A Lie subalgebra L of [gfr ][lfr ][ ](V) is said to be finitary if it consists of elements of finite rank. We study the situation when L acts irreducibly on the infinite-dimensional vector space V and show: if Char [ ] > 7, then L has a unique minimal ideal I. Moreover I is simple and L/I is solvable.

The point of this note is to make an observation concerning the variety M(E) parametrizing line subbundles of maximal degree in a generic stable vector bundle E over an algebraic curve C. M(E) is smooth and projective and its dimension is known in terms of the rank and degree of E and the genus of C (see Section 1). Our observation (Theorem 3·1) is that it has exactly the Chern numbers of an étale cover of the symmetric product SδC where δ = dim M(E).

This suggests looking for a natural map M(E) → SδC; however, it is not clear what such a map should be. Indeed, we exhibit an example in which M(E) is connected and deforms non-trivially with E, while there are only finitely many isomorphism classes of étale cover of the symmetric product. This shows that for a general deformation in the family M(E) cannot be such a cover (see Section 4).

One may conjecture that M(E) is always connected. This would follow from ampleness of a certain Picard-type bundle on the Jacobian and there seems to be some evidence for expecting this, though we do not pursue this question here.

Note that by forgetting the inclusion of a maximal line subbundle in E we get a natural map from M(E) to the Jacobian whose image W(E) is analogous to the classical (Brill–Noether) varieties of special line bundles. (In this sense M(E) is precisely a generalization of the symmetric products of C.) In Section 2 we give some results on W(E) which generalise standard Brill–Noether properties. These are due largely to Laumon, to whom the author is grateful for the reference [9].

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition [Bscr ] with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph Γ[Bscr ] relative to [Bscr ], the bipartite subgraph Γ[B, C] of Γ induced by adjacent blocks B, C of Γ[Bscr ] and a certain 1-design [Dscr ](B) induced by a block B ∈ [Bscr ]. The present paper studies the case where the size k of the blocks of [Dscr ](B) satisfies k = v − 1. In the general case, where k = v − 1 [ges ] 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on [Bscr ]. We prove that [Dscr ](B) contains no repeated blocks if and only if Γ[Bscr ] is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as Γ[Bscr ] for some graph Γ with these properties. We prove further that Γ[B, C] ≅ Kv−1,v−1 if and only if Γ[Bscr ] is (G, 3)-arc transitive.

Let S denote a compact, connected, orientable surface with genus g and h boundary components. We refer to S as a surface of genus g with h holes. Let [Mscr ]S denote the mapping class group of S, the group of isotopy classes of orientation-preserving homeomorphisms S → S.

Let G be a group. G is hopfian if every homomorphism from G onto itself is an automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists a normal subgroup of finite index in G which does not contain g. Every finitely generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian ([2, 3]) if every homomorphism ψ G → G with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots.

By a result of Grossman [5], [Mscr ]S is residually finite. Since [Mscr ]S is also finitely generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian. In this paper, we shall answer a more general question. We say that a group G is ultrahopfian if every homomorphism ψ: G → G with ψ(G) normal in G and G/ψ(G) abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We shall prove the following result.

It is a theorem of Reidemeister [4] and Singer [6] that any two Heegaard splittings of a closed 3-manifold are stably equivalent. In this short note, we give this theorem a simple proof.

In [7], Hitchin showed that the data (∇, Φ), comprising an SU(2) Yang–Mills–Higgs monopole in the Prasad–Sommerfeld limit on ℝ3, encodes faithfully into an auxiliary rank 2 holomorphic vector bundle E˜ over T, the total space of the holomorphic tangent bundle of ℙ1. In this construction ℝ3 is viewed as a subset of H0(ℙ1, [Oscr ](T)) ≅ [Copf ]3.

Generically, the restriction of E˜ to a line is trivial. (The image of a global section ℙz ⊂ T, for z ∈ [Copf ]3, is referred to here as a line on T.) Hence c1(E˜) = 0 and, for all z ∈ [Copf ]3, there exists m ∈ {0} ∪ ℕ such that E˜[mid ]ℙz ≅ [Oscr ](m) [oplus ] [Oscr ](−m). If m [ges ] 1 then ℙz is a jumping line of E˜ of height m. The jumping lines are parameterized by an analytic set J ⊂ [Copf ]3, which is stratified by height. When the monopole has charge k, the height is bounded above by k. In this case we write J = J1 ∪ … ∪ Jk, where Ji parameterizes jumping lines of height i. A priori, some Ji may be empty.

