In this paper, we study quasiconformal extensions of harmonic mappings. Utilizing a complex parameter, we build a bridge between the quasiconformal extension theorem for locally analytic functions given by Ahlfors [‘Sufficient conditions for quasiconformal extension’, Ann. of Math. Stud.79 (1974), 23–29] and the one for harmonic mappings recently given by Hernández and Martín [‘Quasiconformal extension of harmonic mappings in the plane’, Ann. Acad. Sci. Fenn. Math.38 (2) (2013), 617–630]. We also give a quasiconformal extension of a harmonic Teichmüller mapping, whose maximal dilatation estimate is asymptotically sharp.

This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of $p$-groups from different coclass families that have equivalent Quillen categories.

From the mid-1990s onwards, the main focus of L. G. Kovács’ research was on Lie powers. This brief survey presents some of the key results on Lie powers obtained by Kovács and his collaborators, and discusses some subsequent developments and applications of this work.

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

In this paper, for the convolution and convex combination of harmonic mappings, the radii of univalence, full convexity and starlikeness of order $\unicode[STIX]{x1D6FC}$ are explored. All results are sharp. By way of application, the univalent radius and the Bloch constant of the convolution of two bounded harmonic mappings are obtained.

The work of L. G. (Laci) Kovács (1936–2013) gave us a deeper understanding of permutation groups, especially in the O’Nan–Scott theory of primitive groups. We review his contribution to this field.

We present a new construction of crossed-product duality for maximal coactions that uses Fischer’s work on maximalizations. Given a group $G$ and a coaction $(A,\unicode[STIX]{x1D6FF})$ we define a generalized fixed-point algebra as a certain subalgebra of $M(A\rtimes _{\unicode[STIX]{x1D6FF}}G\rtimes _{\,\widehat{\unicode[STIX]{x1D6FF}}}G)$, and recover the coaction via this double crossed product. Our goal is to formulate this duality in a category-theoretic context, and one advantage of our construction is that it breaks down into parts that are easy to handle in this regard. We first explain this for the category of nondegenerate *-homomorphisms and then, analogously, for the category of $C^{\ast }$-correspondences. Also, we outline partial results for the ‘outer’ category, which has been studied previously by the authors.

Given two $C^{\ast }$-correspondences $X$ and $Y$ over $C^{\ast }$-algebras $A$ and $B$, we show that (under mild hypotheses) the Cuntz–Pimsner algebra ${\mathcal{O}}_{X\otimes Y}$ embeds as a certain subalgebra of ${\mathcal{O}}_{X}\otimes {\mathcal{O}}_{Y}$ and that this subalgebra can be described in a natural way in terms of the gauge actions on ${\mathcal{O}}_{X}$ and ${\mathcal{O}}_{Y}$. We explore implications for graph algebras, crossed products by $\mathbb{Z}$, crossed products by completely positive maps, and give a new proof of a result of Kaliszewski, Quigg, and Robertson related to coactions on correspondences.

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

We discuss some of the work of Laci Kovács on representation theory and related topics.

Using a result of Warnaar, we prove a number of single- and multi-sum identities in the spirit of Ramanujan’s partial theta identities, but with partial indefinite binary theta functions in the role of partial theta functions. We also calculate the corresponding residual identities and use a result of Ji and Zhao to recast our identities in terms of indefinite ternary theta functions.

We study the existence of a weak solution of a nonlocal problem

$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$

${\mathcal{L}}_{k}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x1D6FA}$

$\mathbb{R}^{n}$

$n>2s$

$s\in (0,1)$

$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$

$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$

$h:\mathbb{R}\rightarrow \mathbb{R}$

This is a short account of some of the work of L. G. (Laci) Kovács on varieties of groups.

Skew Boolean algebras for which pairs of elements have natural meets, called intersections, are studied from a universal algebraic perspective. Their lattice of varieties is described and shown to coincide with the lattice of quasi-varieties. Some connections of relevance to arbitrary skew Boolean algebras are also established.

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.