In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible number of j-faces of a d-polytope with v vertices is achieved by a cyclic polytope C(v, d).

It is well-known and easy to prove that the maximal line segments on the boundary of a convex domain in the plane are countable. T. J. McMinn [1] has shown that the end-points of the unit vectors drawn from the origin in the directions of the line segments lying on the surface of a convex body in 3-dimensional Euclidean space E3 form a set of σ-finite linear Hausdorff measure on the 2-dimensional surface of the unit ball. A. S. Besicovitch [2] has given a simpler proof of McMinn's result. W. D. Pepe, in a paper to appear in the Proc. Amer. Math. Soc., has extended the result to E4. In this paper we generalize McMinn's result to En by use of Besicovitch's method, proving:

THEOREM 1. If K is a convex body in En, the set S, of end-points of the vectors drawn from origin in the directions of the line segments lying on the surface of K, is a set of σ-finite (n − 2)-dimensional Hausdorff measure on the (n − 1)-dimensional surface of the unit ball.

It is well known that Cantor's ternary set C, constructed on [0, 1], has a difference set, D(C), equal to [0, 1]. If E ⊂ R we define Dk(E)(⊂ Rk-1), by Dk(E) = {(d1, d2, …, dk-1); di ≥ 0, and there is x ∈ E such that x + di ∈ E for all i, 1 ≤ i < k}. Thus D2(E) ≡ D(E) and Dk(E) tells us whether or not a particular set of k real numbers can be translated into E. We call Dk(E) the k-difference set of E. In this work we seek criteria for finding the Besicovitch dimension of Dk(E) (written dim Dk(E))) and in particular, conditions on certain classes of linear sets E that ensure that Dk(E) should contain an open interval in Rk-1.

Suppose K is a convex body in Euclidean n-space En and that all the orthogonal projections of K onto p-dimensional linear subspaces have the same p-dimensional volume, that is K has constant outer p-measure. If p = 1, this means that K is of constant width; if p = n − 1, this means that K has constant brightness. It is known that, when the boundary of K is smooth enough to admit principal radii of curvature R1, …, Rn-1 as functions of the outer normal u, and if we define Fp(u) to be the p-th elementary symmetric function of these radii, then, when K has constant width,

Let P be a d-dimensional convex polytope (briefly, a d-polytope) in d-dimensional euclidean space Ed. Associated with P is a vector f(P), known as the f-vector of P, defined by

where fj(P) is the number of j-faces of P for 0 ≤ j ≤ d − 1 and the superscript T denotes transposition. (We regard f(P) as a column vector, and identify it with the vector

where (e1, …, ed) is some fixed basis of Ed.) Let d be the set of all d-polytopes in Ed, and d be any subset of d. Using tghe notation of [1; §8.1], we donate by f(d) the set of vectors {f(P): P ε d}, and write aff f(d) for the (unique) affine subspace of lowest dimension in Ed which contains all the vectors of f(d). Then it is well-known that

the equation of the hyperplane aff f(d) being that given by the Euler relation between the numbers fj(P) [1; Theorem 8.1.1].

Let G be a locally compact Abelian non-discrete group. Let M(G) be the convolution algebra of Radon measures on G. Let µ be an element of M(G) with its Lebesgue decomposition [1]

into absolutely continuous, purely discontinuous and continuous singular parts. The chief problem one encounters in the study of the invertibility of µ is with the case µs ≠ 0. As observed by Wiener and Pitt [2], the problem can be handled provided µs be “not too large”. In fact, Wiener and Pitt (loc. cit.) proved the following:

Let µ be a Radon measure on R (the real line) such that

whereare the Fourier transforms of µ, µd, and ≑µs≑ is the variational norm of µs. Then, µ has an inverse in M (R).

Erdös and Straus have conjectured that for every integer n > 1,

is soluble in positive integers x, y, z. Schinzel has conjectured that for every a > 0 if n > no(a),

is soluble in positive integers x, y, z.

When the Helmholz equation

is separated in ellipsoidal coordinates [1; §1.6] and the technique of separation of variables applied, there results the ordinary differential equation

known as the ellipsoidal wave equation or Lamé wave equation. In this equation k is the modulus of the Jacobian elliptic function sn z, and is related to the eccentricity of the fundamental ellipse of the ellipsoidal coordinates; a, b are separation constants, and the parameter q is connected with the wave number χ by

l being a real constant, the dimensional parameter of the coordinate system.

A PV-number is defined to be an algebraic integer θ, of modulus greater than one, all of whose conjugates (excluding θ itself) lie inside the unit circle. Salem [1] has shown that the set S of PV-numbers forms a closed subset of the real line.

Closed form solutions are obtained for a class of singular integral equations of the first kind with difference kernels. The kernel function is the sum of a polynomial and a second polynomial multiplied by a logarithm, with the possible addition of a strong singularity. A wider class of kernels have approximate representations in one of the above forms, where, for a specified accuracy, the polynomial orders will depend on the range of any parameter present. For example, the modified Bessel function kernels K0(|γx|), K1(|γx|)sgn x in the interval |x| ≤ 2 can be so expressed using 16th order polynomials with a maximum error 10-5 for real γ in the range 0 < γ ≤ 4.

Let E be a real normed linear space. Let K be a closed convex set containing 0, the origin, as an extreme point. Let A be a linear operator with AK ⊆ K. Stated below are theorems concerning eigenvectors and spectral (partial spectral) radius of A which generalize the well-known theorems of Bonsall [3] and Krein and Rutman [7] on positive operators. Proofs are given in §2.

