We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity “sets in”. We will show how this ‘quasi’ stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of a dual chain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity for absorbing Markov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

This paper considers a cell population model with a general maturation rate. This model is described by a nonlinear PDE. We use the theory of operator semigroups to stud the problem under simple hypotheses on the growth function and the nonlinear term. By showing that a related operator generates a strongly continuous semigroup, we prove the existence of a classical solution of the nonlinear problem and its positivity. It is also proved that under simple hypotheses, the problem generates a semiflow. The invariance of the semiflow is studied as well.

This is a discussion of some numerical integration methods for surface integrals over the unit sphere in R3. Product Gaussian quadrature and finite-element type methods are considered. The paper concludes with a discussion of the evaluation of singular double layer integrals arising in potential theory.

Inf-sup conditions are proven for three finite-difference approximations of the Stokes equa-tions. The finite-difference approximations use a staggered-mesh scheme and the schemes resulting from the backward and the forward differencings.

Line distributions of Stokes flow singularities are used to model the flow around a slender body which is straddling a flat interface between two viscous fluids. Motion of the slender body parallel to the interface and normal to the interface is considered where the axis of symmetry of the slender body is always perpendicular to the undisturbed interface. Asymptotic approximations to the force distributions on the slender body are evaluated and the relative contributions of that part of the slender body in one fluid to the force distribution in the other fluid and of the interface interaction to the force distribution are examined. It is observed that a shielding region exists about the interface which is due to the interaction with that part of the slender body in the other fluid. Finally, for parallel motion, the first order interface deformation is calculated.

In this paper, the results of Freedman and So [13] on global stability and persistence of simple food chains are extended to general diffusive food chains. For global stability of the unique homogeneous positive steady state, our approach involves an application of the invariance principle of reaction-diffusion equations and the construction of a Liapunov functional. For persistence, we use the dynamical system results of Dunbar et al. [11] and Hutson and Moran [29].

We consider a generalised symmetric eigenvalue problem Ax = λMx, where A and M are real n by n symmetric matrices such that M is positive semidefinite. The purpose of this paper is to develop an algorithm based on the homotopy methods in [9, 11] to compute all eigenpairs, or a specified number of eigenvalues, in any part of the spectrum of the eigenvalue problem Ax = λMx. We obtain a special Kronecker structure of the pencil A − λM, and give an algorithm to compute the number of eigenvalues in a prescribed interval. With this information, we can locate the lost eigenpair by using the homotopy algorithm when multiple arrivals occur. The homotopy maintains the structures of the matrices A and M (if any), and the homotopy curves are n disjoint smooth curves. This method can be used to find all/some isolated eigenpairs for large sparse A and M on SIMD machines.

The Einstein field equations have been solved for Bianchi type VI0 spacetimes with viscous fluid source. Four cosmological models are derived. They have nonzero expansion and shear. One of them have nonzero constant shear viscosity coefficient.

In this paper, we consider a coupled, nonlinear, singular (in the sense that the reaction terms in the equations are not Lipschitz continuous) reaction-diffusion system, which arises from a model of fractional order chemical autocatalysis and decay, with positive initial data. In particular, we consider the cases when the initial data for the the dimensionless concentration of the autocatalyst, β, is of (a) O(x−λ) or (b) O(e−σ x) at large x (dimensionless distance), where σ > 0 and λ are constants. While initially the dimensionless concentration of the reactant, α, is identically unity, we establish, by developing the small-t (dimensionless time) asymptotic structure of the solution, that the support of β(x, t) becomes finite in infinitesimal time in both cases (a) and (b) above. The asymptotic form for the location of the edge of the support of β as t → 0 is given in both cases.

Halley's method is a famous iteration for solving nonlinear equations. Some Kantorovich-like theorems have been given. The purpose of this note is to relax the region conditions and give another Kantorovich-like theorem for operator equations.

Linear dynamical systems of the Rayleigh form are transformed by linear state variable transformations , where A and B are chosen to simplify analysis and reduce computing time. In particular, A is essentially a square root of M, and B is a Lyapunov quotient of C by A. Neither K nor C is required to be symmetric, nor is C small. The resulting state-space systems are analysed by factorisation of the evolution matrices into reducible factors. Eigenvectors and eigenvalues are determined by these factors. Conditions for further simplification are derived in terms of Kronecker determinants. These results are compared with classical reductions of Rayleigh, Duncan, and Caughey, which are reviewed at the outset.

In this paper we propose a new affine scaling interior trust region algorithm with a nonmonotonic backtracking technique for nonlinear equality constrained optimisation with nonnegative constraints on the variables. In order to deal with large problems, the general full trust region subproblem is decomposed into a pair of trust region subproblems in horizontal and vertical subspaces. The horizontal trust region subproblem in the algorithm is defined by minimising a quadratic function subject only to an ellipsoidal constraint in a null tangential subspace and the vertical trust region subproblem is defined by the least squares subproblem subject only to an ellipsoidal constraint. By adopting Fletcher's penalty function as the merit function, combining a trust region strategy and a nonmonotone line search, the mixing technique will switch to a backtracking step generated by the two trust region subproblems to obtain an acceptable step. The global convergence of the proposed algorithm is proved while maintaining a fast local superlinear convergence rate, which is established under some reasonable conditions. A nonmonotonic criterion is used to speed up the convergence progress in some highly nonlinear cases.

In this work we use the Discrete Wavelet Transform in watermarking applications for digital BMP images with the objective is to guarantee some level of security for the copyright. We also compare the results with the Discrete Cosine Transform for the same application. Results are obtained from a number of tests, primarily in order to validate the security level and the robustness of the watermark, but also to prove that the original image suffers only very small variations after the watermark is embedded. We also show how to embed the watermark, where to insert it and the capacity supported for inserting an image.

Sparse matrix factorizations of transfer matrices for the interactions round a face model are reviewed. The sparse factors of a more general Ising model containing first, second and third nearest neighbour interactions are also presented. For both models the factorizations are achieved by considering the required auxiliary spin sets as a hierarchy of interacting spins.

The paper contains mainly three theorems involving generating functions expressed in terms of single and double Laplace and beta integrals. The theorems, in turn, yield, as special cases, a number of bilinear and bilateral generating functions of generalized functions particularly general double and triple hypergeometric series. One variable special cases of the generalized functions are important in several applied problems.

Professor Andrew Barbour (Institut für Angewandte Mathematik, Universität Zürich) has pointed out to us that the conditional intensity specified on Page 356 of our paper is incorrect and, consequently, so too is the bound (14) and the expression on Page 358 for the variance of the conditional intensity, given by (15). This variance should be 0 if ω = 1, where recall that ω = ω (r) is uniquely determined by rω = k, while if ω > 1, the variance is given by

A solution is found for a line source or sink beneath a free surface, at a unique squared Froude number of 12.622.