We show how to partition certain classes of finite projective planes into equicardinal arcs. Several partitions of this kind are to be found in the recent literature and they have aroused a certain amount of interest on two counts, as we shall shortly see.

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz–Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as ${\rm Prim}\;C^*(E)$ are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph $\wt{E}$ from E such that $C^*(\wt{E})$ is an AF-algebra and ${\rm Prim}\;C^*(E)$ and ${\rm Prim}\;C^*(\wt{E})$ are homeomorphic.

In this paper we study a mixture of a viscous fluid and an elastic solid using a global Clausius–Duhem entropy inequality. Results are similar in character to those obtained by previous writers using a more restricted type of entropy inequality. Special attention is given to the case of a mixture of a Newtonian fluid and an elastic solid.

For a compact abelian group G, a corner in G × G is a triple of points (x, y), (x, y+d), (x+d, y). The classical corners theorem of Ajtai and Szemerédi implies that for every α > 0, there is some δ > 0 such that every subset A ⊂ G × G of density α contains a δ fraction of all corners in G × G, as x, y, d range over G.

Recently, Mandache proved a “popular differences” version of this result in the finite field case $G = {\mathbb{F}}_p^n$, showing that for any subset A ⊂ G × G of density α, one can fix d ≠ 0 such that A contains a large fraction, now known to be approximately α4, of all corners with difference d, as x, y vary over G. We generalise Mandache’s result to all compact abelian groups G, as well as the case of corners in $\mathbb{Z}^2$.

The relation between the reaction velocity of catalytic dehydrogenation and the rate of passage of alcohol vapour over the catalyst has been found by experiment. At low rates of flow the concentration of the aldehyde and hydrogen in the effluent gas, and therefore over the catalyst, is comparable with that of the reactant. The whole of the surface is thus not covered with the reactant, and the adsorbed reaction products cause marked decrease in the reaction velocity.

When the concentration of the reaction products becomes sufficiently small, that is, when the rate of passage of the alcohol vapour exceeds some limit, the velocity is approximately independent of the rate of flow of reactant. With very high speeds the reaction velocity tends to fall again, but this is probably due to the difficulty of keeping the temperature of the catalyst constant.

The formula previously deduced connecting the partial pressure of the reactant in a mixture with the fractional reaction velocity has been verified for mixtures of alcohol vapour with water, acetone and benzene; and the composition of the adsorbed gas film with the fractional partial pressure of the gases for mixtures of carbon monoxide with oxygen on platinum.

The effect of mixture of vapours on the temperature coefficient of reaction is considered, and equations deduced to show the alteration produced. If the heat of desorption of the reactant is greater than that of the diluent, then the temperature coefficient is diminished, if they are equal there is no alteration, and if less the temperature coefficient is increased. Experiments show that water present in alcohol has little effect on the temperature coefficient of the reaction.

We prove that for an arbitrary Borel measurable function f on the space of all planar lines there exists a set which intersects almost every line [lscr ] in a set of packing dimension f([lscr ]).

Let K be a knot in the 3-sphere S3; let m be a meridian and 1 a longitude of K. In [1] we gave a detailed study of the ‘π-orbifold group’ O(K): π1(S3\K)/〈m2〉 of the knot, where 〈m2〉 is the subgroup normally generated by the square of the meridian. In particular, we found out to what extent the π-orbifold group classifies knots (and links) and determines the symmetry group of a knot. In the present paper, we consider the groups Gn(K): = π1(S3\K)/〈ln〉. Our main results are the following.

A normed space E is said to be series immersed in a Banach space X if for every absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely. The theorem of Dvoretzky and Rogers(1) implies that a normed space E is series immersed in a finite dimensional space if and only if E itself is finite dimensional. In (4) and (9) it was shown that E is series immersed in lp 1 p < if and only if E is isomorphic to a subspace of Lp() for some measure . In particular, E is isomorphic to an inner product space if and only if it is series immersed in a Hilbert space. The property of series immersion was further studied in the papers (7) and (8). The main results in these two papers are conditions on X under which E series immersed in X would imply E locally immersed in X, a condition slightly stronger (formally) than E finitely representable in X.

A subspace G of a locally convex space E has property (b) if for every bounded set B of E the codimension of G in the linear hull of G ∪ B is finite, (5). Extending the results of (5) and (14), we prove that, if the strong dual of E is complete, then subspaces with property (b) inherit the following properties of E: σ-evaluability, evaluability, the property of being Mazur, semibornological and bornological. We also prove that a dense subspace with property (b) of a Mazur space is sequentially dense, and of a semibornological space – dense in the sense of Mackey (locally dense, following M. Valdivia).

