The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities of the quasi-parabolic structure equals one. This gives a new proof that the moduli space of vector bundles of coprime rank and degree is stably rational, a result originally due to Ballico, and the bound on the level is strong enough to deduce rationality in many cases, extending results of Newstead.

The fine topology on ℝn (n[ges ]2) is the coarsest topology for which all superharmonic functions on ℝn are continuous. We refer to Doob [11, 1.XI] for its basic properties and its relationship to the notion of thinness. This paper presents several theorems relating the fine topology to limits of functions along parallel lines. (Results of this nature for the minimal fine topology have been given by Doob – see [10, Theorem 3.1] or [11, 1.XII.23] – and the second author [15].) In particular, we will establish improvements and generalizations of results of Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihl and Seidel [6], Schneider [21], Berman [7], and Armitage and Nelson [4], and will also solve a problem posed by the latter authors.

An early version of our first result is due to Evans [12, p. 234], who proved that, if u is a superharmonic function on ℝ3, then there is a set E⊆ℝ2×{0}, of two-dimensional measure 0, such that u(x, y,·) is continuous on ℝ whenever (x, y, 0)∉E. We denote a typical point of ℝn by X=(X′ x), where X′∈ℝn−1 and x∈ℝ. Let π:ℝn→ℝn−1×{0} denote the projection map given by π(X′, x) = (X′, 0). For any function f:ℝn→[−∞, +∞] and point X we define the vertical and fine cluster sets of f at X respectively by

formula here

and

formula here

Sets which are open in the fine topology will be called finely open, and functions which are continuous with respect to the fine topology will be called finely continuous. Corollary 1(ii) below is an improvement of Evans' result.

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser Ge of every edge e is dihedral of order 4 and the stabiliser Gv of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that Hv is the cyclic subgroup of index 2 in Gv. An analogous result is proved for orientably-regular embeddings.

Let Σ be a compact Riemann surface of genus g, X={x1, …, xn}⊆Σ a finite set of points, and Ω1Σ(log X) be the sheaf of 1-forms, holomorphic over Σ[setmn ]X and generated near xj by dzj/zj for a coordinate zj centred at xj.

The study of reductive group actions on a normal surface singularity X is facilitated by the fact that the group Aut X of automorphisms of X has a maximal reductive algebraic subgroup G which contains every reductive algebraic subgroup of Aut X up to conjugation. If X is not weighted homogeneous then this maximal group G is finite (Scheja, Wiebe). It has been determined for cusp singularities by Wall. On the other hand, if X is weighted homogeneous but not a cyclic quotient singularity then the connected component G1 of the unit coincides with the [Copf ]* defining the weighted homogeneous structure (Scheja, Wiebe, Wahl). Thus the main interest lies in the finite group G/G1. Not much is known about G/G1. Ganter has given a bound on its order valid for Gorenstein singularities which are not log-canonical. Aumann-Körber has determined G/G1 for all quotient singularities.

We propose to study G/G1 through the action of G on the minimal good resolution X˜ of X. If X is weighted homogeneous but not a cyclic quotient singularity, let E0 be the central curve of the exceptional divisor of X˜. We show that the natural homomorphism G→Aut E0 has kernel [Copf ]* and finite image. In particular, this re-proves the rest of Scheja, Wiebe and Wahl mentioned above. Moreover, it allows us to view G/G1 as a subgroup of Aut E0. For simple elliptic singularities it equals (ℤb×ℤb)[rtimes ]Aut0E0 where −b is the self-intersection number of E0, ℤb×ℤb is the group of b-torsion points of the elliptic curve E0 acting by translations, and Aut0E0 is the group of automorphisms fixing the zero element of E0. If E0 is rational then G/G1 is the group of automorphisms of E0 which permute the intersection points with the branches of the exceptional divisor while preserving the Seifert invariants of these branches. When there are exactly three branches we conclude that G/G1 is isomorphic to the group of automorphisms of the weighted resolution graph. This applies to all non-cyclic quotient singularities as well as to triangle singularities. We also investigate whether the maximal reductive automorphism group is a direct product G≃G1×G/G1. This is the case, for instance, if the central curve E0 is rational of even self-intersection number or if X is Gorenstein such that its nowhere-zero 2-form ω has degree ±1. In the latter case there is a ‘natural’ section G/G1[rarrhk ]G of G[rarrhk ]G/G1 given by the group of automorphisms in G which fix ω. For a simple elliptic singularity one has G≃G1×G/G1 if and only if −E0 · E0 = 1.

