In this paper, we study the global attractivity of the zero solution of a particular impulsive delay differential equation. Some sufficient conditions that guarantee every solution of the equation converges to zero are obtained.

The motion of small, near neutrally buoyant tracers in vortex flows of several types is obtained on the basis of Charwat's mathematical model, which is highly non-linear.

The solution method in the non-degenerate case expresses the squared orbital radius r2 as a product AA*, where the complex number A satisfies a second-order linear differential ‘factor equation’, generally with variable coefficients. The angular coordinate is expressed in terms of log(A*/A). Solid-type rotation and sinusoidally perturbed solid-type rotation correspond respectively to constant coefficients and sinusoidal coefficients. The former exactly yields a scalloped spiral tracer motion; the latter yields unstable tracer motion as t → ∞ except when the perturbing frequency and amplitude are rather specially related to the flow and tracer parameters. Free vortex motion is somewhat degenerate for this solution method but can be partially analyzed in terms of solutions of a generalized Emden–Fowler equation. The method can be used for other planar flow problems with a symmetry axis.

The “Hartree hybrid method” has recently been employed in one-dimensional non-linear aortic blood-flow models, and the results obtained appear to indicate that shock-waves could only form in distances which exceed physiologically meaningful values. However, when the same method is applied with greater numerical accuracy to these models, the existence of a shock-wave in the vicinity of the heart is predicted. This appears to be contrary to present belief.

We present a method for solving a class of optimal control problems involving hyperbolic partial differential equations. A numerical integration method for the solution of a general linear second-order hyperbolic partial differential equation representing the type of dynamics under consideration is given. The method, based on the piecewise bilinear finite element approximation on a rectangular mesh, is explicit. The optimal control problem is thus discretized and reduced to an ordinary optimization problem. Fast automatic differentiation is applied to calculate the exact gradient of the discretized problem so that existing optimization algorithms may be applied. Various types of constraints may be imposed on the problem. A practical application arising from the process of gas absorption is solved using the proposed method.

We examine the transmission problem in a two-dimensional domain, which consists of two different homogeneous media. We use boundary integral equation methods on the Maxwell equations governing the two media and we study the behaviour of the solution as the two different wave numbers tend to zero. We prove that as the boundary data of the general transmission problem converge uniformly to the boundary data of the corresponding electrostatic transmission problem, the general solution converges uniformly to the electrostatic one, provided we consider compact subsets of the domains.

Design of an interior point method for linear programming is discussed, and results of a simulation study reported. Emphasis is put on guessing the optimal vertex at as early a stage as possible.

In this paper we resolve the problem of controllability of nonlinear interconnected systems of neutral type. We consider two types of systems, a general one, and one in which some control appears linearly. In each case we insist that each isolated system of the interconnected problem is controlled by its own variables while taking into account the interacting effects. Controllability is proved by assuming some controllability criteria of each isolated system and some growth condition of the interconnecting function. Fixed point and open mapping theorems are used. Examples from economics and engineering are presented.

Pointwise bounds are obtained for the solution of a Dirichlet problem involving the nonlinear Liouville equation in the plane, Illustrative calculations are performed for a square domain.

An inequality involving the logarithmic mean is established. Specifically, we show that

where . Then several generalizations are given.

This paper treats the multi-peg generalization of the Tower of Hanoi problem with n(≥ 1) disks and p(≥ 3) pegs, P1, P2,…, Pp. Denoting by M(n, p) the presumed minimum number of moves required to transfer the tower of n disks from the source peg, P1, to the destination peg, Pp, under the condition that each move transfers the topmost disk from one peg to another such that no disk is ever placed on top of a smaller one, the Dynamic Programming technique has been employed to find the optimality equation satisfied by M(n, p). Though an explicit expression for M(n, p) is given, no explicit expressions for the partition numbers (at which M(n, p) is attained) are available in the literature for p ≥ 5. The values of the partition numbers have been given in this paper.

A class of evolution equations in divergence form is studied in this paper. Specifically, we develop conditions under which the spatial divergence term, the flux, corresponds to the characteristic of a conservation law. The KdV equation is a prominent example of an equation having a flux term that is also a characteristic for a conservation law. We show that the flux term must be self-adjoint. General equations for the corresponding conservation laws and Hamiltonian densities are derived and supplemented with examples. 2000 Mathematics subject classification: primary 35K.

Vacuum asymptotically flat Robinson-Trautman spacetimes are a well known class of spacetimes exhibiting outgoing gravitational radiation. In this paper we describe a method of locating the past apparent horizon in these spacetimes and discuss the properties of the horizon. We show that the past apparent horizon is non-timelike and that its surface area is a decreasing function of the retarded time. A numerical simulation of the apparent horizon is also discussed.

An initial value problem is considered for impulsive functional-differential equations. The impulses occur at fixed moments of time. Sufficient conditions are found for Lipschitz stability of the zero solution of these equations. An application in impulsive population dynamics is also discussed.

The Mach-number series expansion of the potential function for the two-dimensional flow of an inviscid, compressible, perfect, diatomic gas past a circular cylinder is obtained to 29 terms. Analysis of this expansion allows the critical Mach number, at which flow first becomes locally sonic, to be estimated as M* = 0.39823780 ± 0.00000001. Analysis also permits the following estimate of the radius of convergence of the series for the maximum velocity to be made: Mc = 0.402667605 ± 0.00000005, though we have been unable to determine the nature of the singularity of M = Mc. Since Mc exceeds M* by some 1.1%, it follows that this particular “airfoil” can possess a continuous range of shock-free potential flows above the critical Mach number. This result hopefully resolves a 70-year old controversy.

This paper studies a class of impulsive switched systems with persistent bounded disturbance using robust attractor analysis and multiple Lyapunov functions. Some sufficient conditions for internal stability of the systems are obtained in terms of linear matrix inequalities (LMI). Based on the results, a simple approach for the design of a feedback controller is presented to achieve a desired level of disturbance attenuation. Numerical examples are also worked out to illustrate the obtained results.

We apply superadditivity and monotonicity properties associated with the Jensen discrete inequality to derive relationships between the entropy function of a probability vector and a renormalized arbitrary sub-vector. The results are extended to cover other entropy measures such as joint entropy, conditional entropy and mutual information.