This paper is based on a talk given at a joint meeting of the London Mathematical Society and the British Society for the History of Mathematics held in Oxford on 13 May 1995.

We consider the following strengthening of Hadwiger's Conjecture.

Let G be any graph of chromatic number k, S any subset of V(G) which takes all k colours in each proper k-colouring of G. Then there are k pairwise adjacent connected subgraphs of G, each of whose vertex sets has non-trivial intersection with S.

We show that the truth of this conjecture for all graphs of chromatic number [les ]k implies the truth of Hadwiger's Conjecture for all graphs of chromatic number [les ]k + 1. We also show that its truth implies the following statement (which is at first sight even stronger).

For any graph G of chromatic number k and any subset S of V(G), define χ(S; G) to be the least number of colours that can appear on S in any proper k-colouring of G, and h(S; G) to be the largest number of pairwise adjacent connected subgraphs of G each having non-trivial intersection with S. Then χ(S; G) [les ]h(S; G).

We define the number w(S; G) to be the largest cardinality of a subset T of S such that, however T is partitioned into pairs (possibly with one spare element), there are vertex-disjoint paths linking the elements in each pair, none passing through the spare element if it exists. We show that χ(S; G) [les ]([mid ]S[mid ]+w(S; G))/2 for any graph G and subset S of V(G).

Finally, we show that for any graph G, χ(S; G) [les ]h(S; G) whenever χ(S; G) [les ]3.

Every complex manifold carries a canonical orientation, and it is natural to wonder when the underlying topological or smooth manifold carries a complex structure compatible with the other orientation. In [2], Beauville raised this question for compact complex surfaces. He noted that there are a lot of examples, like products of curves, or Hopf surfaces, where the underlying manifold admits an orientation-reversing selfdiffeomorphism. This implies that the signature is zero. Beauville asked if there are any examples of non-zero signature.

A noncommutative version of the Hilbert basis theorem is used to show that certain [Rscr ]-symmetric algebras S[Rscr ](V) are Noetherian. This result applies in particular to the coordinate ring of quantum matrices A[Rscr ](V) associated with an R-matrix [Rscr ] operating on the tensor square of a vector space V, to show that, under a natural set of hypotheses on [Rscr ], the algebra A[Rscr ](V) is Noetherian and its augmentation ideal has a polynormal set of generators. As a corollary we deduce that these properties hold for the generic quantized function algebras Rq [G] over any field of characteristic zero, for G an arbitrary connected, simply connected, semisimple group over [Copf ]. That Rq[G] is Noetherian recovers a result due to Joseph [10], with a different proof.

Let G be a finite group acting faithfully and transitively on a set Ω of size m, and let E = {x1, …, xk} be a generating set for G with x1x2…xk = 1. If x∈G has cycles of length r1, …, rl in its action on Ω, define ind (x) = [sum ]l1(ri−1). Then the genusg = g(G, Ω, E) is defined by

formula here

The norm of a group G is the subgroup of elements of G which normalise every subgroup of G. We shall denote it κ(G). An ascending series of subgroups κi(G) in G may be defined recursively by: κ0(G) = 1 and, for i[ges ]0, κi+1(G)/κi(G) = κ(G/κi(G)). For each i, the section κi+1(G)/κi(G) clearly contains the centre of the group G/κi(G). A result of Schenkman [8] gives a very close connection between this norm series and the upper central series: ζi(G)⊆κi(G) ⊆ζ2i(G).

Let V be a vector space over some division ring D, and G a finitary subgroup of GL(V). If G is locally completely reducible, then the D-G modules V, [V, G] and V/CV(G) need not be completely reducible, even if dimDV is finite. Moreover, if F is a field, then V and V/CV(G) need not be completely reducible. We prove here that if D is a finite-dimensional division algebra and G is locally completely reducible, then [V, G] is always a completely reducible D-G module.

If p is a prime, then a finite group is p-soluble if each of its composition factors is either a p-group or has order coprime to p. For example, soluble groups are p-soluble. However, there are many insoluble groups that are p-soluble. We shall prove the following result.

Let K. denote the graded Koszul complex associated to the regular sequence (x0, …, xn) in the graded polynomial ring A = k[x0, …, xn], [mid ]xi[mid ] = 1 for all i, over an arbitrary field k. Let K′. denote the Koszul complex associated to another regular sequence of homogeneous elements (p0, …, pn) in A. In [5] we have studied ranks of graded chain complex morphisms f.[ratio ]K′.→K′. with the property f0 = id. Let Ωk (respectively, Ω′k) denote the kernel of the Koszul differential d[ratio ]Kk→ Kk−1 (respectively, d′[ratio ] K′k→ K′k−1), and let fk[ratio ] Ω′k→Ωk denote the restriction of fk. The main result was that Rank (fk)>n−k.

