We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x) (log log x) is not asymptotic as x→+∞ to a composition of semialgebraic functions, log and exp.

In this paper we study the value distribution of the least primitive root to a prime modulus, as the modulus varies. For each odd prime number p, we shall denote by g(p) and G(p) the least primitive root and the least prime primitive root (mod p), respectively. Numerical examples show that, in most cases, g(p) and G(p) are very small (cf. §4). We can support this observation by a probabilistic argument [14, §1]. In fact, on the assumption of a good distribution of the primitive residue classes modulo p, we can surmise that

formula here

where π(x) denotes the number of primes not exceeding x.

A method is given to solve any Thue–Mahler equation when the coefficients and variables come from the ring of integers of a number field. The method involves using an algorithm for solving an S-unit equation and a method of reducing the number of exponential variables. The method is used to solve the equation

formula here

In this paper, we consider a basic question in commutative algebra: if I and J are ideals of a commutative ring S, when does IJ=I∩J? More precisely, our setting will be in a polynomial ring k[x0, …, xn], and the ideals I and J will define subschemes of the projective space ℙnk over k. In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J. By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space.

Our main technique in this paper is a study of the deficiency modules of a subscheme V of ℙn. These modules are important algebraic invariants of V, and reflect many of the properties of V, both geometric and algebraic. For instance, when V is equidimensional and dim V[ges ]1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [14, 11, 15, 7], although they do not in general completely determine the even liaison class, except in the case of curves in ℙ3 [14]. On the algebraic side, at least for curves in ℙ3, the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [12]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space ℙn.

We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result: if I and J define subschemes V and Y, respectively, of ℙn, we relate the quotient module (I∩J)/IJ to the cohomology of V, at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product.

In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen–Macaulay, equidimensional subscheme V of any codimension in ℙn, and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V. The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product.

Finally, in Section 3, we give an extension of a surprising result of Amasaki [1] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in ℙ3 (and later gave a natural extension to Buchsbaum codimension 2 subschemes of ℙn [2]). Easier proofs were subsequently given by Geramita and Migliore in [6], based on a determination of the free resolution of the ideal from a resolution of its deficiency module. For Buchsbaum codimension 2 subschemes of ℙn whose intermediate cohomology vanishes, we are able to extend these considerations.

This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized that these algebras and their deformations, which form a class of quadratic graded algebras, have not been studied much and yet are interesting to algebra, arrangement theory and combinatorics.

Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family [Ascr ]=(Ai)i∈I of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of Nash-Williams for any family [ufr ]. We also present a self-contained proof that if [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I, then our condition is necessary and sufficient for the family [ufr ] to have a transversal.

Large classes of integrable Hamiltonian systems can be expressed as systems over coadjoint orbits in a loop algebra defined over a semi-simple Lie algebra [gfr ]. These systems can then be integrated via the classical, symplectic Liouville–Arnold method. On the other hand, the existence of spectral curves as constants of motion allows one to integrate these systems in terms of flows of line bundles on the curves. This note links the symplectic geometry of the coadjoint orbits with the algebraic geometry of these curves for arbitrary semi-simple [gfr ], which then allows us to reconcile the two integration methods.

Let G be a permutation group on a set Ω, and let m and k be integers where 0<m<k. For a subset Γ of Ω, if the cardinalities of the sets Γg\Γ, for g∈G, are finite and bounded, then Γ is said to have bounded movement, and the movement of Γ is defined as move (Γ) =maxg∈G[mid ]Γg\Γ[mid ]. If there is a k-element subset Γ such that move (Γ)[les ]m, it is shown that some G-orbit has length at most (k2−m)\(k−m). When combined with a result of P. M. Neumann, this result has the following consequence: if some infinite subset Γ has bounded movement at most m, then either Γ is a G-invariant subset with at most m points added or removed, or Γ nontrivially meets a G-orbit of length at most m2+m+1. Also, if move (Γ)[les ]m for all k-element subsets Γ and if G has no fixed points in Ω, then either [mid ]Ω[mid ][les ]k+m (and in this case all permutation groups on Ω have this property), or [mid ]Ω[mid ][les ]5m−2. These results generalise earlier results about the separation of finite sets under group actions by B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann, and groups in which all subsets have bounded movement (by the author).

