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

We consider R simultaneous equations of additive type

where the coefficients aij are integers. Artin's conjecture, for additive forms, is that the equations (1) have a non-trivial solution in integers x1,…,xN provided that they have a non-trivial real solution, which is clearly satisfied when k is odd, and

Circularly and transversely polarised (henceforth called circular and transverse) waves have been shown to occur as solutions of non-linear equations governing a wide range of physical phenomena, including finite elasticity (1), magnetohydrodynamics (2), and gyromagnetism (3), but only when the material properties of the medium are isotropic with respect to the direction of wave propagation. This paper is an attempt to unify and generalise these results.

In this paper, we obtain some characterizations for the weighted weak type (1, q) inequality to hold for the Hardy-Littlewood maximal operator in the case 0<q<1; prove that there is no nontrivial weight satisfying one-weight weak type (p, q) inequalities when 0<p≠q< ∞, and discuss the equivalence between the weak type (p, q) inequality and the strong type (p, q) inequality when p≠q.

A one parameter family of algorithms is studied, which contains both the arithmetic-geometric mean of Gauss and its generalization by Borchardt, recently studied by J. and P. Borwein. We prove that the presence of an asymptotic formula for such an algorithm is, in view of the Poisson summation formula, equivalent to the vanishing of certain integrals. In the case of Gauss and Borchardt the latter involve theta functions. Finally, we investigate the question of convergence of the algorithm for complex values, thereby generalizing the corresponding result of Gauss.

The objective of this article are sums S(M)=∑n;ψ(Mf(n/M)) where ψdenotes essentially the fractional part minus ½, f is a C4-function with fn nonvanishing, and summation is extended over an interval of order M. For S(M) an Ω-estimate and a mean-square bound is obtained. Applications to problems concerning the number of lattice points in large planar domains are discussed.