On Generalized Statistical and Ideal Convergence of Metric-Valued Sequences

Das, Pratulananda; Savaş, Ekrem

doi:10.1007/s11253-017-1333-7

Cited by 8 publications

(8 citation statements)

References 18 publications

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“…The first result shows that there is a one-to-one correspondence between topologies on L and the subsets of L consisting of all I-convergent net for certain special types of ideals. The result is in line of Theorem 3.1 in [13]. Proof.…”

Section: Results For Weighted Densitysupporting

confidence: 69%

“…In the articles [25,27] some investigation done that under what condition subsequence of a statistical convergent sequence is also convergent. Following this line of investigation some work done in [13]. We prove the next results in this direction.…”

Section: Results For F-densitymentioning

confidence: 57%

“…As a natural consequence, in [13] Das and Savas investigated the similar type problem that are proposed in [25] for metric valued sequences by considering the notion of natural density of weight , which was introduced in [4].…”

Section: Introductionmentioning

confidence: 99%

“…If (n(k)) is an infinite strictly increasing sequence of natural numbers andx = (x n ) ∈L, then we writex = (x n(k) ) and Kx = {n(k) : k ∈ N}. A subsequencex is called an I-dense subsequence ofx, if Kx is an I-dense subset of N. (cf [13]…”

mentioning

confidence: 99%

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On generalized statistical convergence and boundedness of Riesz space-valued sequences

Pal

Chakraborty

2019

Filomat

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We consider the notion of generalized density, namely, the natural density of weight recently introduced in [4] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Also we consider similar types of results for the case of generalized statistically bounded sequence. Some results are further obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of Riesz spaces extending the recent results in [13].

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Section: Results For Weighted Densitysupporting

confidence: 69%

Section: Results For F-densitymentioning

confidence: 57%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On generalized statistical convergence and boundedness of Riesz space-valued sequences

Pal

Chakraborty

2019

Filomat

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“…After the study [1], the concept of g-density was applied to various problems related to sequences and interesting results were obtained in [4,7,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

Weighted Statistical Convergence of Real Valued Sequences

Adem

Altınok

2020

FU Math Inform

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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.

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Erdős–Ulam ideals vs. simple density ideals

Kwela

2018

Journal of Mathematical Analysis and Applications

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The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one.Consider a function g : ω → [0, ∞) such that limn→∞ g(n) = ∞ and n g(n)does not converge to 0. Then the family Zg = {A ⊆ ω : limn→∞ card(A∩n)= 0} is an ideal called simple density ideal (or ideal associated to upper density of weight g). We compare this class of ideals with Erdős-Ulam ideals. In particular, we show that there are ⊑-antichains of size c among Erdős-Ulam ideals which are and are not simple density ideals (in [12] it is shown that there is also such an antichain among simple density ideals which are not Erdős-Ulam ideals).We characterize simple density ideals which are Erdős-Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős-Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal I, asserts that given B ∈ I and C ⊆ ω with card(C ∩ n) ≤ card(B ∩ n) for all n, we have C ∈ I. This notion is inspired by [3] and is later applied in [12] for a partial solution of [15, Problem 5.8].Finally, we pose some open problems.

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On Generalized Statistical and Ideal Convergence of Metric-Valued Sequences

Cited by 8 publications

References 18 publications

On generalized statistical convergence and boundedness of Riesz space-valued sequences

On generalized statistical convergence and boundedness of Riesz space-valued sequences

Weighted Statistical Convergence of Real Valued Sequences

Erdős–Ulam ideals vs. simple density ideals

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