A Coxeter group $W$ is said to be rigid if, given any two Coxeter systems $(W, S)$ and $(W, S^\prime)$ , there is an automorphism $\rho: W \longrightarrow W$ such that $\rho(S) = S^\prime$ . The class of Coxeter systems $(W, S)$ for which the Coxeter graph $\Gamma_S$ is complete and has only odd edge labels is considered. (Such a system is said to be of type $K_n$ .) It is shown that if $W$ has a type $K_n$ system, then any other system for $W$ is also type $K_n$ . Moreover, the multiset of edge labels on $\Gamma_S$ and $\Gamma_{S^\prime}$ agree. In particular, if all but one of the edge labels of $\Gamma_S$ are identical, then $W$ is rigid.

Let f:M→N be a stable map between orientable 4-manifolds, where M is closed and N is stably parallelisable. It is shown that the signature of M vanishes if and only if there exists a stable map g:M→N homotopic to f which has only fold and cusp singularities. This together with results of Ando and Èliašberg shows that, in this situation, the Thom polynomials are the only obstructions to the elimination of the singularities except for the fold singularity. Also studied are some topological properties (including those of the discriminant set) of stable maps between 4-manifolds with only Ak-type singularities.

A family of transcendental meromorphic functions, fp(z), p ∈ N is considered. It is shown that, if p [ges ] 6, then the Hausdorff dimension of the Julia set of λfp satisfies dim J(λfp) [les ] 1/p, for 0 < λ < 1/6p, and dim J(λfp) [ges ] 1−(30 ln ln p/ln p), for p4p−1/105 ln p < λ < p4p−1/104 ln p. These results are used elsewhere to show that, for each d ∈ (0, 1), there exists a transcendental meromorphic function for which dim J(f) = d.

In the nineteenth century, field theory brilliantly resolved a number of questions that had taxed mathematicians for centuries; for example, ‘The circle cannot be squared’ by straight edge and compass, and solving polynomial equations by radicals is not always possible. These successes have continued to be held up as superb examples of the power of mathematical thought, and are demonstrated at an undergraduate level. The purpose of this article is to provide another such natural example which leads to a concrete realisation of the free group on 2 generators.

Given a non-atomic, finite and complete measure space $(\Omega,\Sigma,\mu)$ and a Banach space $X$, the modulus of continuity for a vector measure $F$ is defined as the function $\omega_F(t) = \sup_{\mu(E)\le t} |F|(E)$ and the space ${V}^{p,q}(X)$ of vector measures such that $t^{-1/p^\prime}\wf (t)\in L^q((0,\mu(\Omega)],dt/t)$ is introduced. It is shown that $V^{p,q}(X)$ contains isometrically $\lpq(X)$ and that $L^{p,q}(X)=V^{p,q}(X)$ if and only if $X$ has the Radon–Nikodym property. It is also proved that ${V}^{p,q}(X)$ coincides with the space of cone absolutely summing operators from $L^{p^\prime,q^\prime}$ into $X$ and the duality ${V}^{p,q}(X^*)= (L^{p^\prime,q^\prime}(X))^*$ where $1/p+1/p^\prime= 1/q+1/q^\prime=1$. Finally, $V^{p,q}(X)$ is identified with the interpolation space obtained by the real method $(V^{1}(X)$, $V^{\infty}(X))_{1/p',q}$. Spaces where the variation of $F$ is replaced by the semivariation are also considered.

The paper considers second-order, strongly elliptic, operators H with complex almost-periodic coefficients in divergence form on Rd. First, it is proved that the corresponding heat kernel is Hölder continuous and Gaussian bounds are derived with the correct small and large time asymptotic behaviour on the kernel and its Hölder derivatives. Secondly, it is established that the kernel has a variety of properties of almost-periodicity. Thirdly, it is demonstrated that the kernel of the homogenization Ĥ of H is the leading term in the asymptotic expansion of t [map ] Kt.

Smooth complete algebraic varieties defined over an algebraically closed field are divided into two classes according to the behaviour of the albanese map. Criteria are given for when a variety belongs to either class. The relationship with the Tate conjecture for abelian varieties is discussed.

The cohomology groups of some sequence algebras are considered. In particular, the amenability of the James space $J_p$ is investigated; this space was first studied by R. C. James and was shown to be a Banach algebra for the pointwise product. Then a full classification of the closed ideals in $J_p$ is given, extending a result of N. J. Laustsen. Finally, some related algebras $\alp$ are considered, and it is established when these Banach algebras are amenable.

A stage in the classification of the periodic simple finitary linear groups by J. I. Hall and others, is a partial reduction to the countable case; specifically a simple group is isomorphic to a finitary linear group if and only if its countable subgroups are isomorphic to finitary linear groups. It is natural to ask, how common is it for a group to have a faithful finitary linear (or possibly only skew linear) representation, whenever each of its countable subgroups has such a representation? In very recent work F. Leinen and O. Puglisi have shown that periodic primitive finitary linear groups are countably recognizable, see [4]. It is easy to see that every abelian group is isomorphic to a finitary linear group, for example over the subfield of the complexes [Copf ] generated by all roots of unity. Thus the first non-obvious case, at the opposite extreme to the simple groups (and also in some sense to the primitive groups), is that of the nilpotent groups. We show the following, which, it should be noted, covers the case of periodic nilpotent groups.

