In this paper, we introduce the concepts of $\mathcal{I}$ and $\mathcal{I^{*}}-$convergence of sequences in gradual normed linear spaces. We study some basic properties and implication relations of the newly defined convergence concepts. Also, we introduce the notions of $\mathcal{I}$ and $\mathcal{I^{*}}-$Cauchy sequences in the gradual normed linear space and investigate the relations between them.

In this paper, we present the notion of quasi-statistical convergence of triple sequences in the neutrosophic normed spaces mainly as a generalization of statistical convergence of triple sequences. We investigate a few principal properties of the newly presented notion and investigate the relationship with statistical convergence of triple sequences in the neutrosophic normed spaces. In the end, we introduce the concept of quasi-statistical Cauchy sequence of triple sequences and show that quasi-statistical Cauchy sequences for triple sequences are equivalent to quasi-statistical convergent triple sequences in the neutrosophic normed spaces.

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Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ {\mathcal{I}} -statistical supremum and ℐ {\mathcal{I}} -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ {\mathcal{I}} -statistical monotonicity and establish the condition under which an ℐ {\mathcal{I}} -statistical monotonic sequence is ℐ {\mathcal{I}} -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ {\mathcal{I}} -statistical convergence of a real-valued sequence.

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The aim of this paper is to we examine the notion of gradually rough I_((λ,μ) )-statistical convergence of double sequences in gradual normed linear spaces (GNLS). In addition, we define the concept of gradually rough I_((λ,μ) )-statistical limit set of double sequences and obtain some algebraic and topological features of this set. Theorems are proved in the light of GNLS theory approach. Results are obtained via different perspective and new examples are established to justify the counterparts and indicate existence of introduced notions. We produce significant results that present several fundamental properties of this notion. The results established in this research work supplies an exhaustive foundation in GNLS and make a significant contribution in the theoretical development of GNLS in literature. The original aspect of this study is the first wholly up-to-date and thorough examination of the features and implementations of new introduced notions in GNLS.

The aim of this paper is to investigate the intuitionistic Nörlund [Formula: see text]-statistically convergent sequence space. We present some intuitionistic fuzzy normed spaces (IFNS) in Nörlund convergent spaces. Moreover, we also put forward several topological and algebraic properties of these convergent sequence spaces.

In this paper, we introduce and investigate the concept of \(A^{\mathcal{I^{K}}}\)-summability as an extension of \(A^{\mathcal{I^{*}}}\)-summability which was recently (2021) introduced by O.H.H.~Edely, where \(A=(a_{nk})_{n,k=1}^{\infty}\) is a non-negative regular matrix and \(\mathcal{I}\) and \(\mathcal{K}\) represent two non-trivial admissible ideals in \(\mathbb{N}\). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that \(A^{\mathcal{K}}\)-summability always implies \(A^{\mathcal{I^{K}}}\)-summability whereas \(A^{\mathcal{I}}\)-summability not necessarily implies \(A^{\mathcal{I^{K}}}\)-summability. Finally, we give a condition namely \(AP(\mathcal{I},\mathcal{K})\) (which is a natural generalization of the condition \(AP\)) under which \(A^{\mathcal{I}}\)-summability implies \(A^{\mathcal{I^{K}}}\)-summability.