If the tangent at a point P on the parabolic curve cy=xn meet the axis of x at M, it is a well-known property that the area between the radius vector OP and the are OP is n times that between the arc OP and the two tangents OM, MP, O being the origin and n > 1. The converse is also true; for taking any point O on a curve as origin and the tangent at O as axis of x, let us seek for the locus of P if the area between OP and the arc OP be n times the area between the arc OP and the tangents OM, MP.

The following is a problem in voting:

A number of n individuals are separately arranged in order of preference by a number of k electors ; it is desired to ascertain the order of preference in the opinion of the electors as a whole.

Without discussing what is the best solution of this problem, there are three plausible methods, and it will appear as the result of a curious theorem that two of them lead to identical results.

The theory of representing continuous linear operators on function spaces in terms of integrals has had a long and fruitful history, beginning with the Riesz representation theorem in 1909. If T is such an operator, then the standard representation is T (f) = ∫f dμ, where the integral is denned in diverse ways, depending on the nature of the set of functions and the nature of T.

One of the many interesting problems discussed by Ramanujan is concerned with the effect of truncating at its maximum term nn/n! the exponential series for en, where n is a positive integer. When n is large, the sum of the first n terms is, roughly speaking, half the sum of the whole series.

We give a characterisation of the Bloch space in terms of an area version of the Nevanlinna characteristic, analogous to Baernstein's description of the space BMOA in terms of the usual Nevanlinna characteristic. We prove analogous results for the little Bloch space and the space VMOA, and give value distribution characterizations for all these spaces. Finally we give valence conditions on a Bloch or little Bloch function for containment in BMOA or VMOA.

In the paper, “Sul sistema di tre forme ternarie quadratiche,” Ciamberlini has derived the complete irreducible system of concomitants for three ternary quadratics and has given a short treatment of their geometrical interpretations. Among the concomitants is the invariant (abc)2 which is symmetrical and linear in the coefficients of each quadratic. The purpose of this note is to give a geometrical interpretation of the invariant, and to extend the result for symmetrical invariants of forms in higher dimensions.

The use of Green's Functions in the Theory of Potential is well known. The function is most conveniently defined, for the closed surface S, as the potential which vanishes over S and is infinite as when r is zero, at the point P(x0, y0, z0), inside the surface. If this is represented by G(P), the solution with no infinity inside S and an arbitrary value V over the surface, is given by

denoting differentiation along the outward drawn normal.

In this paper we introduce a generalised Hankel operator and generalised Erdélyi-Kober operators and deduce some relations between them. The operators are then applied to obtain solutions to some dual integral equations which have applications in diffraction theory.

We construct examples of unit-regular rings R for which K0(R) has torsion, thus answering a longstanding open question in the negative. In fact, arbitrary countable torsion abelian groups are embedded in K0 of simple unit-regular algebras over arbitrary countable fields. In contrast, in all these examples K0(R) is strictly unperforated.

We consider the equation of a one-dimensional viscous heat-conducting compressible gas in the variable domain with the appropriate boundary conditions. We study the large-time behaviour of the solution in the particular case where the displacement of the variable boundary is given by $L(t)=L_0(1+at)^\alpha$ with $0lt\alphalt1$, where $a$ is a positive constant and $L_0$ is the initial amplitude of our domain.

AMS 2000 Mathematics subject classification: Primary 35B40; 76N15

The elements of the abstract number systems termed groups, rings, ideals, modules and algebras are mere symbols arranged in systems by means of consistent and independent postulates which isolate these systems from the complete realm of abstract mathematics. The postulates are usually chosen so as to generalise the special number systems which have been noticed in traditional mathematics and their independence and consistenc}" are usually proved by means of numerical examples. It is suggested in this note that the extents of the consistency and independence of a set of postulates should also be studied

We determine the number of conjugacy classes in the natural quotient groups of the Nottingham group over the p-element field up to the quotient of order p3p+1.

1. In Dr Peddie's problem of a sphere cooling in a well-stirred liquid, the conditions to be satisfied by the temperature v(r, t) are

(i) For every positive t, and every r from 0 to a, v is to be finite and one-valued, and is to possess finite derivatives , satisfying

(ii) For every positive t, and r = a,

where p = ⅓ (capacity of liquid)/(capacity of sphere).

(iii) = a given arbitrary function f(r), for every r less than a.

(iv) = initial temperature of liquid = 0 suppose.