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 ]?

Let Kn ={x∈ℝn[ratio ]xi[ges ]0 for 1[les ]i[les ]n} and suppose that f[ratio ]Kn→Kn is nonexpansive with respect to the [lscr ]1-norm and f(0)=0. It is known that for every x∈Kn there exists a periodic point ξ=ξx∈Kn (so fp(ξ)=ξ for some minimal positive integer p=pξ) and fk(x) approaches {fj(ξ)[ratio ]0[les ]j<p} as k approaches infinity. What can be said about P*(n), the set of positive integers p for which there exists a map f as above and a periodic point ξ∈Kn of f of minimal period p? If f is linear (so that f is a nonnegative, column stochastic matrix) and ξ∈Kn is a periodic point of f of minimal period p, then, by using the Perron–Frobenius theory of nonnegative matrices, one can prove that p is the least common multiple of a set S of positive integers the sum of which equals n. Thus the paper considers a nonlinear generalization of Perron–Frobenius theory. It lays the groundwork for a precise description of the set P*(n). The idea of admissible arrays on n symbols is introduced, and these arrays are used to define, for each positive integer n, a set of positive integers Q(n) determined solely by arithmetical and combinatorial constraints. The paper also defines by induction a natural sequence of sets P(n), and it is proved that P(n)⊂P*(n)⊂Q(n). The computation of Q(n) is highly nontrivial in general, but in a sequel to the paper Q(n) and P(n) are explicitly computed for 1[les ]n[les ]50, and it is proved that P(n)=P*(n)=Q(n) for n[les ]50, although in general P(n)≠Q(n). A further sequel to the paper (with Sjoerd Verduyn Lunel) proves that P*(n)=Q(n) for all n. The results in the paper generalize earlier work by Nussbaum and Scheutzow and place it in a coherent framework.

Let f(x, y) =[sum ]aijxiyj be a real polynomial, and let Δ(f) be the Newton polygon of f, that is, the convex hull of the set of points (i, j) with aij≠0. In this paper, we study the relation between the topology of the real zero locus of f and the Newton polygon Δ(f) of f.

Obviously, the Newton polygon Δ(f) is an integral convex polygon. Here, an integral polygon is a polygon with vertices that are integral points. A polynomial f(x, y) is said to be non-degenerate if the gradient of fγ(x, y) =[sum ](i, j)∈γaijxiyj has no zeros in (C−0)2 for each face γ of Δ(f). If f is non-degenerate, then the zero locus of f can be compactified in a suitable toric surface PΔ(K) (K=R, C) as a non-singular algebraic curve, and we denote the compactifications by Z(R), Z(C). Here, toric surfaces form a class of algebraic surfaces which contains the affine plane, the projective plane, the product of two projective lines, and so on. In Section 1, we give a review of toric varieties.

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.

The existence of the limit, in the uniform topology, of the averages of a finite family of commuting strongly continuous semigroups of bounded linear operators in a Banach space is characterised in terms of infinitesimal generators.

Let G be a permutation group on a finite set Ω. A sequence B=(ω1, …, ωb) of points in Ω is called a base if its pointwise stabilizer in G is the identity. Bases are of fundamental importance in computational algorithms for permutation groups. For both practical and theoretical reasons, one is interested in the minimal base size for (G, Ω), For a nonredundant base B, the elementary inequality 2[mid ]B[mid ][les ][mid ]G[mid ][les ][mid ]Ω[mid ][mid ]B[mid ] holds; in particular, [mid ]B[mid ][ges ]log[mid ]G[mid ]/log[mid ]Ω[mid ]. In the case when G is primitive on Ω, Pyber [8, p. 207] has conjectured that the minimal base size is less than Clog[mid ]G[mid ]/log[mid ]Ω[mid ] for some (large) universal constant C.

It appears that the hardest case of Pyber's conjecture is that of primitive affine groups. Let H=GV be a primitive affine group; here the point stabilizer G acts faithfully and irreducibly on the elementary abelian regular normal subgroup V of H, and we may assume that Ω=V. For positive integers m, let mV denote the direct sum of m copies of V. If (v1, …, vm)∈mV belongs to a regular G-orbit, then (0, v1, …, vm) is a base for the primitive affine group H. Conversely, a base (ω1, …, ωb) for H which contains 0∈V=Ω gives rise to a regular G-orbit on (b−1) V. Thus Pyber's conjecture for affine groups can be viewed as a regular orbit problem for G-modules, and it is therefore a special case of an important problem in group representation theory. For a related result on regular orbits for quasisimple groups, see [4, Theorem 6].

Let K be a field and let V be a vector space of dimension 2m over K. Let ∧V denote the exterior algebra of V and ∧kV its kth exterior power for 0[les ]k[les ]2m. Let f be a non-degenerate alternating bilinear form defined on V×V. The symplectic group Sp2m(K) is the group of all isometries of f and it acts as a group of vector space automorphisms on ∧kV. In the case that K is algebraically closed and 1[les ]k[les ]m, it is known that ∧kV contains a composition factor corresponding to the fundamental weight ωk of a root system of type Cm. We shall refer to the irreducible module for Sp2m(K) given by this composition factor as a fundamental module.

