The purpose of this paper is to prove the following result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping, such that $2T\left(x^2\right)=T\left(x\right)x+xT\left(x\right)$ holds for all $x\in R$. In this case $T$ is left and right centralizer.

We show that if $X$ is a subspace of a linearly ordered space, then $C_k\left(X\right)$ is a Baire space if and only if $C_k\left(X\right)$ is Choquet iff $X$ has the Moving Off Property.

A space $X$ is if $X$ is either weakly pseudocompact or Lindelöf locally compact. We prove that if $X$ is a generalized linearly ordered space, and either (i) each proper open interval in $X$ is truly weakly pseudocompact, or (ii) $X$ is paracompact and each point of $X$ has a truly weakly pseudocompact neighborhood, then $X$ is truly weakly pseudocompact. We also answer a question about weakly pseudocompact spaces posed by F. Eckertson in [Eck].

 Let $F_i=1,...,N$ be affine mappings of ${ℝ}^n$. It is well known that if there exists j ≤ 1 such that for every ${\sigma }_1,...,{\sigma }_j\in 1,...,N$ the composition (1) $F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}$ is a contraction, then for any infinite sequence ${\sigma }_1,{\sigma }_2,...\in 1,...,N$ and any $z\in {ℝ}^n$, the sequence (2)$F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^n$ and any $\sigma ={\sigma }_1,{\sigma }_2,...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_1,{\sigma }_2,...\in \Sigma$ the composition (1) is a contraction....

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d₁\left(x\right)+1/\left(d₁\left(x\right)d₂\left(x\right)\right)+...+1/(d₁\left(x\right)d₂\left(x\right)...d_n\left(x\right))+...$, where $d_j\left(x\right),j\ge 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j+1}\left(x\right)\ge d_j\left(x\right)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $li{m}_{n\to \infty }{d}_n^{1/n}\left(x\right)=e.$He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $di{m}_{H}x\in (0,1]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_H$ to denote...