A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

Sparse grids are the basis for efficient high dimensional approximation and have recently been applied successfully to predictive modelling. They are spanned by a collection of simpler function spaces represented by regular grids. The sparse grid combination technique prescribes how approximations on a collection of anisotropic grids can be combined to approximate high dimensional functions.

In this paper we study the parallelisation of fitting data onto a sparse grid. The computation can be done entirely by fitting partial models on a collection of regular grids. This allows parallelism over the collection of grids. In addition, each of the partial grid fits can be parallelised as well, both in the assembly phase, where parallelism is done over the data, and in the solution stage using traditional parallel solvers for the resulting PDEs. Using a simple timing model we confirm that the most effective methods are obtained when both types of parallelism are used.

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.

This paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

Under the assumptions that the spatial variable is one dimensional and the distributed delay kernel is the general Gamma distributed delay kernel, when the average delay is small, the existence of travelling wave solutions for the population genetics model with distributed delay is obtained by using the linear chain trick and geometric singular perturbation theory. On the other hand, for the population genetics model with small discrete delay, the existence of travelling wave solutions is obtained by employing a technique which is based on a result concerning the existence of the inertial manifold for small discrete delay equations.

Based on the theory of difference equations, we derive necessary and sufficient conditions for the existence of eigenvalues and inverses of Toeplitz matrices with five different diagonals. In the course of derivations, we are also able to derive computational formulas for the eigenvalues, eigenvectors and inverses of these matrices. A number of explicit formulas are computed for illustration and verification.

In this paper we describe an elementary method for calculating the matrix exponential on an arbitrary time scale. An example is also given to illustrate the result.

The problem of surface water wave scattering by two thin nearly vertical barriers submerged in deep water from the same depth below the mean free surface and extending infinitely downwards is investigated here assuming linear theory, where configurations of the two barriers are described by the same shape function. By employing a simplified perturbational analysis together with appropriate applications of Green's integral theorem, first-order corrections to the reflection and transmission coefficients are obtained. As in the case of a single nearly vertical barrier, the first-order correction to the transmission coefficient is found to vanish identically, while the correction for the reflection coefficient is obtained in terms of a number of definite integrals involving the shape function describing the two barriers. The result for a single barrier is recovered when two barriers are merged into a single barrier.

A disease transmission model of SEIR type with exponential demographic structure is formulated, with a natural death rate constant and an excess death rate constant for infective individuals. The latent period is assumed to be constant, and the force of the infection is assumed to be of the standard form, namely, proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The infected individuals are assumed not to be able to give birth and when an individual is removed fromthe I-class, it recovers, acquiring permanent immunity with probability f (0 ≤ f ≤ 1) and dies from the disease with probability 1 − f. The global attractiveness of the disease-free equilibrium, existence of the endemic equilibrium as well as the permanence criteria are investigated. Further, it is shown that for the special case of the model with zero latent period, R0 > 1 leads to the global stability of the endemic equilibrium, which completely answers the conjecture proposed by Diekmann and Heesterbeek.

We study the discrete asymptotic stability conditions of the perturbed system of first-order linear difference equations with periodic coefficients under the assumption that the related unperturbed system is discrete asymptotically stable. These conditions are dependent on the perturbation matrix B(n) itself and a different parameter is given for obtaining some estimates for the solutions of the unperturbed system.

Some generalisations of the Preece theorem involving the product of two Kummer's functions 1F1 are obtained using Dixon's theorem and some well-known identities. Its special cases yield various new transformations and reduction formulae involving Pathan's quadruple hypergeometric function and Srivastava's quadruple hypergeometric function F(4) and triple hypergeometric function F(3). Some known results of Preece, Pathan and Bailey are also obtained as special cases.