The method of the large sieve has played a very important role in number theory. It turns out that estimates of exponential sums are of basic importance for large sieve inequalities. Let

be an exponential sum with complex coefficients c(n). It follows from Theorem 1 of Bombieri and Davenport [1] that

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.

Both S. Bergman [1] and I. N. Vekua [13] have constructed integral operators which map ordered pairs of analytic functions of one complex variable onto solutions of fourth order elliptic equations in two independent variables. Such operators play an important role in the investigation of the analytic properties of solutions to higher order elliptic equations and in the approximation of solutions to boundary value problems associated with these equations. Unfortunately, little progress has been made in developing an analogous theory for elliptic equations in more than two independent variables. Recently, however, Colton and Gilbert [7] constructed integral operators for a class of fourth order elliptic equations with spherically symmetric coefficients in p + 2 (p ≥ 0) independent variables, and at present Dean Kukral [11], a student of R. P. Gilbert, is in the process of trying to extend some recent results of Colton [3, 4, 5] for second order equations in three independent variables to the fourth order case.

The concepts of bounded paracompactness and bounded metacompactness were introduced and studied in [3]. In this paper we define bounded full normality and show that this concept is equivalent to bounded paracompactness.

It is known that countable paracompactness and countable metacompactness are equivalent properties in a normal space. We show that bounded countable metacompactness is equivalent to bounded countable paracompactness in a normal space and that bounded countable paracompactness is equivalent to bounded paracompactness in a hereditarily paracompact space.

The properties of the solution of the differential equation governing the evolution of localised line-centred disturbances to a marginally unstable plane parallel flow were described by Hocking and Stewartson (1972). A corresponding study of the properties when the initial disturbance is point-centred is presented here. A localised burst at a finite time can be produced, for certain values of the coefficients which can be determined analytically. When the equation permits solutions with circular symmetry, two kinds of bursting solutions, as well as solutions which remain finite, are possible, but the boundary between bursting and finite solutions could not be determined analytically.

Let ˜ be an equivalence relation on a topological space X. A point x ε X i s stable with respect to ˜ if it is in the interior of an equivalence class. We may also add, if ambiguity arises, that x is stable under perturbations in X. Let E be a Banach space, and let L(E) be the Banach space of continuous linear endomorphisms of E, with norm given by |T| = sup{ |T(x) | : |x| = 1}. In this paper we discuss stability of elements of L(E) with respect to some natural equivalence relations.

Maxwell's equations within a dielectric and/or a magnetic medium were first developed macroscopically and must be complemented by constitutive relations to obtain solutions. These relations connect D with E (and with B in optically active material) and H with B (and again with E in optically active material). The atomic and molecular theories of quantum mechanics allow a microscopic approach to derive these constitutive relations where the macroscopic electric and magnetic fields are averages of the microscopic fields e and b. In classical electromagnetic theory Lorentz [1] originally showed how to derive the Maxwell's macroscopic equations from electron theory using microscopic fields obeying the Maxwell's equations in vacuo but coupled to electronic and ionic sources. There are two distinct steps in this procedure. The first introduces microscopic polarization fields, both electric amd magnetic, p and m from which microscopic electric displacement vector field d and auxiliary magnetic field h are simply constructed. The resulting equations for the microscopic fields e, b, d, and h are called the atomic field equations. The second step is the statistical one where the macroscopic fields E, B, D, and H are defined as averages of the microscopic fields and these macroscopic fields are then shown to obey the phenomenological macroscopic Maxwell's equations. A historical appraisal may be found in the recent book by de Groot [2].

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.

Problem I was raised (oral communication) by Goffman some three years ago, and I found the example then. Problem II was raised at about the same time by Topsøe; Christensen has given an affirmative answer for spaces satisfying certain additional conditions.

It is easy to see that if the answer to problem I were affirmative then so would be that to problem II; therefore our counter-example for problem II implies the existence of one for problem I. It is also possible that another counter-example for problem I could be found by analysing the construction of Dieudonn6 in [1], which is also concerned (implicitly) with a failure of Vitali's theorem. Nevertheless our construction may be of independent interest.

If G is a finite group with classifying space BG and complex representation ring R(G), one defines the topological filtration on R(G) by means of the natural map Ψ: R(G) → KU(BG).

Certain properties of ascending sequences of quotient rings of points of algebraic varieties were investigated by O. Zariski in [4] and some related problems were discussed in [3].

