The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.

It is shown that an entire function having k distinct entire asymptotic functions of order less than ¼ is of lower order ½k, mean type at least; further that if f is of lower order ½k, mean type, then its order is ½k.

In the paper “A nonlinear complementarity problem for monotone functions”, Bull. Austral. Math. Soc. 20 (1979), 227–231, Nanda and Patel proved that for a monotone function that fixes the origin, the complementarity problem admits a solution. In this note we give a short proof of the same result under weaker assumptions.

Using degree theory we establish the inverse mapping theorem and a Sard-Smale theorem for analytic contractive vector fields in Fréchet spaces. Applying these theorems we obtain Cronin's theorem relating topological degree to the number of zeros of analytic contractive vector fields.

The limiting behaviour of the J∞ jackknife estimator for parameters associated with stochastic processes is shown to depend on the nature of the underlying process through the asymptotic behaviour of the estimator being jackknifed. In particular, the jackknifed versions of certain estimators associated with renewal processes are shown to have an asymptotic normal distribution.

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C0 and (B, D) = [(Lp, Lq).

Let X be a real or complex normed space and L(X) the algebra of all bounded linear operators on X. Suppose there exists a *-algebra B(X) ⊂ L(X) which contains the identity operator I and all bounded linear operators with finite-dimensional range. The main result is: if each operator U ∈ B(X) with the property U*U = UU* = I has norm one then X is a Hilbert space.

A one-parameter group on evolution space which permutes the classical trajectories of a Lagrangian system is called a dynamical symmetry. Following a review of the modern approach to the “symmetry-conservation law” duality an attempt is made to classify such invariance groups according to the induced transformation of the Cartan form. This attempt is fairly successful inasmuch as the important cases of Lie, Noether and Cartan symmetries can be distinguished. The theory is illustrated with a presentation of results for the classical Kepler problem.

The role of convexity in the development of mathematical programming is reviewed. Many recent generalizations of convexity and their applications to optimization theory are discussed.

A well-known inequality relating the areas and squares of the sides of two triangles is generalised to higher-dimensional euclidean spaces. Extension of the results to non-euclidean spaces is also considered.

We show how a well known algorithm to compute solutions of a second order recursion which are unstable both in forward and in backward direction, can be related to a number of other methods. They are: order reduction, invariant imbedding and decoupling based on triangularization. It is shown that these methods in this order form an increasingly general approach to solve the problem. In particular this means that the stability of the first three algorithms can be understood from the theory that has been established for the decoupling algorithm. In this way one does not need to investigate the stability of the large sparse system which is often related to the first method.

The aim of this note is to prove a theorem on the pointwise degree of approximation of continuously differentiable functions by positive linear operators. As can be seen from the applications to Bernstein and Hermite-Fejér operators, our inequality yields better constants and sometimes even a higher degree of approximation than the known general results.

We conjecture that a complete hypersurface of non-positive Ricci curvature in the Euclidean space must be unbounded. We prove this under the additional assumption that all sectional curvatures of the hypersurface are bounded away from negative infinity.

This paper is devoted to duscussion of some functional equations obtained in the theory of delayed differential equations. By means of the Laplace transform distribution solutions of the considered equations are constructed.

Let G be a connected reductive algebraic group defined over a finite field k. The finite group G(k) of k-rational points of G acts on the spherical building B(G), a polyhedron which is functorially associated with G. We identify the subspace of points of B(G) fixed by a regular semisimple element s of G(k) topologically as a subspace of a sphere (apartment) in B(G) which depends on an element of the Weyl group which is determined by s. Applications include the derivation of the values of certain characters of G(k) at s by means of Lefschetz theory. The characters considered arise from the action of G(k) on the cohomology of equivariant sheaves over B(G).

In this article we use real methods to study the Dirichlet and the boundary value problems of an annulus. Then we establish various properties of real Hardy spaces on the annulus, and give some applications.

The author combines the methods used by Yamamuro and Omori to define a differentiation in Sobolev chains and obtain an Inverse Mapping Theorem. He then uses this theorem to give a new proof for a result of Sunada on the local finite-dimensionality of the solution space of a non-linear elliptic differential operator with smooth coefficients.

Let R be a semiprime ring in which for each x in R there exists a positive integer n = n(x) < 1 such that (xy)n = xnyn for all y in R. Then R is commutative.

We prove the existence of an optimal control in Banach spaces for a system characterized by Hammerstein operator equations.

Planar graphs have recently been characterised as those which have no strict elegant odd ring of circuits. Here we generalise that result by showing that its dual yields a theorem that is valid for all graphs.