In a lecture at the Oslo Congress in 1936, Marcel Riesz introduced an important generalisation of the Riemann-Liouville integral of fractional order. Riesz's integral Iaf of order α is a multiple integral in m variables which converges uniformly when the real part of αexceeds m —2 and so represents an analytic function of the complex variable α. This integral is important in the theory of the generalised wave equation, for it provides a direct method of solving Cauchy's initial-value problem. The most recent developments show that it is likely to be also of great importance in quantum electrodynamics.

In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann – Schatten p-classes of compact operators. The ideas developed there can be used in ℓp-spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results.

We study affine invariants of plane curves from the view point of the singularity theory of smooth functions

The object of the following paper is to consider the motion of one or more vortices in a compressible fluid, which is rotating as a whole with uniform angular velocity ω about an axis, taken as axis of z. To save space I shall when possible refer for results to a previous paper in the Proceedings, distinguishing the equations of that paper, Vol. V.; pp. 52–59, by the suffix a.

The theorem is—

The distance between the circumscribed and the inscribed centres of a triangle it a mean proportional between the circumscribed radius and its excess above the inscribed diameter.

The purpose of this paper is to impose conditions on a radical class P so that the P-radical of the ring of n × n-matrices over a ring A is equal to the ring of n×n-matrices over the ring P(A). In (1), Amitsur gave such conditions, but with the stipulation that the radical class P contained all zero-rings (rings in which all products are zero). In what follows, we shall be working within the class of associative rings.

In Kottler's theoretical discussion1 of the diffraction of a plane wave of monochromatic light of wave-length 2π/k by a black halfplane, the function

where (r, θ, z) are cylindrical coordinates, plays an important part. In particular it is necessary to have asymptotic formulae for f (r, θ), valid when r is either very large or very small compared with the wave-length.

1. The subject-matter of this paper is in some sense known; but we will try to organise, explain and reprove it, and to give examples.

We study certain convexity-type properties of homogeneous functions on topological vector lattices, focusing on a concept of 0+-convexity, and using some probabilistic inequalities.

Following the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the subgroups.

In this paper we provide an answer whenever $U$ is quasinilpotent and $\mathfrak{F}$ is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.

AMS 2000 Mathematics subject classification: Primary 20F17; 20D35

This paper deals with a few of the simpler specialisations of the intersections of a plane curve and the envelope of the family to which it belongs. It follows the method adopted by Professor Chrystal in dealing with the p-discriminant of a differential equation of the first order. This method is specially applicable to definite problems; in these it is safer to work out the result than to rely on theory.

The principal result of this paper is a characterisation of those commutative semigroups S which have the property that each character of each subsemigroup of S can be extended to a character of S. This work was partially inspired by the discovery that Theorem 5.65 of (1) is incorrect; it is related to that of Hill,who has obtained a different solution of the problem (2). Warne and Williams proved in (4) that any bounded character defined on an inverse subsemigroup of an inverse semigroup can be extended to the semigroup.

Einstein's fundamental equations of the gravitational field are

where Tμν are the components of the energy tensor and λ is the cosmical constant. In empty space these equations become

which may be reduced to

since G = 4λ, by contraction of (2).