Monotone normality free of T 1 axiom

Monotone normality is usually defined in the class of T ₁ spaces. In this paper new characterizations of monotone normality, free of T ₁ axiom, are provided and it is shown that in this context it is not a hereditary property. Also, a Tietze-type extension theorem for lattice-valued functions for this class of spaces is given.     

SPACES WITH WALLMAN EQUIVALENT INTERMEDIATE SPACES

Abstract

If every infinite closed subset of the Wallman compactification, WX, of a space X must contain at least one element of X, then for any space Y intermediate between X and WX the Wallman compactification WY is homeomorphic to WX. This extends a property which characterizes normality inducing spaces. In the case where X is not normal, however, this is not a characterization, since there are nonnormal spaces for which all intermediate spaces are Wallman equivalent, but have infinite closed subsets contained in WX/X.     

New types of generalized closed sets in bitopological spaces

In this paper, we introduce a new type of closed sets in bitopological space (X, τ₁, τ₂), used it to construct new types of normality, and introduce new forms of continuous function between bitopological spaces. Finally, we proved that the our new normality properties are preserved under some types of continuous functions between bitopological spaces.     

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly^? subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that,1) on weakly subparacompact spaces, countable compactness = compactness, ω₁-compactness = Lindelöfness;2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and3) on weakly subparacompact regular T₁-spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness.The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wicke?s factorization of paracompactness (over Hausdorff spaces) along with Krajewski?s are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.     

On the uniform asymptotic joint normality of sample quantiles

Summary Uniform (or type (B) _d ) asymptotic normality of the joint distribution of an increasing number of sample quantiles as the sample size increases is investigated in both cases where the basic distributions are equal and are unequal. Under fairly general assumptions, sufficient conditions are derived for the asymptotic normality of sample quantiles. Type (B) _d asymptotic normality is a strictly stronger notion than the usual one which is based on the convergence in law, and the results obtained in this article will be helpful to widen the applicability of results on asymptotic normality of sample quantiles to related statistical inferences.     

Weighted composition operators from F（p,q, s） spaces to Bers-type spaces in the unit ball

This paper deals with the boundedness and compactness of the weighted composition operators from the F（p, q, s） spaces, including Hardy space, Bergman space, Qp space, BMOA space, Besov space and α-Bloch space, to Bers-type spaces Hv^∞（ or little Bers-type spaces Hv,o∞ ）, where v is normal.     

Discontinuous extensions and almost resolvability

It is proven that in large classes of topological spaces each real-valued continuous function on aG _δ-setG has an extension to the whole space which is continuous exactly at the points ofG. AmongG _δ-spaces this property characterizes the almost resolvable normal spaces.     

Monotone insertion of lattice-valued functions

Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality. This research was supported by the MEyC and FEDER under grant MTM2006-14925-C02-02/ and by UPV05/101     

Some categorical properties of the functors O τ and O R of weakly additive functionals

In the paper, the spaces of weakly additive τ-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive τ-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive τ-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented.     

A class of Banach sequence spaces analogous to the space of Popov

Hagler and the first named author introduced a class of hereditarily l ₁ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l _p Banach spaces for 1 ⩽ p < ∞. Here we use these spaces to introduce a new class of hereditarily l _p (c ₀) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l ₁ Banach spaces failing the Schur property.     

Normality-Type Properties and Covariant Functors

Topological properties similar to normality are considered in subspaces of products and powers of topological spaces, of spaces of closed subsets, and of spaces having the form (X), where is a normal functor. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 57–98, 2003.     

Improvements of goodness-of-fit statistics for sparse multinomials based on normalizing transformations

We consider multinomial goodness-of-fit tests for a specified simple hypothesis under the assumption of sparseness. It is shown that the asymptotic normality of the PearsonX ² statistic (X _k ² ) and the log-likelihood ratio statistic (G _k ² ) assuming sparseness. In this paper, we improve the asymptotic normality ofX _k ² andG _k ² statistics based on two kinds of normalizing transformation. The performance of the transformed statistics is numerically investigated.     

ON CERTAIN INITIAL COMPLETIONS IN FUZZY TOPOLOGY

Abstract

Following a lead given by I.W. Alderton, it is shown that the MacNeille completion and the universal initial completion coincide for the categories of zero-dimensional fuzzy T₀-topological spaces, T₀-fuzzy closure spaces, 2T ₀-fuzzy bitopological spaces, and T ₁-fuzzy topological spaces and that these turn out to be respectively the categories of zero-dimensional fuzzy topological spaces, fuzzy closure spaces, fussy bitopological spaces, and fuzzy R ₀ topological spaces.     

On Groups in Which Normality Is a Transitive Relation

A subgroup H of a group G is said to be weakly normal if H ^g  = H whenever g is an element of G such that H ^g  ≤ N _G (H). There is a strictly relation between weak normality and groups in which normality is a transitive relation ( T-groups). In [Ballester-Bolinches, A., Esteban-Romero, R. (2003). On finite T-groups. J. Aust. Math. Soc. 75:181–191] it is proved that a finite group G is a soluble T-group if and only if every subgroup of G is weakly normal. In this article, we extend the above result to infinite groups having no infinite simple sections. Moreover, it will be shown that every locally graded non-periodic group, all of whose subgroups are weakly normal, is abelian.     

Quasi-contractions on a nonnormal cone metric space

Ilić and Rakočević [6] proved a fixed point theorem for quasi-contractive mappings on cone metric spaces when the underlying cone is normal. Recently, Z. Kadelburg, S. Radenović, and V. Rakočević obtained a similar result without using the normality condition but only for a contractive constant λ ∈ (0, 1/2) [8]. In this note, using a new method of proof, we prove this theorem for any contractive constant λ ∈ (0, 1).     

Cantor series constructions contrasting two notions of normality

Rényi (Mat Lapok 7:77–100, 1956) made a definition that gives a generalization of simple normality in the context of Q-Cantor series. In Mance ), a definition of Q-normality was given that generalizes the notion of normality in the context of Q-Cantor series. In this work, we examine both Q-normality and Q-distribution normality, treated in Lafer (Normal numbers with respect to Cantor series representation, 1974) and S̆alát (Czechoslovak Math J 18(93):476–488, 1968). Specifically, while the non-equivalence of these two notions is implicit in Lafer (Normal numbers with respect to Cantor series representation. Washington State University, 1974), in this paper, we give an explicit construction witnessing the nontrivial direction. That is, we construct a base Q as well as a real x that is Q-normal yet not Q-distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base Q as well as a real x that is both Q-normal and Q-distribution normal.     

A Family of Nonnormal Cayley Digraphs

We call a Cayley digraph Γ = Cay(G, S) normal for G if G _R , the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p ² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found. Received February 23, 1998, Revised September 25, 1998, Accepted October 27, 1998     

Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M<1 and for each k>1 there are cones with normal constant M>k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.     

Baire functions and spaces of Baire functions

The paper considers (1) the tightness of spaces of Baire functions and their subspaces endowed with the topology of pointwise convergence; (2) Z _σ-mappings of K-analytic spaces; (3) K _σ-analytic spaces (Tychonoff spaces that are Z _σ-images of K-analytic spaces). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003.