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.

In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p- adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Mahler's measure of a polynomial can be written as a logarithmic integral over the torus. We propose a definition when the underlying group is an elliptic curve. Having reviewed some of the classical results in the toral case, we take some first steps towards realising elliptic analogues. In particular, we focus on elliptic analogues of Kronecker's theorem and Lehmer's problem. We wish to stress the fundamental role played by Jensen's formula in both the toral and elliptic formulations of these results.

In the Kourovka Notebook [7] Khukhro posed a conjecture on the structure of finite p-groups admitting an automorphism of order p.

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).

It is proved that there is a certain degree of independence between stationary reflection phenomena at different cofinalities.

Let g be a primitive root modulo a prime p. It is proved that the triples (gx, gy, gxy), x, y = 1, …, p−1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε>0 be fixed. Then

formula here

uniformly for any integers a, b, c with gcd(a, b, c, p) = 1. Incomplete sums are estimated as well.

The question is motivated by the assumption, often made in cryptography, that the triples (gx, gy, gxy) cannot be distinguished from totally random triples in feasible computation time. The results imply that this is in any case true for a constant fraction of the most significant bits, and for a constant fraction of the least significant bits.

The non-existence set forth in the title is proved. It is known that for numerical Campedelli surfaces the algebraic fundamental group is of order [les ]9 and that the dihedral group of order 8 cannot occur. Therefore the quaternionic group is the only non-abelian algebraic fundamental group in this range.

Let a=(a1, a2, a3, …) be an arbitrary infinite sequence in U=[0, 1). Let

formula here

Van der Corput [5] conjectured that d(a, n) (n=1, 2, …) is unbounded, and this was proved in 1945 by van Aardenne-Ehrenfest [1]. Later she refined this [2], obtaining

formula here

for infinitely many n. Here and later c1, c2, … denote positive absolute constants.

In 1954, Roth [8] showed that the quantity

formula here

is closely related to the discrepancy of a suitable point set in U2.

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.

It is well known that the multiplicity of a complex zero ρ=β+iγ of the zeta-function is O(log[mid ]γ[mid ]). This may be proved by means of Jensen's formula, as in Titchmarsh [7, Chapter 9]. It may also be seen from the formula for the number N(T) of zeros such that 0<γ<T,

formula here

due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and

formula here

We assume here that T is not the ordinate of a zero; with appropriate definitions of N(T) and S(T) the formula is valid for all T. We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT), (Cramér [2]), and on the Riemann Hypothesis

formula here

(Littlewood [5]). These results are over 70 years old.

formula here

Because the multiplicity problem is hard, it seems worthwhile to see what can be said about the number of distinct zeros in a short T-interval. We obtain the following result, which is independent of any unproved hypothesis.

One of the most useful results in modular representation theory of finite groups is Green's indecomposability theorem [4]. In order to state it in a simple form, let us fix a complete discrete valuation ring [Oscr ] of characteristic 0 with algebraically closed residue field [ ] of characteristic p≠0. Unless stated otherwise, all our modules will be free of finite rank over [Oscr ] or [ ], respectively. In its most popular form, Green's theorem says the following.

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.

Let Δ be a finite quiver, that is, a finite directed graph, k be a commutative ring, and kΔ be the semigroup ring of paths in Δ. In this paper we compute the Hochschild homology of the following classes of algebras:

(1) kΔ/[mfr ]n, where [mfr ] is the arrow ideal;

(2) kΔ/I where I is an ideal generated by quadratic monomials.

In Section 2 we establish the notation, and recall a projective resolution of kΔ0, the degree 0 part in kΔ, over a monomial algebra which is due to Anick and Green. This resolution is then used in Section 3 as the building block in the construction of a resolution of a truncated or quadratic monomial algebra A, over its enveloping algebra (in the Hochschild sense), Ae. The fact that a monomial algebra possesses a fine grading then enables us to compute the homology.

The results obtained extend results by Liu and Zhang [3] and Geller, Reid and Weibel [2].

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 aim of this paper is to determine the minimal polynomials of p-elements in irreducible representations of quasi-simple groups with a cyclic Sylow p-subgroup. We consider both the characteristic 0 and characteristic p cases; we have no results for representations over fields of other characteristics. It suffices to describe the cases where the degree of the polynomial is less than the order of the element.

An interpretation is given of the Shintani descent from the points of a connected algebraic group over a finite field to its group of points over a subfield in terms of a Shintani descent between two different rational structures over the same finite field, using restriction of scalars. From this a generalization is deduced of a result of Gyoja's on the Shintani descent of Deligne–Lusztig characters of a finite group of Lie type.

If σ is an automorphism and δ is a σ-derivation of a ring R, then the subring of invariants is the set R(δ)={r∈R[mid ]δ(r)=0}. The main result of this paper is ‘let R be a semiprime ring with an algebraic σ-derivation δ such that R(δ) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.