The stability equation of the asymptotic suction boundary layer profile, using a linear stability analysis, is transformed into a generalised hypergeometric equation. The solution of the stability problem may thus be written formally in terms of the relevant generalised hypergeometric functions. An asymptotic analysis is carried out on these functions for large values of the Reynolds number, and the asymptotic representation of the solutions shown to agree with that given by the usual Orr–Sommerfeld analysis.

§0. We begin by stating a theorem of Jarník [1]. Let m be a positive integer, 0 < β < m, and Ψ(q) a positive function of integers q > 0. Let E be the set of points (x1, …, xm) for which the inequality

admits infinitely many solutions q ≥ 1. Then, supposing

Jarník proves that E has infinite Hausdorff β-measure.

The fluid mechanical problem with which we are concerned is the behaviour of fluid occupying the half-space x > 0 above a rotating disc which is coincident with the horizontal plane x = 0 and rotating about its axis which remains fixed. Studying rotationally symmetric solutions of this problem, von Kármán [1] (see also [2; p. 93]; [3; p. 133]) reduced it to the solution of two simultaneous equations in functions f(x), g(x) which may, with suitable normalisation, be written in the form

with the boundary conditions

§1. Preliminaries. A Cauchy process in d-dimensional Euclidean space, Rd, is a stochastic process, Xt(ω), with stationary independent increments and with a continuous transition density, p(t, y − x) defined by

and

where m, the isotropic measure, is a probability measure on Sd, the unit sphere in Rd, such that when d > 1 the support of m is not contained in any d − 1 dimensional subspace. In (2) w is given by

where . It follows that for each t > 0 and y we have p(t, y) > 0 and that for each t > 0 p(t, y) is a bounded and continuous function of y. Xt(ω) can be considered as being a standard Markov process (for a full description of the definition of such a process see Chapter 1 of [1]) and in particular we can assume that the sample functions of Xt(ω) are right continuous and have left limits. We can also assume that Xt(ω) enjoys the strong Markov property. We write Px and Ex for probabilities and expectations conditional on X0(ω) = x, and we write P for P0.

A method of “inner and outer expansions” employed by Cox and Brenner (1967) is used to calculate the hydrodynamic force experienced by either of two identical small solid spheres, approaching each other, at low Reynolds number, with the same velocity along their line of centres, for the limiting case when the gap width between them tends to zero. Numerical results are compared with those furnished by the exact solution and an improved expression is then obtained for a similar problem arising from the motion of a small sphere towards an identical stationary sphere, in the limit of small gap widths.

A transversely isotropic elastic material can transmit three body waves in each direction, a quasi-longitudinal (QL) wave, a quasi-transverse (QT) wave, and a purely transverse wave. When the material is able to conduct heat the properties of small amplitude QL and QT waves are modified and we consider here the analysis of such thermo-elastic interactions in plane harmonic disturbances. The modified QL and QT waves are both found to exhibit frequency-dependent dispersion and damping of the kind known to affect dilatational waves in isotropic heat-conducting elastic materials, and in addition we show that the particle paths in the associated motions are ellipses with their axes inclined to the wave normal. This latter effect is peculiar to body waves travelling in anisotropic heat-conducting elastic materials and seems not to have been studied in detail hitherto. Numerical results referring to the propagation of plane harmonic body waves in a single crystal of zinc are presented and discussed.

In [1] Fröhlich considers the kernel D(Z(Γ)) of the map of class-groups C(Z(Γ)) → C(), Γ a finite abelian group, the maximal order in the rational group ring Q(Γ). We obtain under mild hypotheses a non-trivial lower bound for the cardinality k(Γ) of the finite group D(Z(Γ)) when Γ is the cyclic group of order 2pn, p an odd prime. In fact, let f be the smallest positive integer such that 2f ≡ 1 mod.pn, If 2|f then k(Γ) > 1 for pn ≠ 3, 32, 5 and k(Γ) is divisible by primes ≠ 2, p except possibly when pn is a Fermat prime or when pn = 32. The latter result contrasts with the fact that D(Z(Γ)) is a p-group if Γ is a p-group [1; Theorem 5].

Let Γ be a finite group of order n and let K be a field whose characteristic does does not divide n. The group ring K(Γ) is then an involution algebra if we define for y є Γ and extend by linearity, so that ¯ is trivial on K. A subalgebra T of K(Γ) is said to be an S-ring on Γ (see [4]) if there exists a decomposition

of Γ into non-empty, pairwise disjoint subsets Fi with the properties that the elements of K(Γ) form a K-basis of T and that for each τi there exists a τj such that .

Generating functions for the number of linearly independent invariants of a set of tensors under a given group of transformations are given by the theory of group representations. For the full and proper orthogonal groups these generating functions are in the form of definite integrals. The classical theory of algebraic invariants gives generating functions for the number of invariants of tensors under two-dimensional unimodular transformations, these generating functions being algebraic expressions. Because of a correspondence between the two-dimensional unimodular group and the three-dimensional proper orthogonal group, the corresponding generating functions are equivalent. The main result of this paper is an explicit demonstration of this equivalence. In addition, algebraic generating functions for the three-dimensional full orthogonal group are obtained and the use of the algebraic generating functions illustrated by applying them to a third order symmetric tensor.