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.

Let $L/K$ be a finite Galois extension of fields whose Galois group is a nonabelian simple group. It is shown that $L/K$ admits exactly two Hopf–Galois structures.

Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand–Kirillov dimension 2 over a field $F$ all have transcendence degree 1 over $F$. Finally, centralizers of elements in a finitely graded Goldie domain of Gelfand–Kirillov dimension 2 over an algebraically closed field are considered. In this case, it is shown that the centralizer of a non-scalar element is an affine commutative domain of Gelfand–Kirillov dimension 1.

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.The author was partially supported by NSF grant #DMS 9802558.

This paper proves bounds for the commutator width of a wreath product of two groups. As a corollary, it is shown that the commutator width of finite perfect linear groups of dimension 15 is unbounded. It follows that the covering number of these groups is unbounded. On the other hand, the commutator width of iterated wreath products of nonabelian finite simple groups is bounded by an absolute constant.

Algebraic higher-rank actions on connected groups are often remarkably rigid in their topological and measurable structure. In contrast to this, the author of this paper constructs uncountably many closed invariant sets and uncountably many invariant measures with positive entropy for irreducible algebraic ${\mathbb{Z}^d}$-actions on zero-dimensional groups.

The set of vectors tangent to complete simple geodesics at a Weierstrass point is investigated, and interpreted as an infinitesimal version of the Birman–Series set. Three new identities for the lengths of simple geodesics are deduced. Fundamental use is made of the properties of the elliptic involution.

Let $p$ be a monic complex polynomial of degree $n$, and let $K$ be a measurable subset of the complex plane. Then the area of $p(K)$, counted with multiplicity, is at least $\pi n ( \Area (K)/\pi )^n$, and the area of the pre-image of $K$ under $p$ is at most $\pi^{1-1/n} (\Area (K) )^{1/n}$. Both bounds are sharp. The special case of the pre-image result in which $K$ is a disc is a classical result, due to Pólya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.

This paper presents an algebraic approach to Mislin's theorem that control of $p$-fusion is equivalent to inducing a mod-$p$ cohomology isomorphism. There are consequences for the cohomology of $p$-permutation modules.

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.

Let $p$ be an $m$-homogeneous polynomial on a complex Banach space, and let $(x_n)_n$ be a bounded sequence such that when evaluated in polynomials of degree less than $m$, it converges to zero, but $p(x_n)=1$. It is proved here that there exists a basic sequence $(y_k)_k$ equivalent to a subsequence $(x_{n_k})_k$, for which $p(\sum_{k=1}^{\infty}a_ky_k)=\sum_{k=1}^{\infty}a_k^m$.

The paper proves a result on the existence of finite flat scheme covers of Deligne–Mumford stacks. This result is used to prove that a large class of smooth Deligne–Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to a classical question on Brauer groups of schemes.Research of the first author supported in part by the Wolfgang Paul program of the Humboldt Foundation; research of the second author supported in part by the University of Bologna, funds for selected research topics.

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 minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with $\pm1$ coefficients and degree at most 72 are also determined.

The authors solve the following problem: for which connected Lie groups is the group of homotopy classes of self maps commutative?

An ovoid of ${\rm PG}(3,q)$ can be defined as a set of $q^2+1$ points with the property that every three points span a plane, and at every point there is a unique tangent plane. In 2000, M. R. Brown proved that if an ovoid of ${\rm PG}(3,q)$, $q$ even, contains a conic, then the ovoid is an elliptic quadric. Generalising the definition of an ovoid to a set of $(n-1)$-spaces of ${\rm PG}(4n-1,q)$, J. A. Thas introduced the notion of pseudo-ovoids or eggs: a set of $q^{2n}+1$$(n-1)$-spaces in ${\rm PG}(4n-1,q)$, with the property that any three egg elements span a $(3n-1)$-space and at every egg element there is a unique tangent $(3n-1)$-space. In this paper, a proof is given that an egg in ${\rm PG}(4n-1,q)$, $q$ even, contains a pseudo-conic (that is, a pseudo-oval arising from a conic of ${\rm PG}(2,q^n)$) if and only if the egg is classical (that is, arising from an elliptic quadric in ${\rm PG}(3,q^n)$).

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.

Based on the classification of contact structures on the 3-torus and classical contact geometric methods, it is shown that there exist essential loops in the space of contact structures on torus bundles over the circle.

A real-valued function $f$ defined on an open, convex set $D$ of a real normed space is called $(\varepsilon,\delta)$-midconvex if it satisfies $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2} + \varepsilon|x-y| + \delta, \quad\hbox{for } x,y\in D.$$ The main result of the paper states that if $f$ is locally bounded from above at a point of $D$ and is $(\varepsilon,\delta)$-midconvex, then it satisfies the convexity-type inequality $$f(\lambda x+(1-\lambda) y) \leq \lambda f(x)+(1-\lambda)f(y)+2\delta +2\varepsilon \varphi(\lambda)|x-y| \quad\hbox{for } x,y\in D, \, \lambda\in[0,1],$$ where $\varphi:[0,1]\to{\mathbb R}$ is a continuous function satisfying $$\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda)) \le\varphi(\lambda)\le 1.4\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda))$$. The particular case $\varepsilon=0$ of this result is due to Ng and Nikodem (Proc. Amer. Math. Soc. 118 (1993) 103–108), while the specialization $\varepsilon=\delta=0$ yields the theorem of Bernstein and Doetsch (Math. Ann. 76 (1915) 514–526).

The class $P_n(I)$ of matrix monotone functions of order $n$, defined on an open interval $I$, is considered in this paper. Explicit examples are given to prove that if $I$ is different from the whole real line, then $P_{n+1}(I)$ is a proper subset of $P_n(I)$ for each natural number $n$. The subhomogeneous C*-algebras are then characterized in terms of matrix monotone functions.