This paper discusses robust stochastic stability and stabilization of time-delay discrete Markovian jump singular systems with parameter uncertainties. Based on the restricted system equivalent (RES) transformation, a delay-dependent linear matrix inequalities condition for time-delay discrete-time Markovian jump singular systems to be regular, causal and stochastically stable is established. With this condition, problems of robust stochastic stability and stabilization are solved, and delay-dependent linear matrix inequalities are obtained. A numerical example is also given to illustrate the effectiveness of this method.2000Mathematics subject classification: primary 39A12; secondary 93C55.

The problem of an anisotropic elastic slab containing two arbitrarily-oriented coplanar cracks in its interior is considered. Using a Fourier integral transform technique, we reduce the problem to a system of simultaneous finite-part singular integral equations which can be solved numerically. Once the integral equations are solved, relevant quantities such as the crack energy can be readily computed. Numerical results for specific examples are obtained.

In oil reservoirs, the less-dense oil often lies over a layer of water. When pumping begins, the oil-water interface rises near the well, due to the suction pressures associated with the well. A boundary-integral formulation is used to predict the steady interface shape, when the oil well is approximated by a series of sources and sinks or a line sink, to simulate the actual geometry of the oil well. It is found that there is a critical pumping rate, above which the water enters the oil well. The critical interface shape is a cusp. Efforts to suppress the cone by using source/sink combinations are presented.

Using Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

The low-velocity impact of two convex surfaces comprised of identical material, which approach each other along the direction of the normal at first contact, and obey a J2 = k2 plastic yield condition, is shown for very early times to satisfy the following conditions: the interior surface which separates the two bodies is equivalent to either the locus of points formed by the intersecting curves resulting from moving the two bodies towards each other along their normal; or to the locus of points formed from the level surfaces (suitably parametrized) drawn about each body at the time of first contact. This separating surface lies midway between the geometrical overlap of the two approaching surfaces for times sufficiently short for inertial effects not to significantly affect the approaching velocities.

In this paper, we study the existence of periodic solutions of the NDDE (neutral differential difference equation):

where τ > 0 and c is a real number. We obtain a sufficient condition under which (*) has at least k nonconstant oscillatory periodic solutions.

Often in oil reservoirs a layer of water lies under the layer of oil. The suction pressure due to a distribution of oil wells will cause the oil-water interface to rise up towards the wells. Given a particular distribution of oil wells, we are interested in finding the flow rates of each well that maximise the total flow rate without the interface breaking through to the wells. A method for finding optimal flow rates is developed using the Muskat model to approximate the interface height, and a version of the Nelder-Mead simplex method for optimisation. A variety of results are presented, including the perhaps nonintuitive result that it is better to turn off some oil wells when they are sufficiently close to one another.

A relationship between the fractal geometry and the analysis of recursive (divide-and-conquer) algorithms is investigated. It is shown that the dynamic structure of a recursive algorithm which might call other algorithms in a mutually recursive fashion can be geometrically captured as a fractal (self-similar) image. This fractal image is defined as the attractor of a mutually recursive function system. It then turns out that the Hausdorff-Besicovich dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modelled. That is, if the Hausdorff D-dimensional measure of the image is finite then it serves as the constant of proportionality and the time complexity is of the form Θ(nD), else it implies that the time complexity is of the form Θ(nD logpn), where p is an easily determined constant.

In the loop representation theory of non-perturbative quantum gravity, gravitational states are described by functionals on the loop space of a 3-manifold. In the order to gain a deeper insight into the physical interpretation of loop states, a natural question arises: to wit, how are gravitations related to loops? Some light will be shed on this question by establishing a definite relationship between loops and 3-geometries of the 3-manifold.

In this paper, we consider the system governed via the coefficients of a semilinear elliptic equation and give the necessary conditions for optimal control. Furthermore, we obtain the necessary conditions for an optimal domain in a domain optimization problem.

The new optima and equilibria discussed in the preceding two papers are compared with the results of bargaining experiments between two and three players performed by Fouraker and Siegel. Experiments where players have complete or incomplete information are considered. There is clear evidence that the new optima are operating, and that traditional optima–Cournot, Pareto and competitive (threat)–are less satisfactory in explaining the course of the whole bargaining process.

We consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The two best methods, one presented here for the first time, apparently, and a second, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes equations and incompressible Navier-Stokes equations at low Reynolds number.

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.