We define the total character $\tau$ of a finite group $G$ as the sum of all its irreducible characters. A question of K. W. Johnson asks whether the total character of a finite group can be expressed as a polynomial with integer coefficients in some irreducible character $\chi$ of $G$. We show that in the case of dihedral groups of twice odd order the question has an affirmative answer and we give the explicit polynomial.

A "double'' version of the C-S inequality in any complex pre-Hilbert space is given, along with some numerical applications.

Let ${\mathcal P}$ denote the set of all Pythagorean triples $\{ (a,b,c) \in {\mathbb Z} ^3 : a^2 + b^2 = c^2 \}$, and let ${\mathcal P}_n = \{ (a,b,c) \in {\mathcal P} : c-b = n \}$, for $n \ne 0$, and ${\mathcal P} _0 = \{ (0,j,j) : j \in {\mathbb Z} \}$. It is shown that the ring operations defined by A. Grytczuk on ${\mathcal P} _n$'s are determined by shifts and an injection acting from suitable subsets of ${\mathbb Z} + i {\mathbb Z}$ into ${\mathbb Z} ^3$ (Section 2), and that all ${\mathcal P} _n$'s are distributive lattices (Theorem 2). The ring and the lattice structures of $\Pi = \{ (a,b,c) \in {\mathcal P} : a = 2xy,\ b = x^2 - y^2,\ c = x^2 + y^2 \}$ and some of its subsets are discussed in Theorems 3, 4, and 5.

Starting with a continuous map $\pi \colon E \to B$, we define several categories of spaces of fiber transferring maps with respect to $\pi$ and investigate the extent to which some basic properties of $\pi$ are inherited by the induced map $p$ on the above mentioned spaces.

In this paper the Littlewood-Ross paradox is discussed. Some questions are raised regarding the most common resolution of this paradox and an alternative resolution is proposed.