Equifacetal simplices, all of whose codimension one faces are congruent to one another, are studied. It is shown that the isometry group of such a simplex acts transitively on its set of vertices and, as an application, equifacetal simplices are shown to have unique centres. It is conjectured that a simplex with a unique centre must be equifacetal. The notion of the combinatorial type of an equifacetal simplex is introduced and analysed, and all possible combinatorial types of equifacetal simplices are constructed in even dimensions.

Every homogeneous convex body in ℝd (d≥2) put to sit on a horizontal hyperplane finds a position of stable equilibrium. A cube has 2d such positions and an ellipsoid with pairwise distinct axis-lengths has 2. How many positions of stable equilibrium have most convex bodies?

The term “most” is understood in the Baire category sense. For various other results on most convex bodies, see [2], [4].

Suppose throughout that l, an (n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ]. Let

Suppose that is a distribution of N points in the unit square U = [0, 1]2. For every measurable set B in U, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2

In the following pages there will be found an account of the properties of a certain class of local rings which are here termed semi-regular local rings. As this name will suggest, these rings share many properties in common with the more familiar regular local rings, but they form a larger class and the characteristic properties are preserved under a greater variety of transformations. The first occasion on which these rings were studied by the author was in connection with a problem concerning the irreducibility of certain ideals, but about the same time they were investigated in much greater detail by Rees [7] and in quite a different connection. In his discussion, Rees made considerable use of the ideas and techniques of homological algebra. Here a number of the same results, as well as some additional ones, are established by quite different methods. The essential tools used on this occasion are the results obtained by Lech [3] in his important researches concerning the associativity formula for multiplicities. Before describing these, we shall first introduce some notation which will be used consistently throughout the rest of the paper.

Let ℱ be a set of m subsets of X = {1,2,…, n}. We study the maximum number λ of containments Y ⊂ Z with Y, Z ∊ ℱ. Theorem 9. , if, and only if, ml/n → 1. When n is large and members of ℱ have cardinality k or k–1 we determine λ. For this we bound (ΔN)/N where ΔN is the shadow of Kruskal's k-cascade for the integer N. Roughly, if m ∼ N + ΔN, then λ ∼ kN with infinitely many cases of equality. A by-product is Theorem 7 of LYM posets.

Let q be a natural number. When the multiplicative iroup (ℤ/qℤ)* is a cyclic group, its generators are called primitive roots. Note that the generators are also elements with the maximum order if (ℤ/qℤ)* is cyclic. Thus, when (ℤ–qℤ)* is not a cyclic goup, we then call an element with: he maximal possible order a primitive root, which was initially introduced by R. Carmichael [1].

Littlewood posed the following problem: for each pair of real numbers, θ and φ, and each ε > 0, is there a positive integer n such that

where ‖α‖ denotes the difference between a and the nearest integer.

In a conversation with Professor R. G. Stoneham, he mentioned the estimate which he had found for a new type of exponential sum, namely

where p is a prime, g a primitive root mod p, pp × b, and l ≤ X ≤ p − 2. I found a proof and sent it to him. In correspondence with him and Professor D. A. Burgess,‡ I learnt that the estimate arose in Stoneham's work on normal numbers, in which they had been interested for some time. A proof of (1) will appear as a lemma in a paper by Stoneham [1] which has been submitted for publication. In fact, each of them has found several proofs of (1), and in particular, Burgess had found quite a simple one which is practically the same as my proof. However, my presentation led to an interesting generalization.

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size $B$ with each component having no prime divisors below $B^{1/u}$, where $u$ equals $c_{0}n^{3/2}$, $n$ is the number of variables and $c_{0}$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $n^{3/2}$ to the logarithmic $(\log n)(\log \log n)^{-1}$. Our main new ingredients are the generalization of the Brüdern–Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport–Birch version of the circle method.

