Methods of Functional Analysis
and Topology

We introduce a concept of antisymmetric Arens regularity of Banach spaces and prove that a Banach space is Arens regular if and only if it is both symmetrically Arens regular and antisymmetrically Arens regular. We show that if a Banach space is nonreflexive, then it is not antisymmetrically Arens regular and provide an example of a symmetrically Arens regular Banach space which is not antisymmetrically Arens regular.

We prove sufficient conditions for the faithfulness of the family of cylinders generated by $\widetilde{Q}$ -expansions for the Hausdorff-Besicovitch dimension calculation. We also represent the conjecture about necessary and sufficient conditions for such a family of cylinders to be faithful. Based on these new results we study fine fractal properties of random variables with independent $\widetilde{Q}$-digits and prove exact formulae for the calculation of the Hausdorff dimension of the corresponding probability measure.

In this manuscript we define $A$-double density of a given point in $\mathbb{R}$ for double sequences. We state the relation between double density of a point and spliced double sequences and also examine limit points in view of density of a point.

In this paper two lemmas for arbitrary general functions are given and then, by iteration,we express them as a continued fraction after that by specializing the parameters the continued fraction representation for the generalized tenth order mock theta functions are obtained.