The analytic continuation of the monopole to [Copf ]3 has singularities over J. To see this recall how the monopole data are recovered from E˜: very briefly, E˜ induces a sheaf [Escr ] = π2*ε*E˜ over [Copf ]3 which is locally free away from J2 ∪ … ∪ Jk, (π2 and ε are defined in Section 2). A holomorphic connection and Higgs field are defined in [Escr ] over [Copf ]3 null planes that cut out a given direction (see [1, 7, 9]). On restriction to ℝ3, [Escr ] gives a rank 2, SU(2) bundle and the holomorphic connection and Higgs field give the monopole data. It is easy to see that the flat connections are singular at points of J: for example, an analogous situation is described in [10].

L2 estimates of the Kohn–Hörmander type are obtained for the ∂u operator on relatively compact Stein subdomains of complex manifolds. This is done by using a result of Hörmander locally, patching with cohomology of coherent analytic sheaves with bounds to get global solutions of the ∂u equation and then finishing with standard Hilbert space methods to obtain weak elliptic estimates.

In a recent paper [5], the classical Bernoulli and Euler polynomials were expressed as finite sums involving the Hurwitz zeta function. The object of this sequel is first to give several remarkably shorter proofs of each of these summation formulas. Various generalizations and analogues, which are relevant to the present investigation, are also considered.

We prove global and local weighted stability theorems for representations of abelian semigroups on Banach spaces under countable spectral conditions and certain growth assumptions on the weights. In the global case, we use a limit semigroup construction together with an extension theorem for weighted semigroup representations. In the local case, we adapt a definition of Albrecht to introduce a local spectrum of a representation which is no larger than the usual notion of local spectrum for representations of Z+ and R+, and we establish the corresponding local stability theorem. We give an application to the asymptotic theory of functions on abelian semigroups.

We introduce non-homogeneous equilibrium states which include the classical equilibrium states for subshifts of finite type, Riesz products and G-measures. We prove a rather precise estimate for ∥Pnf∥∞ where Pn are the averaging operators. Applications are given to estimate ∥Lngf∥∞ (Lg being a transfer operator on a subshift of finite type) and then to prove a central limit theorem for f(Tnx) (T being the shift), to study the almost everywhere convergence of some lacunary series with respect the equilibrium state and then to compute the Hausdorff dimension of the equilibrium state etc.

A function f, analytic in the unit disc Δ, is said to be a Bloch function if supz∈Δ(1 − [mid ]z[mid ]2)[mid ]f′(z)[mid ] < ∞. In this paper we study the zero sequences of non-trivial Bloch functions. Among other results we prove that if f is a Bloch function with f(0) ≠ 0 and {zk} is the sequence of ordered zeros of f. then

formula here

and

formula here

We will also prove that (ii) is best possible even for the little Bloch space [Bscr ]0. To this end we construct a function f ∈ [Bscr ]0 whose zero sequence {zk} satisfies

formula here

We also consider analogous problems for some other related spaces of analytic functions.

A necessary condition for the existence and uniqueness of mild solutions of first- and second-order differential equations on the real line is given. It is known that the first-order problem is related to the operator equation AX − XD = −δ0. Here, it is shown how the second-order problem on the real line is connected to the operator equation AX − XD2 = −δ0 and also to some other operator equations.

Based on the characterization of periodic eigenvalues using rotation numbers, we analyse the second and the third periodic eigenvalues of one-dimensional Schrödinger operators with certain step potentials. This gives counter-examples to the Alikakos–Fusco conjecture on the second periodic eigenvalues. Using this simple model, we can also construct infinitely many resonance pockets, which are much like calabashes emanating from a cane, of one-parameter Hill's equations.

We prove sharp L2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first space derivatives, using only the Hardy inequality. These imply L2 boundary decay properties of the heat kernel and spectral density. We deduce bounds on the rate of convergence of the eigenvalues when the region is slightly reduced in size. It is remarkable that several of the bounds do not involve the space dimension.

It is shown that if we know the Love's strain function due to the existence of any arbitrary axisymmetric singularity in the elastic whole space, then the corresponding Love's function due to the existence of the same singularity in the interior of one of two dissimilar semi-infinite solids separated by a dissimilar thick layer can be obtained by differentiation and integration of that for the elastic whole space. This discovery rests on the principle that to every singularity in the layered space there will correspond an infinite system of images. Remarkably, the number of images is finite when the phases have equal rigidities but different compressibilities. The analysis thus brings under one general theorem the solutions due to such influencing singularities as a normal point force, a normal point force doublet, an infinitesimal prismatic dislocation loop, a centre of dilatation and a dilatation doublet. From the theorem, we readily infer that the elastic field at large distances from any arbitrary axisymmetric influencing singularity becomes nearly that for two perfectly bonded semi-infinite solids without an interface layer.