Working over an algebraically closed, complete, non-Archimedean, nontrivially valued field of characteristic zero, it is shown that (i) any subanalytic subset of a one-dimensional semialgebraic set is semialgebraic, (ii) any one-dimensional subanalytic set is semianalytic, and (iii) any subanalytic subset of a two-dimensional semianalytic set is semianalytic.

We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL2(ℤ). This completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E7 and E8 over fields of odd characteristic occur as Galois groups of geometric extensions of ℚab(t), where ℚab denotes the maximal Abelian extension field of ℚ. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.

Consider an analytic germ f:(Cm, 0)→ (C, 0) (m[ges ]3) whose critical locus is a 2-dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [8], respectively one [12]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f:(C4, 0)→(C, 0), defined by f(x1, x2, x3, x4) = x1x2x3x4 has the homotopy type of S1×S1×S1. On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition; see for example [9–12, 17, 18].

In the last few years different types of ‘bouquet theorems’ have appeared. Some of them deal with germs f:(X, x)→(C, 0) where f defines an isolated singularity. In some cases, similarly to the Milnor case [8], F has the homotopy type of a bouquet of (dim X−1)-spheres, for example when X is an icis [2], or X is a complete intersection [5]. Moreover, in [13] Siersma proved that F has a bouquet decomposition F∼F0∨Sn∨…∨Sn (where F0 is the complex link of (X, x)), provided that both (X, x) and f have an isolated singularity. Actually, Siersma conjectured and Tibăr proved [16] a more general bouquet theorem for the case when (X, x) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F∼∨iFi, where the Fi are repeated suspensions of complex links of strata of X. (If (X, x) has the ‘Milnor property’, then the result has been proved by Lê; for details see [6].)

In our situation, the space-germ (X, x) is smooth, but f has big singular locus. Surprisingly, for dim Sing f−1(0)[les ]2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma's paper [12], where dim Sing f−1(0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as explained in Corollary 2.4). In the present paper, we attack the dim Sing f−1(0) = 2 case. In our investigation some results of Zaharia are crucial [17, 18].

We continue an investigation of complementation in closed ideals of commutative group algebras. The main result here is a sufficient condition for a closed ideal of a commutative group algebra to have a Banach space complement. This involves the comparison of uniformities on the hull of the ideal.

The problem of completing a partially prescribed Toeplitz matrix to a positive definite Toeplitz matrix is considered. The main result is a characterization of the patterns of specified entries that guarantee the existence of a positive definite Toeplitz completion of a partial positive definite Toeplitz matrix. This characterization resolves affirmatively a conjecture of C. R. Johnson and L. Rodman.

Let A be a regular local ring with quotient field K. Assume that 2 is invertible in A. Let W(A)→W(K) be the homomorphism induced by the inclusion A[rarrhk ]K, where W( ) denotes the Witt group of quadratic forms. If dim A[les ]4, it is known that this map is injective [6, 7]. A natural question is to characterize the image of W(A) in W(K). Let Spec1(A) be the set of prime ideals of A of height 1. For P∈Spec1(A), let πP be a parameter of the discrete valuation ring AP and k(P) = AP/PAP. For this choice of a parameter πP, one has the second residue homomorphism ∂P:W(K)→W(k(P)) [9, p. 209]. Though the homomorphism ∂P depends on the choice of the parameter πP, its kernel and cokernel do not. We have a homomorphism

formula here

A part of the so-called Gersten conjecture is the following question on ‘purity’. Is the sequence

formula here

exact? This question has an affirmative answer for dim(A)[les ]2 [1; 3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogel on the question of purity. However, in the literature, there is no proof for purity even for dim(A) = 3. One of the consequences of the main result of this paper is an affirmative answer to the purity question for dim(A) = 3.

