A few definite integrals involving more than two Legendre functions in the integrand have been considered by Ferrers, Adams, Dougall, Nicholson and Bailey. We take for example the following integrals.

Let T(α1, α2, …, αn−1) be the n-ary transformation which takes the sequence {si}, (i = 0, 1, …), into the sequence {s'i} where

with sm = 0 when m is a negative integer, and where α1, …, αn an are real numbers with sum unity. In a previous note (1) conditions were found on α and β for the ternary transformation T(α, β) to be equivalent to convergence. A method is given here for treating the similar problem for the general n-ary transformation.

The Takai duality theorem has proved to be a fundamental tool in the theory of crossed products of C*-algebras. It was inspired by Takesaki's duality theorem for crossed products of von Neumann algebras [7], so it is not surprising that the original proof [6] depended heavily on spatial techniques. Here we shall prove Takai's theorem by exploiting the universal properties of crossed products.

The following two modes of generation of the wave-surface are pretty generally known.

(a) A given ellipsoid (surface of elasticity) is cut by any central plane π along an ellipse of semiaxes λ1 and λ2. If π varies, the two pairs of planes π∲ parallel to π at distances k2/λ1, k2/λ2 (k = constant) from it envelope the wave-surface W1 represented by the tangential equation

if the tangential coordinates u1 depend on Σux +1 = 0.

This is a sequel to our previous paper [4] where we initiated a study of inverse eigenvalue problems for matrices in the multiparameter setting. The one parameter version of the problem under consideration asks for conditions on a given n × n symmetric matrix A and on n given real numbers s1≦s2≦…≦sn under which a diagonal matrix V can be found so that A + V has sl,…,sn as its eigenvalues. Our motivation for this problem and our method of attack on it in [4]p comes chiefly from the work of Hadeler [5] in which sufficient conditions were given for existence of the desired diagonal V. Hadeler's approach in [5] relied heavily on the Brouwer fixed point theorem and this was also our main tool in [4]. Subsequently, using properties of topological degree, Hadeler [6] gave somewhat different conditions for the existence of the diagonal V. It is our desire here to follow this lead and to use degree theory to give some results extending those in [6] to the multiparameter case.

Let w = w(a0, a1, an–1) be a word in the free group freely generated by a0, a1, …, an–1; let wi, denote the word w(ai, ai+1, …, ai+n–1), where the subscriptsj in aj are reduced modulo n; and let

Consider a class C of projective R-modules, where R is a commutative ring with identity, which satisfies the conditions of (2), namely that C is closed under the operations of direct sum and isomorphism and C contains the zero module. Following (2) a module M is said to have C-cotype n (respectively C-type n) if it has a projective resolution … → Pn → P0 → M → 0 with Pi ∈ C for i>n (respectively Pi ∈ C for i≦n). Let S be the class of modules of C-cotype −1, equivalently of C-type infinity. It is assumed throughout that S is a Serre Class. We define an abelian category of modules with the property that C-cotype is homological dimension in while in the case C = 0, S is just the category of R-modules. It follows that all categorical results on homological dimension also hold for cotype.

Adopting the notation of Barnes1 and Fox, let us write

If q > p — l, the series on the right of (1) represents an integral function, while if q = p — 1, the series converges only inside or on the circle | x |=1.

Chasles in his Aperçu, Historique sur l'origine et le développement des Méthodes en Géométrie (seconde édition, 1875, pp. 214–215) makes the following statement:

“Essays of the same kind as the geometry of the rule and that of the compasses, and which hold, so to speak, the mean between the two, had long previously engaged the attention of famous mathematicians. Cardan first of all in his book De Subtilitate had resolved several of Euclid's problems by the straight line and a single aperture of the compasses, as if one had in practice only a rule and invariable compasses. Tartalea was not long in following his rival on this field, and extended this mode of treatment to some new problems. (General trattato di numeri et misure; 5ta parte, libra terzo; in-fol. Venise, 1560). Finally, a learned Piedmontese geometer, J.–B. de Benedictis, made it the object of a treatise entitled: Resolutio omnium Euclidis problematum, aliorumque ad hoc necessario inventorum, una tantummodo circini data apertura; in-4°. Venise, 1553.”

If y=z + xy(y) where x and z are independent, Lagrange's series gives the expansion of any function of y in terms of x. The coefficients of the powers of x may be found thus. Let ø(y) be the function to be expanded. Then by Taylor's theorem

Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) module. It is shown that if M satisfies a mild uniqueness condition on essential closures of certain submodules, then the existence of an infinite independent set of submodules of M implies the existence of a larger independent set on some quotient of M modulo a directed union of direct summands. This provides new characterisations of injective (or quasi-continuous) modules of finite Goldie dimension. These results are then applied to the study of indecomposable decompositions of quasi-continuous modules and nonsingular CS modules.

In (4) J. F. C. Kingman and A. P. Robertson introduced the idea of thin sets in certain ℒ1 spaces. Thin sets are extreme cases of sets which are not total, and so the problem naturally arises of partitioning a measure space relative to a given set of integrable functions in such a way that on each element of the partition, the set of functions is either thin or total in a sense which is made precise below. In the present note, such partitions are obtained in §2 for finite or totally σ-finite measure spaces. In §3 the basic ideas are reformulated in terms of Radon measures on locally compact spaces, leading to an extension of the results of §2 in this context.