The purpose of this paper is to answer some questions posed by Doob [2] in 1965 concerning the boundary cluster sets of harmonic and superharmonic functions on the half-space D given by D = ℝn−1 × (0, +∞), where n[ges ]2. Let f[ratio ]D→[−∞, + ∞] and let Z∈δD. Following Doob, we write BZ (respectively CZ) for the non-tangential (respectively minimal fine) cluster set of f at Z. Thus l∈BZ if and only if there is a sequence (Xm) of points in D which approaches Z non-tangentially and satisfies f(Xm)→l. Also, l∈CZ if and only if there is a subset E of D which is not minimally thin at Z with respect to D, and which satisfies f(X)→l as X→Z along E. (We refer to the book by Doob [3, 1.XII] for an account of the minimal fine topology. In particular, the latter equivalence may be found in [3, 1.XII.16].) If f is superharmonic on D, then (see [2, §6]) both sets BZ and CZ are subintervals of [−∞, + ∞]. Let λ denote (n−1)-dimensional measure on δD. The following results are due to Doob [2, Theorem 6.1 and p. 123].

Let f be a 1-periodic C1-function whose Fourier coefficients satisfy the condition [sum ]n[mid ]n[mid ]3[mid ] fˆ(n[mid ]2<∞. For every α∈R\Q and m∈Z\{0}, we consider the Anzai skew product T(x, y) = (x+α, y+mx+f(x)) acting on the 2-torus. It is shown that T has infinite Lebesgue spectrum on the orthocomplement L2(dx)⊥ of the space of functions depending only on the first variable. This extends some earlier results of Kushnirenko, Choe, Lemańczyk, Rudolph, and the author.

A famous problem of Zariski, stated in 1971 [21], is to decide whether the classical algebro-geometric multiplicity mo(f) of a complex hypersurface f−1(0) at an isolated singularity O is an invariant of the topological type of the embedded germ. That is, if f[ratio ]Cn, O→C, 0 and g[ratio ]Cn, O→C, 0 are two germs of polynomial functions, and there is a homeomorphism ϕ defined between two open neighbourhoods U and V of O in Cn such that ϕ(U) = V, ϕ(U∩f−1(0)) = V∩g−1(0) and ϕ(O) = O, then is it always the case that mo(f) = mo(g)? Here the multiplicity mo(f) is defined to be the intersection number of f−1(0) with a generic complex line L passing through O in Cn, that is, the number of points of intersection of f−1(0) with a line which is an arbitrarily small perturbation of L. If f is reduced, then mo(f) is equal to the order of the polynomial f(z) at O.

In this paper we are going to study the solutions of a complex T-periodic Riccati equation

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and we are going to determine all the possible dynamics of this type of equation. Also, we can say that generically there are two different T-periodic solutions.

It is known [10, 11] that if T is an integral operator with an extended totally positive kernel, then T has a countably infinite family of simple, positive eigenvalues. We prove a similar result for a rather larger class of kernels and, writing the eigenvalues of T in decreasing order as (λn)n∈IN, we use results obtained in [4] and [5] to give a formula for the ratio λn+1/ lambda;n analogous to that given in [3] for the case of a strictly totally positive matrix, and to the spectral radius formula

formula here

This may be regarded as a generalisation of inequalities due to Hopf [8, 9].

It is shown that for every compact group G, L1(G)ˆ is unique and minimal among all the closed subsets I of M(G)** such that I is a proper (≠0, ≠M(G)**) algebraic ideal, and such that I is solid with respect to absolute continuity; that is, n∈L1(G)ˆ whenever n∈M(G)** and n[Lt ]μ∈L1(G)ˆ.

Let p be a point of a Lorentzian manifold M. We show that if M is spacelike Osserman at p, then M has constant sectional curvature at p; similarly, if M is timelike Osserman at p, then M has constant sectional curvature at p. The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [3]; we use a different approach to this case.

1. Definition of the A-polynomial

The A-polynomial was introduced in [3] (see also [5]), and we present an alternative definition here. Let M be a compact 3-manifold with boundary a torus T. Pick a basis λ,μ of π1T, which we shall refer to as the longitude and meridian. Consider the subset RU of the affine algebraic variety R = Hom (π1M, SL2[Copf ]) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic subset of R, since one just adds equations stating that the bottom-left entries in certain matrices are zero. There is a well-defined eigenvalue map

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given by taking the top-left entries of ρ(λ) and ρ(μ).

Klass has established decoupling inequalities for discrete time processes with independent increments. We extend his result to continuous time processes with independent increments. This extension enables us to obtain BDG inequalities for exponential functions instead of moderate functions.