A system of openly disjoint discs in the plane is said to form a 6-neighboured circle packing if every is tangent to at least 6 other elements of (It is evident that such a system consists of infinitely many discs.) The simplest example is the regular circle packing all of whose circles are of the same size and have exactly 6 neighbours. L. Fejes Tóth conjectured that the regular circle packing has the interesting extremal property that, if we slightly “perturb” it, then there will necessarily occur either arbitrarily small or arbitrarily large circles. More precisely, he asked whether or not the following “zero or one law” (cf. [3], [6]) is valid: If is a 6-neighboured circle packing, then

where r(C) denotes the radius of circle C, inf and sup are taken over all C ∊

In a previous paper (2), the inner product of two S-functions {λ}.{μ} was defined for (λ),(μ) partitions of the same integer n. Briefly, the ordinary product {λ}.{μ} of two S-functions corresponds to the analysis of the direct product of two corresponding representations of the full linear group, while the inner product {λ}.{μ} corresponds to the analysis of the direct product of two representations of the corresponding symmetric group.

The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. Sce, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments.

To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n ⩾ 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however.

Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,\,p,\,n)$ were constructed recently in the work of Ariki and Koike [AK], Broué and Malle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,\,p,\,n)$ given by Ariki and Koike [AK] and Ariki [Ari].

The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.

1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t, and let the absolute value of the determinant of Q be denoted by |det Q|. In this paper we study the problem of determining the maximum of |det Q| for fixed t and v. It turns out that this problem is closely related to the v, k, λ problem, which has been extensively studied of late.

Let n be a finite affine plane with n points on a line. We denote by G(II) the graph whose vertices are all points and lines of II, with two vertices adjacent if and only if one is a point, the other is a line, and the point and line are incident. Let L(11) denote the line graph of G(II), i.e., the vertices of L(II) are the edges of G(II), and two vertices of L(II) are adjacent if the corresponding edges of G(II) are adjacent. It is clear that L(II) is a regular, connected graph with n2(n + 1) vertices and valence 2n — 1.

Let $\Xi $ be a discrete set in ${{\mathbb{R}}^{d}}$. Call the elements of $\Xi $centers. The well-known Voronoi tessellation partitions ${{\mathbb{R}}^{d}}$ into polyhedral regions (of varying volumes) by allocating each site of ${{\mathbb{R}}^{d}}$ to the closest center. Here we study allocations of ${{\mathbb{R}}^{d}}$ to $\Xi $ in which each center attempts to claim a region of equal volume $\alpha $.

We focus on the case where $\Xi $ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation.

The model exhibits a phase transition in the appetite $\alpha $. In the critical case $\alpha \,=\,1$ we prove a power law upper bound on $X$ in dimension $d\,=\,1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha <1$ and $\alpha \,>1$we prove exponential upper bounds on $X$.

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

Given a set of vertices which have each been assigned one of the colours C1, C2, …, Cm, with nj vertices Cj, a formula is derived for the number of oriented trees on these vertices, having a designated root, and subject to any number of restrictions of the form “no arc goes from a vertex of colour Ci to a vertex of colour Cj”. The formula is based on a combinatorial construction which defines a correspondence between such trees and certain sequences.

In an important paper (7), long overlooked, J. H. Redfield dealt with several aspects of enumerative combinatorial analysis. In a previous paper (1) I showed the relation between a certain repeated scalar product of a set of permutation characters of a symmetric group and Redfield's composition of his group reduction functions. Here I consider, from a group representational point of view, Redfield's idea of a range-correspondence and its application to enumeration of linear graphs. The details of the application of these ideas to more general enumerations are also given.

Let [Pn(k)] be the class of all subsets Pn(k) of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1. Denote by dk(n, r) the maximum number of times a given distance r can occur among points of a set Pn(k).Put

In other words Dk(n) denotes the maximum number of times the same distance can occur between n suitably chosen points in k-dimensional space.

Let P be a finite set with p objects oj, j = 1, 2, … , p, and let {Si}, i = 1, 2, … , n, be a family of n subsets of P. The incidence matrix A = (aij) for the family {Si} is defined by the rules: aij = 1 if 0j, ∈ Si and aij = 0 if 0j ∉ Si. Then, if AAT = B = (bij) (where AT denotes the transpose of A), it is easy to see that bij = |Si ⋂ Sj|, i = 1, … , n, j = 1, … , n, so that the elements of B are integers with bii ⩾ bij ⩾ 0.

Analytical study of the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray–Hopf solution of the Boussinesq system are established as the regularizing parameter $\alpha$ vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.

1. Introduction. In [4] Arhangel'skiĭ proved the remarkable result that every regular space which is hereditarily a Lindelöf p-space has a countable base. As a consequence of the main theorem in this paper, we obtain an analogue of Arhangel'skiĭs result, namely that every regular space which is hereditarily an ℵi-compact strong ∑-space has a countable net. Under the assumption of the generalized continuum hypothesis (GCH), the main theorem also yields an affirmative answer to Problem 2 in Arhangel'skiĭs paper.

In § 3 we introduce and study a new cardinal function called the discreteness character of a space. The definition is based on a property first studied by Aquaro in [1], and for the class of T1 spaces it extends the concept of Kicompactness to higher cardinals.

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebra A (i.e., one with no finite non-zero central projections), then the symmetries may be chosen from A. A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1 factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

A space X is called countdbly metacompact (countably paracompact) if every countable open cover has a point finite (locally finite) open refinement. According to Hodel [5], a space X is called countably subparacompact if every countable open cover has a σ-discrete closed refinement. It is well-known (see Mansfield [10] and Dowker [4]) that in normal spaces all of the preceding notions are equivalent. Also, according to Hodel [5], a countably subparacompact space is countably metacompact and the reverse implication is false.

Let

Then if p is a prime > 3, Glaisher [4] proved

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if

(1)

where

It is known (see [3, p. 96]) that (1) is equivalent to

(2)

We study positive measures on 𝔹n satisfying that . for any ƒ ∈ where is the Hardy-Sobolev space in the unit ball. We obtain several computable sufficient conditions as well as some necessary conditions and establish their sharpness. We study the same problem for Besov-Sobolev spaces and give some applications to multipliers.