This paper discusses real-time language recognition by 1dimensional one-way cellular automata (OCAs) and two-way cellular automata (CAs), focusing on limitations of the parallel computation power. To clarify the limitations, we investigate real-time recognition of cyclic strings of the form uk with u ∈ {0, 1}+ and k ≥ 2. We show a version of pumping lemma for recognizing cyclic strings by OCAs,...

A π,z J(z) = lim t>0,t→0 e −αt E z,π [J(z(t))] − J(z) t Lemma 1. Let J ∈ J satisfy J(0, a, b) = 0. Let τ = inf{t : J(z t) = 0}. E τ 0 e −αt e −π(z)/r a b π(z) + A π,z J(z) dt = J π (z 0) − J(z 0) Proof: Define H π J(z) = e −π(z)/r a b π(z) + A π,z J(z) Lemma 6 later verifies that this definition is in agreement with our previous definition (in [2]) provided J ∈ J. Let τ be a stopping time of th...

and Applied Analysis 3 2. Main Results It will be derived that properties of E are closely connected with the corresponding secondorder differential equation ( 1 r t v′ t )′ p t v t 0 Ev as the following theorem says. Theorem 2.1. Let v t be a positive solution of Ev . Then E can be written as ( v2 t r t ( 1 v t y′ t )′)′ q t v t y σ t 0. E Proof. The proof follows from the fact that 1 v t ( v2...

Let T be a finite entropy, aperiodic automorphism of a nonatomic probability space. We give an elementary proof of the existence of a finite entropy, countable generating partition for T . In this short article we give a simple proof of Rokhlin’s countable generator theorem [Ro], originating from considerations in [Se] which use standard techniques in ergodic theory. We hope that these consider...

In this paper we prove a general result establishing a priori L estimates for solutions of RiemannHilbert Problems (RHP’s) in terms of auxiliary information involving an associated “conjugate” problem (see Conjugation Lemma 1.39 below). We then use the result to obtain uniform estimates for a RHP (see Theorem 1.48) that plays a crucial role in analyzing the long-time behavior of solutions of th...

Koenig’s infinity lemma states that an infinite rooted tree in which every node has finite degree has an infinite path. A variation of this lemma about mappings from one tree to another is presented in this note. Its proof utilizes Koenig’s lemma, and Koenig’s lemma follows from this variation.

and Applied Analysis 3 The rest of this paper is organized as follows. In Section 2, we give notations, definition, propositions, and lemma to be used in the proof of the main results. Delay-dependent sufficient conditions for uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity are presented in Section 3. Numerical examples illustrated the obtained results and are giv...