Transitive and series transitive maps on [formula omitted

For a nonempty set S, Banach spaces X and Y over a field K = R or C, and a nonempty open set \(U\,\subset\,X\), let G, F, and H be vector spaces of functions acting from S into X,from U into Y, and from S into Y, respectively. Usually, but not always, F, G, and H will be normed spaces. Given a function \((u,s)\longmapsto,\psi(u,s)\) from U × S into Y, and a function g : \(S\,\rightarrow\,U\), the Nemytskii operator \(N\psi\) is defined by \(N\psi(g)(s)\quad:=\quad,N\psi\,g)(s)\quad:=\quad\psi(g(s),s)\qquad s\,\epsilon\,S.\) (6.1) Other authors call such an operator a superposition operator, e.g. Appell and Zabrejko [3].We use the term Nemytskii operator (as many others have) partly to distinguish it from the two-function composition operator \((F,G)\mapsto FoG\) to be treated in Chapter 8. Recall also that for a linear operator A we write Ax := A(x). We will often apply this rule also when A is a Nemytskii operator.KeywordsBanach SpaceBanach AlgebraMinkowski InequalityNonempty Open SubsetPoint PartitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For a continuous selfmap $f$ of a compact metric space $X$ westudy the set of its continuous extensions $F$ on the space$X\times I$, where $I$ is a compact interval. In particular, wehave solved an open problem (raised in [Ll. Alseda, S. Kolyada, J.Llibre, and L. Snoha, Entropy and periodic points for transitivemaps, Trans. Amer. Math. Soc. 351 (1999)]) by provingthat any continuous transitive map $f$ on $X$ can be extended to acontinuous transitive triangular map $F=(f,g_x)$ on $X\times I$without increasing topological entropy.

Chaos for induced hyperspace maps

Computable topological abelian groups

In this paper, we are concerned with the coupled nonlinear Schrödinger system [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are nonnegative continuous potentials, and [Formula: see text] is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as [Formula: see text], when [Formula: see text] and [Formula: see text] achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.

Bour ded point evaluations for nonminimal measures

In this paper, we first define Teodorescu operator related to the -Dirac operator in Clifford analysis. Then we discuss the existence, q-times integrability, differentiability and mapping properties of Teodorescu operator in a nonempty open set , which is bounded connected.

Definition: A subset Y of a space X is said to be dense in X if every nonempty open set in X contains at least one point in Y.

In this paper we prove a form of the Zilber's trichotomy conjecture for reducts of algebraically closed valued fields of characteristic 0 which are expansions of the valued vector space structure. We prove first that a non-modular reduct of a nontrivially valued algebraically closed field containing the valued vector space structure defines a non-semilinear curve. Then we show that the expansion of such a reduct by a non-semilinear curve defines multiplication on a nonempty open set.

Densities and liftings for derived algebras

Throughout the present paper, {Xn} denotes a sequence of random quantities which are regarded as exchangeable, and which are assessed with a probability measure P(•) which is a member of the mixture-exponential family. To be precise, it will be presumed that the assessment P(•) for any finite subsequence (X1,…,Xn) can be represented using the product of an identical non-degenerate parametric measure for each Xi, Pθ (•)= P(•θ=θ), determined by $$ {\rm{d}}{{\rm{P}}_{\rm{\theta }}} = \exp \{ {\rm{\theta x}} - {\rm{M}}({\rm{\theta }})\} {\rm{d\mu }} $$ (1.1) μ being a σ-finite measure on the class B of Borel sets of IR. It will always be assumed that the interior X° of the convex hull X of the support of μ (in symbols:suρp(μ))is a nonempty open set (interval) in IR and that {Pθ;θeΘ} is a regular exponential family (cf. Barndorff - Nielsen 1978, p.116). The latter condition implies that Θ = {θ:M(θ)<∞ } is an open interval in IR. Moreover, we will suppose that the set of the logically possible values of θ coincides with θ. Given such a particular frame, the present paper deals with the choice of a prior for (1.1); an excellent treatment of the same topic is included in Diaconis and Ylvisaker (1979, 1985). Our approach bases itself on the obvious remark that the choice of a prior establishes the strength of the dependence among the elements of the sequence {Xn} and, consequently, the strength of the influence exercised by experience on our future predictions. This subjective standpoint is skilfully expounded in de Finetti (1937).

Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form. For example, we can show that if the system $dx/dz = P(x,y)$, $dy/dz = Q(x,y)$ has a solution $(x(z),y(z))$ satisfying a liouvillian relation then either $x(z)$ and $y(z)$ are algebraically dependent or the system has a liouvillian first integral of the form $F(x,y) = \smallint RQ dx - RP dy$ where $R = \exp (\smallint U dx + V dy)$ and $U$ and $V$ rational functions of $x$ and $y$ . We can also reprove an old result of Ritt stating that a second order linear differential equation has a nonconstant solution satisfying a liouvillian relation if and only if all of its solutions are liouvillian.

Definition: A subset Y of a space X is said to be dense in X if every nonempty open set in X contains at least one point in Y.

Bau-Sen Du introduced a notion of chaos which is stronger than Li-Yorke sensitivity. A TDS (X, f) is called chaotic if there is a positive e such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not e-asymptotic. In this article, we show that a TDS (T, f) is transitive but not mixing if and only if (T, f) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos.

We give a new formula for the topological entropy of a continuous map f of the interval to itself with f2 topologically transitive. This leads to stronger forms of well-known results about horseshoes and periodic orbit patterns for continuous maps of the interval in the special case of topologically transitive maps.