Multiwavelets possess some nice features that uniwavelets do not. A consequence of this is that multiwavelets provide interesting applications in signal processing as well as in other fields. As is well known, there are perfect construction formulas for the orthogonal uniwavelet. However, a good formula with a similar structure for multiwavelets does not exist. In particular, there are no effective methods for the construction of multiwavelets with a dilation factor a (a ≥ 2, a ∈ Z). In this paper, a procedure for constructing compactly supported orthonormal multiscaling functions is first given. Based on the constructed multiscaling functions, we then propose a method of constructing multiwavelets, which is similar to that for constructing uniwavelets. In addition, a fast numerical algorithm for computing multiwavelets is given. Compared with traditional approaches, the algorithm is not only faster, but also computationally more efficient. In particular, the function values of several points are obtained simultaneously by using our algorithm once. Finally, we give three examples illustrating how to use our method to construct multiwavelets.

The Sharpe-Lotka-McKendrick (or von Foerster) equations for an age-structured population, with a nonlinear term to represent overcrowding or competition for resources, are considered. The model is extended to include a growth term, allowing the population to be structured by size or weight rather than age, and a general solution is presented. Various examples are then considered, including the case of cell growth where cells divide at a given size.

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.

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.

We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.

In this paper we have introduced extensions νυ(α, x; b) and Γν(α, x; b) of the generalized Gamma functions γ(α x; b) and Γ(α, x; b) considered recently by Chaudhry and Zubair. These extensions are found useful in the representations of the Laplace and K-transforms of a class of functions. We have also defined a generalization of the inverse Gaussian distribution. The cumulative and the reliability functions of the generalized inverse Gaussian distribution are expressed in terms of these functions. Some useful properties of the functions are also discussed.

The present paper proves criteria for oscillation of the soultions of functional differential equations of the type

where λ, τ > 0.

In this paper we propose a systematic theoretical procedure for the constructive approximation of non-linear operators and show how this procedure can be applied to the modelling of dynamical systems. We extend previous work to show that the model is stable to small disturbances in the input signal and we pay special attention to the role of real number parameters in the modelling process. The implications of computability are also discussed. A number of specific examples are presented for the particular purpose of illustrating the theoretical procedure.

Heterogeneity in pulmonary microvascular blood flow (perfusion) provides an early indicator of lung disease or disease susceptibility. However, most computational models of the pulmonary vasculature neglect structural heterogeneities, and are thus not accurate predictors of lung function in disease that is not diffuse (spread evenly through the lung). Models that do incorporate structural heterogeneity have either neglected the temporal dynamics of blood flow, or the structure of the smallest blood vessels. Larger than normal oscillations in pulmonary capillary calibre, high oscillatory stress contribute to disease progression. Hence, a model that captures both spatial and temporal heterogeneity in pulmonary perfusion could provide new insights into the early stages of pulmonary vascular disease. Here, we present a model of the pulmonary vasculature, which captures both flow dynamics, and the anatomic structure of the pulmonary blood vessels from the right to left heart including the micro-vasculature. The model is compared to experimental data in normal lungs. We confirm that spatial heterogeneity in pulmonary perfusion is time-dependent, and predict key features of pulmonary hypertensive disease using a simple implementation of increased vascular stiffness.

In a previous paper the authors have shown that the classical barrier function has an O(r) rate of convergence unless the problem is degenerate when it reduces O(r½). In this paper a modified barrier function algorithm is suggested which does not suffer from this problem. It turns out to have superior scaling properties which make it preferable to the classical algorithm, even in the nondegenerate case, if extrapolation is to be used to accelerate convergence.

We present an extension of vendor-managed inventory (VMI) problems by considering advertising and pricing policies. Unlike the results available in the literature, the demand is supposed to depend on the retail price and advertising investment policies of the manufacturer and retailers, and is a random variable. Thus, the constructed optimization model for VMI supply chain management is a stochastic bi-level programming problem, where the manufacturer is the upper level decision-maker and the retailers are the lower-level ones. By the expectation method, we first convert the stochastic model into a deterministic mathematical program with complementarity constraints (MPCC). Then, using the partially smoothing technique, the MPCC is transformed into a series of standard smooth optimization subproblems. An algorithm based on gradient information is developed to solve the original model. A sensitivity analysis has been employed to reveal the managerial implications of the constructed model and algorithm: (1) the market parameters of the model generate significant effects on the decision-making of the manufacturer and the retailers, (2) in the VMI mode, much attention should be paid to the holding and shortage costs in the decision-making.

In this paper, we consider denumerable state continuous time Markov decision processes with (possibly unbounded) transition and cost rates under average criterion. We present a set of conditions and prove the existence of both average cost optimal stationary policies and a solution of the average optimality equation under the conditions. The results in this paper are applied to an admission control queue model and controlled birth and death processes.

Using the fact that a differentiable quasi-convex function is also pseudo-convex at every point x of its domain where ∇f(x) ≠ 0 recent results relating different forms of convexity and invexity are strengthened.

Sufficient conditions are obtained for the existence of a unique linearly asymptotically stable positive periodic solution of an ecosystem model of two species competition in a periodic environment with time lags in interspecific interactions. It is shown that if the self-regulating intraspecific interaction effects are strong enough and act without time delays then time delays of any length in the interspecific interactions cannot destabilise an otherwise stable ecosystem in a periodic environment.

Using an estimate on the group velocity we give an independent proof of the existence of time translations for a large class of short range interactions. We demonstrate that these systems satisfy a strong form of causal propagation and that space-time algebras in suitable space-like directions are disjoint. Finally we derive criteria for dispersion of the interaction in terms of the algebraic density of the orbit of local subalgebras under the evolution or under the associated group of shifts. In this sense the Heisenberg and X-Y models are dispersive but the Ising model is not.

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals, high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. In pourous media such as rocks, fissures provide a system of high-flow paths. Recently, based on a continuum approach, these phenomena have been modelled, resulting in coupled systems of partial differential equations of parabolic type for the concentrations in bulk and in the high-diffusivity paths. This theory assumes that each point of the medium is simultaneously occupied by more than one diffusion or flow path. Here a simple discrete random walk model of diffusion in a medium with double diffusivity is given.

A class of mixed control-state constrained optimal control problems for elliptic partial differential equations arising, for example, in Lavrentiev-type regularized state constrained optimal control is considered. Its numerical solution is obtained via a primal-dual activeset method, which is equivalent to a class of semi-smooth Newton methods. The locally superlinear convergence of the active-set method in function space is established, and its mesh independence is proved. The paper contains a report on numerical test runs including a comparison with a short-step path-following interior-point method and a coarse-to-fine mesh sweep, that is, a nested iteration technique, for accelerating the overall solution process. Finally, convergence and regularity properties of the regularized problems with respect to a vanishing Lavrentiev parameter are considered. 2000 Mathematics subject classification: primary 65K05; secondary 90C33.

The principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

A fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the exact numerical analysis of Pullin and Grimshaw [12], and they are found to agree rather favourably.

Polynomial identities for the generators of a simple basic classical Lie superalgebra are derived in arbitrary representations generated by a maximal (or minimal) weight vector. The infinitesimal characters occurring in the tensor product of two finite dimensional irreducible representations are also determined.