Let $R$ be an arbitrary ring and $(-)^+=\operatorname{Hom}_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.

In this paper we define lower, upper, and symmetric completeness and discuss closure of the sets in product and direct sums. In particular, we introduce suitable bases for these topologies, which leads us to investigate completeness of the direct sum and its components. Some results obtained about $X$-topologies and polars of the neighborhoods.

We prove that a finitely generated group contains a sequence of non-trivial elements that converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian. As a consequence of the methods used, we show that a finitely generated group satisfies Chu duality if and only if it is virtually abelian.

The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by translation in (measure/category) ``large'' sets has two generalizations. Miller replaces the translated sequence by a ``sequence homotopic to the identity''. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows from the functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version.

In this paper we consider the stepping-stone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow on the circle and the marginal distribution of this model. We then give a new representation for the stepping-stone model using Arratia flow and circular coalescing Brownian motion. Such a representation enables us to carry out some explicit computations. In particular, we find the distribution for the first time when there is only one type left across the circle.

We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$ \left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t), $$ where $$ f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds. $$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$ \|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t), $$ where $$ f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds. $$