The transformation of Continued Fractions into one another is a subject in which very little work has so far been done. Beyond the simple transformations given in works on elementary algebra, few transformation-theorems are known ; the best known being those connected with the “contraction” and “extension” of Continued Fractions, and the transformations of Euler, Bauer and Muir.

When the curvature of a plane curve continuously increases or diminishes (as is the case with a logarithmic spiral, for instance) no two of its circles of curvature can intersect one another.

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfies

for all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.

In [8], Rooney defines a class of complex-valued functions ζ each of which is analytic in a vertical strip α(ζ)< Res < β(ζ) in the complex s-plane and satisfies certain growth conditions as |Im s| →∞ along fixed lines Re s = c lying within this strip. These conditions mean that the functions

fulfil the requirements of the one-dimensional Mihlin-Hörmander theorem (see [6, p. 417]) and so can be regarded as Fourier multipliers for the Banach spaces . Consequently, each function gives rise to a family of bounded operators W[ζ,σ] σ ∈(α(ζ),β(ζ)), on , 1<p<∞.

The linear differential equation of the second order

is not in general integrable by any method at present available. At the same time, several equations of this type have been integrated, either in terms of finite functions or by means of expansions in series. Some properties of the integrals of the general equation have also been obtained. It is the object of this paper to develop some general properties of these integrals, which throw some light on the nature of the solutions, even if not obtainable in explicit terms.

We generalize the classical example, due to Abraham, of a train algebra that is not special train, to non necessarily commutative right nil algebras of index n.

This note presents a proof of the following proposition:

Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.

The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:

Since a normed linear space is known to be Euclidean if the parallelogram law:

is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

The theorem, “If upon the sides of a triangle as diagonals parallelograms be described, whose sides are parallel to two given lines, then the other three diagonals will intersect in the same point,“ occurs in Hutton's Course of Mathematics, 12th ed., vol. II., p. 191.

This method is associated with the well-known construction of a triangle equal in area to a given polygon.

In Fig. 1 let ABCDD1C1B1A be the polygonal circuit enclosing the area, and let DD1, be any side not adjacent to A.

In [10] Segal shows that the groups of units in certain ordinary cohomology rings are the zeroth terms of generalised cohomology theories. Geometric methods then give a multiplicative transfer on these groups of units for fibrations with finite fibres; see Kahn and Priddy [6] and Adams ([1], 4). On the other hand Evens [5] by manipulations with cochains has constructed a multiplicative transfer in the cohomology of a group G and a subgroup H of finite index. Now it is well known that the algebraic cohomology of G and H can be identified with the topological cohomology of their classifying spaces BG and BH, and that there is a fibration BH→BG with finite fibres. This suggests thatEvens' algebraic transfer and the geometric transfer derived from Segal's work may be related. In the present paper I confirm this by constructing a common generalisation; I also describe some of its properties.

We study the relationship between the dual of the K-space defined by means of a polygon and the J-space generated by the dual N-tuple. The results complete the research started in [4]. Special attention is paid to the case when the N-tuple is formed by Banach lattices