We consider a continuous-time Markov chain in which one cannot observe individual states but only which of two sets of states is occupied at any time. Furthermore, we suppose that the resolution of the recording apparatus is such that small sojourns, of duration less than a constant deadtime, cannot be observed. We obtain some results concerning the poles of the Laplace transform of the probability density function of apparent occupancy times, which correspond to a problem about generalised eigenvalues and eigenvectors. These results provide useful asymptotic approximations to the probability density of occupancy times. A numerical example modelling a calcium-activated potassium channel is given. Some generalisations to the case of random deadtimes complete the paper.

This note gives a new strong stationary time (SST) for reversible finite Markov chains. A modification of the initial distribution is represented as a mixture of distributions which have eigenvector interpretations, and for which good simple SSTs exist. This provides some insight into the relationship between SSTs and eigenvalues. Connections to duality and the threshold phenomenon are discussed.

The surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.

Every invertible n-by-n matrix over a ring R satisfying the first Bass stable range condition is the product of n simple automorphisms, and there are invertible matrices which cannot be written as the products of a smaller number of simple automorphisms. This generalizes results of Ellers on division rings and local rings.

Let R be a not necessarily commutative local ring, M a free R-module, and π ∈ GL(M) such that B(π) = im(π –1)is a subspace of M. Then π = σ1…σtρ, where σi are simple mappings of given types, ρ is a simple mapping, B(sgr;i) and B(ρ) are subspaces and t ≤ dim B(π).

The classification of spaces of matrices of bounded rank is known to depend upon ‘primitive’ spaces, whose structure is considerably restricted. A characterisation of an infinite class of primitive spaces is given. The result is then applied to obtain a complete description of spaces whose matrices have rank at most 3.

The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.

A weak canonical form is derived for vector spaces of m × n matrices all of rank at most r. This shows that the structure of such spaces is controlled by the structure of an associated ‘primitive’ space. In the case of primitive spaces it is shown that m and n are bounded by functions of r and that these bounds are tight.

If G, H and B are groups such that G × B ≃ H × B, G/[G, G]. Z(G) is free abelian and B is finitely generated abelian, then G ≃ H. The equivalence classes of triples (Vξ,A) where Vand A are finitely generated free abelian groups and ξ: V⊗ V → A is a bilinear form constitute a semigroup B undera natural external orthogonal sum. This semigroup B is cancellative. A cancellation theorem for class 2 nilpotent groups is deduced.

R. Paré and W. Schelter (1978) have extended the Cayley-Hamilton theorem by showing that for each n<1 there is an integer k such that all n x n matrices over any (possibly noncommutative) ring satisfy a monic polynomial of degree k. We give a lower bound for this degree, namely π(n), which is defined as the shortest possible length of a sequence with entries from {1, 2, …, n}.

The Moore-Penrose inverse of a general bordered matrix is found under various conditions. The Moore-Penrose inverses obtained by Hall and Hartwig (1976) are shown to be special cases of these more general results.

A link between norms from quadratic fields and —det (AB−BA) for 2 × 2 matrices is reformulated via central polynomials and thereby generalized.