It is proved that the automorphism group of a countable structure cannot be a free uncountable group. The idea is that instead of proving that every countable set of equations of a certain form has a solution, it is proved that this holds for a co-meagre family of appropriate countable sets of equations.

An $\varepsilon$ -entropy theoretical solution is given to an analytic function-theoretical version of the 13th problem formulated by D. Hilbert.

It is proved that in groups with all subgroups subnormal every nilpotent subgroup is contained in a normal nilpotent subgroup.

For each finitely presented group $Q$ , a short exact sequence $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ is produced, such that $G$ is residually finite and $N$ is finitely generated.

This paper provides a unified and elementary proof of four Thompson–Wielandt-like theorems.

By means of a simplification in a significant step of Zelmanov's solution of the restricted Burnside problem, the known bounds for the order of finite $m$ -generator groups of exponent $n$ are improved.

A relationship is established between the factorization of $24 n + 1$ and the 5-divisibility of $Q(n)$ , where $Q(n)$ is the number of partitions of $n$ into distinct parts. As an application, an abundance of infinite families of congruences for $Q(n)$ modulo powers of 5 are explicitly exhibited.

This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as $l^1(q)$ , and this definition is extended to all height $m$ polynomials as \[ l^m(q):= \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n,\, \epsilon_i \in {\bb Z},\, \vert\epsilon_i\vert \leqslant m,\, y \ne 0). \] A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of $l^m(q)$ when $q$ is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows.

THEOREM. Let $q$ be a quadratic Pisot number that satisfies a polynomial of the form $p(x) = x^2 - ax \pm 1$ , with conjugate $r$ . Let $q$ have convergents $\{C_k/D_k\}$ and let $k$ be the maximal integer such that \[ \vert D_k r - C_k\vert \leqslant m \frac{1}{1-\vert r\vert}; \] then \[ l^m (q) = \vert D_k q - C_k \vert. \]

A value related to $l(q)$ is $a(q)$ , the minimal non-zero value when all ${\pm}1$ polynomials are evaluated at $q$ . Formally, this is \[ a(q) := \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^2 + \cdots + \epsilon_n q^n,\, \epsilon_i = {\pm} 1,\, y \ne 0). \] An open question concerning how often $a(q) = l(q)$ is also answered in this paper.

The authors study the algebra of uniformly continuous holomorphic symmetric functions on the ball of ${\cal L}_p$ , investigating in particular the spectrum of such algebras. To do so, they examine the algebra of symmetric polynomials on ${\cal L}_p$ -spaces, as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the points in ${\cal L}_p$ up to a permutation.

Akemann and Anderson made a conjecture about ‘paving’ projections in finite-dimensional matrix algebras which, if true, would settle the well-known Kadison–Singer problem. Their conjecture is falsified in this paper by an explicit sequence of counterexamples.

The behaviour of the Ricci curvature along rays in a complete open manifold is examined.

Let $X$ and $Y$ be affine nonsingular real algebraic varieties. A general problem in real algebraic geometry is to try to decide when a continuous map $f : X \rightarrow Y$ can be approximated by regular maps in the space of ${\cal C}_0$ mappings from $X$ to $Y$ , equipped with the ${\cal C}_0$ topology. This paper solves this problem when $X$ is the connected component containing the origin of the real part of a complex Abelian variety and $Y$ is the standard 2-dimensional sphere.

A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor is a transitive set to which every positive nearby orbit converges; likewise, every negative nearby orbit converges to a repeller. It is shown in this paper that a generic $C^1$ vector field on a closed $n$ -manifold has either infinitely many homoclinic classes, or a finite collection of attractors (or, respectively, repellers) with basins that form an open-dense set. This result gives an approach to use in proving a conjecture by Palis. A proof is also given of the existence of a locally residual subset of $C^1$ vector fields on a 5-manifold having finitely many attractors and repellers, but infinitely many homoclinic classes.

This paper deals with better-than-uniform approximation by superharmonic functions on non-compact sets. An application of the main result improves recent work of GowriSankaran and Nersessian concerning approximation with an error bound that decays arbitrarily quickly at infinity.

Let $f$ be meromorphic of finite order in the plane, such that the second derivative $f^{\prime\prime}$ has finitely many zeros. Then $f$ has finitely many poles. This result was conjectured by the author in 1996, and an example shows that the theorem is sharp.

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras.

A quantum deformation of the classical conjugation action of ${\rm GL}(N,{\bb C})$ on the space of $N\times N$ matrices $M(N,{\bb C})$ is defined via a coaction of the quantum general linear group ${\cal O}({\rm GL}_q(N,{\bb C}))$ on the algebra of quantum matrices ${\cal O}(M_q(N,{\bb C}))$ . The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of ${\cal O}(M_q(N,{\bb C}))$ generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.

Write $G^{\ast} = G^{{\cal L}{\cal U}{\cal C}}\setminus G$ where $G^{{\cal L}{\cal U}{\cal C}}$ is the largest semigroup compactification of the locally compact group $G$ . Then the set of points of $G^{\ast}$ which are right cancellable in $G^{{\cal L}{\cal U}{\cal C}}$ is large; in fact it has an interior in $G^{\ast}$ which is dense in $G^{\ast}$ . Corollaries are given about the number of left ideals in $G^{{\cal L}{\cal U}{\cal C}}$ and the size of right ideals in the algebra ${{\cal L}{\cal U}{\cal C}}(G)^{\ast}$ .