Permutations that have no fixed points have been known for a very long time as ‘derangements’. Under that heading Rouse-Ball [10, p. 46] puts the matter in the following charming way: ‘Suppose you have written a letter to each of n different friends, and addressed the n corresponding envelopes. In how many ways can you make the regrettable mistake of putting every letter into a wrong envelope?’ He traces this problem back to 1713 and since then it has occurred in one form or another in many elementary texts on probability theory (see for example [12, p. 57] and references cited by Rouse-Ball). Let pn be the probability that all the letters are put into the wrong envelope. It is well known that pn→e−1 as n→∞, and that, moreover, convergence is very fast. In fact, pn =[sum ]n0(−1)r/r!, so that [mid ]pn−e−1[mid ]<1/(n+1)!.

A derangement can be thought of as a fixed-point-free element of the symmetric group Sym(n). In this paper we turn our attention to eigenvalue-free elements of finite linear groups. Since an eigenvector of a linear transformation X in a vector space V corresponds to a fixed point of X in the projective space whose points are the 1-dimensional subspaces of V, eigenvalue-free invertible matrices correspond to derangements in projective groups.

On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem \[-\Delta _pu=\lambda u^{\alpha }-a(x)u^q\quad \mbox{in}\;\Omega,\qquad u|_{\partial \Omega }=0,\] where $\Omega \subset {\mathbb{R}}^N$ ($N\geq 2$) is a bounded smooth domain, $p>1$ and λ is a parameter, α, q are given constants such that $p-1<\alpha <p^*-1$, $\alpha <q$, $p^*=Np/(N-p)$ if $N > p$ and $p^*=\infty $ when $N\leq p$, and $a(x)$ is a continuous nonnegative function. Making use of the Leray–Schauder degree of a compact mapping and a priori estimates, the paper finds that the problem above possesses at least one positive solution. It also discusses the corresponding perturbed problem, where $a(x)$ is replaced by $a(x)+\epsilon$, $\epsilon>0$. The results are strikingly different from those obtained for the case $\alpha=p-1$.

The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitutionsubshifts, measure-theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. Partial answers are given to this question.

The relationship between the connections on two Riemannian manifolds that are related by a Riemannian submersion are used to characterise parallel translation on shape space in terms of its behaviour on the pre-shape sphere and so to derive a concrete procedure for the unrolling, or development, of curves in shape space onto tangent spaces. This will enable one to adapt to shape spaces the standard statistical procedures for fitting curves to Euclidean data.

The sector of analyticity of the Ornstein–Uhlenbeck semigroup is computed on the space $L^p_\mu \,{:=}\, L^p (\R^N; \mu)$ with respect to its invariant measure $\mu$. If $A\,{=}\,\Delta + Bx\cdot \nabla$ denotes the generator of the Ornstein–Uhlenbeck semigroup, then the angle $\theta_2$ of the sector of analyticity in $L^2_\mu$ is ${\pi}/{2}$ minus the spectral angle of $BQ_\infty, Q_\infty$ being the matrix determining the Gaussian measure $\mu$. The angle of analyticity in $L^p_\mu$ is then given by the formula \[\cot {\theta_p} = \frac{\sqrt{(p-2)^2+p^2(\cot\theta_2)^2}}{2\sqrt{p-1}}.\]

For each Artin group, the reduced $\eltwo$-cohomology of (the universal cover of) its ‘Salvetti complex’ is computed. This is a CW-complex which is conjectured to be a model for the classifying space of the Artin group. In the many cases when this conjecture is known to hold the calculation describes the reduced $\eltwo$-cohomology of the Artin group.

1. Some groups with the small index property

In this paper we shall use the methods of the paper [6] by Hodges, Hodkinson, Lascar and Shelah to show that the free group of countably infinite rank and certain relatively free groups of countably infinite rank have the small index property.

It is important and interesting to study harmonic functions on a Riemannian manifold. In an earlier work of Li and Tam [21] it was demonstrated that the dimensions of various spaces of bounded and positive harmonic functions are closely related to the number of ends of a manifold. For the linear space consisting of all harmonic functions of polynomial growth of degree at most d on a complete Riemannian manifold Mn of dimension n, denoted by [Hscr ]d(Mn), it was proved by Li and Tam [20] that the dimension of the space [Hscr ]1(M) always satisfies dim[Hscr ]1(M) [les ] dim[Hscr ]1(ℝn) when M has non-negative Ricci curvature. They went on to ask as a refinement of a conjecture of Yau [32] whether in general dim [Hscr ]d(Mn) [les ] dim[Hscr ]d(ℝn) for all d. Colding and Minicozzi made an important contribution to this question in a sequence of papers [5–11] by showing among other things that dim[Hscr ]d(M) is finite when M has non-negative Ricci curvature. On the other hand, in a very remarkable paper [16], Li produced an elegant and powerful argument to prove the following. Recall that M satisfies a weak volume growth condition if, for some constant A and ν,

formula here

for all x ∈ M and r [les ] R, where Vx(r) is the volume of the geodesic ball Bx(r) in M; M has mean value property if there exists a constant B such that, for any non- negative subharmonic function f on M,

formula here

for all p ∈ M and r > 0.