We develop a deformation theory for k-parameter families of pointed marked graphs with fixed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), computations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn).

We study the relation between the growth of a subharmonic function in the half space and the size of its asymptotic set.

A function f defined on a domain D has an asymptotic value b∈[−∞, ∞] at a∈δD if there exists a path γ in D ending at a such that u(p) tends to b as p tends to a along γ. The set of all points on δD at which f has an asymptotic value b is denoted by A(f, b).

G. R. MacLane [10, 11] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower [8, 9] studied the analogous class for subharmonic functions. Many theorems have since been proved having the following character: for a function f of a given growth, if A(f, +∞) is a small set then f has nice boundary behavior on a large set. See [1, 3–7] and the references therein.

For α>0, let [Mscr ]α be the class of subharmonic functions u in IRn+1+ ≡{(x, y)[ratio ]x∈IRn, y>0} satisfying the growth condition

formula here

for some constant C(u) depending on u. Denote by [Fscr ](u) the Fatou set of u, which consists of points on δIRn+1+ where u has finite vertical limits. For β>0, denote by Hβ the β-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].

Our goal in this paper is to determine conditions on a potential V which ensure that an operator such as

formula here

acting on L2(RN) defines a semigroup in Lp(RN) for various values of p including p=1. The operator is defined as a quadratic form sum. That is, we put

formula here

for f∈C∞c (all integrals are on RN and are with respect to Lebesgue measure), and note that the closure of the form is non-negative and has domain equal to the Sobolev space Wm,2. We then assume that the potential has quadratic form bound less than 1 with respect to Q0, and define

formula here

This form is closed and is associated with a semibounded self-adjoint operator H in L2 (see [17, p. 348; 5, Theorem 4.23]). One can then ask whether the semigroup e−Ht defined on L2 for t[ges ]0 is extendable to a strongly continuous one-parameter semigroup on Lp for other values of p, and if so whether one can describe the domain and spectrum of its generator.

Relating geometric notions like type and cotype from local Banach space theory with some important topological invariants for Fréchet spaces (for example the density condition), we prove a version of Maurey's celebrated extension theorem for the category of Fréchet spaces, and apply it to Grothendieck's problème des topologies.

The paper gives new examples of [Ascr ]-structures on certain SO(3)-principal fibre bundles.

Let P be a polyhedron in H3 of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of Isom H3. Let Γ+(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ+(P) is a Kleinian group of finite covolume. Γ+(P) is called a polyhedral group.

As discussed in [12] and [13] for example (see §2 below), associated to a Kleinian group Γ of finite covolume is a pair (AΓ, kΓ) which is an invariant of the commensurability class of Γ; kΓ is a number field called the invariant trace-field, and AΓ is a quaternion algebra over kΓ. It has been of some interest recently (cf. [13, 16]) to identify the invariant trace-field and quaternion algebra associated to a Kleinian group Γ of finite covolume since these are closely related to the geometry and topology of H3/Γ.

In this paper we give a method for identifying these in the case of polyhedral groups avoiding trace calculations. This extends the work in [15] and [11] on arithmetic polyhedral groups. In §6 we compute the invariant trace-field and quaternion algebra of a family of polyhedral groups arising from certain triangular prisms, and in §7 we give an application of this calculation to construct closed hyperbolic 3-manifolds with ‘non-integral trace’.

Using simplicial methods developed in an earlier note, the paper constructs topological quantum field theories using an algebraic model of a homotopy n-type as initial data, generalising constructions of Yetter for n=1 and n=2.

We prove Landesman–Lazer type existence conditions for the solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equations of the form x″+cx′+f(t, x)=0. The proofs use upper and lower solutions techniques together with some limiting arguments.

Combining Le Cam's operator approach with Bergström's identity, estimates are obtained for convolutions of compound measures. The results are exemplified by consideration of asymptotic expansions for Compound Poisson approximation in Le Cam's theorem, approximation of nearly homogeneous portfolios and convergence to Compound Poisson laws.

It is shown that if S is an aperiodic random walk on the integers, S* is the Markov chain that arises when S is killed when it leaves the non-negative integers, and H+ is the renewal process of weak increasing ladder heights in S, then there is a 1[ratio ]1 correspondence between functions which are non-negative and superregular for S* and H+. This allows all the regular functions for S* to be described, and thus a result due to Spitzer to be completed for the recurrent case. This result is then applied to give a ratio limit theorem for Px(τ* =n)/P0{τ*=n}, where τ* is the lifetime of S*, in the case when S drifts to −∞, and the right-hand tail of its step distribution is ‘locally sub-exponential’.