We briefly outline our main result.

This paper is centred around a single question: can a minimal left ideal L in G[Lscr ][Uscr ][Cscr ], the largest semi-group compactification of a locally compact group G, be itself algebraically a group? Our answer is no (unless G is compact). In deriving this conclusion, we obtain for nearly all groups the stronger result that no maximal subgroup in L can be closed. A feature of our work is that completely different techniques are required for the connected and totally disconnected cases. For the former, we can rely on the extensive structure theory of connected, non-compact, locally compact groups to derive the solution from the commutative case, using some reduction lemmas. The latter directly involves topological dynamics; we construct a compact space and an action of G on it which has pathological properties. We obtain other results as tools towards our main goal or as consequences of our methods. Thus we find an extension to earlier work on the relationship between minimal left ideals in G[Lscr ][Uscr ][Cscr ] and H[Lscr ][Uscr ][Cscr ] when H is a closed subgroup of G with G/H compact. We show that the distal compactification of G is finite if and only if the almost periodic compactification of G is finite. Finally, we use our methods to show that there is no finite subset of G[Lscr ][Uscr ][Cscr ] invariant under the right action of G when G is an almost connected group or an IN-group.

The positivity of the partial sums of the series of Legendre polynomials is a classical result of Fejér. The paper presents corresponding results for both even and odd Legendre polynomials with best possible leading constants. These results have applications to demonstrating positivity of Cotes numbers at Jacobi abscissas.

Spatial concavity properties of non-negative weak solutions of the filtration equations with absorption ut = (φ(u))xx−ψ(u) in Q = R×(0, ∞), φ′[ges ]0, ψ[ges ]0 are studied. Under certain assumptions on the coefficients φ, ψ it is proved that concavity of the pressure function is a consequence of a ‘weak’ convexity of travelling-wave solutions of the form V(x, t) = θ(x−λt+a). It is established that the global structure of a so-called proper set [Bscr ] = {V} of such particular solutions determines a property of B-concavity for more general solutions which is preserved in time. For the filtration equation ut = (φ(u))xx a semiconcavity estimate for the pressure, vxx[les ](t+τ)−1θ″(ξ), due to the B-concavity of the solution to the subset [Bscr ] of the explicit self-similar solutions θ(x/√(t+τ)) is proved.

The analysis is based on the intersection comparison based on the Sturmian argument of the general solution u(x, t) with subsets [Bscr ] of particular solutions. Also studied are other aspects of the B-concavity/convexity with respect to different subsets of explicit solutions.

Let [Oscr ] be a complete discrete valuation ring with unique maximal ideal J([Oscr ]), let K be its quotient field of characteristic 0, and let k be its residue field [Oscr ]/J([Oscr ]) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the [mid ]G[mid ]-th roots of unity, where [mid ]G[mid ] is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R∈{[Oscr ], k}. We consider the following question: when are A and B Morita equivalent? Actually, we deal with ‘naturally Morita equivalent blocks A and B’, which means that A is isomorphic to a full matrix algebra of B, as studied by B. Külshammer. However, Külshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Külshammer. Moreover, in the case H is normal in G, we get the same results as Külshammer; however, he uses the results of E. C. Dade, and we do not.

