Nonempty Open Set Research Articles

Growth conditions for analytic functions in unbounded open sets

Let G be a non-empty open set in the complex plane ℂ with at least two finite boundary points. Let be a continuous function that is analytic in G. Let μ be a non-decreasing non-negative function defined for t ≥ 0 such that μ(2t) ≤ 2μ(t) for all t ≥ 0. Suppose that |f(z 1) − f(z 2)| ≤ μ(|z 1 − z 2|) for a fixed z 1 ∈ ∂G and for all z 2 ∈ ∂G. Suppose that for each unbounded component D of G, if any, there is a positive number q such that f(z) = O(|z| q ) as z → ∞ in D. Then at least one of the following holds: i. For all z 2 ∈ G we have |f(z 1) − f(z 2)| ≤ Cμ(|z 1 − z 2|) where C = 3456. ii. The set G contains a neighbourhood of infinity, so that G has exactly one unbounded component, and f has a pole at infinity. Furthermore, if |f(z 1) − f(z 2)| ≤ μ(|z 1 − z 2|) for all z 1, z 2 ∈ ∂G, then |f(z 1) − f(z 2)| ≤ Cμ(|z 1 − z 2|) for all with C = 3456 except perhaps when G contains a neighbourhood of infinity, so that G has exactly one unbounded component, and f has a pole at infinity.

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ℝ does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, Sr be the image of S under the translation of ℝ by a rational number r and F = {Sr: r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ℝ possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ⊃ O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.

Holomorphic extension theorem for tempered ultrahyperfunctions

In this article we are concerned with the space of tempered ultrahyperfunctions corresponding to a proper open convex cone. A holomorphic extension theorem (the version of the celebrated Edge-of-the-Wedge Theorem) will be given for this setting. As application, a version is also given of the principle of determination of an analytic function by its values on a non-empty open real set. The article finishes with the generalization of holomorphic extension theorem à la Martineau.

On support points and support functionals of convex sets

Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set Σ1(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set Σ 1(K) is always uncountable, which answers a question posed by L. Zajicek.

On metric spaces and local extrema

The main aim of this paper is to give a positive answer to a question of Behrends, Geschke and Natkaniec regarding the existence of a connected metric space and a non-constant real-valued continuous function on it for which every point is a local extremum. Moreover we show that real-valued continuous functions on connected spaces such that every family of pairwise disjoint non-empty open sets is of size < | R | are constant provided that every point is a local extremum.

On the ultradistributions of Beurling type

Sea un conjunto abierto no vac´ýo del espacio euclideo . En este articulo se demuestra que si S es una ultradistribucion en , perteneciente a una clase de tipo Beurling que sea estable frente a operadores diferenciales, entonces S se puede representar en la formaP 2Nk0 D f , donde f es una funcion compleja definida en que es Lebesgue medible y esencialmente acotada en cada subconjunto compacto de . Tambi´en se obtienen otros resultados de estructura de ciertas ultradistribuciones.

On closed sets with convex projections under somewhere dense sets of directions

Let k, n ∈ N with k < n and let g k (R n ) denote the Grassmann manifold consisting of all k-dimensional linear subspaces in R n . In an earlier paper the authors showed that if the projections of a nonconvex closed set C C R n are convex and proper for projection directions from some nonempty open set P ⊂ g k (R n ), then C contains a closed copy of an (n―k―1)-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that P is somewhere dense in g k (R n ).

ON THE LYAPUNOV SPECTRUM OF INFINITE DIMENSIONAL RANDOM PRODUCTS OF COMPACT OPERATORS

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic probability which is positive on non-empty open sets, we conclude that there is a C0-residual subset of cocycles within which, for almost every x, either the Oseledets–Ruelle's decomposition along the orbit of x is dominated or all the Lyapunov exponents are equal to -∞.

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ < Δ ( X ) , where Δ ( X ) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.

On the structure of the ultradistributions of Beurling type

Let O be a nonempty open set of the k-dimensional euclidean space Rk. In this paper, we give a structure theorem on the ultradistributions of Beurling type in O. Also, other structure results on certain ultradistributions are obtained, in terms of complex Borel measures in O.

