Let M be an ω-categorical structure (that is, M is countable and Th(M) is ω-categorical). A nice enumeration of M is a total ordering [pr ] of M having order-type ω and satisfying the following. Whenever ai, i<ω, is a sequence of elements from M, there exist some i<j<ω and an automorphism σ of M such that σ(ai) = aj and whenever b[pr ]ai, then σ(b)[pr ]aj.

Such enumerations were introduced by Ahlbrandt and Ziegler in [1] where they showed that any Grassmannian of an infinite-dimensional projective space over a finite field (or of a disintegrated set) admits a nice enumeration; this combinatorial property played an essential role in their proof that almost strongly minimal totally categorical structures are quasi-finitely axiomatisable.

Recall that if M is ω-categorical and a is a k-tuple of distinct elements from M (with tp(a) non-algebraic), then the Grassmannian Gr(M; a) is defined as follows. The domain of Gr(M; a) is the set of realisations of tp(a) in Mk, modulo the equivalence relation xEy if x and y are equal as sets. This is a 0-definable subset of Meq, and now the relations on Gr(M; a) are by definition precisely those which are 0-definable in the structure Meq. (In particular, Gr(M; a) is also ω-categorical.)

Notice that it is by no means clear that if M admits a nice enumeration, then so do Grassmannians of M. However, there is a strengthening of the notion of nice enumeration for which this is the case.

This paper deals with a formula satisfied by ‘r-reduced’ Schur functions. Schur functions originally appear as irreducible characters of general linear group over the complex number field. In this paper they are considered as weighted homogeneous polynomials with respect to the power sum symmetric functions. More precisely, for a Young diagram λ of size n, the Schur function indexed by λ reads

formula here

where χλ(v) is the character value of the irreducible representation Sλ of the group algebra ℚ[Sfr ]n, evaluated at the conjugacy class of the cycle type v = (1v12v2 … nvn). Setting tjr = 0 for j = 1, 2, … in Sλ(t), we have the r-reduced Schur function S(r)λ(t). The set of all r-reduced Schur functions spans the polynomial ring P(r) = ℚ[tj; j[nequiv ]0 (mod r)]. We show that a good choice of basis elements leads to an explicit description of all other r-reduced Schur functions involving the Littlewood–Richardson coefficients.

The formula has not only a purely combinatorial meaning, but also nice implications in two different fields. One is about the basic representation of the affine Lie algebra A(1)r−1. We show that the basis in the main theorem gives in turn a weight basis of the basic A(1)r−1-module realised in P(r). The other implication is about modular representations of the symmetric group. Our explicit formula implies that the determination of the decomposition matrices reduces to that for the basic set we give in this paper.

The paper is organised as follows. In Section 1 we introduce generalised Maya diagrams and associated r-reduced Schur functions. In Section 2 we discuss combinatorics of Young diagrams. Section 3 is devoted to the main theorem. In Section 4 we describe weight vectors of the basic A(1)r−1-module. In Section 5 the formula is translated into that in the modular representation theory.

We characterise all maximal convex [lscr ]-subgroups of A(Ω), the automorphism group of a doubly transitive chain Ω of countable coterminality. We then determine, up to isomorphism, all doubly transitive actions of the [lscr ]-group A(Ω). Finally, assuming the Continuum Hypothesis, we show that there exists an unbounded chain of proper prime subgroups of A(Ω).

A famous Diophantine equation is given by

formula here

For integers k[ges ]2 and m[ges ]2, this equation only has the solutions x = −j (j = 1, …, m), y = 0 by a remarkable result of Erdős and Selfridge [9] in 1975. This put an end to the old question of whether the product of consecutive positive integers could ever be a perfect power (except for the obviously trivial cases). In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679]) showed that (1) has no solution with x[ges ]0 in the case k = 2 and m = 3. In 1857, Liouville [18] derived from Bertrand's postulate that for general k[ges ]2 and m[ges ]2, there is no solution with x[ges ]0 if one of the factors on the right-hand side of (1) is prime. By use of the Thue–Siegel theorem, Erdős and Siegel [10] proved in 1940 that (1) has only trivial solutions for all sufficiently large k[ges ]k0 and all m. This was closely related to Siegel's earlier result [30] from 1929 that the superelliptic equation

formula here

has at most finitely many integer solutions x, y under appropriate conditions on the polynomial f(x). The ineffectiveness of k0 was overcome by Baker's method [1] in 1969 (see also [2]).

In 1955, Erdős [8] managed to re-prove the result jointly obtained with Siegel by elementary methods. A refinement of Erdős' ideas finally led to the above-mentioned theorem as follows.

Let [ ] be an algebraic number field of degree n over the rationals, and denote by Jk the subring of [ ] generated by the kth powers of the integers of [ ]. Then G[ ](k) is defined to be the smallest s[ges ]1 such that, for all totally positive integers v∈Jk of sufficiently large norm, the Diophantine equation

formula here

is soluble in totally non-negative integers λi of [ ] satisfying

formula here

In (1.2) and throughout this paper, all implicit constants are assumed to depend only on [ ], k, and s. The notation G[ ](k) generalizes the familiar symbol G(k) used in Waring's problem, since we have Gℚ(k) = G(k).