Let F be a meromorphic function in the complex plane. We investigate the behaviour of the iterates of F in a Baker domain B. In particular, we describe the dynamics of the orbits with the help of conformal conjugacies; that is, we determine a function φ which is univalent in a large simply connected subdomain of B such that φ(F(z))=T(φ(z)) holds throughout B. Here T is either a parabolic or hyperbolic Möbius transformation mapping either a half plane or [Copf ] onto itself. This functional equation is always solvable in a Baker domain if F has only finitely many poles. Moreover, there is an example of a function with infinitely many poles where one cannot find an appropriate conformal conjugacy in an invariant Baker domain.

Asymptotic cones were first used by Gromov in [6], where he constructed limit spaces of nilpotent groups in order to prove that groups with polynomial growth are virtually nilpotent. Gromov does not use the term asymptotic cone, which was introduced later by Van den Dries and Wilkie in [12], when they gave a nonstandard interpretation of Gromov's results. Ultrafilters appear in [12] for the first time in this context. Later, Gromov gave an extensive treatment of asymptotic cones in [7]. Since then several authors have used asymptotic cones to obtain interesting results, for instance, in identifying quasi-isometry classes of 3-manifolds [8] or relating asymptotic cones with Dehn functions of finitely presented groups (see [2] and [11], the results of which are stated in Section 2).

The purpose of this paper is to develop some of the results stated in [7], in particular those describing the asymptotic cone of the Baumslag–Solitar groups and of Sol. According to [2], these spaces are not simply connected, since their Dehn functions are exponential, so our primary goal is to study their fundamental groups. It will be proved that these fundamental groups are uncountable and nonfree (Section 9), by constructing subgroups isomorphic to the fundamental group of the Hawaiian earring. These subgroups are constructed by finding subspaces in the asymptotic cones which are homotopically equivalent to the Hawaiian earring, and which induce injections in the fundamental group level (Section 8). Crucial to the proof of these facts is the computation of the covering dimension of these asymptotic cones (Section 7), which is done using a more general theorem on dimensions of spaces which admit certain maps into well-known spaces (Section 6).

Let us suppose that

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with

formula here

converges for s>0. If the series converges to sum A for s = 0, then f(s)→A as s→0+, but the converse will not hold without an additional, Tauberian, hypothesis. The usual form of such a hypothesis is a condition on {an}, but in the ‘high-indices’ theorem of Hardy and Littlewood [4], the hypothesis is a condition on {λn}. They showed that if

formula here

then the existence and finiteness of lims→0+f(s) imply the convergence of [sum ]an.

The condition (1.3) is usually referred to as the ‘Hadamard gap condition’ and a series (1.1) such that {λn} satisfies (1.3) is called a lacunary series, although this terminology is occasionally used for series satisfying other gap conditions.

Zygmund has shown [15] that if the function f(s), defined as in (1.1) by a lacunary series, is of bounded variation in (0, ∞), then [sum ]an converges absolutely. Clearly, the converse implication holds.

The result of Zygmund has the form

formula here

an uppercase letter with subscripts here denoting, as it will throughout this paper, a constant determined by the indicated subscripts.

The result of Zygmund and that of Hardy and Littlewood are the extreme cases, corresponding to m = 1 and m = ∞ of the inequality

formula here

which holds for m[ges ]1 and which, with more general results, is to be found in the first part of this paper [8].

In all of these results, the interval of integration can be reduced to a finite interval (0, Bq,λ1).

For an irrational rotation α of the circle group T=R/Z and a piecewise absolutely continuous function f@[ratio ]T→R, the unitary operator Vh(x)=e2πif(x)h(x+α) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0<[mid ]κ[mid ]<1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0<[mid ]κ[mid ]<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O(1/n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.

The question of extendability of the action of a cyclic group of automorphisms of a compact Riemann surface is considered. Particular attention is paid to those cases corresponding to Singerman's list of Fuchsian groups which are not finitely-maximal, and more generally to cases involving a Fuchsian triangle group. The results provide partial answers to the question of which cyclic groups are the full automorphism group of some Riemann surface of given genus g>1.