The objectjof this paper is to extend to algebraic number fields some of the recent results proved by the large sieve method. In particular we prove generalizations of Bombieri's form of the large sieve inequality [1; Theorem 1] and of the theorems of Davenport and Halberstam [2] and Bombieri [1, 4], on the average distribution of primes in arithmetic progressions.

A graph consists of a set of vertices some pairs of which are joined by a single edge. A tree is a graph with the property that each pair of vertices is connected by precisely one path, i.e., a sequence of distinct vertices joined consecutively by edges. The complexity c of a graph G(n, k) with n vertices and k edges is the number of trees with n vertices which are subgraphs of G(n, k). The distribution of c over the class of all graphs G(n, k) is of physical interest because it throws light on the classical many-body problem. (See, e.g. [9].) Ford and Uhlenbeck [3] gave numerical data which suggested that the distribution of c tends to normality for increasing n if k is near No moments higher than the first were known in general and they remarked in [4] that even “the second would be worth knowing”. The main object in this paper is to derive a formula for the second moment of c.

Let K be a convex body in Euclidean space Rd, d≥2, with volume V(K) = 1, and n ≥ d +1 be a natural number. We select n independent random points y1, y2, …, yn from K (we assume they all have the uniform distribution in K). Their convex hull co {y1, y2, …, yn} is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope

It is easy to see that if U: Rd → Rd is a volume preserving affine transformation, then for every convex body K with V(K) = 1, m(K, n) = m(U(K), n).

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.

In ring theory there is the following theorem (cf. [1], p. 39):

If R is a ring satisfying the descending chain condition for left ideals, then the following three conditions are equivalent: (i) R is primitive, (ii) R is simple, and (iii) R is isomorphic to the complete ring of linear transformations in a finite dimensional vector space over a division ring.

Artin, in 1927, conjectured that for any given non-zero integer a other than — 1 or a perfect square there exist infinitely many primes for which a is a primitive root. He also conjectured that the number of primes not exceeding x, denoted by Na(x), for which a is a primitive root is given by the asymptotic formula

where A(a) is a constant depending on a.

Let AN be an N-point set in the unit square and consider the discrepancy function

where x = (x1, x2) ∈ [0,1;]2, and |[0, x)]| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of Schmidt [Irregularities of distribution. VII. Acta Arith. 21 (1972), 45–50] on the L∞ norm of DN. We show that necessarily

The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have

whenever N = 2n for some positive integer n. This estimate depends upon variants of the Chang–Wilson–Wolff inequality [S.-Y. A. Chang, J. M. Wilson and T. H.Wolff, Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv.60(2) (1985), 217–246]. We also provide similar estimates for the BMO norm of DN.

For a fixed integer k ≥ 2 suppose we have functions Li (x) satisfying the following conditions.

Positive definite quadratic forms are associated with pointlattices in the following way. If x ∊ Zm and H = MT for a real m × m matrix M, then xTHx is the square of the distance from the origin to the point Mx of MZm (equally, to W Mx ∊ W MZm, for orthogonal W).

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated lower bound theorem (LBT) for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; and we propose the balanced analog of the generalized lower bound conjecture (GLBC) and establish some related results. We close with constructions of balanced manifolds with few vertices.

In this note we prove:

Theorem. Let Λ be a normal number field of absolute degree n. Then there exist infinitely many normal number fields K with the following properties.

(i) The Galois group of K over the rational field Q is the symmetric group Sn on n symbols.

Let be a finitely generated subgroup of SL (2, ℱ), where ℱ is the ring; of holomorphic functions on the open unit disc Δ. For each point z0 in Δ we can evaluate all matrix entries of at z0, to obtain a subgroup {z0} of SL (2, ℂ) and a surjective representation → {z0}. If this representation is not faithful, then contains a nontrivial element W such that W evaluated; at z0 is trivial. But W can evaluate to the identity only on a countable subset) of Δ, and there are only countably many choices for W in Consequently there are at most countably many points zk in Δ such that {zk} is not isomorphic to Δ. Our main result can now be stated as follows.

This paper is a sequel to [1], but can be read independently. However, we do quote one of the results of [1] in the proof of our theorem here.

As in [1], we define the continuantK(a1, a2, …, an) formed from the positive integers a1 …,an to be the denominator qn of the continued fraction

where pn/qn is in lowest terms.

