The purpose of this article is to illustrate how the theorem of Lévy about conformal invariance of Brownian motion can be used to obtain information about boundary behaviour and removable singularity sets of analytic functions. In particular, we prove a Frostman–Nevanlinna–Tsuji type result about the size of the set of asymptotic values of an analytic function at a subset of the boundary of its domain of definition (Theorem 1). Then this is used to prove the following extension of the classical Radó theorem: If φ is analytic in B\K, where B is the unit ball of ℂ;n and K is a relatively closed subset of B, and the cluster set of φ at K has zero harmonic measure w.r.t. φ(B\K)\≠Ø, then φ extends to a meromorphic function in B (Theorem 2).

The derivation of the 4-nodal Cubic Surface and the Quartic Surface, of which it is a particular case, are well known: certain new results of interest from the point of view of symmetry, and extension to n-fold space, are provided by the symbolic algebra. In particular a simple proof is given, in § 2, of the theorem that a symmetry exists between the four vertices of the tetrahedron and the fifth point whose locus is the cubic surface, and this property can be extended to the case of n + 1 points in n-fold space with one additional point.

A basic set (formerly basic sequence) ℬ is a set of pairs (a, b) of positive integers satisfying

(1) if (a, b) ∈ ℬ, then (b, a) ∈ ℬ,

(2) (a, bc) ∈ ℬ if and only if (a, b) ∈ ℬ and (a, c) ∈ ℬ,

(3) (1, k ∈ ℬ, k = 1, 2, ….

Some familiar examples of basic sets are

, where Sk = {(1, k), (k, 1)},

= {(a, b)| a and b are relatively prime positive integers},

ℒ = {(a, b)| a and b are any positive integers}.

The following rational method of dealing with the reduction of a singular matrix pencil to canonical form has certain advantages. It is based on the principle of vector chains, the length of the chain determining a minimal index. This treatment is analogous to that employed by Dr A. C. Aitken and the author in Canonical Matrices (1932) 45–57, for the nonsingular case. In Theorems 1 and 2 tests are explicitly given for determining the minimal indices. Theorem 2 gives a method of discovering the lowest row (or column) minimal index. Theoretically it should be possible to state a corresponding theorem for each of these indices, not necessarily the lowest, and prior to any reduction of the pencil. This extension still awaits solution.

Let xir (i = 1,…., q; r = 1, …., m; q ≦ m) be random variables which have an elementary probability law p(x11, …., xqm) Let

§ 1. In the Life and Letters of Lewis Carroll occurs the following extract from his Diary:—

“Dec. 19 (Sun).—Sat up last night till 4 a.m., over a tempting problem, sent me from New York, ‘to find 3 equal rational-sided rt.-angled ∆&s.’ I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three.' (v. page 343.)

Let G be a group and let K be a field. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be the vector space over K with basis elements ; let α: G ×G → K be a 2-cocycle and define a multiplication on Kt(G) by

extending this by linearity to Kt(G) yields an associative algebra. We are interested in information concerning the Jacobson radical of Kt(G), denoted by JKt(G).

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.

Let

be a series for which the Abelian generator,

converges whenever s > 0. Then the same is true of the Lambertian generator,

and vice versa.

There is now a huge volume of literature dealing with exact and approximate solutions of the equations governing the motion of an inviscid, incompressible fluid. Major difficulties in finding exact solutions arise when the fluid is bounded by a socalled "free" surface on which the pressure is constant because the equation of such a surface is not known in advance. A boundary condition of this type arises when a mathematical analysis of water wave theory is attempted. Different approximations are, of course, appropriate to different circumstances and here we deal with a well known approximation for waves in shallow water which will be described in detail in the next section.The main ideas underlying this particular approximation have been known for some time but it is probably true to say that they were first brought into real prominence by Stoker [3] in a long article in the Communications of Pure and Applied Mathematics. The main ideas in this article were later developed in considerable detail in Stoker's book “Water Waves” [4]. More recently this approximation has been used by Sewell and Porter [2] as one of several examples in fluid and solid mechanics which are given a geometrical interpretation in terms of “constitutive surfaces” which has close links with catastrophe theory. An excellent account of this interesting approach, which lists many associated references, is given by Sewell [1].

The importance of matric algebras in Function Theory and in Physics (Birtwistle—The new Quantum Mechanics; and Courant and Hilbert—Methoden der mathematischen Physik) has resulted in comprehensive works on finite matrices. Very little progress has, however, been made in the necessary algebras of infinite matrices.

With the issue of the Third Part of the Third Volume, Mr Cantor completes his History of Mathematics, in accordance with the plan he sketched out for himself when he undertook the work. That the labour involved in collecting material and in reducing it to shape would be great, Mr Cantor doubtless well knew; but in all probability his most liberal estimate of the demands likely to be made upon his energies has been far exceeded; in any case, one can readily understand the feelings of satisfaction with which he writes the preface to the concluding volume.

The theorems which furnish in C.F. form the roots of a quadratic equation, and the similar process which leads to particular integrals of an ordinary differential equation of the second order, may be applied to certain types of difference equation. The types which suggest themselves for examination are

the bilinear equation, and

a special form of the linear equation; the coefficients are functions of r, and s is constant.

We assume the reader to be familiar with the basic definitions of near-rings, N-subgroups etc. as presented, for example, in (4). Throughout, N will denote a left near-ring (i.e. a, b, c ∈ N imply a(b + c) = ab + ac) in which 0n = 0 for each n ∈ N. We say that N is strictly semiprime if A2 implies A = (0) where A is an N-subgroup of N. An N-subgroup A is nilpotent if An for some positive integer n and an element a ∈ N is nil if an = 0 for some n. An element a ∈ N is regular if ax = 0 or xa = 0 implies x = 0.

Let a rigid body suffer a displacement to a new position, where it will be denoted by

Then any point A belonging to will take up a position A′ in . Let B be the point of F which coincides with A′. Then corresponding to B in there will be a point B′ in . Let this again coincide with C in , and so on. We have thus a sequence or chain of points whose relation may be thus indicated

This paper builds on the work of V. P. Snaith [5] (in particular Section 6) and gives a more explicit determination of the result.

Recently J. E. McLure has given a satisfactory account of Dyer-Lashof operations in X-theory. For any E∞-space Y and for any r≧2 there is a map

with specific properties (see [3], Theorem 1). Earlier, L. Hodgkin [2] and V. P. Snaith [5] constructed an operation

where Ind(Y) = {xp\x∈Kα(Y,Z/p)} is the indeterminancy of .

One of the well known properties of the number 7 is that when is reduced to a decimal, the periods of two digits are obtained to infinity by successive doubling. It is interesting to find for what other numbers this property is true.

Let n be any number, and r the base of notation, then