Acta Math Hungar Research Articles

Generalization of weakly clopen and strongly [formula omitted]- b-continuous functions

In this paper, we introduce a new class of functions called weakly BR-continuous function including the classes of strongly θ - b-continuous functions due to Park [Park JH. Strongly θ - b -continuous functions. Acta Math Hungar 2006;110(4):347–59] and weakly clopen functions due to Son et al. [Son MJ, Park JH, Lim KM. Weakly clopen functions. Chaos, Solitons & Fractals 2007;33:1746–55]. Characterizations and properties of the class of weakly BR-continuous functions are investigated. Moreover, the relationships among weakly BR-continuous functions, strongly θ - b-continuous functions, weakly clopen functions and the related functions are investigated.

On the minimum number of completely 3-scrambling permutations

A family P = { π 1 , … , π q } of permutations of [ n ] = { 1 , … , n } is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x 1 , … , x k ∈ [ n ] , permutations π i 's in P produce all k ! possible orders on π i ( x 1 ) , … , π i ( x k ) . Let N * ( n , k ) be the minimum size of such a family. This paper focuses on the case k = 3 . By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. 2 log 2 e log 2 n ⩽ N * ( n , 3 ) ⩽ 2 log 2 n + ( 1 + o ( 1 ) ) log 2 log 2 n . We also prove the existence of lim n → ∞ N * ( n , 3 ) / log 2 n = c 3 . Determining the value c 3 and proving the existence of lim n → ∞ N * ( n , k ) / log 2 n = c k for k ⩾ 4 remain open.

Weakly clopen functions

Abstract We introduce a new class of functions called weakly clopen function which includes the class of almost clopen functions due to Ekici [Ekici E. Generalization of perfectly continuous, regular set-connected and clopen functions. Acta Math Hungar 2005;107:193–206] and is included in the class of weakly continuous functions due to Levine [Levine N. A decomposition of continuity in topological spaces. Am Math Mon 1961;68:44–6]. Some characterizations and several properties concerning weakly clopenness are obtained. Furthermore, relationships among weak clopenness, almost clopenness, clopenness and weak continuity are investigated.

Almost strongly θ-precontinuous functions☆

Abstract In this paper, we introduce a new class of functions called almost strongly θ -precontinuous function which is a generalization of almost strongly θ -continuous functions due to Noiri and Kang [Noiri T, Kang SM. On almost strongly θ -continuous functions. Indian J Pure Appl Math 1984;15(1):1–8] and strongly θ -precontinuous functions due to Noiri [Noiri T. Strongly θ -precontinuous functions. Acta Math Hung 2001;90(4):307–16]. Some characterizations and several properties concerning almost strongly θ -precontinuous functions are obtained. The relationships between almost strongly θ -precontinuity and other types of continuity are also given.