### A class of conservative four-dimensional matrices.

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In this paper, we define some classes of double sequences over $n$-normed spaces by means of an Orlicz function. We study some relevant algebraic and topological properties. Further some inclusion relations among the classes are also examined.

First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space. Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter...

Double sequences are important extension of the ordinary notion of a sequence. In this article we formalized three types of limits of double sequences and the theory of these limits.

In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In...

In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.

In 1939 Agnew presented a series of conditions that characterized the oscillation of ordinary sequences using ordinary square conservative matrices and square multiplicative matrices. The goal of this paper is to present multidimensional analogues of Agnew’s results. To accomplish this goal we begin by presenting a notion for double oscillating sequences. Using this notion along with square RH-conservative matrices and square RH-multiplicative matrices, we will present a series of characterization...

The notion of oscillation for ordinary sequences was presented by Hurwitz in 1930. Using this notion Agnew and Hurwitz presented regular matrix characterization of the resulting sequence space. The primary goal of this article is to extend this definition to double sequences, which grants us the following definition: the double oscillation of a double sequence of real or complex number is given P-lim sup(m,n)→∞;(α,β)→∞|S m,n-S α,β|. Using this concept a matrix characterization of double oscillation...