Steady potential flow about a thin wing, flying in air above a dynamic water surface, is analysed in the asymptotic limit as the clearance-to-length ratio tends to zero. This leads to a non-linear integral equation for the one-dimensional pressure distribution beneath the wing, which is solved numerically. Results are compared with established “rigid-ground” and “hydrostatic” theories. Short waves lead to complications, including non-uniqueness, in some parameter ranges.

We present an approximate relation for the effective slip length for flows over mixed-slip surfaces with patterning at the nanoscale, whose minimum slip length is greater than the pattern length scale.

In this paper we consider the asymptotics of extinction for the nonlinear diffusion reaction equation

with non-negative initial data possessing finite support. For t > 0, both solution and support vanish as t → T and x → x0. With T as the extinction time we construct the asymptotic solution as τ = T – t → 0 near the extinction point x0 using matched expansions. Taking x0= 0, we first form an outer expansion valid when η =xt–(m–p)/2 (1–p) = 0(1). This is nonuniformly valid for large |η| and has to be replaced by an intermediate expansion valid for |x| = O(τ−1/l0) where l0 is an even integer greater than unity. If p + m ≥ 2 this expansion is uniformly valid while for p + m < 2, there are regions near the edge of the support where diffusion becomes important. The zero order solution in these inner regions is discussed numerically.

Continental shelf waves are examined for side band instability. It is shown that a modulated shelf wave is described by a nonlinear Schrödinger equation, from which the stability criterion is derived. Long shelf waves are stable to side band modulations, but as the wavenumber is increased there are regions of instability (in wavenumber space). A change of stability occurs at each long wave resonance, defined by the condition that the group velocity of the shelf wave equals a long wave speed. Equations describing the long wave resonance are derived.

Consider the nonlinear neutral equation

where pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

Segal's unitarizing complex structure J is shown, in the Fermi-Dirac case, to be the orthogonal component in the polar decomposition of the real skew adjoint generator of classical dynamics. It is proven that in the Bose-Einstein case, the classical symplectic dynamics cannot be unitarized unless the generator is similar to a real skew adjoint operator.

With the classical Hamiltonian strictly positive, J is the pseudo-orthogonal component in the polar decomposition of the generator, using spectral theory in Krein space with indefinite metric. Thus, J can be expressed simply in terms of the projection E(0) onto the subspace of classical solutions with negative frequency. This complements the physicists' experience that conceptual difficulties arise when dynamically invariant separation of positive and negative frequency solutions is impossible.

We treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

We investigate two mean–variance optimization problems for a single cohort of workers in an accumulation phase of a defined benefit pension scheme. Since the mortality intensity evolves as a general Markov diffusion process, the liability is random. The fund manager aims to cover this uncertain liability via controlling the asset allocation strategy and the contribution rate. In order to have a more realistic model, we study the case when the risk aversion depends dynamically on current wealth. By solving an extended Hamilton–Jacobi–Bellman system, we obtain analytical solutions for the equilibrium strategies and value function which depend on both current wealth and mortality intensity. Moreover, results for the constant risk aversion are presented as special cases of our models.

A duality theory for a class of fractional programs is developed. A fractionalprogram which is non-convex is convexified using a one-to-one transformation. The resulting convex equivalent is then dualized with generalized geometric programming duality.

The problem of determining the temperature, displacement and stress fields around a single crack in an anisotropic slab is considered. The problem is reduced to Fredholm integral equations which may be solved numerically.

We obtain a generalized discrete Hilbert and Hardy-Hilbert inequality with non-conjugate parameters by means of an Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our findings with existing results. We improve some earlier results and apply the results to some special homogeneous functions.

Simple chemical reactions can be described by the Michaelis-Menten response curve relating the velocity V of the reaction and the concentration [S] of the substrate S. To handle more complicated reactions without introducing general polynomial response curves, the rate constants can be considered to be scale dependent. This leads to a new response curve with characteristic sigmoidal shape. But not all sigmoidal curves can be accurately fit with three parameters. In order to get an accurate fit, the lower part of the ∫ shaped curve cannot be too shallow and the upper part can't be too steep. This paper determines an exact mathematical expression for the steepness and shallowness allowed.

Steady plane inviscid symmetric vortex streets are flows defined in the strip R × (0, b) and periodic in x with period 2a in which the flow in (−a, a) × (0, b) is irrotational outside a vortex core on which the vorticity takes a prescribed constant value. A family of such vortex street flows, characterised by a variational principle in which the area |Aα| and the centroid yc of the vortex core Aα are fixed, will be considered. For such a family, indexed by a parameter α, suppose that the cores Aα become small in the sense that

Asymptotic estimates on functionals such as flux constant and speed are obtained.

High tuberculosis (TB) prevalence in Papua New Guinea (PNG) is a serious public health concern. The epidemic in this region is exacerbated by the presence of drug-resistant TB strains as well as HIV infection. This presents a public health threat not only locally but also to Australia due to the high potential for cross-border transmission between PNG’s Western Province and the Australian Torres Strait Islands. We present two mathematical models of TB in the Western Province: a simple model of the underlying TB dynamics, and a detailed model which accounts for the additional effects of HIV and drug resistance. The detailed model is used to make quantitative predictions about the impact of expanding the TB case detection rate under the Directly Observed Treatment, Short-course treatment regimen. This paper provides a framework for future investigation into the economic costs and public health benefits of potential TB interventions in this region, with the eventual aim of providing recommendations to guide policy makers in both PNG and Australia.

The main result of this paper is that the oscillation and nonoscillation properties of a nonlinear impulsive delay differential equation are equivalent respectively to the oscillation and nonoscillation of a corresponding nonlinear delay differential equation without impulse effects. An explicit necessary and sufficient condition for the oscillation of a nonlinear impulsive delay differential equation is obtained.

This is an application of the characteristic identity satisfied by matrices whose elements are also elements of a semi-simple Lie algebra. Generalized eigenvectors are determined for matrices consisting of generators of GL(n), O(n) and Sp(n), and it is shown how to resolve the identity into idempotents constructed from such eigenvectors. By this means rather general functions of the matrices may be defined. It is also shown how to determine traces of such functions, in terms of the invariants of the Lie algebra.

The electrostatic field of a set of arbitrarily located circular disks is considered. A set of governing integral equations is derived by a new method. It is shown that some integral characteristics can be found without solving the integral equations. The upper and lower bounds for the total charge are found from a set of linear algebraic equations whose coefficients are defined by simple geometric characteristics of the system. Examples considered show sufficient sharpness of the estimations.

In this paper, we present sufficient conditions for global optimality of a general nonconvex smooth minimisation model problem involving linear matrix inequality constraints with bounds on the variables. The linear matrix inequality constraints are also known as “semidefinite” constraints which arise in many applications, especially in control system analysis and design. Due to the presence of nonconvex objective functions such minimisation problems generally have many local minimisers which are not global minimisers. We develop conditions for identifying global minimisers of the model problem by first constructing a (weighted sum of squares) quadratic underestimator for the twice continuously differentiable objective function of the minimisation problem and then by characterising global minimisers of the easily tractable underestimator over the same feasible region of the original problem. We apply the results to obtain global optimality conditions for optinusation problems with discrete constraints.

Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.