Let $\mathrm{R}$ be a ring, ${\sigma }_1$ an automorphism of $\mathrm{R}$ and ${\delta }_1$ a ${\sigma }_1$-derivation of $\mathrm{R}$. Let ${\sigma }_2$ be an automorphism of $O_1\left(R\right)=R[x;{\sigma }_{1,}{\delta }_1]$, and ${\delta }_2$ be a ${\sigma }_2$-derivation of $O_1\left(R\right)$. Let $S\subseteq Z\left(O_1\right(R\left)\right)$, the center of $O_1\left(R\right)$. Then it is proved that ${\sigma }_i$ is identity when restricted to $S$, and ${\delta }_i$ is zero when restricted to $S$; $i=1,2$. The result is proved for iterated extensions also.

Let $X_i,i\ge 1$ be a sequence of random variables with continuous distribution functions and let $\left\{N\right(t),t\ge 0\}$ be a random counting process. Denote by $X_{i:N\left(t\right)},i\le N\left(t\right)$ the $i$-th lower order statistics of $X_1,\dots ,X_{N\left(t\right)},t\ge 0$ and define a point process in $\mathrm{ℝ}$ by ${\mathbf{M}}_{t,m}(·):=\sum _{i=1}^{N\left(t\right)}\mathbf{1}(X_{N\left(t\right)-m+1:N\left(t\right)}-X_i\in ·),m\in \mathrm{\mathbb{N}}$. In this paper we derive distributional and asymptotical results for ${\mathbf{M}}_{t,m}(·)$. For special marginals of the point process we retrieve some general results for the number of $m$-th near-extremes.

Let a mapping $f:X\to Y$ between continua $X$ and $Y$ be given. We shall prove: a) if the induced mapping $2^f:2^X\to 2^Y$ is light, then $w\left(X\right)=w\left(Y\right)$. In particular, if $Y$ is metrizable, then $X$ is metrizable, b) if the induced mapping $C\left(f\right):C\left(X\right)\to C\left(Y\right)$ is light and $X$ is a $\mathrm{D}$-continuum, then $w\left(X\right)=w\left(Y\right)$.

Giuga has conjectured that if the sum of the $(n-1)$-st powers of the residues modulo $n$ is $-1\left(\text{mod }n\right)$, then $n$ is $1$ or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if $\phi \left(n\right)$ divides $n-1$, with $\phi$ being Euler’s function, must it be true that $n$ is $1$ or prime. No examples are known, but a composite number with this property must be a Carmichael number. We show that there are infinitely many Carmichael numbers $n$ that are not counterexamples to Giuga’s conjecture and also do not satisfy $\phi \left(n\right)|n-1$.