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

An integral of the generalized Riemann type is developed which inverts the Schwarz derivative of a continuous function.

For every Hausdorff function h, the paper gives a construction of a compact metric space of infinite diameter-based h-packing measure which has no subsets of positive finite measure. It then indicates how such a construction can be modified to provide the same result for a radius-based packing measure in the case of certain Hausdorff functions which do not satisfy a doubling condition. This is in surprising contrast to another result which shows that for a particular radius-based definition of packing measure no such example can be found.

A well-known theorem of Hurwitz states that if τ[ratio ]S→S is a conformal self-mapping of a compact Riemann surface of genus g[ges ]2, then it has at most 2g+2 fixed points and that equality occurs if and only if τ is a hyperelliptic involution.

In this paper we consider this problem for a K-quasiconformal self-mapping f[ratio ]S→S. The result we obtain is that the number of fixed points (suitably counted) is bounded by 2+g(K1/2+K−1/2), and that this bound is sharp. We see that when K=1, that is, when f is conformal, our result agrees with the classical one.

A construction is described of a 153-dimensional representation of the triple cover of O'Nan's sporadic simple group, over the field of order 4. This is then extended to a 306-dimensional representation of 3·O'N[ratio ]2 over GF(2). A similar construction in characteristic 3 gives an unrelated representation of the O'Nan group in 154 dimensions over GF(3), which extends to O'N[ratio ]2 provided that the field is extended to GF(9).

We introduce the notion of the finite dimensional approximation property (the FDAP) and prove that if a subset X of a linear metric space has the FDAP, then every non-empty convex subset of X is an AR.

As an application we show that every needle point space X contains a dense linear subspace E with the following properties:

(i) E contains a non-empty compact convex set with no extreme points;

(ii) all non-empty convex subsets of E are AR.

In this paper we construct quasinilpotent operators without nontrivial invariant subspaces. As well as being interesting in its own right, we believe that this is a sensible (perhaps necessary) first step for addressing the difficult problem of constructing topologically simple Banach algebras.

An ovoid in a 3-dimensional projective geometry PG(3, q) over the field GF(q), where q is a prime power, is a set of q2+1 points no three of which are collinear. Because of their connections with other combinatorial structures ovoids are of interest to mathematicians in a variety of fields; for from an ovoid one can construct an inversive plane [3], a generalised quadrangle [11], and if q is even, a translation plane [13]. In fact the only known finite inversive planes are those arising from ovoids in projective spaces (see [2]). Moreover, there are only two classes of ovoids known, namely the elliptic quadric and, for q even and not a square, the Tits ovoids; these will be described in the next section. If the field order q is odd, then it was shown by Barlotti and Panella (see [3, 1.4.50]) that the only ovoids are the elliptic quadrics. This paper contains a geometrical characterisation of the two known classes of ovoids.

In [2] we discussed almost complex curves in the nearly Kähler S6. These are surfaces with constant Kähler angle 0 or π and, as a consequence of this, are also minimal and have circular ellipse of curvature. We also considered minimal immersions with constant Kähler angle not equal to 0 or π, but with ellipse of curvature a circle. We showed that these are linearly full in a totally geodesic S5 in S6 and that (in the simply connected case) each belongs to the S1-family of horizontal lifts of a totally real (non-totally geodesic) minimal surface in [Copf ]P2. Indeed, every element of such an S1-family has constant Kähler angle and in each family all constant Kähler angles occur. In particular, every minimal immersion with constant Kähler angle and ellipse of curvature a circle is obtained by rotating an almost complex curve which is linearly full in a totally geodesic S5.

Our object of study is the natural tower which, for any given map f[ratio ]A→B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E∗, yielding, for example, an analogue of Quillen's plus-construction for E∗. We discuss in detail the case of ordinary homology with coefficients in ℤ/p or ℤ[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f[ratio ]A→pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(ℤ[1/p], 1).

We discuss Gaussian generalized random fields indexed by smooth sections of vector bundles with respect to Markov properties. We propose a new set-up which is suitable for the present question and within which new phenomena are detected naturally. In particular, we give a counterexample to the belief that locality in the RKHS implies the germ Markov property. We also prove the close connection between the Markov property and cokernels of local operators. Furthermore, we prove the Markov property for a very degenerate Gaussian random field.