In this paper, we investigate two problems concerning a class of higher order uniformly elliptic operators with measurable coefficients. In our first result, we prove a stability estimate for the resolvent of such an operator under perturbation of its principal coefficients in Lp spaces. Our second result is a regularity theorem concerning the smoothing properties of the resolvent, and it provides a sufficient condition under which the first result can be applied.

The main result in this paper, Theorem 1.2, generalizes a theorem of Zerner [26] concerning sufficient conditions for the holomorphic continuability of a solution of a linear holomorphic partial differential equation across a point of a hypersurface, on one side of which it is holomorphic. The point of the new theorem is, roughly speaking, that it applies also to regular solutions of partial differential equations whose coefficients may have certain kinds of singularities. This enables us to deduce some new results (see §2) on elliptic partial differential equations in ℝ2[ratio ]Theorem 2.1 extends a result of Vekua on the size of the domain of holomorphy of solutions to elliptic equations, in the case where singularities are permitted in the coefficients; Theorem 2.2 is of an apparently novel type, showing (roughly) that under certain conditions the solution to Cauchy's problem is real-analytic in a domain whose size depends only on the principal part of the operator, which is assumed to be the Laplacian, and the Cauchy data on the real axis. (Results of this kind are very delicate, as we shall illustrate in §4 with a simple counterexample.) Theorem 2.2 is new and non-trivial even for equations with analytic coefficients, in which case though, Theorem 1.2 is not needed for the proof.

There is extensive literature concerning approximately finite-dimensional (AF) C*-algebras. For example see [1, 5]. In recent years study has focused on non-self-adjoint subalgebras of AF C*-algebras ([4, 8, 9], for example) and this paper continues that theme. We define T(m, n) to be the block upper triangular subalgebra of Mm+n with self-adjoint part equal to Mm [oplus ] Mn. The class of algebras that are considered here are norm closures of ascending chains of algebras of the form T(m, n) and are referred to as uniformly T2 algebras. This class of algebras properly contains the matroid C*-algebras of Dixmier.

Weak slice conditions are introduced and imbeddings of Sobolev spaces in various Lipschitz-type spaces are investigated.

The theory of higher-dimensional shifts of finite type is still largely an open area of investigation. Recent years have seen much activity, but fundamental questions remain unanswered. In this paper we consider the following basic question. Given a shift of finite type (SFT), under what topological mixing conditions are we guaranteed the existence of Bernoulli (or even $K$, mixing, or weakly mixing) invariant measures?

Let Γ be an anisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalues of the fractal differential operator

formula here

acting in the classical Sobolev space W˚12(Ω) where Ω is a bounded C∞ domain in the plane ℝ2 with Γ⊂Ω. Here −Δ is the Dirichlet Laplacian in Ω and trΓ is closely related to the trace operator trΓ.

Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C(Ω, E). If a ∈ C(Ω) is not constant on any open subset of Ω and aX ⊂ X, then it is shown that there is at most one F-space topology on X that makes the multiplication by a continuous. Furthermore, if [Ufr ] is a subset of C(Ω) which separates strongly the points of Ω and [Ufr ]X ⊂ X, then it is proved that there is at most one F-space topology on X that makes the multiplication by a continuous for each a ∈ [Ufr ].

These results are applied to the study of the uniqueness of the F-space topology and the continuity of translation invariant operators on the Banach space L1(G, E) for a noncompact locally compact group G and a Banach space E. Furthermore, the problems of the uniqueness of the F-algebra topology and the continuity of epimorphisms and derivations on F-algebras and some algebras of vector-valued functions are considered.

The normalized positive definite class functions $\C G\,{\longrightarrow}\,\C$ are determined for all those direct limits $G$ of finite alternating groups $\Alt(\Omega_i)$ for which the embeddings $\Alt(\Omega_i)\,{\longrightarrow}\,\Alt(\Omega_j)$ are natural in the sense that every non-trivial $\Alt(\Omega_i)$-orbit in $\Omega_j$ is natural.

We present a method for computing the 3-point genus zero Gromov–Witten invariants of the complex flag manifold $G/B$ from the relations of the small quantum cohomology algebra $QH^\ast G/B$ ($G$ is a complex semisimple Lie group and $B$ is a Borel subgroup). In [3] and [9], at least in the case $G={\rm GL}_n{\bf C}$, two algebraic/combinatoric methods have been proposed, based on suitably designed axioms. Our method is quite different, being differential geometric in nature; it is based on the approach to quantum cohomology described in [7], which is in turn based on the integrable systems point of view of Dubrovin and Givental.

We prove that if G is a non-trivial finite group, then the integral group ring $\mathbb{Z} G$ possesses a superdecomposable pure-injective module.

The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function $R$ such that the Julia set of $R$ is $\bar{{\bb C}}$ . Further, it is shown that under mild conditions on another real rational function $\tilde{R}$ with possibly non-empty Fatou set, the Julia set of $\tilde{R} \circ R$ is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is $\bar{{\bb C}}$ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.