Methods of Functional Analysis
and Topology

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.

Let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$ into itself. Given $A,B\in \mathcal{L}(H)$, the generalized derivation $\delta_{A,B}\in \mathcal{L}(\mathcal{L}(H))$ is defined by $\delta_{A,B}(X)=AX-XB$. For hyponormal operators $A$ and $B$, L.A. Fialkow gave necessary conditions for which the range of $\delta_{A,B}$ is norm closed in $\mathcal{L}(H)$ [13]. In this paper, we present necessary or sufficient conditions under which the generalized derivation $\delta_{A,B}$ has closed range, where $A$ and $B$ are $p$-hyponormal operators. We give a characterization for the range of $\delta_{A,B}$ to have closed range in the case when $A$ and $B$ are cyclic subnormal operators.

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.