A well-known theorem of Schauder asserts that a continuous self-map T of a closed convex subset K of a normed linear space X, such that the closure of T(K) is compact, has at least one fixed point. No general method is known for a computation of the fixed point(s) of T.

Estimates involving polynomials can often naturally be given in terms of the discriminants of these polynomials or of the resultants of pairs of polynomials. Since the discriminant of a polynomial with multiple zeros vanishes as does the resultant of two polynomials with common zeros these results become trivial when applied to such polynomials or pairs of polynomials. Therefore it is often necessary to exclude polynomials with multiple zeros from a given investigation. In theoretical studies of a measure-theoretical nature this often does not affect the results; however for the purpose of constructing polynomials with specified properties it can be an advantage if it is not necessary to restrict the attention to polynomials without multiple zeros.

The vertex-connectivity and the edge-connectivity of a graph involve minimum sets of vertices and edges, respectively, whose removal results in a disconnected graph. However, the mixed case of separating sets consisting of both vertices and edges appears to have been overlooked. Such considerations might apply to vulnerability problems, such as that of disrupting a railway network with both tracks and depots being destroyed. Depending on the relative costs, a particular combination of tracks and depots might be optimal for the purpose.

It was proved in a recent paper† that if au α1, …, αn denote non-zero algebraic numbers and if‡ logαn, …, log αn and and 2πi are linearly independent over the rationals then log α1, …, log αn are linearly independent over the field of all algebraic numbers. Further it was shown that if α1 …, αn are positive real algebraic numbers other than 1 and if β1, …, βn denote real algebraic numbers with 1, β1 …, βn linearly independent over the rationals then is transcendental.

In 1933, K. Borsuk [1] established the well-known result that if n closed sets cover Sn−1 then at least one set contains antipodal points, where Sn−1 is the surface of the ball Tn of centre O and unit diameter in Rn. This result prompted H. Hadwiger [2] to make a still unresolved conjecture which, in the spirit of B. Griinbaum's survey [3], we state as follows: Let r be the largest integer such that whenever r closed sets cover Sn−1 at least one set realizes all distances between 0 and 1. Then r = n.

Introduction and result. Let

be analytic in |z| < 1. The Hankel determinants are defined by

Let Λ be a lattice in 3-dimensional space which provides a double covering for spheres of unit radius. By this we mean that if X is any point of space, there are at least two distinct lattice points P, Q such that XP ≤ 1, XQ ≤ 1. Let d(Λ) be the determinant of Λ. We shall prove that

where

Truesdell and Noll [1; sections 22, 27, 34] have discussed the concepts of material uniformity and homogeneity in continuum mechanics. A body is said to be materially uniform if, roughly speaking, all the particles composing the body are of the same material and homogeneous if there exists a global reference configuration which can be taken as a natural state for the whole body. To make the ideas precise for elastic materials, consider a small neighbourhood of each particle X and suppose that a reference configuration κ is chosen for each . Then during the motion, the deformation gradients may be calculated at each point X relative to the local reference configurations k. The stress at X is a function of these deformation gradients and if the stress relation does not depend explicitly on X the body is said to be materially uniform. If each local reference configuration κ can be taken as the configuration of its associated set of particles in some global reference configuration for the whole body, the body is said to be homogeneous. In general, however, the configurations κ need not fit together to form a global reference configuration. The body is then said to contain a distribution of dislocations.

In the present paper the researches initiated in the two earlier papers of this series are continued, and, by suitable generalizations of the techniques employed therein, solutions are obtained to some further well known problems from the theory of transcendental numbers. It will be proved, for example, that a non-vanishing linear form, with algebraic coefficients, in the logarithms of algebraic numbers, cannot be algebraic. This implies, in particular, that π + log α is transcendental for any algebraic number α ≠ 0, and also eα π+ß is transcendental for all algebraic numbers α, β with β ≠ 0.

It has been pointed out to us by Professor L. Schoenfeld that there is a fallacy in the proof of Theorem 3 of our paper “The values of a trigonometrical polynomial at well spaced points” [ Mathematika, 13 (1966), 91–96]. The fallacy occurs in the appeal to Theorem 1 at the end of the proof. If this is to be made explicitly, we must not only put n = dn′ but also put q = dq′ but then the sum over m goes from 1 to q instead of from 1 to q′, and if one allows for this the final result becomes much weakened.