For subsets of R + = [0, ∞) we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at 0 sets is strongly porous if and only if these sets are coherently porous. This result leads to a characteristic property of the intersection of all maximal ideals contained in the family of strongly porous at 0 subsets of R + . It is also shown that the union of a set A ⊆ R + with arbitrary strongly porous at 0 set is porous at 0 if and only if A is lower porous at 0 .
For subsets of R + we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function µ : N → R + provides an extension of the concept of porosity numbers on subsets of N. The main results describe interconnections between porosity numbers of a set, features of the scaling funtions and the geometry of so-called pretangent spaces to this set. 1991 Mathematics Subject Classification. 2010MSC. Primary 28A05, Secondary 54E35, 30D40. Key words and phrases. local upper porosity, local lower porosity, set of porosity numbers, porosity of subsets of N, pretangent space. (M. Küçükaslan)
Functions defined in the form ``$g:\mathbb{N}\to[0,\infty)$ such that $\lim_{n\to\infty}g(n)=\infty$ and $\lim_{n\to\infty}\frac{n}{g(n)}=0$'' are called weight functions. Using the weight function, the concept of weighted density, which is a generalization of natural density, was defined by Balcerzak, Das, Filipczak and Swaczyna in the paper ``Generalized kinsd of density and the associated ideals'', Acta Mathematica Hungarica 147(1) (2015), 97-115.In this study, the definitions of $g$-statistical convergence and $g$-statisticalCauchy sequence for any weight function $g$ are given and it is proved that these two concepts are equivalent. Also some inclusions of the sets of all weight $g_1$-statistical convergent and weight $g_2$-statistical convergent sequences for $g_1,g_2$ which have the initial conditions are given.
In this paper, we are going to define λ-statistical supremum and λ-statistical infimum for real valued sequencex = (xn) n∈N by considering λ-statistical upper and lower bounds, respectively. After giving some basic properties of these new notations, then we will give a necessary and sufficient condition for to existance of λ-statistical convergence of the real valued sequence.
In this paper by using natural density real valued bounded sequence space l1 is extented and statistical bounded sequence space l st 1 is obtained. Besides the main properties of the space l st 1 , it is shown that l st 1 is a Banach space with a norm produced with the help of density. Also, it is shown that there is no matrix extension of the space l1 that its bounded sequences space covers l st 1 . Finally, it is shown that the space l1 is a non-porous subset of l st 1 .
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