By extending the Hardy–Littlewood circle method to number fields, Siegel [8, 9] initiated a line of research (see [1–4, 11]) which generalized existing methods for treating G(k). This typically led to upper bounds for G[ ](k) of approximate strength nB(k), where B(k) was the best contemporary upper bound for G(k). For example, Eda [2] gave an extension of Vinogradov's proof (see [13] or [15]) that G(k)[les ](2+o(1))k log k. The present paper will eliminate the need for lengthy generalizations as such, by introducing a new and considerably shorter approach to the problem. Our main result is the following theorem.

Three questions concerning the distribution of the numbers of points on elliptic curves over a finite prime field are considered. First, the previously published bounds for the distribution are tightened slightly. Within these bounds, there are wild fluctuations in the distribution, and some heuristics are discussed (supported by numerical evidence) which suggest that numbers of points with no large prime divisors are unusually prevalent. Finally, allowing the prime field to vary while fixing the field of fractions of the endomorphism ring of the curve, the order of magnitude of the average order of the number of divisors of the number of points is determined, subject to assumptions about primes in quadratic progressions.

There are implications for factoring integers by Lenstra's elliptic curve method. The heuristics suggest that (i) the subtleties in the distribution actually favour the elliptic curve method, and (ii) this gain is transient, dying away as the factors to be found tend to infinity.

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.

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.

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.

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.

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).

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.

This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary has M isolated, non-degenerate critical points. Then it is shown that for any positive integer M[les ]M there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.

Estimates for the Dirichlet eigenfunctions near the boundary of an open, bounded set in euclidean space are obtained. It is assumed that the boundary satisfies a uniform capacitary density condition.

An asymptotically perturbed cylinder is a manifold which, to the exterior of a compact set, is of the form R×M, where M is a compact manifold (with or without boundary), and for which the metric approaches the product metric as the axial variable tends to infinity. The properties of eigenfunctions of the Laplacian on asymptotically perturbed cylinders, with either Dirichlet or Neumann boundary conditions if the boundary is non-empty, are studied. For a large class of asymptotic perturbations, the eigenfunctions are proved to decay faster than the reciprocal of any polynomial as the axial variable tends to infinity. The decay estimates are then used to prove that the eigenvalues are of finite multiplicity, and can accumulate at the thresholds only from below. The results are shown to apply to a large class of acoustic and quantum waveguides.

For each permutation f of the set of positive integers, all triples s, t, u are determined such that t and u are the lower and upper limits of the sequence of partial sums of the ‘f-rearrangement’ [sum ]af(n) of some real series [sum ]an with sum s.

Given a smooth, compactly supported hypersurface S in ℝn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f∈[Sscr ], the Schwartz class of functions, by

formula here

where dσ is Lebesgue measure on S. We can now define a maximal average operator,

formula here

In the case where S is Sn−1, the sphere of unit radius in ℝn, we are looking at Stein's spherical maximal function, as treated in his paper [10]. Stein proved that M is a bounded operator on the Lp spaces if and only if p>n/(n−1) when n>3. Subsequently, Bourgain [2] showed that if S is any compactly supported smooth curve with non-vanishing Gaussian curvature, then M will be bounded on Lp, if and only if p>2, thus dealing with the case of the circular maximal function in the plane. (For related results, see also [6] and [7]). In the case where S is a curve whose curvature vanishes to order at most m−2 at a single point, Iosevich [4] showed that M is bounded on Lp for p>m, and unbounded if p = m. If we study curves given by Γ(s) = (s, γ(s)+1), s∈[0, 1], for some suitably smooth γ, where γ(0) = γ′(0) = … = γ(m−1)(0) ≠ γ(m)(0)>0, then we can reinterpret his results as follows. Define

formula here

for Schwartz functions f. Iosevich proved that

formula here

for p>2. If we note that κ(s), the curvature of the curve Γ(s) is approximately 2−k(m−2) whenever s∈[2−k, 21−k], then we have that the operator

formula here

is bounded on Lp for some p>2, if σ is sufficiently large, since

formula here

which is finite so long as σ>(m/p−1) (m−2)−1. If we want to choose σ independent of m>2, the type of the curve, such that Mσ is bounded on Lp for some fixed p>2, then clearly we can take σ = 1/p. In this paper we show that Mσ will be bounded on Lp for p>max{σ−1, s} for a class of infinitely flat, convex curves in the plane. Counterexamples will show that this is the best possible result, in that there exist flat curves for which Mσ is unbounded for 2<p[les ]σ−1.

Part of the Choquet theory of integral representations on a universal cap is presented here in a slightly improved form. This makes possible a rapid proof of a generalised Bochner theorem for non-separable locally compact groups and of two extensions to such groups of the classical Lévy–Cramér continuity theorem concerning the convergence of sequences of probabilities.