The main argument of the report TR06-133 is in error. The paper claims to prove the result of the title by reduction from the (∃, k)-pebble game, shown to be EXPT IME-complete by Kolaitis and Panttaja. This note shows that the principal lemma is incorrect by providing a simple counterexample. 1 The counter-example The main theorem of the paper depends on the following claim, stated as Lemma 5.3...

a 1993 result of j. llibre, and m. misiurewicz, (theorem a [5]), states that if a continuous map f of a graph into itself has an s-horseshoe, then the topological entropy of f is greater than or equal to logs, that is h( f ) ? logs. also a 1980 result of l.s. block, j. guckenheimer, m. misiurewicz and l.s. young (lemma 1.5 [3]) states that if g is an a-graph of f then h(g) ? h( f ). in this pap...

If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent. 1 Background and overview A complete deterministic finite automaton (DFA) is a triple A = 〈Q,Σ, δ〉, where Q and Σ are finite sets called the state set and the input a...

In this paper, we provide solution formulae for higher order discrete two-dimensional (2D) autonomous systems. We first consider a special type of 2D autonomous systems. These systems are described by equations that satisfy a certain special property: the module of equations contains elements of the form (σn 2 + an−1(σ1)σ n−1 2 + · · · + a0(σ1))wj = 0, for each dependent variable wj , where ai(...

We discuss various properties of a ring over which each simple module is Σ-injective. We shall consider associative rings with identity. Our modules will be unital right modules unless stated otherwise. The class of right V rings was introduced by Villamayor [20]. A ring R is called a right V ring if each simple right Rmodule is injective. It is a well-known unpublished result due to Kaplansky ...