Direct complements in a module $M$ are said to be almost (essentially) unique if, whenever $M = A \oplus B = A \oplus C$, $(B+C)/B$ is small in $M/B$ ($B \cap C$ is essential in $B$). The module $M$ is said to be a DCAU-module (DCEU-module) if direct complements of $M$ are almost (essentially) unique. We determine the structure of both DCAU- and DCEU-modules over discrete valuation rings. When $R$ is a non-local Dedekind domain, we describe the structure of the torsion part of a DCAU-$R$-module $M$ (and of a DCEU-$R$-module $N$) which turns out to be a direct summand of $M$ (of $N$). Moreover, we investigate the class of rings $R$ for which every right DCAU-$R$-module is DCEU. A ring of this type will be called a right AE-ring. Analogous to this class of rings, we shed some light on right EA-rings (i.e., rings $R$ for which every right DCEU-$R$-module is DCAU). Among other results, we show that every right AE-ring is right Bass and every commutative EA-ring is perfect. We provide examples to delineate the concepts and results.

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup $K_{(2,2)}$ of the Thompson group $F$, each containing Jones's 3-colorable subgroup $\mathcal{F}$, can be characterized as stabilizer subgroups. Additionally, we show that the $\vec{F}$-index, an elementary knot invariant introduced thanks to Jones's construction of knots from Thompson groups, may increase by at most 3 after changing the orientation of a knot.

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a $u$-$S$-Artinian module if there is $s\in S$ such that any descending chain of submodules of $M$ is $S$-stationary with respect to $s$. The notion of $u$-$S$-Artinian modules is characterized in terms of ($S$-MIN)-conditions and $u$-$S$-cofinite properties. A ring $R$ is said to be $u$-$S$-Artinian if $R$ itself is a $u$-$S$-Artinian module. It is then shown that any $u$-$S$-semisimple ring is $u$-$S$-Artinian. It is proved that a ring $R$ is $u$-$S$-Artinian if and only if $R$ is $u$-$S$-Noetherian, the $u$-$S$-Jacobson radical $\operatorname{Jac}_S(R)$ of $R$ is $S$-nilpotent and $R/\operatorname{Jac}_S(R)$ is a $u$-$S/\operatorname{Jac}_S(R)$-semisimple ring. Besides, some examples are given to distinguish Artinian rings, $u$-$S$-Artinian rings, and $S$-Artinian rings.

We establish several sufficient conditions under which a locally-integrable function $f:\mathbb{R}^n \to \mathbb{R}$ represents a positive-definite distribution. In particular, we consider functions of the form $f(\|x\|)$, where $\|\cdot\|$ is a fixed norm in $\mathbb{R}^n$.