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

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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].

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).

Let C be a smooth proper curve of genus 2 over an algebraically closed field k. Fix a Weierstrass point ∞in C(k) and identify C with its image in its Jacobian J under the Albanese embedding that uses ∞ as base point. For any integer N[ges ]1, we write JN for the group of points in J(k) of order dividing N and J*N for the subset of JN of points of order N. It follows from the Riemann–Roch theorem that C(k)∩J2 consists of the Weierstrass points of C and that C(k)∩J*3 and C(k)∩J* are empty (see [3]). The purpose of this paper is to study curves C with C(k)∩J*5 non-empty.

Consider an analytic germ f:(Cm, 0)→ (C, 0) (m[ges ]3) whose critical locus is a 2-dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [8], respectively one [12]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f:(C4, 0)→(C, 0), defined by f(x1, x2, x3, x4) = x1x2x3x4 has the homotopy type of S1×S1×S1. On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition; see for example [9–12, 17, 18].

In the last few years different types of ‘bouquet theorems’ have appeared. Some of them deal with germs f:(X, x)→(C, 0) where f defines an isolated singularity. In some cases, similarly to the Milnor case [8], F has the homotopy type of a bouquet of (dim X−1)-spheres, for example when X is an icis [2], or X is a complete intersection [5]. Moreover, in [13] Siersma proved that F has a bouquet decomposition F∼F0∨Sn∨…∨Sn (where F0 is the complex link of (X, x)), provided that both (X, x) and f have an isolated singularity. Actually, Siersma conjectured and Tibăr proved [16] a more general bouquet theorem for the case when (X, x) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F∼∨iFi, where the Fi are repeated suspensions of complex links of strata of X. (If (X, x) has the ‘Milnor property’, then the result has been proved by Lê; for details see [6].)

In our situation, the space-germ (X, x) is smooth, but f has big singular locus. Surprisingly, for dim Sing f−1(0)[les ]2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma's paper [12], where dim Sing f−1(0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as explained in Corollary 2.4). In the present paper, we attack the dim Sing f−1(0) = 2 case. In our investigation some results of Zaharia are crucial [17, 18].

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.

A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all $4$ -dimensional quadratic division algebras over a square-ordered field $k$ is shown to be equivalent to the problem of finding normal forms for all pairs $(X, Y)$ of $3 \times 3$ matrices over $k, X$ being antisymmetric and $Y$ being positive definite, under simultaneous conjugation by ${\rm SO}_3(k)$ . A solution is derived for the subproblem of this matrix pair problem defined by requiring $Y+Y^t$ to be orthogonally diagonalizable. The classifying list is given in terms of a $9$ -parameter family of configurations in $k^3$ , formed by a pair of points and an ellipsoid in normal position.

Each $4$ -dimensional quadratic division algebra $A$ over a square-ordered field $k$ is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism $\alpha$ of its purely imaginary hyperplane. Calling A diagonalizable in case $\alpha$ is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable $4$ -dimensional quadratic division $k$ -algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those $4$ -dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real $4$ -dimensional quadratic division algebras.

Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a $4$ -parameter family of pairs of definite $3 \times 3$ matrices over $k$ , embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.

It is proved that quadratic systems of autonomous differential equations in the plane admit forty-five topologically distinct phase configurations at infinity. The related conditions on the coefficients are given in affine invariant form.

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

Some subnormal embeddings of groups into groups with additional properties are established, and in particular, embeddings of countable groups into two-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, H. Neumann and G. Higman on embeddings of that type. Through the consideration of subnormal embeddings of finite groups into finite groups with additional properties, a result of H. Heineken and J. Lennox is illustrated.

It is proved that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof is that the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F).

Let A be a Banach algebra, and let D: A → A* be a continuous derivation, where A* is the topological dual space of A. The paper discusses the situation when the second transpose D**: A** → (A**)* is also a derivation in the case where A** has the first Arens product.

A computer construction of the Monster is used to prove that PSL(2, 59) is a subgroup of the Monster.

Conjecture: Let [Mscr ] and [Nscr ] be two matroids (possibly of infinite ranks) on the same set S. Then there exists a set I independent in both [Mscr ] and [Nscr ], which can be partitioned as I=H∪K, where sp[Mscr ](H)∪sp[Nscr ](K)=S. This conjecture is an extension of Edmonds' matroid intersection theorem to the infinite case. We prove the conjecture when one of the matroids (say [Mscr ]) is the sum of countably many matroids of finite rank (the other matroid being general). For the proof we have also to answer the following question: when does there exist a subset of S which is spanning for [Mscr ] and independent in [Nscr ]?

All rings in this paper are commutative, with identity. If the ring R is either finitely generated, or semi-local (by which I mean Noetherian and possessing only finitely many maximal ideals), then R has only a finite number of ideals of each finite index. Thus it makes sense to study the function n[map ]an(R), where an(R) denotes the number of ideals of index n in R. In an earlier note [10], I determined an(R) for certain 2-dimensional domains R. The results to be discussed here are less precise, but more general; in particular, they deal with submodules of a finitely generated module. This makes it possible to translate them into results about the growth of normal subgroups in certain metabelian groups, the original motivation for this work: for example, we determine the finitely generated metabelian groups with ‘polynomial normal subgroup growth’ (Theorem 6.1).

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.

The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F' = f almost everywhere and there exists a differentiable function H such that F – H is absolutely continuous. They strengthen this definition by proving that F – H can have arbitrary small variation. Nevertheless, their definition still needs a choice of a differentiable function.

Estimates are given for the exponential sum $\sum_{x=1}^p \exp(2\pi i f(x)/p)$, $p$ a prime and $f$ a nonzero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod $p$, namely the smallest $\gamma$ such that $f(x_1)+\cdots +f(x_{\gamma})\equiv N$ (mod $p$) is solvable in integers for any $N$.

The integral closure and the Buchsbaum-Rim multiplicity are computed of a wide class of submodules of $\mathcal O_n^p$ through suitable Newton polyhedra. The result thus obtained is an extension to submodules of $\O^p$ of the works of Yoshinaga and Saia on the characterization of Newton non-degenerate functions and ideals, respectively.

Sufficient conditions are found for stochastic convolution integrals driven by a Wiener process in a Hilbert space to belong to the Orlicz space exp $L^2$ ; standard exponential tail estimates follow from these results. Proofs are based on the extrapolation theory and are rather simple.

The paper gives an explicit classification of the finite groups which are homogeneous in the sense of Fraïssé.