to extend Piyavskii’s algorithm for Lipschitz global optimization to the larger domain of continuous (not-necessarilyLipschitz) global optimization.

Abstract: Non-uniformly sampled systems are widely found in industry. In these systems the process output is sampled and the control input is generated at non-uniformly distributed time instants. In this paper, an optimal residual generator is developed for fault detection in non-uniformly sampled systems. In the direct approach used here, the intersample behavior of fault and disturbance is ca...