Clifford analysis with higher order kernel over unbounded domains

In this article we study Clifford analysis with higher order kernel over unbounded domains. First we derive a higher order Cauchy-Pompeiu formula for the functions with rth order continuous differentiability over an unbounded domain whose complementary set contains non-empty open set. Then we obtain higher order Cauchy integral formula for k-regular functions and prove Cauchy inequality. Based on the higher order Cauchy integral formula, we define higher order Cauchy-type integrals and discuss boundary behaviours of the higher order Cauchy-type integrals.

A three critical points theorem revisited

In this paper the following result is proved: Let X be a reflexive real Banach space; I ⊆ R an interval; Φ : X → R a sequentially weakly lower semicontinuous C 1 functional, bounded on each bounded subset of X , whose derivative admits a continuous inverse on X ∗ ; J : X → R a C 1 functional with compact derivative. Assume that lim ‖ x ‖ → + ∞ ( Φ ( x ) + λ J ( x ) ) = + ∞ for all λ ∈ I , and that there exists ρ ∈ R such that sup λ ∈ I inf x ∈ X ( Φ ( x ) + λ ( J ( x ) + ρ ) ) < inf x ∈ X sup λ ∈ I ( Φ ( x ) + λ ( J ( x ) + ρ ) ) . Then, there exist a non-empty open set A ⊆ I and a positive real number r with the following property: for every λ ∈ A and every C 1 functional Ψ : X → R with compact derivative, there exists δ > 0 such that, for each μ ∈ [ 0 , δ ] , the equation Φ ′ ( x ) + λ J ′ ( x ) + μ Ψ ′ ( x ) = 0 has at least three solutions in X whose norms are less than r .

Dynamics of two-dimensional Blaschke products

AbstractIn this paper we study the dynamics on $\mathbb {T}^2$ and $\mathbb {C}^2$ of a two-dimensional Blaschke product. We prove that in the case when the Blaschke product is a diffeomorphism of $\mathbb {T}^2$ with all periodic points hyperbolic then the dynamics is hyperbolic. If a two-dimensional Blaschke product diffeomorphism of $\mathbb {T}^2$ is embedded in a two-dimensional family given by composition with translations of $\mathbb {T}^2$, then we show that there is a non-empty open set of parameter values for which the dynamics is Anosov or has an expanding attractor with a unique SRB measure.

Dispersion of volume under the action of isotropic Brownian flows

We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and–under slightly stronger assumptions–asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive.

Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y $$ \subseteqq $$ C p (X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C p (Y) has the cardinality at most κ, that is p(C p (Y)) ≦ κ. Besides, it was proved that w(Y) = p(C p (Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C p (Y)) = ℵ0. This gives answer to a question of O. Okunev and V. Tkachuk.

Category version of the Poincaré recurrence theorem

We prove a category version of the Poincaré recurrence theorem for a non-invertible map of a Baire space which is continuous, nearly feebly open and has no nonempty open wandering set.

Sur les automorphismes analytiques des variétés hyperboliques

Results of H. Cartan about holomorphic automorphisms on bounded domains are generalized to the case of hyperbolic manifolds in the sense of Kobayashi. In this setting, we give an identity theorem together with its topological version. We show also that a sequence of automorphisms which converges uniformly on some nonempty open set, has a limit on the whole space which is an automorphism. At the end of the paper, conditions are given for the sequence of iterates of a self holomorphic map in order to be an automorphism.

Generalization of a theorem of Gonchar

Let X and Y be two complex manifolds, let D⊂X and G⊂Y be two nonempty open sets, let A (resp. B) be an open subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Under a geometric condition on the boundary sets A and B, we show that every function locally bounded, separately continuous on W, continuous on A×B, and separately holomorphic on (A×G)∪(D×B) “extends” to a function continuous on a “domain of holomorphy” $\widehat{W}$ and holomorphic on the interior of $\widehat{W}$.

Asymptotic direction for random walks in random environments

In this paper we study the existence of an asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient contains a non-empty open set, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.

Centers and shore points of a dendroid

A bottleneck in a dendroid is a continuum that intersects every arc connecting two nonempty open sets. Every dendroid contains a point called a center which is contained in arbitrarily small bottlenecks. A subset A of a dendroid is a shore set if for every ε > 0 there is a continuum in D ∖ A with Hausdorff distance from D less than ε. If a shore set has only one point it is called a shore point. This paper explores the relationship between center points and shore points in a dendroid. We show that if a dendroid contains a strong center, then any finite union of the arc components of the set of shore points is a shore set.