Measurable time-restricted sensitivity

Measurable Sensitivity

We introduce the notions of measurable and strong measurable sensitivity, which are measure-theoretic versions of the conditions of sensitive dependence on initial conditions and strong sensitive dependence on initial conditions, respectively. Strong measurable sensitivity is a consequence of light mixing, implies that a transformation has only finitely many eigenvalues, and does not exist in t...

Time-Restricted Sequence Generation

The classes of sequences generated by timeand spacerestricted multiple counter machines are compared to the corresponding classes generated by similarly restricted multiple tape Turing machines. Special emphasis is placed on the class of sequences generable by machines which operate in real time. Real-time Turing machines are shown to be strictly more powerful than real-time counter machines. A...

Measurable Utility and the Measurable Choice Theorem

Impulsive Tracking Control for Non-measurable State with Time-delay

In this paper, we discuss the problem by utilizing impulsive control and Lyapunov function methods, which is about the state of disturbed systems with time-delay tracking the state of reference systems. Sufficient conditions for the solvability of the tracking control problem are given for the measurable state and the non-measurable state of system respectively. This impulsive control law based...

Real Analytic Solutions of Parabolic Equations with Time-measurable Coefficients

We use Bernstein’s technique to show that for any fixed t, strong solutions u(t, x) of the uniformly parabolic equation Lu := aij (t)uxixj−ut = 0 in Q are real analytic in Q(t) = {x : (t, x) ∈ Q}. Here, Q ⊂ Rd+1 is a bounded domain and the coefficients aij(t) are measurable. We also use Bernstein’s technique to obtain interior estimates for pure second derivatives of solutions of the fully nonl...