In this paper we present a computational method for solving a class of time-lag optimal control problems with restricted phase coordinates.

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.

The square-root fixed-interval discrete-time smoother has been used extensively in discrete recursive estimation since it was first developed by Rauch, Tung and Streibel [10]. Various people, for example Bierman [2], [3], have recognized the inherent instability in employing this kind of smoother in its original form; they have investigated implementing the recursion more stably. Bierman's paper [3] is one such contribution. In this paper we plan to present a more comprehensive development of Bierman's approach, and to show that this algorithm can be implemented more stably as a square-root smoother. Throughout this paper the fixed-interval discrete-time smoother will be referred to as the RTS smoother. Numerical results are given for the usual form of the RTS smoother, Bierman's algorithm and our square-root formulation of his algorithm. These confirm that the square-root formulation is more desirable than Bierman's algorithm, which performs better than the usual implementation of the RTS smoother.

Bounds are presented for the modulus of the complex growth rate p of an arbitrary oscillatory perturbation, neutral or unstable, in some double-diffusive problems of relevance in oceanography, astrophysics and non-Newtonian fluid mechanics.

We consider in this paper a topology (which we call the A-topology) on Minkowski space, the four-dimensional space–time continuum of special relativity and derive its group of homeomorphisms. We define the A-topology to be the finest topology on Minkowski space with respect to which the induced topology on time-like and light-like lines is one-dimensional Euclidean and the induced topology on space-like hyperplanes is three- dimensional Euclidean. It is then shown that the group of homeomorphisms of this topology is precisely the one generated by the inhomogeneous Lorentz group and the dilatations.

A numerical method for calculation of the generalized Chakalov-Popoviciu quadrature formulae of Radau and Lobatto type, using the results given for the generalized Chakalov-Popoviciu quadrature formula, is given. Numerical results are included. As an application we discuss the problem of approximating a function f on the finite interval I = [a, b] by a spline function of degree m and variable defects dv, with n (variable) knots, matching as many of the initial moments of f as possible. An analytic formula for the coefficients in the generalized Chakalov-Popoviciu quadrature formula is given.

Rumours have become part of our daily lives, and their spread has a negative impact on a variety of human affairs. Therefore, how to control the spread of rumours is an important topic. In this paper, we extend the classic Maki–Thompson model from a deterministic framework to a stochastic framework with a forgetting mechanism, because real-world person-to-person communications are inevitably affected by random factors. By constructing suitable stochastic Lyapunov functions, we show that the asymptotic behaviour of the stochastic rumour model is governed by the basic reproductive number. If this number is less than one, then the solution of the stochastic rumour model oscillates around the rumour-free equilibrium under extra mild conditions, indicating the extinction of the rumour with a probability of one. Otherwise, the solution always fluctuates around the endemic equilibrium under certain parametric restrictions, implying that the rumour will continually persist. In addition, we discuss a possible intervention strategy that stops the spread of rumours by strengthening the intensity of white noise, which is very different from the deterministic rumour model without white noise. Also, numerical simulations are conducted to support our analytical results.

A three-dimensional barotropic and baroclinic model is developed to simulate currents and temperature changes induced by tropical cyclones traversing the continental shelf and slope region of the Australian North West Shelf. The model is based on a layered, explicit, finite difference formulation using the nonlinear primitive equations with an embedded entrainment scheme; a mixed-surface-layer interface is defined, which is allowed to shift from one interface to another, depending on the strength of a storm. The model has been tested by simulating the currents and temperature changes induced by tropical cyclones Orson and Ian. The modelled currents and temperatures agreed well with the available measured records except near the seabed. It has been found that the pre-storm currents have very little influence on the peak of the storm-induced currents and the currents in the wake of a tropical cyclone. The model contained no coefficients which must be calibrated for a particular application and clearly illustrated the importance of the baroclinic effects on the storm-induced response over the North West Shelf of Australia.

Necessary conditions for optimal controls are given for non-linear systems with time delayed effects in both control and state variables.

