In 1933 Henri Cartan proved a fundamental theorem in Nevanlinna theory, namely a generalization of Nevanlinna's second fundamental theorem. Cartan's theorem works very well for certain kinds of problems. Unfortunately, it seems that Cartan's theorem, its proof, and its usefulness, are not as widely known as they deserve to be. To help give wider exposure to Cartan's theorem, the simple and general forms of the theorem are stated here. A proof of the general form is given, as well as several applications of the theorem.

A proof is given to show that the operator Hilbert space $\operatorname{OH}$ does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge's recent result, which admits such an embedding in the non-semi-finite case.

The metric entropy of absolute convex hulls of sets in Hilbert spaces is studied for the general case when the metric entropy of the sets is arbitrary. Under some regularity assumptions, the results are sharp.

This paper describes – in terms of potential-theoretic properties – the necessary and sufficient conditions required for a bounded domain satisfying the capacity density condition to be, respectively: a John domain, a uniform domain and an inner uniform domain.

It is proved that, given a convex polytope $P$ in $\mathbb{R}^n$, together with a collection of compact convex subsets in the interior of each facet of $P$, there exists a smooth convex body arbitrarily close to $P$ that coincides with each facet precisely along the prescribed sets, and has positive curvature elsewhere.

A short proof is given that if $E$ is a super-reflexive Banach space, then $\mc B(E)$, the Banach algebra of operators on $E$ with composition product, is Arens regular. Some remarks are made on necessary conditions on $E$ for $\mc B(E)$ to be Arens regular.

The aim of this paper is to exhibit a real Paley–Wiener space sitting inside the Schwartz space, and to give a quick and simple proof of a Paley–Wiener-type theorem. A simple and elementary proof of a theorem postulated by H. H. Bang is also given.

The purpose of this paper is to show how the methods of motivic integration of Kontsevich, Denef–Loeser (Invent. Math. 135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290) and Looijenga (Astérisque 276 (2002) 267–297) can be adapted to prove the McKay–Ruan correspondence, a generalization of the McKay–Reid correspondence to orbifolds that are not necessarily global quotients.

Consider the family of exponential maps $\Ek(z)=\exp(z)+\kappa$. This paper shows that any unbounded Siegel disk $U$ of $\Ek$ contains the singular value $\kappa$ on its boundary. By a result of Herman, this implies that $\kappa\in \partial U$ if the rotation number is diophantine.

A natural topology on the space of left orderings of an arbitrary semi-group is introduced here. This space is proved to be compact, and for free abelian groups it is shown to be homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases.

In this paper, half-dimensional Einstein–Kähler submanifolds or locally reducible Kähler submanifolds are classified in a quaternion projective space.

The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach towards proving the well-known virtually Haken conjecture. As instructive examples, Seifert fibered 3-manifolds are considered in this paper. The author shows that a compact orientable Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over either the circle or a surface other than the 2-sphere or, equivalently, if it has infinite fundamental group.

Let $s:\Q\,{\longrightarrow}\,\Q$ be the Dedekind sum, given by $s(h/k)=\sum_{\nu=1}^{k-1}({\nu/k}\,{-}\,{1/2})(\{{h\nu/k}\}\,{-}\,{1/2})$ when $\gcd(h,k)\,{=}\,1$. Then for every rational $\alpha\,{\ne}\,1/12$ there are infinitely many rational $x$ such that $s(x)\,{=}\,\alpha x$. Also, the fixed points of $s$ are dense in the real line.

The methods of $p$-adic analysis are used to prove a congruence for $(pn)!/(p^{n}n!)$ modulo a power of a prime $p$.

Barry Johnson made major contributions to the theory of Banach algebras, by stimulating research on automatic continuity and cohomology in these algebras. His research on the continuous Hochschild cohomology of Banach and operator algebras led to major developments in these areas, and to a recognition of ‘amenability’ as more than simply a group-theoretic idea, but also one that is widely applicable in modern analysis.