This paper is concerned with the geometry of a measure μ, and in particular with the relationship between various .s-dimensional densities of μ, the geometry of the support of μ and the question of whether s is an integer.

In this paper we continue our study of the values taken by Euler's ø-function begun in [1]–[3]. Let ør(n) be the iterated ø-function, that is ør(n) = ø{ør-1(n} where ø1 = ø1. Let

One of the oldest questions concerning primitive roots is that of the actual order df magnitude of the least positive primitive root m(p) to a large prime modulus p. During the last sixty years the interest of several mathematicians has been attracted by this problem. The history of the major results is too well known to need elaboration. The best known bound for m(p) is due to Wang [8] and Burgess [1] who independently obtained

In the part (16-3) of his extensive study on measurability in Banach spaces, Talagrand [12] considered the Banach space C(K) of continuous functions on a dyadic topological space K. He proved that C(K) is realcompact in its weak topology, if, and only if, the topological weight of K is not a twomeasurable cardinal (Theorem 16-3-1). Then he asked for an alternative to a rather complicated proof presented there (p. 214) and posed the problem whether C(K) is measure-compact whenever the weight of K is not a realmeasurable cardinal (Problem 16-3-2).

Wave propagation in an elastic medium rotating with a constant angular velocity Ω about an axis is studied by the method of Lighthill [1]. The waves are created by a concentrated periodic force of fixed frequency ω at the origin. The changes in the wave pattern and the decay of amplitude are discussed when 2Ω. increases from zero to a value greater than ω.

§1. An initial and boundary value problem. In this article we study the solution of an initial and boundary value problem of dynamic linear thermoelasticity in which both inertia and the coupling between mechanical and thermal effects are retained.

In this paper we continue our investigation of the topological filtration on the complex representation ring R(G) of a finite group, see [4] and [5]. To recall the basic definitions from (1): let

map a k-dimensional representation ζ to the (flat) vector bundle over the classifying space BG associated to the universal G-bundle by the G-structure on Ck. Then, if denotes the (2k − l)-skeleton of BG,

Our investigation starts from the question: is the even dimensional cohomology of the non-abelian group of order p3 and exponent p generated by Chern classes? From the computation of the complete cohomology ring in [8] one quickly sees that the essential problem is to express elements of the form cor (γk), γ ∈ H2 (k, ℤ), K a subgroup of index p, in terms of Chern classes. For a more general pair of groups (K ≤ G) it is known, see [7] that the best for which one can hope is a description of some multiple of cor (γk) in this way. Our first theorem shows that, under suitable hypotheses (satisfied in particular by the example of order p3) the numerical factor may be removed. Thus

Reid (1974) derived “first approximations” to solutions of the Orr–Sommerfeld equation, which are uniformly valid in a full neighbourhood of a critical point. This paper shows that such approximations may be calculated to higher order, and makes a first step towards placing the theory on a rigorous basis by providing error bounds for the dominant-recessive approximations. These are obtained by generalizing methods discussed by Olver (1974) for second order linear ordinary differential equations.

The self-complement index s(G) of a graph G is the maximum order of an induced subgraph of G whose complement is also induced in G. This new graphical invariant provides a measure of how close a given graph is to being selfcomplementary. We establish the existence of graphs G of order p having s(G) = n for all positive integers n < p. We determine s(G) for several families of graphs and find in particular that when G is a tree, s(G) = 4 unless G is a star for which s(G) = 2.

The object of this paper is to prove the following:

Theorem. Every perfect septenary quadratic form assumes its minimum value at a set of 7 points with integer co-ordinates whose determinant is 1.

This is true also, as shown by Rankin [1], with n ≤ 6 in place of 7. The proof will be shortened considerably by using the weaker result obtained in [1] for n = 7, and we shall also use the following classical results, see, e.g., [2], for Hermite's constant γn:

In the present paper a simple technique will be developed for the arithmetical determination of certain class group components and class number factors in finite number fields. This technique is based on classical theories (Hilbert's work on inertia groups, the theory of absolutely Abelian fields as class fields of congruence groups, absolute class fields of number fields). In keeping with the traditional approach to the subject we shall use here the language of ideal theory. The only non-classical concepts to be used (which, however, are of fundamental importance) are those of the inertia groups and the congruence groups associated to p-adic fields. We shall also give some illustrations of the use of our technique in some special cases. Further applications will follow in subsequent papers.

§1. Introduction. Most prominent among the classical problems in additive number theory are those of Waring and Goldbach type. Although use of the Hardy–Littlewood method has brought admirable progress, the finer questions associated with such problems have yet to find satisfactory solutions. For example, while the ternary Goldbach problem was solved by Vinogradov as early as 1937 (see Vinogradov [16], [17]), the latter's methods permit one to establish merely that almost all even integers are the sum of two primes (see Chudakov [4], van der Corput [5] and Estermann [7]).

It is proved that there are infinitely many integers n such that n and n + 1 have the same number of distinct prime divisors.

This paper arises from an attempt to generalise the following result of [1].

THEOREM (Armitage-Fröhlich). If K is a cyclic extension of Q of degree lr, where 2 has even order mod l, then

The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture.

We prove that if 3|d, then the d-dimensional balls are m-divisible for every m large enough. In particular, the three-dimensional balls are m-divisible for every m≥22.

Let $L(s,\unicode[STIX]{x1D712})$ be the Dirichlet $L$-function associated to a non-principal primitive character $\unicode[STIX]{x1D712}$ modulo $q$ with $3\leqslant q\leqslant 400\,000$. We prove a new explicit zero-free region for $L(s,\unicode[STIX]{x1D712})$: $L(s,\unicode[STIX]{x1D712})$ does not vanish in the region $\mathfrak{Re}\,s\geqslant 1-1/(R\log (q\max (1,|\mathfrak{Im}\,s|)))$ with $R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$.

Dyadic shifts $D\oplus T$ of point distributions $D$ in the $d$-dimensional unit cube $U^{d}$ are considered as a form of randomization. Explicit formulas for the $L_{q}$-discrepancies of such randomized distributions are given in the paper in terms of Rademacher functions. Relying on the statistical independence of Rademacher functions, Khinchin’s inequalities, and other related results, we obtain very sharp upper and lower bounds for the mean $L_{q}$-discrepancies, $0<q\leqslant \infty$. The upper bounds imply directly a generalization of the well-known Chen theorem on mean discrepancies with respect to dyadic shifts (Theorem 2.1). From the lower bounds, it follows that for an arbitrary $N$-point distribution $D_{N}$ and any exponent $0<q\leqslant 1$, there exist dyadic shifts $D_{N}\oplus T$ such that the $L_{q}$-discrepancy ${\mathcal{L}}_{q}[D_{N}\oplus T]>c_{d,q}(\log N)^{(1/2)(d-1)}$ (Theorem 2.2). The lower bounds for the $L_{\infty }$-discrepancy are also considered in the paper. It is shown that for an arbitrary $N$-point distribution $D_{N}$, there exist dyadic shifts $D_{N}\oplus T$ such that ${\mathcal{L}}_{\infty }[D_{N}\oplus T]>c_{d}(\log N)^{(1/2)d}$ (Theorem 2.3).

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.