We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on $0<x<L(t)$. A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with $L(t)>L_{\text{c}}$ will eventually spread as $t\rightarrow \infty$, whereas solutions where $L(t)\ngtr L_{\text{c}}$ will vanish as $t\rightarrow \infty$. In one dimension it is well known that the critical length is $L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate $L_{\text{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\text{c}}$ depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

Gamma-Weibull variates with five parameters are defined by multiplication of gamma and Weibull densities and renormalising. Sums of independent such variates are distributed as combinations of products of gammas and confluent hypergeometric functions and are explicitly determined. Sums of independent non-identical Weibulls arise as a special case. These variates can be used to model moderately extreme scenarios between gamma and Weibull that occur in many natural applications. All results are exact.

In this work the asymptotic behavior of the partial sums of the divergent asymptotic moment series , where μi are the moments of the weight functions w(x) = xαe−x, α > −1, and w(x) = xαEm (x), α > −1, m + α > 0, on the interval [0, ∞), is analyzed. Expressions for the converging factors are derived by the author for the infinite range integras with w(x) as given above.

The point form of the conservation of energy equation is used to give simple and direct proof of results concerning the mean energy flux vector for systems of sinusoidal small amplitude waves in linear conservative systems. No constitutive equation is used explicitly.

The problem of transient two-dimensional transport by diffusion and advection of a decaying contaminant in two adjacent porous media is solved using a boundary-integral method. The method requires the construction of appropriate Green's functions. Application of Green's theorem in the plane then yields representations for the contaminant concentration in both regions in terms of an integral of the initial concentration over the region's interior and integrals along the boundaries of known quantities and the unknown interfacial flux between the two adjacent media. This flux is given by a first-kind integral equation, which can be solved numerically by a discretisation technique. Examples of contaminant transport in fractured porous media systems are presented.

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.

We solve a minimization problem in liver kinetics posed by Bass, et al., in this journal, (1984), pages 538–562. The problem is to choose the density functions for the location of two enzymes, in order to minimize the concentration of an intermediate form of a substance at the outlet of the liver. This form may be toxic to the rest of the body, but the second enzyme renders it harmless. It seems natural that the second enzyme should be downstream from the first. However, we can show that the minimum problem is sometimes solved by an overlap of the supports of the two density functions. Even more surprising is that, for certain forms of the kinetic functions and high levels of transformation of the first enzymatic reaction, some of the first enzyme should be located downstream from all the second enzyme. This suggests that the first reaction should be relatively slow.

The Poiseuille flow of a generalized Maxwell fluid is discussed. The velocity field and shear stress corresponding to the flow in an infinite circular cylinder are obtained by means of the Laplace and Hankel transforms. The motion is caused by the infinite cylinder which is under the action of a longitudinal time-dependent shear stress. Both solutions are obtained in the form of infinite series. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases. Finally, the influence of the material and fractional parameters on the fluid motion is brought to light.

We investigate the strict lower subdifferentiability of a real-valued function on a closed convex subset of Rn. Relations between the strict lower subdifferential, lower subdifferential, and the usual convex subdifferential are established. Furthermore, we present necessary and sufficient optimality conditions for a class of quasiconvex minimization problems in terms of lower and strict lower subdifferentials. Finally, a descent direction method is proposed and global convergence results of the consequent algorithm are obtained.

In the paper we study the conditions under which multiconnection networks are nonblocking. A multiconnection network deals with the connections of pairs {(T1, T2)} where T1 is a subset of the input terminals and T2 is a subset of the output terminals. We investigate networks composed of digital switching matrices. Such networks can be treated as a very general case encompassing many kinds of networks used in practice as well as studied theoretically.

We present four routing strategies and then develop conditions under which multiconnection networks are nonblocking when each of these strategies is used. We also show that the obtained conditions reduce to known results for some values of network parameters.

In this, the first of three papers, we examine conditions, derived previously, which specify the equilibrium solutions of an adjustment process for N players engaged in a game with continuous (in fact, continuously differentiable) payoff functions, where each player's strategy is to choose a single real number. It is equivalent to the basic form of quantity-variation competition between N firms. The conditions are related to a new optimum which takes account of the ability of firms, or coalitions of firms, to discipline another firm that tries to increase its own profit. Closely related optima are also introduced and analysed. The new optima occupy N-dimensional regions in the strategy space, and contain the optima of Cournot, Pareto, von-Neumann and Morgenstern, and Nash as special cases.