In §§2, 3 a simple expression in finite terms is found for the small transverse displacement of a thin plane elastic plate due to a transverse force applied at an arbitrary point of the plate. The plate is clamped along, and is bounded internally by, the parabola CDE shown in Fig. 1.

In [10] the following generalization of the Erdös–Ko–Rado [6] theorem was proved:

If 1 ≤ h ≤ m/2 andare r sets of h-subsets of {1,…, m} such that

then

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.

Let f(n) be a real-valued additive arithmetic function. For each positive rational integer n set

For the duration of this paper we shall assume that as n → ∞ both

A prime p > 2 is called irregular, if it divides the numerator of at least one of the Bernoulli numbers B2, B4, …, Bp – 3 (in the even suffix notation). The study of irregular primes has its origin in the famous theorem of Kummer which states that p divides the class number of the p-th cyclotomic field, if, and only if, p is irregular. Carlitz [1] has given the simplest proof of the fact that the number of irregular primes is infinite.

The assertions about the congruence properties of the coefficients Ci, Di, in formula (3) of the paper (Mathematika, 16 (1969), 101–105) referred to above are patently false. They are, moreover, irrelevant, though the succeeding argument is so condensed and badly expressed that the true and the false in the paper cannot easily be distinguished. In short, the paper ought to be re-written. My thanks are due to the critic who first complained of the mistakes to the Editors and who thereby took upon himself the arduous task of refereeing this Corrigendum. His patient criticisms led to many improvements of the arguments alluded to in my paper, as well as of those which are presented here, and I am most grateful to him for his help.

A triangle, in the sense of Schubert [1], is an entity in the plane S2 consisting of

(a) an ordered triad of points (P1, P2, P3);

(b) an ordered triad of lines (l1, l2, l3) connected with the points Pi by the incidence relations Pi ⊂ lj (i ≠ j); and

(c) a three base-point net Φof conies with the Pi as base-points.

A section of the non-negative orthant by an affine subspace is a polyhedral set. A technique, analogous to that of Gale diagrams, is described which enables one to determine the facial structure of such a polyhedral set.

Let (R, ) be a commutative Noetherian local ring. We investigate conditions for a non-finitely generated R-module M to have a system of parameters. We prove that if

then any system of parameters for R/AnR (M) is a system of parameters for M. As an application we characterize by means of systems of parameters those balanced big Cohen–Macaulay R-modules M for which SuppR (M) = suppR (M).

Let $\unicode[STIX]{x1D6E4}\subseteq \operatorname{PSL}(2,\mathbf{R})$ be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\unicode[STIX]{x1D6E4}$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $\text{e}^{(2/3)R}$ is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$ taking $z=w$ and averaging over $z$ in a certain way the error term can be improved to $\text{e}^{(7/12+\unicode[STIX]{x1D716})R}$. Here we show such an improvement for a general $\unicode[STIX]{x1D6E4}$; our error term is $\text{e}^{(5/8+\unicode[STIX]{x1D716})R}$ (which is better than $\text{e}^{(2/3)R}$ but weaker than the estimate of Risager and Petridis in the case $\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$). Our main tool is our generalization of the Selberg trace formula proved earlier.

We obtain the formula for the twisted harmonic second moment of the L-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean-value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet L-functions.

A diophantine system is studied from which is deduced an analogue of van der Corput's A5-process in order to bound analytic exponential sums of the form . The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a “ninth derivative test” for exponential sums which is essential for giving new exponent pairs in [3].

The relationship between the topological dimension of a separable metric space and the Hausdorff dimensions of its homeomorphic images has been known for some time. In this note we consider topological and packing dimensions, and show that if X is a separable metric space, then

where and denote the topological and packing dimensions of X, respectively.

Linear stability of an incompressible triple-deck flow over a wall roughness is considered for disturbances of high frequency. The wall roughness consists of two relatively short obstacles placed far apart on an otherwise flat surface. It is shown that the flow is unstable to feedback or global mode disturbances. The feedback loop is formed by algebraically decaying disturbances propagating upstream and weakly growing Tollmien-Schlichting waves travelling downstream and as such represents an interaction between modes from continuous and discrete spectra of the corresponding parallel-flow problem. An example of growth rate calculation for a specific roughness is considered.

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$, $(i=1,2,3)$, attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$, give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “$3$-systems”.

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.

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

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.