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In normed linear space settings, modifying the sequential definition of continuity of an operator by replacing the usual limit "lim" with arbitrary linear regular summability methods G we consider the notion of a generalized continuity ((G1,G2)-continuity) and examine some of its consequences in respect of usual continuity and linearity of the operators between two normed linear spaces. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a series of four-dimensional matrix characterizations of statistically asymptotically equivalent double and single sequences. These characterizations begins with the presentation of definitions for asymptotically equivalent and asymptotically statistical equivalent of multiple L for double and single sequences. The type of questions we provided answers for are those of the following sort. What are the necessary and sufficient conditions on the entries A that ensure the preservation of statistically asymptotically equivalent rate of convergence. [Copyright &y& Elsevier]

This paper presents multidimensional characterizations of equivalence classes of double sequences. These characterizations will be analyzed by presenting theorems of the following type: If A is convergent preserving over $${\{x \in S^{\prime\prime}_{0} : x \;{\rm equivalent \; to}\; b\}}$$ then there exists a $${b^{\prime} \in S^{\prime\prime}_{0}}$$ such that b converges faster than b′ and A is convergent preserving over $${\{x \in S^{\prime\prime}_{0} : x \; {\rm equivalent \; to}\; b^{\prime}\}.}$$ [ABSTRACT FROM AUTHOR]

Abstract: Following the line of recent works of Das et al. (2011)  and Savas and Das (2011)  we apply the notion of ideals to strong matrix summability, and in this note, we introduce the notion of strong -summability with respect to an Orlicz function. We make certain investigations on the classes of sequences arising out of this new summability method. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to present the following multidimensional category theorem for double sequences via four-dimensional matrix transformations. The set of second category double subsequences of the double sequence is summable by a four-dimensional RH-regular matrix if and only if is -convergent. This theorem is established using multidimensional sliding hump constructions for bounded and unbounded doubles sequences. [Copyright &y& Elsevier]

Abstract: In this paper, following the line of Savas and Das (2011) , we provide a new approach to two well-known summability methods by using ideals, introduce new notions, namely, -statistical convergence and -lacunary statistical convergence, investigate their relationship, and make some observations about these classes. [Copyright &y& Elsevier]

Abstract: In this work, we generalize the lacunary almost bounded variation sequence spaces to lacunary almost -bounded variation sequence spaces. Also some topological results and inclusion relations of such spaces have been discussed. [Copyright &y& Elsevier]