The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor.

The aim of this paper is to determine the minimal polynomials of p-elements in irreducible representations of quasi-simple groups with a cyclic Sylow p-subgroup. We consider both the characteristic 0 and characteristic p cases; we have no results for representations over fields of other characteristics. It suffices to describe the cases where the degree of the polynomial is less than the order of the element.

It is shown that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a consequence, a parametrization of (1,1)-knots by 4-tuples of integers is obtained. Moreover, using a representation of (1,1)-knots by the mapping class group of the twice-punctured torus, an algorithm is provided which gives the parametrization of all torus knots in $\mathbf{S}^3$.

In a previous paper, the authors laid the foundations of a theory of Schatten–von Neumann classes [Gfr ]p (0<p[les ]∞) of trilinear forms in Hilbert space. This paper continues that research. In the n-dimensional case, it is shown that the best constant dˆ that relates the Hilbert–Schmidt norm of a form with its bounded norm behaves like n. Some results are also obtained in the quasi-Banach case (0<p<1), and for two-bounded forms. Finally, the domination problem is investigated in the trilinear set-up.

A general approach is proposed to prove that the combination of expansion with bounded distortion yields strong rigidity of conjugacies.

We study the following question: given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $\Omega$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and ${\mathbb Z}^d$. We exhibit upper bounds for $\int f$ assuming geometric properties of $\Omega$ of two types: (a) packing properties of $\Omega$ and (b) spectral properties of $\Omega$. Several examples and applications of the main theorems are shown. In particular, we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating $\int_{\Omega}f$ over possibly dispersed sets solely in dependence of the given measure $m:=|\Omega|$ of $\Omega$. In this respect we show that in ${\mathbb R}$ and ${\mathbb Z}$ the integral is maximal for intervals.

Throughout this paper, we adopt the same notations as in [1, 6, 8] such as the Möbius group M(ℝn), the Clifford algebra Cn−1, the Clifford matrix group SL(2, Γn), the Clifford norm of

formula here

and the Clifford metric of SL(2, Γn) or of the Möbius group M(ℝn)

formula here

where [mid ]·[mid ] is the norm of a Clifford number and

formula here

represents fi ∈ M(ℝn), i = 1, 2, and so on. In addition, we adopt some notions in [6, 12]: the elementary group, the uniformly bounded torsion, and so on. For example, the definition of the uniformly bounded torsion is as follows.

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arc-transitive graphs. Some of these graphs have arisen as examples in studies of arc-transitive graphs with complete quotients and arc-transitive graphs which ‘almost cover’ a 2-arc transitive graph.

The question is addressed of when all pure-projective modules are direct sums of finitely presented modules. It is proved that this is the case over hereditary noetherian rings. Partial results are obtained for uniserial rings. Some of the methods are model-theoretic, and the techniques developed using these may be of interest in their own right.

Some elementary results are given on restricted sums $2^{\wedge}{\cal A}$ and $3^{\wedge}{\cal A}$ of a subset ${\cal A}\subset {\bb Z}/n{\bb Z}$ in the case when the cardinality of ${\cal A}$ , is close to $n/2$ . In the last section, some of these results are used to derive some new values of a function related to the Erdős–Ginzburg–Ziv problem.

The space of admissible vector fields, consistent with the structure of a network of coupled dynamical systems, can be specified in terms of the network's symmetry groupoid. The symmetry groupoid also determines the robust patterns of synchrony in the network – those that arise because of the network topology. In particular, synchronous cells can be identified in a canonical manner to yield a quotient network. Admissible vector fields on the original network induce admissible vector fields on the quotient, and any dynamical state of such an induced vector field can be lifted to the original network, yielding an analogous state in which certain sets of cells are synchronized. In the paper, necessary and sufficient conditions are specified for all admissible vector fields on the quotient to lift in this manner. These conditions are combinatorial in nature, and the proof uses invariant theory for the symmetric group. Also the symmetry groupoid of a quotient is related to that of the original network, and it is shown that there is a close analogy with the usual normalizer symmetry that arises in group-equivariant dynamics.

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), we shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [8].