Additive reducts of algebraically closed valued fields

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.

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.

Transitive and series transitive maps on [formula omitted

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.

Modifiers connect L-fuzzy sets to topological spaces

In [8], Császár introduced some generalized open sets in generalized topological spaces, namely, μ-semi-open, μ-preopen, μ-β-open, and μ-α-open. The primary purpose of this paper is to study some new properties and characterizations of these concepts as we also introduce and study a new generalized open set in generalized topological spaces, namely, μ-b-open. We also investigate some relationships between these concepts.

The Neumann-to-Dirichlet map in two dimensions

In this paper we first prove a nonexistence result on iterative roots, which presents several sufficient conditions for identifying self-maps on arbitrary sets that have no iterative roots of any order. Then, using this result, we prove that when X is [0,1]m , Rm or S1 every non-empty open set of the space of continuous self-maps on X endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order n⩾2 . This, in particular, proves that the complement of , the set of non-iterates, is dense in for these X.