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

In this note we shall be mainly concerned with convex bodies of constant width in En that are invariant under the group of congruences that leave invariant a regular simplex with its centre of gravity at the origin. We first show that there are many such convex bodies. This follows, by showing that any set S ot diameter 1 that is invariant under a group of congruences about the origin, is contained in a convex body of constant width 1 that is invariant under the group.

If ω, ω′ are two points in the extended complex plane then the distance between their stereographic projections on the Riemann sphere is

with an obvious interpretation if ω or ω′ is ∞. is called the distance between ω and ω′ in the spherical or in the chordal metric.

A one parameter solution of th equation

was given by Ryley in 1825. Others were found by Richmond and myself. In the Landau memorial volume [1] recently published, there appears a joint paper, with the above title, of Davenport and Landau dating from 1935. They prove that if n is a positive integer, then positive rational solutions of (1) exist with denominators of order of magnitude O(n2). Their proof depended upon a two parameter solution of (1) due to Richmond and is very complicated.

Let R be a commutative ring with identity, and let U be a unitary commutative R-algebra with identity. In [1] Gilmer defines the (l/n)th power (n a positive integer) of a valuation ideal R when R is a domain. Sections 2, 3 of the present note are devoted to the study of an extension of this notion to positive rational powers of an arbitrary R-submodule of U.

Let fn be a positive-definite n-ary quadratic form, with real coefficients. By the minimum of fn, denoted by min fn, is meant as usual the least value of fn(x1, …, xn) for integers xi not all 0. A minimum point of fn is a point x = (x1, …, xn), with integer coordinates, at which fn takes its minimum value. Let Δ (> 0) be the determinant of a set of n minimum points of fn; then in [1] it was proved that

where γn is the Hermite constant. Enough is known about γn to deduce from (1.1), as in [1], that

The following theorem has been proved by Bambah, Rogers and Zassenhaus [1],

Theorem A. Let K be a closed convex domain with a centre. Let

be points such that:

(i) the polygon A0 A1 … An is a Jordan polygon bounding a closed domain π of area a(π);

(ii) for each r, 0 ≤ r ≤ n, there is a point common to K + An-1 and K + Ar;

(iii) the points An+1, …. An+m are in the interior of π;

(iv) for each point X of π, there exists an Ar, 1 ≤ r ≤ n + m, such that X ∈ K + Ar and the line segment XAr is in π. Then

where t(K) is the area of the largest triangle contained in K.

When a viscous incompressible fluid with uniform conditions at upstream infinity flows past a paraboloid of revolution, the boundary-layer equations admit of a similarity solution which leads to the ordinary differential equation (see Miller [1])

with boundary conditions

Let

be a quadratic form with integral coefficients, and suppose the equation

has a solution in integers x1…, xn, not all 0. It was proved by Cassels [2] that there is such a solution, which satisfies the estimate

where F = max|fij|. It was later observed by Birch and Davenport [1] that the result can be stated in a slightly more general form. Let

be a quadratic form which assumes only integral values at the points (x1 …, x2) of an n-dimensional lattice Λ of determinant Δ. Suppose there is some point of Λ, other than the origin, at which ø = 0. Then there is such a point for which also

where Φ = max |øij|.