Let A1A2A3A4, be a planar convex quadrangle with diagonals A1A3 and A2A4. Is there a quadrangle B1B2B3B4 in Euclidean space such that A1A3 < B1B3, A2A4 < B2B4 but AiAj > BiBj for other edges?

The answer is “no”. It seems to be obvious but the proof is more difficult. In this paper we shall solve similar more complicated problems by using a higher dimensional geometric inequality which is a generalisation of the well-known Pedoe inequality (Proc. Cambridge Philos. Soc.38 (1942), 397–398) and an interesting result by L.M. Blumenthal and B.E. Gillam (Amer. Math. Monthly50 (1943), 181–185).

We derive interior second derivative estimates for solutions of equations of Monge-Ampère type which extend those of Pogorelov for the case of affine boundary values. A key ingredient in our method is the existence of a strong solution of the homogeneous Monge-Ampère equation.

In this paper we have established some fixed point theorems in complete and compact metric spaces.

Let r be a given prime. Then a monoid M is the endomorphism monoid of a field of characteristic r if and only if either M is a finite cyclic group or M is a right cancellative monoid and M has an element of infinite order in its centre. The main lemma is the technical base of the present and other papers.

In this article we use a theorem of T.G. Kurtz to prove a central limit theorem for geodesic random walks.

We present some results concerning the structure of polynomially normal operators. It is shown, among other things, that if Tn is normal for some n > 1, then T is quasi–similar to a direct sum of a normal operator and a compact operator and if p(T) is normal with T essentially normal, then T can be written as the sum of a normal operator and a compact operator. Utilizing the direct integral theory of operators we finally show that if p(T) is normal and T*T commutes with T + T*, then T must be normal.

In a large university with a wide range of student options it is often impossible to cater for all allowable student courses of study within the constraints of a practical timetable, and a decision must be made to exclude some options. The entropy of choice available to students is used to develop a measure of timetable efficiency which balances the desirability of having both popular and rarer options available against the need to deny some students of their chosen courses. This efficiency gives a way for ranking the merit of different timetable exclusions. A simple numerical example is used for illustrations.

In a Banach lattice or the hermitian part of a C*-algebra, every element a admits a decomposition a = a+ − a− such that and N(−a) = ‖a−‖, where N is the canonical half-norm of the positive cones. In general ordered Banach spaces, this property is related to the order structure of the duality map and the metric projectability of the positive cones, and it turns out to be equivalent to an “orthogonal” decomposability.

It is shown that if a group G is a product of two normal subgroups, each of which is locally soluble-of-finite-rank, then G is locally soluble if and only if it is locally of finite rank.

It is well-known that every sufficiently large positive integer is the sum of seven cubes. Both proofs of this result, due to Linnik and Watson, are ineffective. Here we show that Watson's proof may be made effective.

The main result of this paper is that the left- and right-quotient rings at a hereditary link closed set of prime ideals of a semiprime fully bounded Noetherian (FBN) ring coincide. This was a result already known for nonsemiprime FBN rigns, but a question left open in the semiprime case. A cornerstone of our approach is that the torsion theory determined by a link-closed hereditary set of prime ideals in an FBN ring is “nice”, but not necessarily perfect. Some conditions which do produce perfect torsion theories are investigated.

Rao, Rodabaugh, Wilke and Zemmer [J. Combin. Theory Ser. A. 11 (1971), 72–92] constructed a number of new VW systems called C-systems from the exceptional near–fields and established that they coordinatize translation planes not isomorphic to generalized André planes. In this paper the translation complement of the plane coordinatized by the C-system I–1 has been found. This plane has the interesting property that its translation complement divides the ideal points into two orbits of lengths 10 and 16. Further, the translation complement contains a subgroup isomorphic to SL(2,5) and therefore one of the exceptional Walker's planes of order 25 [H. Luneberg, Translation Planes, Springer-Verlag (1980), pp.235–244] is indeed the C–plane corresponding to the C–system I–1, which was discovered in 1969.

Let (A, B) denote the class of certain p-valent starlike functions. Recently G. Lakshma Reddy and K.S. Padmanabhan [Bull. Austral. Math. Soc. 25 (1982), 387–396] have shown that the function g defined by

belongs to the class (A, B) if f ∈ (A, B). The technique used by them fails when c is any positive real number. In this paper, by employing a more powerful technique, we improve their result to the case when c is any real number such that c ≥ −p(1+A)/(1+B).

For an abstract mathematical programming problem involving quasidifferentiable cone-constraints we obtain necessary (and sufficient) optimality conditions of the Kuhn-Tucker type without recourse to a constraint qualification. This extends the known results to the non-differentiable setting. To obtain these results we derive several simple conditions connecting various concepts in generalized convexity not requiring differentiability of the functions involved.

Sublinear elliptic differential inequalities with variable coefficients are studied. Sufficient conditions are given that all solutions are oscillatory in exterior domains. Riccati inequalities are used to establish sublinear oscillation criteria.

It is known that diagonal forms over a finite field have non-trivial zeros if the number of variables, or the size of the field, is large enough. We consider diagonal forms of degree up to five in cases where the size hypotheses are not satisfied. There are finitely many fields not covered by the known results, but a direct computational test of all possible equations is impractical. We describe means of cutting down considerably on the number of fields and the number of equations for which there exist diagonal forms, of degree up to 5 and in 3 variables, with no non-trivial zero.

This article develops survey lectures for general mathematical audiences which the author delivered at the 27th Annual Meeting of the Australian Mathematical Society at the University of Queensland, 1983, and at the 10th Austrian Congress of Mathematicians in Innsbruck, 1981. The central theme of these lectures was the use of probabilistic methods in the study of linear elliptic-parabolic differential equations of second order.

The starting point will be an orientative discussion of the role of Brownian motion in classical potential theory. It will then be discussed that, given an elliptic-parabolic differential operator L of a certain type, there exists a uniquely determined diffusion process which is linked with L formally in the same way in which Brownian motion is linked with the Laplace operator. The fundamental results of K. Itô, D.W.Stroock and S.R.S. Varadhan will be in the centre of this part of the paper. We will then proceed to the discussion of more refined problems of the same type for differentiable manifolds. A glimpse at stochastic Riemannian geometry will then close our tour d'horizon.

Let B be the class of functions ω(z) regular in |z| < 1 and satisfying ω(0) = 0, |ω(z)|<1 in |z|<1. We denote by P(A, B), −1 ≤ B < A ≤1, the class of functions p(z) = l+p1z+… regular in |z| < 1 and such that p(z) = [1+Aω(z)]/[1+Bω(z)] for some ω(z) ∈ Β. This paper establishes sharp lower and upper bounds on |z| = r<1 for the functional

where p(z) varies in P(A, B). The results are then used to study certain geometric properties of the corresponding class of meromorphic starlike univalent functions

It is shown that there exists a metric under which a diffeomorphism f on a Riemannian manifold M becomes an isometry, provided that the dynamical system generated by f is of characteristic 0± and all its orbits are closed. Furthermore, it is shown that the foliation given by the suspension of f is parallel in this case.

A class of translation planes of order p2r, where r is an odd natural number and p is a prime, p ≥ 7, p ≢ ± (mod 10) is constructed. A salient feature shared by all these planes is that one ideal point is fixed by the translation complement and the remaining ideal points are divided into at least two orbits, one of which is of length pr.