Let $\unicode[STIX]{x1D6FA}$ be a domain in $\mathbb{R}^{m}$ with nonempty boundary. In Ward [‘On essential self-adjointness, confining potentials and the $L_{p}$-Hardy inequality’, PhD Thesis, NZIAS Massey University, New Zealand, 2014] and [‘The essential self-adjointness of Schrödinger operators on domains with non-empty boundary’, Manuscripta Math.150(3) (2016), 357–370] it was shown that the Schrödinger operator $H=-\unicode[STIX]{x1D6E5}+V$, with domain of definition $D(H)=C_{0}^{\infty }(\unicode[STIX]{x1D6FA})$ and $V\in L_{\infty }^{\text{loc}}(\unicode[STIX]{x1D6FA})$, is essentially self-adjoint provided that $V(x)\geq (1-\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA}))/d(x)^{2}$. Here $d(x)$ is the Euclidean distance to the boundary and $\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$ is the nonnegative constant associated to the $L_{2}$-Hardy inequality. The conditions required for a domain to admit an $L_{2}$-Hardy inequality are well known and depend intimately on the Hausdorff or Aikawa/Assouad dimension of the boundary. However, there are only a handful of domains where the value of $\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$ is known explicitly. By obtaining upper and lower bounds on the number of cubes appearing in the $k\text{th}$ generation of the Whitney decomposition of $\unicode[STIX]{x1D6FA}$, we derive an upper bound on $\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x1D6FA})$, for $p>1$, in terms of the inner Minkowski dimension of the boundary.

Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.

We study the free Lie ring of rank $2$ in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length $3$. This is the quotient $L/([\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime })$ with $c\geqslant 2$ where $L$ is the free Lie ring of rank $2$, $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })$ is the $c$th term of the lower central series of the derived ideal $L^{\prime }$ of $L$, and $L^{\prime \prime \prime }$ is the third term of the derived series of $L$. We show that the quotient $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })+L^{\prime \prime \prime }/[\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime }$ is a direct sum of a free abelian group and a torsion group of exponent $c$. We exhibit an explicit generating set for the torsion subgroup.

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that

$$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$

This note furnishes an example showing that the main result (Theorem 4) in Toumi [‘When lattice homomorphisms of Archimedean vector lattices are Riesz homomorphisms’, J. Aust. Math. Soc. 87 (2009), 263–273] is false.

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ , which is closely related to the cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{n}(x)$ and to primitive prime divisors of $q^{n}-1$ . Our definition of $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$ , we provide an algorithm for determining all pairs $(n,q)$ with $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)\leq cn^{k}$ . This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$ -group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$ -adic vector space associated with  $M$ . This leads to our second variation, which is a study of the finite linear groups that can arise.

It is shown that for many branch groups $G$ the action on the ambient tree can be interpreted in $G$ , in the sense of first-order model theory.