Let

be a system of differential equations in which X1, … Xn are analytic functions independent of t, expansible in convergent series of powers of x1, … xn. Suppose further that not all the functions X1, … Xn vanish when x1 = … = xn = 0. It will be shown in §§ 1–3 that these equations possess (n — 1) independent integrals independent of t, expansible in convergent series of powers of x1, … xn; i.e. functions

such that

identically. In § 4 it is shown how the series for ϕ1,…ϕn−1 may be most directly constructed, and in § 5 is briefly considered the corresponding problem when the origin is a singular point of the equations, i.e. when the expansions of X1,…Xn all begin with terms of the first or higher degrees.

An expression is obtained for the amplitude of the waves radiated to infinity by a line source of sinusoidally-varying strength situated in water over a beach of angle α (0 < α ≤ π). It is shown that, for certain positions of the source, this amplitude is zero. Equations for the loci of these positions, and approximations to their solutions in particular cases, are derived.

There appear to be two main approaches for developing complex splines. One of these, which has been in use for quite some time, consists in defining splines on the boundary of a given region which are then extended into the interior by Cauchy's integral formula (see e.g. [1]). The other approach, which is of a more recent origin, is motivated in spirit by the theory of finite elements (see e.g. [10], p. 320) and is contained in [8] and [9]. Observing that the foregoing extension into the interior is not easy to execute numerically, certain continuous piecewise non-holomorphic functions, called complex planar splines have been studied in [8] and [9]. The choice of non-holomorphic functions is justified, since if we take the pieces to be holomorphic functions like polynomials, then by the well known identity theorem ([5], p. 132, theorem 60) the continuity of such a piecewise function implies that all the pieces represent just one holomorphic function. Thus, we shall consider polynomials in z and its conjugate z¯ of the form

which are generally non-holomorphic functions. The number

will be called the degree of q. For simplicity we also write q(z) for q(z, z¯).

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. Then there exists a number ℳ such that, corresponding to any real numbers a, b, c, there exist rational integers u, v, w for which

Some of Ramanujan's original discoveries about hypergeometric functions and their relation to modular integrals, especially Eisenstein series of negative weight, are still not very well understood. These discoveries take the form of identities that he recorded, without proof, as entries in his notebooks. In the following sections I will introduce some of these entries, discuss their status, give new proofs of several of them and also provide new results of a similar nature.

Consider X and Y to be independent and non-degenerate random variables. If X and Y are non-negative then it follows from (6) that for all given values z of X + Y, the expected values of X and X2 are of the type λz and λ′ z2, respectively, if and only if X and Y have the gamma distributions with the same scale parameter. This is an improvement over Lukacs's result (3). Since the characterization problems for the Poisson, binomial, negative binomial and normal distributions resemble the corresponding problem for the gamma distribution it is quite reasonable to expect the characterizations similar to the above for these distributions. This is established in what follows.

In a recent article [13] a series of vanishing theorems was obtained for the (co)homology of locally nilpotent groups. These results assert that if (co)homology vanishes in low dimensions (0 or 1), then it vanishes in all dimensions, provided that the module satisfies an appropriate finiteness condition.

En prenant connaissance d'un mémoire récent de M. G. Doetsch(2), mon attention a été ramenée sur quelques remarques de nature élémentaire que la lecture de la note de M. Hardy, citée ci-dessus, m'avait suggérées et qui me paraissent apporter quelque clarté sur l'origine des résultats obtenus par ces auteurs relativement à la composition des noyaux de Fourier. Des remarques analogues sont contenues dans un article de M. Hermann Kober qui paraitra prochainement(3) et dont M. Hardy m'a obligeamment communiqué les épreuves. La voie que j'ai suivie diffère de celle de M. Kober; je l'expose brièvement ci-dessous.

A complete system (CS) [Jscr]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr] is a CS of surfaces for M, then any CS equivalent to [Jscr] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

The class of nilpotent semigroups was introduced, via a semigroup identity, independently in [8] and [9], cf. [11], Every nilpotent cancellative semigroup S was shown to have a group of classical fractions G which is nilpotent and of the same nilpotency class as S. Groups, and linear groups in particular, satisfying certain related semigroup identities, introduced in [19], have been recently studied in [1], [15] and [18]. In particular, finitely generated residually finite groups satisfying a semigroup identity must be almost nilpotent, [18]. On the other hand, it was recently shown in [14] that a finitely generated linear semigroup S ⊆ Mn(K), over a field K, with no free non-commutative subsemigroups satisfies an identity and for every maximal subgroup H of Mn(K) the subgroup gp(S ∩ H) of H generated by S ∩ H is almost nilpotent. A natural question that arises here is to decide which of these semigroups are nilpotent. Because of the powerful classical theory of nilpotent linear groups, cf. [21], one can also ask whether such semigroups can be approached via group theoretical methods. We note that the very special case of nilpotent connected algebraic monoids has been recently considered in [4]. In [5] the structure of semigroup algebras of nilpotent semigroups was studied, in particular via prime Goldie homomorphic images, leading naturally to nilpotent subsemigroups of the matrix monoids Mn(D) over division rings D.

