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).

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

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.

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.

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.

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.

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 chain rule is given for differentiating a multivariate function of a multivariate function. In the univariate case this chain rule reduces to Faa de Bruno's formula.

Using this, a simple procedure is given to obtain the rth order multivariate Hermite polynomial from the rth order univariate Hermite polynomial.

A theorem on Nemitsky operators between Sobolev spaces, contained in a 1973 paper of Marcus and Mizel, is extended to Orlicz-Sobolev spaces.

A survey over three different approaches to model hysteresis phenomena is given. The starting point for mathematical models are

(a) memory,

(b) population, or

(c) spatial distribution.

In any case the hysteresis effect is interpreted using an averaging procedure over the inner state of the system in consideration.

If R is (n(n−1)/2)-torsion free ring with 1 and satisfies the identity (xy)n = xnyn, then R is commutative provided that n = 4k.

Let Z(K) denote the center of a ring K. A ring R is compressible if Z(eRe) = eZ(R) for each idempotent e of R. In response to a question of S. Berberian, G. Bergman has constructed a (non-commutative) integral domain, satisfying a polynomial identity, for which the 2×2 matrix ring over the domain is not compressible. In contrast to Bergman's example, we show that the ring of nxn matrices over any commutative ring is always compressible.

Criteria for an analytic function f defined in |z| < 1 to belong to B0, the class of Bloch functions satisfying , and criteria for a meromorphic function g defined in |z| < 1 to belong to N0, namely, to satisfy are obtained in terms of the area and the length of the images of hyperbolic disks and hyperbolic circles, respectively.

Sufficient conditions are obtained for a not necessarily scalar system of the form

to be nonoscillatory.