The basic properties of polytopes and their faces are derived from a set of axioms which are satisfied, in particular, by polytopes in Euclidean, hyperbolic or (hemi-)spherical space. The underlying space is not assumed to be either dense or unbounded.

The basic elementary results about convex sets are derived successively from various properties of segments. The complete set of properties is shown to form a natural set of axioms characterising the convex sets in a real vector space.

Pseudo-autonomous linear differential equations are defined. A linear differential equation with bounded coefficient matrix is pseudo-autonomous if and only if it is almost reducible. A linear differential equation with recurrent coefficient matrix is pseudo-autonomous if and only if it has pure point spectrum.

It is shown that for any almost periodic linear differential system asymptotic stability and uniform stability together imply uniform asymptotic stability.

The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions. On the basis of these results a theory of realisations is developed, which similarly generalises the theory of state space realisations of a matrix of rational functions.

Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.