The initial structure of wing-body interaction in supersonic flow was first investigated by Nielsen (1951), who analysed the steady supersonic inviscid flow past a proto-type wing-body combination consisting of an unswept thin wing at incidence and lying approximately in the axial plane of a non-lifting cylindrical body with a circular cross-section. In particular, he obtained the first two terms in the series expansion of the velocity potential at the start of the root chord where the wing and body meet. More recently, further theoretical studies which relate the problem to similar problems in diffraction theory have been carried out by Stewartson (1966), Jones (1967), Waechter (1969), and Clark (1970). In addition, Stewartson (1968) has extended Nielsen's formula for the following cases:

(a) the lifting wing with rounded leading edge,

(b) the flat plate wing at incidence with supersonic leading edge,

(c) the flat plate wing at incidence with subsonic leading edge in both symmetrical and antisymmetrical cases.

Let G and H be Hausdorff locally compact groups. By R(G, H) we denote the space of continuous homomorphisms of G into H equipped with the compact-open topology, namely that which is generated by subsets of the form

where K is any compact subset of G and U is any open subset of H. Further, let R0(G, H) be the subset of R(G, H) consisting of elements r satisfying the conditions that the quotient space H/r(G) is compact and r is proper, i.e., the action of G on H defined as the action by left translations of r(G) on H is proper. It has been shown by H. Abels [2] that if G and H are such that at least one of them has a compact defining subset then R0(G,H) is open in R(G,H). Moreover, for each r0 ∈ R0(G,H) there exists a neighbourhood M and a compact subset F of H such that r(G) F = H for all r ∈ M and for each compact subset K of H the union is a relatively compact subset of G. It is furthermore shown in [1] that if G contains no small subgroups and H is connected then the subset of R0(G, H) consisting of isomorphisms of G into H is open in R(G, H). These results, in the case in which H is a connected Lie Group and G is a discrete group, have been established by Weil in [5] and [6] appendix 1.

The theory of the rotational motion of slender particles subjected to Brownian couples is examined on the basis of two different sets of assumptions. The first set, which models the stochastic couples by a smooth diffusive type of expression, is well established in practice. The second set, which uses the theory of Wiener processes, is equally well established in principle. For the problem used for illustration, the two sets of assumptions are shown to yield the same results in an asymptotic sense as the inertia of the particle becomes progressively of relatively less importance.

Trees are basic in graph theory and its applications to many fields, such as chemistry, electric network theory, and the theory of games. König [7; 47-48] gives an interesting historical account of independent discoveries of trees by Kirchhoff, Cayley, Sylvester, Jordan, and others who were working in a variety of fields.

There are many equivalent ways of defining trees, the most common being: A tree is a graph which is connected and has no circuits. Figure 1 shows the trees with up to six vertices; those with up to 10 vertices can be found in Harary [5; 233–234]. Some other possible definitions of a tree are the following: (1) A tree is a graph which is connected and has one more vertex than edge. (2) A tree is a graph which has no circuits and has one more vertex than edge. For these and other characterizations see Berge [4; 152ff.] and Harary [5; 32ff.]. A less common definition is by induction: The graph consisting of a single vertex is a tree, and a tree with n + 1 vertices is obtained from a tree with n vertices by adding a new vertex adjacent to exactly one other vertex.

In a paper of the same title [3] Ch. Pommerenke and the author proved several results concerning the distances of Fekete points. In the present paper I will show that the same methods can be adapted to give an answer to a problem which we could not solve at the time.

Let E be a continuum and n ≥ 4 a given positive integer. A system of points z1, …, zn ∈ E that maximizes

is called a system of Fekete points. Such a system may not be unique.

Plummer (see [2; p. 69]) conjectured that the square of every block is hamiltonian, and this has just been proved by Fleischner [1]. It was shown by Karaganis [3] that the cube of every connected graph, and hence the cube of every tree, is hamiltonian. Our present object is to characterize those trees whose square is hamiltonian in three equivalent ways.

We follow the terminology and notation of the book [2]. In particular, the following concepts are used in stating our main result. A graph is hamiltonian if it has a cycle containing all its points. The graph with the same points as G, in which two points are adjacent if their distance in G is at most 2, is denoted by G2 and is called the square of G. The subdivision graph S(G) is formed (Figure 1) by inserting a point of degree two on each line of G.

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ0 the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.

Let a 3-connected graph G be embedded in an orientable surface of genus g in such a way that the connected components of its complement in the surface are topological discs. We denote by vk(G) the number of k-valent vertices of G and by pk(G) the number of k-gonal faces of the map defined by the embedded graph. For the numbers vk(G) and pk(G), it follows from Euler's formula that