The 24 classes of even-valued, unimodular, positive quadratic forms in 24 variables have been determined by Niemeier [6]. One class is distinguished from the rest by the property that its forms have arithmetic minimum 4, rather than 2. The 24-dimensional lattice Λ corresponding to this form-class can therefore be identified with one found earlier by Leech [4], which has been studied extensively in connection with sporadic simple groups (e.g. [1]).

Among Schubert's many experiments in the application of a symbolic calculus to problems of enumerative geometry, some special attention is due to his long memoir entitled “Anzahlgeometrische Behandlung des Dreiecks” [1]. For one thing, he is dealing here with a simple, though not elementary, kind of geometric variable, the triangle in a fixed plane, so that the paper gives a clear insight into his general method; and, for another, there is contained in this paper, as was recently suggested by Freudenthal ([2], p. 19), an apparently miraculous device, the introduction of “infinitesimal triangles”, which we can now recognize (§4) as having had the effect of desingularizing the triangle domain in which the calculus was to operate. The principal target of Schubert's investigations was the discovery of Bézout-type formulae for the number of triangles common to two algebraic systems Σr and Σ6-r (r = 1, 2, 3) of complementary dimensions, the systems being supposed to intersect in only a finite number of triangles, and the multiplicities of these triangles being assumed to be suitably defined. His systems, also, had to be “normal” i.e. they could only contain such sub-systems of degenerate triangles as were of the dimensions he regarded as normal. He found, by his methods, that “normal” system Σ1 and Σ5 are each characterized (in so far as intersection numbers are concerned) by 7 projective characters, systems Σ2 and Σ4 by 17 such characters, and systems Σ3 by 22 such characters.

The main purpose of this paper is to evaluate the Whitehead group K1 (Zπ) of the quaternion group π of order 8. We show that the natural mapping of the units ± π of Zπ into K1(Zπ) induces an isomorphism between K1(Zπ) and ± V, where V, Klein's 4-group, is the commutator quotient group of π.

The theorem of Aleksandrov-Fenchel-Jessen states that two convex bodies in n-dimensional Euclidean space En which, for some p l, , n - l 007D;, have equal area measures of order p (see Section 2 for a definition) differ only by a translation. Two independent proofs were given by Aleksandrov 1 and by Fenchel and Jessen 18 see also Busemann 5 (p. 70) and LeichtweiG 25 (p. 319), 26. If the boundaries of the two bodies are sufficiently smooth and of everywhere positive curvatures, then the assumption of the theorem is equivalent to saying that at points with parallel outer normals the p-th elementary symmetric functions of the principal radii of curvature of both boundary hypersurfaces are the same. For this case, Chern 6 gave a uniqueness proof by means of an integral formula.

General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ + Mδc, in terms of those of the measures dμ and (x − c)2dμ. In particular, these relations allow us to show that Nevai's class M(0, 1) is closed under adding a mass point, as well as obtain several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

In this paper we shall set out the generalization, for n-dimensional space Sn, of some recent results about complete quadrics and complete collineations in S2, S3 and S4. For the results about complete conies in S2, originally introduced by Study [1], we refer the reader to papers by Severi ([2], [3]), van der Waerden [4], Semple [5]; for those about complete quadrics in S3, to Semple ([6], [7]); for the extension to S4 to Alguneid [8]; for the general concept of complete collineations in Sn, and for results in S2 and S3, to Semple [9].

We shall give a simple new proof of the following known theorem [1, 2].

Theorem. The upper density of a packing of translates of a convex disc cannot exceed the density of the densest lattice-packing of these discs.

In [] and [2] this theorem is proved for centrally symmetric discs. The general case can be reduced to this one by applying to the discs the known construction of central symmetrization. Our proof goes in a reverse way. We shall give a direct proof for a special family of asymmetric discs whose centrally symmetric images exhaust the family of centrally symmetric convex discs. Using the properties of the symmetrization, this implies the validity of the theorem for all centrally symmetric discs, and consequently for all convex discs. This procedure of going from a special case to the general one, by applying the symmetrization twice, is illustrated by the following example. The validity of the theorem for a Reuleaux triangle implies its validity for a circle, which implies its validity for any disc of constant width.