The object of this paper is two-fold: first to collect together the known facts about combinatorial cobordism in general, and then to calculate the groups for the first 8 dimensions. As in (29), we shall denote the unoriented and oriented cobordism groups in dimension n by and Ωn, and will distinguish the combinatorial from the differential case by affixes c, d.

This paper investigates the ‘general position’ properties which ANR's may possess. The most important of these is the disjoint discs property of Cannon (5), which plays a vital role in recent striking characterizations of manifolds (5, 9, 12, 18, 19, 22). Also considered are the property Δ of Borsuk(2) (which ensures an abundance of dimension-preserving maps), and the vanishing of local homology groups up to a given dimension (cf. (9)). Our main results give relations between these properties, and clarify their behaviour under the stabilization operation of taking cartesian product with the real line. In the last section these results are applied to give partial solutions to questions about homogeneous ANR's.

Let be the complex vector space consisting of all complex-valued functions of a non-negative real variable t. For each positive number u, the shift operator Iu is the mapping of into itself defined by the formula

A linear operator T which maps a subspace of into itself is said to be a V-operator (13) if:

(a) for each x in , the complex-conjugate function x* is in ;

(b) both and \ are invariant under the shift operators;

(c) every shift operator commutes with T.

(Property (a) ensures that every function x in can be uniquely expressed as x1 + ix2, where x1 and x2 are real functions in .)

An asymptotic estimate is obtained for the error in approximation of functions by partial sums of spherical harmonic expansions. The precise constant in the main term is found and the order of growth of the remainder term is given.

An identity for a rational function is used, in conjunction with generating functions for a certain class of polynomials, to derive infinite sets of identities for these polynomials. Identities are given for the polynomials of Legendre, Gegenbauer, Hermite, Tchebycheff and L. Carlitz. A novel type of classification for these polynomials is indicated. Next, infinite sequences of identities for various Bessel functions and for power functions are derived. New identities for trigonometric functions are also given, as well as brief miscellanea on: functions resembling Dirac's delta function, integral transforms and a functional equation.

A $genus$ (in the sense of Hirzebruch [4]) is a multiplicative invariant of cobordism classes of manifolds. Classical examples include the numerical invariants given by the signature and the $\widehat{A}$- and Todd genera. More recently genera were introduced which take as values modular forms on the upper half-plane, $\frak{h}=\{\,\tau\;|\;\mathrm{Im}(\tau)>0\,\}$. The main examples are the elliptic genus $\phi_{ell}$ and the Witten genus $\phi_W$; we refer the reader to the texts [7] or [9] for details.

Energy of the exterior and the interior fields of a particle, charged or otherwise, embedded in an empty space is calculated, using Møller's tetrad formulation of the energy-momentum complex in general relativity. It is found that the exterior field contains positive gravitational energy, and in addition, the electromagnetic energy if the particle is a charged one and that the sum total of the energies of the interior and the exterior fields is equivalent to the Newtonian mass of the particle.

The so-called ‘clock paradox’ is concerned with the difference in the time-intervals reckoned by two observers in relative motion for the lapse of time between two encounters. In this paper the problem is treated purely by general relativity by considering a particular example in which the two observers are attached to two test-particles moving freely in the field of a gravitating mass; one of these makes complete revolutions in a circular orbit while the other moves radially outwards and inwards. The time-interval between two successive encounters is shorter in the reckoning of the former than in that of the latter. The difference is found to agree qualitatively with a naïve application of special relativity.

The study of the Riccati equation

plays an essential part in the ‘large parameter’ theory of the inhomogeneous van der Pol equation; see for example Littlewood(1), (2). The crucial result is Lemma B of (1), restated and proved as Lemma 5 of (2); for the present paper the relevant parts of it are as follows:

Lemma 1. Let z = z(x) be the solution of (1·1) which satisfies the initial condition z = 0 at x = 0, and assume α > 0. Then there is a unique β0 = β0(α) with the property that

(i) if β > β0 then z → − ∞ as x → + ∞;

(ii) if β < β0 then z → + ∞ at a vertical asymptote x = x0(α,β);

(iii) if β = β0 then z ≥ 0 in 0 ≤ x < + ∞ and z = x + β0 + o(1) as x → + ∞.

Moreover, β0(α) is a continuous monotone increasing function of α.

1.Introduction. To every closed subalgebra S of a regular algebra R(D) of functions tending to zero at infinity on a discrete space D we define the Rudin spectrum of S to be the partition of D consisting of the equivalence classes of the equivalence relation .

Recent work of Atiyah (1) on the Grothendieck rings of classifying spaces of finite groups has yielded, among many other interesting results, a spectral sequence relating the simple integer cohomology of a finite group to its representation ring. He has determined the first differential operator of the spectral sequence which affects the p-part of the cohomology, and the question naturally arises, when is it the only non-zero one. My object in this note is to show that this is so for a large class of groups.