The purpose of this paper is to introduce a new class functions called, subweakly b-continuous functions. Also, we obtain its characterizations and its basic properties.

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It is well known that the meaning of the convergence in posets stings the interest of many investigators such as R. F. Anderson, J. C. Mathews and V. Olej˘cek (see, for example [13,14]). Among others, the notions of the order-convergence and of the o2-convergence in posets were studied in details, presenting necessary and sufficient conditions under of which these convergences are topological. Many researchers give a special attention to the study of these convergences in different posets, inserting new knowledge in the classical theory of posets’s convergence. In this paper, we introduce the ideal-order-convergence in posets, proving results which are based on this notion. We insert topologies in posets and we study their properties. We also give a sufficient and necessary condition for the ideal-order-convergence in a poset to be topological. The introduction of a weaker form of the ideal-order-convergence in posets, called ideal-o2-convergence, completes our study.

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Analysis of variance (ANOVA) is tailored for independent observations. Recently, there has been considerable demand for the ANOVA of high-dimensional and dependent observations in many fields. Thus, it is important to analyze the differences among big data’s averages of areas from all over the world, such as the financial and manufacturing industries. However, the numerical accuracy of ANOVA for such observations has been inadequately developed. Thus, herein, we study the Edgeworth expansion of distribution of ANOVA tests for high-dimensional and dependent observations. Specifically, we present the second-order approximation of classical test statistics proposed for independent observations. We also provide numerical examples for simulated high-dimensional time-series data.

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