Throughout this paper, r will denote a prime. Our object in this paper and another is to describe the finite irreducible subgroups of L(r):=SL(r, [Copf ]), where [Copf ] denotes the field of complex numbers. The cases for small degrees are known (see [14] for r=2 and r=3; [1] and [23–25] for r=5 and r=7; [13, 18]). The imprimitive irreducible subgroups of L(r) are necessarily monomial because r is a prime, and we describe these in another paper [4]. In this paper, we show how the classification of finite simple groups can be used to classify the finite primitive groups of prime degree. Much of the necessary work has already been done by V. Landazuri and G. M. Seitz (see Lemma 1.3 below).

The first nontrivial DCE (2-computably enumerable) Turing approximation to the class of computably enumerable degrees is obtained. This depends on the following extension of the splitting theorem for the DCE degrees. For any DCE degree ${\bf a}$ and any computably enumerable degree ${\bf b}$ , if ${\bf b} < {\bf a}$ , then there are DCE degrees ${\bf x_0}, {\bf x_1}$ such that ${\bf b} < {\bf x_0}, {\bf x_1} < {\bf a}$ and ${\bf a} = {\bf x_0} \lor {\bf x_1}$ . The construction is unusual in that it is incompatible with upper cone avoidance.

Fundamental solutions to second-order parabolic and elliptic equations with lower-order terms are studied. New sufficient conditions on the coefficients of the lower-order terms are given that guarantee the validity of two-sided Gaussian bounds on the fundamental solution to the parabolic equation, locally and globally in time. As an application, the problem of existence and non-existence of positive solutions to a class of semi-linear equations is studied.

Recently Bergelson and Leibman proved an extension to van der Waerden's theorem involving polynomials. They also generalised the Hales–Jewett theorem in a similar way. The paper presents short and purely combinatorial proofs of those results.

Commutative rings for which every ideal can be written as a finite product of invertible and prime ideals are characterized.

Let [Gscr ] be a family of functions analytic in a domain D in the complex plane. It is proved that [Gscr ] is a normal family, provided that for each f∈[Gscr ], there exists k = k(f) > 1 such that the kth iterate fk has no repulsive fixpoint in D. A new proof of a result of Bergweiler and Terglane concerning the dynamics of entire functions is also given.

It is shown that there exists a non-separable reflexive Banach space on which every bounded linear operator is the sum of a scalar multiple of the identity operator and an operator of separable range. There is a strong sense that such a Banach space has as few operators as its linear and topological properties allow.

We consider bihomogeneous polynomials on complex Euclidean spaces that are positive outside the origin and obtain effective estimates on certain modifications needed to turn them into squares of norms of vector-valued polynomials on complex Euclidean space. The corresponding results for hypersurfaces in complex Euclidean spaces are also proved. The results can be considered as Hermitian analogues of Hilbert's seventeenth problem on representing a positive definite quadratic form on $\mathbb{R}^n$ as a sum of squares of rational functions. They can also be regarded as effective estimates on the power of a Hermitian line bundle required for isometric projective embedding. Further applications are discussed.

Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3-dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [17, 22], monoidal categories with additional structure [24], and modular functors [14, 23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalisation of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites $f\,{\circ}\,F$ with isolated singularities is studied, where $f\,{:}\,Y\,{\longrightarrow}\,\mathbb{C}$ is a function with (possibly non-isolated) singularity and $F:X\,{\longrightarrow}\, Y$ is a map into the domain of $f$, and $F$ only is deformed. The corresponding $T^1(F)$ is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that \[\tau=\mu(f\circ F)-\beta_0+\beta_1\], where $\tau$ is the length of $T^1(F)$ and $\beta_i$ is the length of $\mbox{Tor}_i^{\cal{O}_Y}({\cal O}_Y{/}J_f,\cal{O}_X)$. This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When $f$ has Cohen–Macaulay singular locus (for example when $f$ is the determinant function), relations between $\tau$ and the rank of the vanishing homology of the zero locus of $f\circ F$ are obtained.

The distribution of binomial coefficients in residue classes modulo prime powers and with respect to the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions depending on blocks of digits with respect to q-ary expansions are established.

Certain linear operators from a Banach algebra $A$ into a Banach $A$-bimodule $X$, which are called approximately local derivations, are studied. It is shown that when $A$ is a ${\rm C^*}$-algebra, a Banach algebra generated by idempotents, a semisimple annihilator Banach algebra, or the group algebra of a SIN or a totally disconnected group, bounded approximately local derivations from $A$ into $X$ are derivations. This, in particular, extends a result of B. E. Johnson that ‘local derivations on ${\rm C^*}$-algebras are derivations’ and provides an alternative proof of it.