The name ‘cardinal function’ was given to

by E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined by

form a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.

In this paper we study the asymptotic behaviour of large deformations of nonlinearly elastic, noncircular rings under internal hydrostatic pressure. These rings can undergo flexure, extension, and shear. Their governing equations are the same as those for the inflation of cylindrical shells.

By using Bragg and Williams' approximations and Bethe's interaction energies between nearest neighbours, it has been shown that, for alloys of the AB3 type, (1) the maximum value of the critical temperature Tc must occur when c = ½ (2) for all temperatures within the range 0 < T < (Tc)max there is always a concentration range within which the alloy exists in a two-phase state, the one phase being partially ordered and the other disordered. This two-phase region is small at all temperatures and vanishes both at T = 0 and at T = (Tc)max. In particular for T/(Tc)max = 0·82, which is the value of Tc for c = ¼, the limits of the two-phase region are c = 0·245 and c = 0·256. The theoretical phase diagram has therefore qualitatively the shape, given in Fig. 5. For a fixed concentration there is thus a certain temperature range in which a disordered phase and a partially ordered phase exist in equilibrium with one another. The temperature at which the disordered phase completely disappears is that usually given by experimental determinations of the critical temperature. It is somewhat higher than Tc, but its maximum value also occurs when c = ½, so that the discrepancy between theory and experiment for this type of superlattice still remains.

Recent papers by Néel (8) and Lawton and Stewart (7) have provided an explanation of the process of magnetization in a single crystal, using the well-known ideas of the domain theory but bringing out the important role played by the demagnetizing field. These authors applied their ideas to two shapes of single crystals of iron, long rods and oblate spheroids, but in the latter case the only crystal orientation worked out in detail was that in which the equatorial plane was (100). The case of oblate spheroids of iron and iron-silicon where the equatorial plane is a (110) plane of the crystal, and the magnetic field is applied in this plane, are considered in detail in this paper. Calculations are made for In, the component of magnetization perpendicular to the applied field He, and comparison made with experimental measurements. This particular problem was considered by Bozorth and Williams (4) who calculated the torque per unit volume (i.e. the product InHe) experienced by the crystal in a magnetic field. Their calculations are, however, incomplete, because their picture of the magnetization process is not a valid one in moderate fields. Moreover, the demagnetizing field was taken into account only in assessing the size of the field H acting within the crystal, but not its direction.

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.

The joint distribution of the n(n− l)/2 distances between n normally distributed points in d dimensions is studied. Moment generating functions and probability density functions are obtained. It is shown that when n = d the squared distances are jointly exponentially distributed subject only to the constraint that a valid n point configuration is prescribed. In the case n = d = 3 the distributions of the ordered distance are obtained explicitly.

The star diagram of a diagram [λ] was first obtained by Robinson (5) who denned and studied it, thus giving a unique characterization of the hook structure of [λ]. Staal (6) continued Robinson's work and obtained some important results. Recently, Littlewood (2) published a paper on the modular representations of the symmetric group in which he defined ‘congruent partitions’, ‘p-quotients’, etc., relating to the diagram [λ]. Littlewood's ideas, though superficially different from those of Robinson and Staal, are in essence equivalent to them.

The paper introduces concepts of entropy and asymptotic entropy for finite quasigroups. A quasigroup is abelian if and only if its entropy is maximal. It is a З-quasigroup if and only if its asymptotic entropy is maximal.

Average or global spectral properties of the Fibonacci and Lucas polynomials recently introduced in the theory of one-dimensional Ising chains are investigated. In addition, several partial sums of these polynomials are shown to have a compact form.

Suppose (en) is a basis of a Banach space E, and that (e′n) is the dual sequence in E′. Then if (e′n) is a basis of E′ in the norm topology (i.e. (en) is shrinking) it follows that E′ is norm separable: it is easy to give examples of spaces E for which this is not so. Therefore there are plenty of spaces which cannot have a shrinking basis. This leads one to consider whether it might not be reasonable to replace the norm topology on E′ by one which is always separable (provided E is separable). Of course, the weak*-topology σ(E′, E) is one possibility (Köthe (17), p. 259); then it is trivial that (e′n) is a weak*-basis of E′. However, if the weak*-topology is separable, then so is the Mackey topology τ(E′, E) on E′, and so we may ask whether (e′n) is a basis of (E′,τ(E′, E)).