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.

In a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.

It is known that strong uniqueness can be used to prove second order convergence of the generalised Gauss-Newton algorithm. Formally this algorithm includes sequential linear programming as a special case. Here we show that the second order convergence result extends when the sequential linear programming algorithm is formulated appropriately. Also this discussion provides an example which shows that the assumption of Lipschitz continuity is necessary for the second order convergence result based on strong uniqueness.

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.

Vacuum field equations in a scalar-tensor theory of gravitation, proposed by Ross, are obtained with the aid of a static plane-symmetric metric. A closed form exact solution to the field equations in this theory is presented which can be considered as an analogue of Taub's empty space-time in Einstein's theory.

Formulating a dust-filled spherically symmetric metric utilizing the 3 + 1 formalism for general relativity, we show that the metric coefficients are completely determined by the matter distribution throughout the spacetime. Furthermore, the metric describes both inhomogeneous dust regions and also vacuum regions in a single coordinate patch, thus alleviating the need for complicated matching schemes at the interfaces. In this way, the system is established as an initial boundary value problem, which has many benefits for its numerical evolution. We show the dust part of the metric is equivalent to the class of Lemaitre-Tolman-Bondi (LTB) metrics under a coordinate transformation. In this coordinate system, shell crossing singularities (SCS) are exhibited as fluid shock waves, and we therefore discuss possibilities for the dynamical extension of shell crossings through the initial point of formation by borrowing methods from classical fluid dynamics. This paper fills a void in the present literature associated with these collapse models by fully developing the formalism in great detail. Furthermore, the applications provide examples of the benefits of the present model.