The nonlinear interaction equations describing vortex-Rayleigh wave interactions in highly curved boundary layers are derived. These equations describe a strongly nonlinear interaction between an inviscid wave system and a streamwise vortex. The coupling between the two structures is quite different from that found by Hall and Smith [13] in the absence of wall curvature. Here the vortex is forced over a finite region of the flow rather than in the critical layer associated with the wave system. When the interaction takes place the wave system remains locally neutral as it moves downstream and its self interaction drives a vortex field of the same magnitude as that driven by the wall curvature. This modification of the mean state then alters the wave properties and forces the wave amplitude to adjust itself in order that the wave frequency is constant. Solutions of the interaction equations are found for the initial stages of the interaction in the case when the wave amplitude is initially small. Our analysis suggests that finite amplitude disturbances can only exist when the vortex field is nonzero at the initial position where the interaction is stimulated.

Let M be a convex body, i.e., a compact, convex set with non-empty interior, in n-dimensional Euclidean space En. A chord [a, b] of M is said to be an affine diameter of M, if, and only if, there exists a pair of (different) parallel supporting hyperplanes of that body, each containing one of the points a, b. The following result of Eggleston (cf. [1] and [2]) is well-known. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to exactly three affine diameters. In [3] this result is sharpened. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to at least two, but a finite number of affine diameters. A natural problem for the n-dimensional case, based on Eggleston's result, is the following (cf. also [4]). Is it true that the n-dimensional simplex is the only convex body in En such that through each interior point pass precisely 2n − l affine diameters? For the case of convex polytopes, i.e., convex bodies with a finite number of extreme points, we shall give a positive answer to this question.

In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymptotic expansions for these solutions. A fundamental illustration of this technique may be found in Hille [6] wherein the linearly independent solutions of a second order homogeneous differential equation were described by single termed asymptotic expansions. For the dominant solution, this result was successively extended by Waltman in [8] for a second order equation and in [9] for an nth order equation. A further generalization of these results appears in [3] where a complete nth order nonhomogeneous nonlinear differential equation was considered; again, asymptotic representations were given to describe the behaviour of the solutions of the differential equation. Moore and Nehari [7], Wong [10], [11], and Hale and Onuchic [2], also use asymptotic representations in discussing the behaviour of the solutions of certain differential equations. All of the above results are essentially perturbation problems with the unperturbed linear differential equation having the form y(n) = h(t) for some n and h(t).

Let K be any finite (possibly trivial) extension of ℚ, the field of rational numbers. Let denote the ring of integers of K, and let M ⊆ be a full module in K thus a free ℤ-module of rank [K : ℚ] contained in ; ℤ denoting the ring of rational integers. Regarding as an abelian group, the index (: M) is finite. Suppose that m1, …, mk is a ℤ-basis for M and let a ∊ Then the polynomial

(the xi; being indeterminates) will be called a full-norm polynomial; here NK/ℚ denotes the norm mapping from K to ℚ. Apart from constant factors, such a polynomial f(x) is necessarily irreducible in ℤ[x].

Filters and אּ-complete filters can be used to produce set-theoretic extensions of direct sums and direct products. They can be applied to generalize theorems in module theory which involve these. For example, the theorem, stating that a ring is noetherian, if, and only if, direct sums of injectives are injective, can be generalized, provided we replace noetherian by Xa-noetherian and direct sums by אּ- complete filter sums with a suitable property.

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\leqslant K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound

$$\begin{eqnarray}|\{(a_{1}+1)(a_{2}+1)\cdots (a_{k}+1):a_{i}\in A\}|\geqslant \frac{|A|^{k}}{(8k^{4})^{kK}}.\end{eqnarray}$$

$K$

$c\log |A|$

$c=c(k)$

$\unicode[STIX]{x1D6EC}$

We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^{n}$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in works by Sawyer [Mathematika34(1) (1987), 69–76] and Wisewell [Mathematika51(1–2) (2004), 155–162]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one over $\mathbb{F}_{\ell }\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ presented by Dummit and Hablicsek [Mathematika59(2) (2013), 257–266].

We prove asymptotic formulas for the number of integers at most $x$ that can be written as the product of $k~({\geqslant}2)$ distinct primes $p_{1}\cdots p_{k}$ with each prime factor in an arithmetic progression $p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$, $(a_{j},q)=1$ $(q\geqslant 3,1\leqslant j\leqslant k)$. For any $A>0$, our result is uniform for $2\leqslant k\leqslant A\log \log x$. Moreover, we show that there are large biases toward certain arithmetic progressions $(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$, and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.