A construction of the Monster simple group is described implicitly as $196882\,{\times}\,196882$ matrices over the field of 3 elements.

Pansu has shown that the growth function of every virtually nilpotent group Γ with respect to any finite generating set has asymptotics γ(n)∼αnd, where d is the degree of growth of Γ. The paper refines his result in the special case of 2-step nilpotent groups to obtain γ(n)=αnd +O(nd−1).

Büchi's problem asked whether there exists an integer $M$ such that the surface defined by a system of equations of the form

$$x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2,\quad n=2,\dotsc, M-1,$$

has no integer points other than those that satisfy $\pm x_n=\pm x_0+n$ (the $\pm$ signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system $Q=(q_1,\dotsc,q_r)$ of integral quadratic forms and an arbitrary $r$-tuple $B=(b_1,\dotsc,b_r)$ of integers, whether $Q$ represents $B$ (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171–194). Thus it would imply the following strengthening of the negative answer to Hilbert's tenth problem: the positive-existential theory of the rational integers in the language of addition and a predicate for the property ‘$x$ is a square’ would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi's problem remains open.

In this paper we prove the following:

1. an analogue of Büchi's problem in rings of polynomials of characteristic either 0 or $p\geq17$ and for fields of rational functions of characteristic 0; and

2. an analogue of Büchi's problem in fields of rational functions of characteristic $p\geq19$, but only for sequences that satisfy a certain additional hypothesis.

Let $F$ be a field of characteristic either 0 or at least 17 and let $t$ be a variable. Let $L_{t}$ be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property ‘$x$ is a square’ and symbols for multiplication by each element of the image of $\mathbb{Z}[t]$ in $F[t]$. Let $R$ be a subring of $F(t)$, containing the natural image of $\mathbb{Z}[t]$ in $F(t)$. Assume that one of the following is true:

1. $R\subset F[t]$;

2. the characteristic of $F$ is either 0 or $p\geq19$.

Troels Jørgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in ‘most’ cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty (see Theorem 3.1). We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups (see Corollary 3.3). These results are extensions of similar results for purely loxodromic groups which can be found in [4]. Thurston [32] previously established these results in the case when the Kleinian groups are freely indecomposable (see also Ohshika [24, 25, 27]). Using different techniques from ours, Ohshika [26] has proven versions of these results for purely loxodromic function groups.

An ovoid in a 3-dimensional projective geometry PG(3, q) over the field GF(q), where q is a prime power, is a set of q2+1 points no three of which are collinear. Because of their connections with other combinatorial structures ovoids are of interest to mathematicians in a variety of fields; for from an ovoid one can construct an inversive plane [3], a generalised quadrangle [11], and if q is even, a translation plane [13]. In fact the only known finite inversive planes are those arising from ovoids in projective spaces (see [2]). Moreover, there are only two classes of ovoids known, namely the elliptic quadric and, for q even and not a square, the Tits ovoids; these will be described in the next section. If the field order q is odd, then it was shown by Barlotti and Panella (see [3, 1.4.50]) that the only ovoids are the elliptic quadrics. This paper contains a geometrical characterisation of the two known classes of ovoids.

In the study of discrete groups it is important to find conditions for a group to be discrete. Given a discrete subgroup of Möbius transformations containing a parabolic element with fixed point ∞, a classical result, called Shimizu's lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ∞. Recently Parker [8] has shown that if a discrete subgroup G of PU(1, n; C) contains a Heisenberg translation g, then any element of G not sharing a fixed point with g has an isometric sphere whose radius is bounded above by a function of the translation length of g at its centres. Parker's theorem is considered as a generalization of Shimizu's lemma. In [1] Basmajian and Miner have independently obtained qualitatively similar results for discrete subgroups of PU(1, 2; C) by using their stable basin theorem.

The purpose of this paper is twofold. First we improve the stable basin theorem, and second we show that under some conditions Parker's theorem yields the discreteness conditions of Basmajian and Miner for groups with a Heisenberg translation. The latter answers a question posed in [8].

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.

Let d be a fixed positive integer, and ε>0. We show that there is a natural number M such that if G is any graph with m[ges ]M edges and maximum degree at most d, then the harmonious chromatic number h(G) satisfies

formula here

In the Kourovka Notebook [7] Khukhro posed a conjecture on the structure of finite p-groups admitting an automorphism of order p.