In this paper we give the structure of an ℵi-complete ℓ-group and the epicomplete objects in the category Aℓ.

We give a short proof the local boundedness of a monotone operator as an easy consequence of the continuity of an associated convex function.

We prove, under very general structure conditions, that continuous viscosity subsolutions of nonlinear second-order elliptic equations possess second order superdifferentials almost everywhere. Consequently we deduce the twice differentiability almost everywhere of viscosity solutions. The main idea of the proof is the backwards use of the Aleksandrov maximum principle as invoked in a previous work of Nadirashvili on sequences of solutions of linear elliptic equations.

By considering the concept of weak minima, different scalar duality results are extended to multiple objective programming problems. A number of weak, strong and converse duality theorems are given under a variety of generalised convexity conditions.

This paper establishes several conditions on the parameters A, B, b for the exact radius of convexity of the class

where

k=1,2,3,…, −1 ≤ B ≤ A ≤ 1, 0 ≤ b ≤ 1.

In this paper we prove the convexity and the compactness of the cores of targets for neutral control systems. We make use of a weak compactness argument; but in the crucial part where we establish the boundeduess of the cores of the target we make use of the notion of asymptotic direction from convex Set Theory. Let En be n-dimensional Euclidean space. We prove that the core of the target H = L + E (where L = {x ∈ En | Mx = 0}, M is a constant in m × n matrix and E is a compact, convex set containing 0) of the neutral system

is convex, and is compact if, and only if, the system

is Euclidean controllable.

With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be complete but do admit a wide class of limits. Accordingly, we introduce a variety of particular 2-categorical limits of practical importance, and show that certain of these suffice for the existence of indexed lax- and pseudo-limits. Other important 2-categories fail to admit even pseudo-limits, but do admit the weaker bilimits; we end by discussing these.

We define the notion of a convex function between semigroups and show that for such functions one obtains not only Jensen's Inequality but a new and powerful companion inequality. Taken together, these two inequalities can be easily applied to give trivial derivations of many of the classical inequalities as well as many new inequalities.

In this paper we deal with classes of infinitely differentiable functions known in the literature as Carleman classes. Our main result is a characterisation of those Carleman classes that are closed under superposition. This result enables us to give a complete solution to a problem that has been considered by Gevrey, Cartan and Bang.

The author would like to include some information omitted from the paper by a publishing error.

The paragraph following Lemma 2.1 on page 458.