On the <Emphasis Type="Italic">σ</Emphasis>-countable compactness of spaces of continuous functions with the set-open topology

For a Tychonoff space X, we obtain a criterion of the σ-countable compactness of the space of continuous functions C(X) with the set-open topology. In particular, for the class of extremally disconnected spaces X, we prove that the space C _λ(X) is σ-countably compact if and only if X is a pseudocompact space, the set X(P) of all P-points of the space X is dense in X, and the family λ consists of finite subsets of the set X(P).     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

An Extension of the Birkhoff Integrability for Multifunctions

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.     

The free idempotent generated locally inverse semigroup

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.     

Maximal regularity of second order delay equations in Banach spaces

Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：
u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈R
to have maximal regularity in HSlder continuous function spaces C^α （R, X）, where X is a Banach space, A is a closed operator in X, F, G ∈L（C（[-r, 0], X）, X） are delay operators for some fixed r 〉 0.     

On nonregular ideals and <Emphasis Type="Italic">z</Emphasis>°-ideals in <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>)

The spaces X in which every prime z°-ideal of C(X) is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces X, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime z°-ideal in C(X) is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in C(X) a z°-ideal? When is every nonregular (prime) z-ideal in C(X) a z°-ideal? For instance, we show that every nonregular prime ideal of C(X) is a z°-ideal if and only if X is a ?-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.     

Uniform continuity in {\omega_\mu}-metric spaces and uc {\omega_\mu}-metric extendability

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the \({\omega_\mu}\)-metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.

     

Approximation of maps into spheres by regulous maps

Let X be a compact real algebraic set of dimension n. We prove that every Euclidean continuous map from X into the unit n-sphere can be approximated by a regulous map. This strengthens and generalizes previously known results.     

Symplectic <Emphasis Type="Italic">S</Emphasis><Subscript>5</Subscript> action on symplectic homotopy K3 surfaces

Let X be a symplectic homotopy K3 surface and G = S ₅ act on X symplectically. In this paper, we give a weak classification of the G action on X by discussing the fixed-point set structure. Besides, we analyse the exoticness of smooth structures of X under the action of G.     

On Hyperbolic Metric and Invariant Beltrami Differentials for Rational Maps

Given a rational map R, we consider the complement of the postcritical set \(S_R\). In this paper we discuss the existence of invariant Beltrami differentials supported on an R invariant subset X of \(S_R\). Under some geometrical restrictions on X, we show the absence of invariant Beltrami differentials with support intersecting X. In particular, we show that if X has finite hyperbolic area, then X cannot support invariant Beltrami differentials except in the case where R is a Lattès map.     

Classification of finite groups satisfying a minimal condition

If H is a subgroup of a finite group G then we denote the normal closure of H in G by H ^G . We call G a PE-group if every minimal subgroup X of G satisfies N ^G (X) ∩ X ^G = X. The authors classify the finite non-PE-groups whose maximal subgroups of even order are PE-groups.     

Hereditary normality of a space of the form ℱ(<Emphasis Type="Italic">X</Emphasis>)

Assuming the continuum hypothesis we construct an example of a nonmetrizable compact set X with the following properties(1) X ⁿ is hereditarily separable for all n ∈ ?(2) X ⁿ \ Δ _n is perfectly normal for every n ∈ ?, where Δ _n is the generalized diagonal of X ⁿ , i.e., the set of points with at least two equal coordinates(3) for every seminormal functor ? that preserves weights and the points of bijectivity the space ? _k (X) is hereditarily normal, where k is the second smallest element of the power spectrum of the functor ?; in particular, X ² and λ ₃ X are hereditarily normal.Our example of a space of this type strengthens the well-known example by Gruenhage of a nonmetrizable compact set whose square is hereditarily normal and hereditarily separable.     

On the Problem of the Optimal Choice of Record Values

Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X₁ < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X₁ > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x₁, x₂, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X _r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= lim_n→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.     

A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field X on a closed three dimensional manifold M, one may try to decompose (M, X) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection \(\{S_1,\dots ,S_n\}\) of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets \(\Lambda _1,\dots ,\Lambda _m\) of the connected components \(V_1,\dots ,V_m\) of \(M-(S_1\cup \dots \cup S_n)\) satisfy the following properties:

• each \(\Lambda _i\) is a compact invariant locally maximal transitive set for X;
• the collection \(\{\Lambda _1,\dots ,\Lambda _m\}\) is canonically attached to the pair (M, X) (i.e. it can be defined independently of the collection of tori and Klein bottles \(\{S_1,\dots ,S_n\}\));
• the \(\Lambda _i\)’s are the smallest possible: for every (possibly infinite) collection \(\{S_i\}_{i\in I}\) of tori and Klein bottles transverse to X, the \(\Lambda _i\)’s are contained in the maximal invariant set of \(M-\cup _i S_i\).

To a certain extent, the sets \(\Lambda _1,\dots ,\Lambda _m\) are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition \(V_1,\dots ,V_m\), equipped with the restriction of the Anosov vector field X, are “almost unique up to topological equivalence”.

     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.