Recently Dragomir and Goh have produced some interesting new bounds relating to entropy measures in information theory. We establish several refinements of their results.

Recent hypercircie estimates for non-linear equations are extended to include a new class of boundary value problems of monotone type. The results are illustrated by the boundary value problem for the equilibrium-free surface of a liquid with prescribed contact angle.

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.

The long term asymptotic behaviour of a population is evaluated where the age specific fertility behaviour is allowed to change with time. Thus in this article the behaviour of a population is determined with a time dependent net maternity function. It is shown that methods used when the net maternity function was independent of time are still applicable if the change with time is explicit only for the initial population. Further, using the fact that for realistic situations the net maternity function is non-zero over a finite interval α < x < β, it is shown that traditional methods can again be used if the time dependence is associated with ages less than α, the minimum age of childbearing. Recent extensions of Cerone and Keane to include exponential time dependence are utilized and models are presented which are piecewise defined, allowing general and exponential time dependence for the parent and new-born populations respectively. The Sharpe-Lotka single sex determinstic population model is used as the basis for the analysis.

Over the past two decades considerable effort has been devoted to problems of stochastic stability, stabilisation, filtering and control for linear and nonlinear systems with Markovian jump parameters, and a number of results have been achieved. However, due to the exponential distribution of the Markovian chain, there are many restrictions on existing results for practical applications. In the present paper, we study systems whose jump parameters are semi-Markovian rather than fully Markovian. We consider only linear systems with semi-Markovian jump parameters, and also study systems with phase-type semi-Markovian jump parameters, because the family of phase-type distributions is dense in the families of all probability distributions on [0, +∞). Some stochastic stability results are obtained. An example is given to show the potential of the proposed techniques.

Let jν, denote the first positive zero of Jν. It is shown that jν/(ν + α) is a strictly decreasing function of ν for each positive α provided ν is sufficiently large. For each α lowe bounds on ν are given to assure the monotonicity of jν/(ν + α). From this it is shown that jν > ν + j0 for all ν > 0, which is both simpler and an improvement on the well known inequality Jν ≥ (ν (ν + 2))1/2.

Several generalizations are given of the Gauss-Winckler inequality for the moments of a probability distribution.

In this paper the relations between semi-infinite programs and optimisation problems with finitely many variables and constraints are reviewed. Two classes of convex semi-infinite programs are defined, one based on the fact that a convex set may be represented as the intersection of closed halfspaces, while the other class is defined using the representation of the elements of a convex set as convex combinations of points and directions. Extension to nonconvex problems is given. A common technique of solving a semi-infinite program computationally is to derive necessary conditions for optimality in the form of a nonlinear system of equations with finitely many equations and unknowns. In the three-phase algorithm, this system is constructed from the optimal solution of a discretised version of the given semi-infinite program. i.e. a problem with finitely many variables and constraints. The system is solved numerically, often by means of some linearisation method. One option is to use a direct analog of the familiar SOLVER method.

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:

In summary, we prove that, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible set M[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible set M[ H, G] is stable (perturbations of H and G produce homeomorphic feasible sets) if and only if MFCQ holds; under a stability condition, two lower level sets of f with a Kuhn-Tucker point between them are homotopically related by attachment of a k-cell (k being the stationary index in the sense of Kojima).

An N-map Iterated Fuzzy Set System (IFZS), introduced in [4] and to be denoted as (w, Φ), is a system of N contraction maps wi: X → X over a compact metric space (X, d), with associated “grey level” maps øi: [0, 1] → [0, 1]. Associated with an IFZS (w, Φ) is a fixed point u ∈ f*(X), the class of normalized fuzzy sets on X, u: X → [0, 1]. We are concerned with the continuity properties of u with respect to changes in the wi, and the φi. Establishing continuity for the fixed points of IFZS is more complicated than for traditional Iterated Function Systems (IFS) with probabilities since a composition of functions is involved. Continuity at each specific attractor u may be established over a suitably restricted domain of φi maps. Two applications are (i) animation of images and (ii) the inverse problem of fractal construction.

The paper discusses equilibrium solutions and solutions with period two and period three for the difference equation

where Q and A are real, positive parameters. The equation was used by Bier and Bountis [1] as an example of a difference equation whose iteration diagram can show bubbles of finite length rather than the successive bifurcations usually expected. The paper examines in more detail what kind of solution can occur for given values of Q and A and establishes a series of critical curves which demarcate the regions in the (Q, A) plane where solutions of period two or period three occur and the subregions where these periodic solutions are stable. This makes it easy to see how Q and A can be combined into a one-parameter equation which gives a bubble, or a series of bubbles, in the iteration diagram.

A simple model for underground mineral leaching is considered, in which liquor is injected into the rock at one point and retrieved from the rock by being pumped out at another point. In its passage through the rock, the liquor dissolves some of the ore of interest, and this is therefore recovered in solution. When the injection and recovery points lie on a vertical line, the region of wetted rock forms an axi-symmetric plume, the surface of which is a free boundary. We present an accurate numerical method for the solution of the problem, and obtain estimates for the maximum possible recovery rate of the liquor, as a fraction of the injected flow rate. Limiting cases are discussed, and other geometries for fluid recovery are considered.

Here we discuss the development of the laminar flow of a viscous incompressible fluid from the entry to the fully developed situation in a straigbt circular pipe. Uniform entry conditions are considered and the analysis is based on the method of matched asymptotic expansions.

We consider an ordinary differential equation arising in the study of the Ricci flow on R2. The existence and uniqueness of solutions of this equation are derived. We then study the asymptotic behaviour of these solutions at ±∞.