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).

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.

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.

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.

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.

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.

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

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].

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 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

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.

One of the main open problems in the theory of Asplund spaces is whether every Asplund space admits a Fréchet differentiable bump function. This problem is also open for C(K) Asplund spaces, where it is unknown even for C∞-Fréchet smooth bump (a general Asplund space does not always admit C2-Fréchet smooth bump – it suffices to consider ℓ3/2[DGZ2]).

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].

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.

Asymptotic analysis of the Hele-Shaw flow with a small moving obstacle is performed. The method of solution utilizes the uniform asymptotic formulas for Green’s and Neumann functions recently obtained by V. Maz’ya and A. Movchan. The theoretical results of the paper are illustrated by numerical simulations.

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

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

$K$

$c\log |A|$

$c=c(k)$

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It is often helpful to compute the intrinsic volumes of a set of which only a pixel image is observed. A computationally efficient approach, which is suggested by several authors and used in practice, is to approximate the intrinsic volumes by linear combinations of the pixel configuration counts. However, we will show that when this approach is used for the computation of an intrinsic volume other than volume or surface area, an asymptotic error of 100% of the correct value cannot be avoided. As a consequence we derive that estimators which ignore the data and return constant values are optimal with respect to a natural criterion which has already been applied successfully for the estimation of the surface area.

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.