A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the graph's end-structure. Using uncountable combinatorics, a representation theorem for embeddability in this class is proved, which asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ>[Nfr ]1 onto set inclusion over all subsets of reals or cardinality λ or less. As corollaries we obtain: (1) the complexity of the class in every regular uncountable λ>[Nfr ]1 is at least

formula here

(2) a characterisation of graphs in the class for which some invariant of the graph has to be inherited by one of fewer than λ subgraphs whose union covers G. Some continuity properties of the homomorphism in the representation theorem are explored and are used to extend the first corollary to all singular cardinals below the first fixed point of second order. The first corollary shows that, contrary to what Shelah has shown for the class of all graphs, the relations of embeddability in the class under discussion is not independent of negations of the generalised continuum hypothesis.

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 ]?

The theory of heights for rational points on arithmetic elliptic curves is becoming well known. An important fact in the basic theory is the relationship between the naïve and the canonical height of a rational point; in fact, they differ by a uniformly bounded amount. The paper provides a generalisation of this fact from the point of view of heights of polynomials, rather than heights of points. This concept is extended to polynomials in several variables.

Given the Diophantine equation a2+b2+c2=3abc, a solution triple of natural numbers (a, b, c) can be arranged in ascending order so that a[les ]b[les ]c. Then, given the largest element c, one can ask whether this uniquely determines the triple. This is referred to as the Markoff conjecture. The paper proves that, if c is prime, then there is indeed only one triple that solves the equation with c as the largest element. The proof uses only standard algebraic number theory, but it was prompted by geometric considerations.

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.

The paper studies projective stationary sets. The Projective Stationary Reflection Principle is the statement that every projective stationary set contains an increasing continuous ∈-chain of length ω1. It is shown that, if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also, this principle is equivalent to the Strong Reflection Principle. The paper shows that the saturation of the nonstationary ideal on ω1 is equivalent to a certain kind of reflection.

We compare the classical semidirect product V[midast ] W of pseudovarieties of semigroups with the recently found ‘regular’ semidirect product [Vscr ] [midast ]e [Wscr ] of e-pseudovarieties of regular semigroups. Applications of our findings include a substantial class of additions to the list of pseudovarieties generated by classes of regular semigroups and for which the membership problem is solved. For example, we compute the pseudovarieties generated by all locally inverse and locally orthodox semigroups, respectively, thereby answering a question raised by Almeida and Kaďourek.

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.

It is shown that the general plane section of a double structure on an integral curve C⊆P3 has a connected numerical character (that is, its Hilbert function is of decreasing type). The paper gives applications, particularly to the stability of the normal bundle of space curves.

All groups and rings in this paper are finite, and p denotes a prime number.

R. Shepherd [28] and C. R. Leedham-Green and S. McKay [16] showed that any p-group of maximal class contains a subgroup of class at most 2 the index of which is bounded above by a function of p. These papers gave rise to a program for the classification of p-groups that used the notion of coclass proposed by C. R. Leedham-Green and M. F. Newman [21] in 1980. They made several conjectures. The strongest conjecture, Conjecture A, asserted that every p-group of coclass r contains a subgroup of class at most 2 the index of which is bounded above by a function depending only on p and r. These conjectures were proved in a long series of papers by C. R. Leedham-Green, S. McKay, S. Donkin, W. Plesken, A. Shalev and E. Zelmanov (cf. [3, 16–20, 27]). In a recent paper, A. Shalev [26] gave a proof of Conjecture A for all primes p with explicit bounds.

A family [Fscr ] of functions from ℝn to ℝ is k-point separating if, for every k-subset S of ℝn, there is a function f∈[Fscr ] such that f is one-to-one on S. The paper shows that, if the functions are required to be linear (or smooth), then a minimum k-point separating family [Fscr ] has cardinality n(k−1). In the linear case, this result is extended to a larger class of fields including all infinite fields as well as some finite fields (depending on k and n). Also, some partial results are obtained for continuous functions on ℝn, including the case when k is infinite. The proof of the main result is based on graph theoretic results that have some interest in their own right. Say that a graph is an n-tree if it is a union of n edge-disjoint spanning trees. It is shown that every graph with k[ges ]2 vertices and n(k−1) edges has a non-trivial subgraph which is an n-tree. A determinantal criterion is also established for a graph with k vertices and n(k−1) edges to be an n-tree.

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.

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).

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 u be a solution of the differential equation u″+Ru=0, where R is rational. Newton's method of finding the zeros of u consists of iterating the function f(z) =z−u(z)/u′(z). With suitable hypotheses on R and u, it is shown that the iterates of f converge on an open dense subset of the plane if they converge for the zeros of R. The proof is based on the iteration theory of meromorphic functions, and in particular on the result that, if the family of K-quasiconformal deformations of a meromorphic function f depends on only finitely many parameters, then every cycle of Baker domains of f contains a singularity of f−1. This result, together with classical results of Hille concerning the asymptotic behaviour of solutions of the above differential equations, is also used to study their value distribution. For example, it is shown that, if R is a rational function which satisfies R(z)∼amzm as z→∞ and has only k distinct zeros where k<(m+2)/2, then δ(0, u)[les ]k/(m+2−k)<1.

The paper shows that the distinction between tangential sequences and tangential curves, which is known to be relevant for the boundary behaviour of harmonic functions in Euclidean half spaces, is also meaningful and relevant in the discrete setting of the potential theory on a tree associated to a very regular, nearest neighbour random walk. In particular, the paper first defines the class of tangential curves on a tree, and then proves that, for any family of tangential curves which have the same order of tangency and are associated (and converging) to points in the boundary, there is a bounded harmonic function which, for (almost) every point in the boundary, has positive oscillation along the corresponding curve. This result does not hold if, in place of tangential curves, tangential sequences are admitted.

The tree is not assumed to be homogeneous. In particular, the results do not depend on the existence of isometries.

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].

We develop a theory of modules of type FP∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted H[Fscr ] and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an h[Fscr ]-group and Mis a ℤG-module of type FP∞ then M has finite projective dimension over ℤH for all torsion-free subgroups H of G. We also show that if G is an h[Fscr ]-group of type FP∞ and M is a ℤG-module which is ℤF-projective for all finite subgroups F of G, then M has finite projective dimension over ℤG. Both of these results have as a special case the striking fact that if G is an h[Fscr ]-group of type FP∞ then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite h[Fscr ]-group of type FP∞ has finite virtual cohomological dimension.

Let G be a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p[ges ]0. In this paper, we classify all pairs (X, Y) of reductive subgroups of G which have a dense (X, Y)-double coset in G. In fact, we show that there is a dense (X, Y)-double coset in G precisely when G=XY is a factorisation. The possible factorisations that can occur have recently been determined in [11, Theorem A].

We now state our main result.

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.