In 1979 Montgomery and Vaughan [3] showed that for each k > 0

the sum being over the non-principal characters χ modulo q. As a corollary they deduced that, if 0 < θ < 1 then there is a constant c(θ) such that at least θφ(q) of the non-principal characters χ modulo q satisfy

Let k denote a fixed natural number with k > 2, let ℛs(n) denote the number of representations of n as the sum of sk-th powers of natural numbers, and let

with

denote the corresponding singular series.

In the last three decades there appeared a number of elementary proofs of the prime number theorem (PNT) in the literature (see [3] for a survey). Most of these proofs are based, at least in part, on ideas from the original proof by Erdős [5] and Selberg [12]. In particular, one of the main ingredients of the Erdős-Selberg proof, Selberg's formula

(where p and q run through primes) appears, in some form, in almost all these proofs.

Large sieve inequalities have been developed and applied to a host of arithmetical problems since their inception by Linnik in 1941. Such inequalities provide mean square estimates for a trigonometric polynomial over a set of well-spaced points. In particular, let x ∈ ℝ and let

E. Cartan's famous isoparametric hypersurface in S4 with three distinct constant principal curvatures is geometrically a parallel hypersurface of the Veronese surface, and topologically it is an 8-fold quotient of the 3-sphere. In the present paper we describe a polyhedral analogue with only 15 vertices. Combinatorially this is an 8-fold quotient of the boundary complex of the 600-cell, and geometrically it is a quite regular subcomplex of a certain almost convex simplicial 4-sphere in E5. The euclidean symmetry group of this embedding is isomorphic to the icosahedral group A5 acting transitively on the 15 vertices.

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most , as n/d → ∞. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thorn from real algebraic geometry.

The purpose of this paper is to give some natural examples of Borel-inseparable pairs of coanalytic sets in Polish spaces.

A Polish space is a topological space homeomorphic to a separable complete metric space. In this paper, all spaces are uncountable Polish spaces. A pointset is analytic (or ) if it is the continuous image of a Borel set (in any space), or equivalently, the projection of a Borel set, and is coanalytic (or ) if it is the complement of an analytic set. The class of analytic sets is closed under countable unions and intersections, images and preimages by Borel measurable functions, and projections; it is not closed under complements, hence there is an analytic set which is not Borel.

The usual method of dealing with delay differential equations such as

is the method of steps [1, 2]. In this, y(x) is assumed to be known for − α < x < 0, thereby defining over 0 < x < α. As a result of integration, the value of y is now known over 0 < x < α, and the integration proceeds thereon by a succession of steps.

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions (x, y) → x° y, x\y, x/y: M × M → M (respectively, U × U → M for an open neighbourhood U of e in M) such that the following conditions are satisfied: (i) x ° e = e ° x = e, (ii) x ° (x\y) = y, and (iii) (x/y)° y = x for all x, y ε M (respectively, U). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L(M) in e is equipped with an anticommutative bilinear operation (X, Y) →[X, Y] and a trilinear operation (X, Y, Z) →〈X, Y, Z〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L(M) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L(M) by defining X ° Y = (exp|B)−1((exp X)° (exp Y)) for X and Y in a neighbourhood C of 0 in B such that (exp C) ° (exp C) ⊂ V. Similarly, we transport / and \.

It is well-known that if R is a commutative ring with identity, M is a Noetherian R-module and I is an ideal of R such that M/IM has finite length, then the function n → lR (M /InM) is a polynomial function for n large (cf. [3], p. II-25), where lR denotes length as an R-module. In this note we are concerned with the function

where a1, … , ar is a multiplicity system for has finite length.

In a series of papers we have analysed the embedding of certain groups H as normal subgroups of absolutely irreducible skew linear groups G, see [7], [8], [9] and [10]. Here we drop the absolutely irreducibility assumption on G. If H is locally finite we derive relatively strong conditions on G, although not as strong as when G is absolutely irreducible. If H is abelian, very much in contrast to the absolutely irreducible case, we show that nothing can be said. The phrase “bounded by an integer-valued function of n only” we abbreviate to “n-bounded”.

We continue the study of approximate subdifferentials initiated in [3], this time for functions on arbitrary locally convex spaces. The complexity of the infinite dimensional theory is in particular determined by the fact that various approaches and definitions which are equivalent in the finite dimensional situation are, in general, no longer equivalent if the space is infinite dimensional.

In a recent paper Taylor and Tricot [10] introduced packing measures in ℝd. We modify their definition slightly to extend it to a general metric space. Our main concern is to show that in any complete separable metric space every analytic set of non-σ-finite h-packing measure contains disjoint compact subsets each of non-σ-finite measure. The corresponding problem for Hausdorff measures is discussed, but not completely resolved, in Rogers' book [7]. We also show that packing measure cannot be attained by taking the Hausdorff measure with respect to a different increasing function using another metric which generates the same topology. This means that the class of pacing measures is distinct from the class of Hausdorff measures.

The following theorem shows that when packing unit spheres in a large box the spheres occupy at most about 0.7784 of the volume of the box. This improves Rogers' bound [2], which is approximately 0.7796. In the most efficient known packing the ratio is about 0.7405. The box in the theorem could be any bounded solid. Then the (l + 2)(m + 2)(n + 2) becomes the volume of a larger solid all of whose boundary points are at least one unit away from the original solid. At the start of the proof the even larger box could be replaced with a large ball concentric with and radius five units larger than some ball containing the original solid.

Let UK = [0, 1)K be the K-dimensional unit cube, where K ≥ 2. Suppose that we have a distribution ℘ of N points in UK. For × = ( x1, … , xK) ε UK, let A(x) denote the box

and write

Note that since N is the cardinality of ℘ and x1 … xK is the K-dimensional volume of A(x), the term Nx1 … xK represents the “expected number” of, points of ℘ in A(x).

Let

be a real quadratic form in n variables with integral coefficients (i.e., 2fij ε ℤ, fiiε ℤ.) and determinant D ≠ O. A well-known theorem of Cassels [1] states that if the equation f = 0 is properly soluble in integers x1 … , xn then there is a solution satisfying

where F = max |fij and we use the «-notation with an implicit factor depending only on n. More recently it has been shown that f has n linearly independent zeros x1 …, xn satisfying

(see [2, 3 and 6])