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.

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.

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.

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.

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).

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.

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

In this paper it is shown that if p(z) is a polynomial of degree n satisfying then

The result is best possible.

Let (T, T, μ) be a σ-finite measure space and X a Suslin space. Let A be a class of normal integrands on T × X. We discuss the existence of an essential supremum of A, namely, a normal integrand l with

where A0 is a countable subclass of A, and, for each α ∈ A,

In this way we obtain an extension of the classical essential supremum concept. The applications include a result on measurable selectors of nonmeasurable multifunctions.

We show that a semigroup satisfying a heterotypical identity of which at least one side has no repeated variable is saturated and find sufficient conditions on a homotypical identity which is not a permutation identity and of which at least one side has no repeated variable, to ensure that any semigroup satisfying the identity is saturated.

Sufficient conditions which are verifiable in a finite number of arithmetical steps are derived for the existence and global asymptotic stability of a feasible steady state in an integro-differential system modelling the dynamics of n competing species in a constant environment with delayed interspecific interactions. A novel method involving a nested sequence of “asymptotic” upper and lower bounds is developed.

Techniques from martingale theory are used to obtain the Skorokhod embedding of reverse martingales in Brownian motion. This result is then used to obtain a functional law of the iterated logarithm for reverse martingales.