Nonlinear convective roll cells that develop in thin layers of magnetized ferrofluids heated from above are examined in the limit as the wavenumber of the cells becomes large. Weakly nonlinear solutions of the governing equations are extended to solutions that are valid at larger distances above the curves of marginal stability. In this region, a vortex flow develops where the fundamental vortex terms and the correction to the mean are determined simultaneously rather than sequentially. The solution is further extended into the nonlinear region of parameter space where the flow has a core-boundary layer structure characterized by a simple solution in the core and a boundary layer containing all the harmonics of the vortex motion. Numerical solutions of the boundary layer equations are presented and it is shown that the heat transfer across the layer is significantly greater than in the conduction state.

In two-dimensional flow past a body close to a free surface, the upwardly diverted portion may separate to form a splash. We model the nose of such a body by a semi-infinite obstacle of finite draft with a smoothly curved front face. This problem leads to a nonlinear integral equation with a side condition, a separation condition and an integral constraint requiring the far-upstream free surface to be asymptotically plane. The integral equation, called Villat's equation, connects a natural parametrisation of the curved front face with the parametrisation by the velocity potential near the body. The side condition fixes the position of the separation point, whereas the separation condition, known as the Brillouin-Villat condition, imposes a continuity relation to be satisfied at separation. For the described flow we derive the Brillouin-Villat condition in integral form and give a numerical solution to the problem using a polygonal approximation to the front face.

A quasi-Newton method (QNM) in infinite-dimensional spaces for identifying parameters involved in distributed parameter systems is presented in this paper. Next, the linear convergence of a sequence generated by the QNM algorithm is also proved. We apply the QNM algorithm to an identification problem for a nonlinear parabolic partial differential equation to illustrate the efficiency of the QNM algorithm.

The Schatten-von Neumann property of a pseudo-differential operator is established by showing that the pseudo-differential operator is a multiplier defined by means of an admissible wavelet associated to a unitary representation of the additive group Rn on the C*-algebra of all bounded linear operators from L2(Rn) into L2(Rn). A bounded linear operator on L2(R) arising in the Landau, Pollak and Slepian model in signal analysis is shown to be a wavelet multiplier studied in this paper.

An approximate analysis, based on the standard perturbation technique, is described in this paper to find the corrections, up to first order to the reflection and transmission coefficients for the scattering of water waves by a submerged slender barrier, of finite length, in deep water. Analytical expressions for these corrections for a submerged nearly vertical plate as well as for a submerged vertically symmetric slender barrier of finite length are also deduced, as special cases, and identified with the known results. It is verified, analytically, that there is no first order correction to the transmitted wave at any frequency for a submerged nearly vertical plate. Computations for the reflection and transmission coefficients up to O(ε), where ε is a small dimensionless quantity, are also performed and presented in the form of both graphs and tables.

This paper deals with the study of a general class of nonlinear variational inequalities. An existence result is given, and a perturbed iterative scheme is analyzed for solving such problems.

A discrete spatial model of a multi-species environment is formulated, and the behaviour of the system is studied. The model is used to explore stability and resilience of biological systems and discuss how they are dependent on spatial scale chosen.

The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline is studied. A previously derived fourth order nonlinear evolution equation is used to find a spectral transport equation for a narrow band of surface gravity wave trains. This equation is used to study the stability of an initially homogeneous Lorentz shape of spectrum to small long wave-length perturbations for a range of spectral widths. The growth rate of the instability is found to decrease with the increase of spectral widths. It is found that the fourth order term in the evolution equation produces a decrease in the growth rate of the instability. There is stability if the spectral width exceeds a certain critical value. For a vanishing bandwidth the deterministic growth rate of the instability is recovered. Graphs have been plotted showing the variations of the growth rate of the instability against the wavenumber of the perturbation for some different values of spectral width, thermocline depth, angle of perturbation and wave steepness.

In this paper we consider the flow of an incompressible, inviscid and homogeneous fluid over two obstacles in succession. The flow is assumed irrotational and solutions are sought in which a hydraulic fall occurs over the first obstacle with supercritical flow over the second. The method used to solve the problem is capable of calculating flows over topography of any shape.

We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.

Many gravity driven flows can be modelled as homogeneous layers of inviscid fluid with a hydrostatic pressure distribution. There are examples throughout oceanography, meteorology, and many engineering applications, yet there are areas which require further investigation. Analytical and numerical results for two-layer shallow-water formulations of time dependent gravity currents travelling in one spatial dimension are presented. Model equations for three physical limits are developed from the hydraulic equations, and numerical solutions are produced using a relaxation scheme for conservation laws developed recently by S. Jin and X. Zin [6]. Hyperbolicity of the model equations is examined in conjunction with the stability Froude number, and shock formation at the interface of the two layers is investigated using the theory of weakly nonlinear hyperbolic waves.

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 give a simple and transparent proof for the square-root method of solving the continuous-time algebraic Riccati equation. We examine some benefits of combining the square-root method with other solution methods. The iterative square-root method is also discussed. Finally, paradigm numerical examples are given to compare the square-root method with the Schur method.

Consider a general class of constrained optimal control problems in canonical form. Using the classical control parameterization technique, the time (planning) horizon is partitioned into several subintervals. The control functions are approximated by piecewise constant or piecewise linear functions with pre-fixed switching times. However, if the optimal control functions to be obtained are piecewise continuous, the accuracy of this approximation process greatly depends on how fine the partition is. On the other hand, the performance of any optimization algorithm used is limited by the number of decision variables of the problem. Thus, the time horizon cannot be partitioned into arbitrarily many subintervals to reach the desired accuracy. To overcome this difficulty, the switching points should also be taken as decision variables. This is the main motivation of the paper. A novel transform, to be referred to as the control parameterization enhancing transform, is introduced to convert approximate optimal control problems with variable switching times into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times. The transformed problems are essentially optimal parameter selection problems and hence are solvable by various existing algorithms. For illustration, two non-trivial numerical examples are solved using the proposed method.

An analytical expression for the voltage response to current stimulation at relatively short and long times is used to obtain estimates of the passive electrical constants of a neuron that is electrotonically coupled at the soma and dendritic terminals to other neurons in a neural network.

Professor John Mahony, F.A.A., was a talented and unusual Australian applied mathematician (cf. Fowkes and Silberstein [6]), trained in Manchester in the early 1950s under James Lighthill and Richard Meyer. He may be best remembered today for his early work on multiple scales ([8]), for the soliton equation named after him and his collaborators Brooke Benjamin and Jerry Bona ([2]) and for the many students and colleagues he influenced positively. This note concerns certain illustrative examples listed in the three-part paper Stiff Systems of Ordinary Differential Equations by John and his then postdoc John Shepherd, published in the Journal of the Australian Math. Society (Series B) ([9]). After skimming their eighty-seven pages, it is hard to tell how thoroughly they understood the behavior of solutions to their sample problems (though these descriptions remain the most compelling parts of the papers). I can now admit that, sometime in the late 1970s, I recommended that (perhaps an early version of some of) these papers not be published in (I think) a SIAM journal. I am now glad Series B accepted them. Indeed, with regard to Mahony ([8]), Fowkes and Silberstein ([6]) reported “It is likely, in fact, that the JAMS paper was rejected by other more prestigious journals. This was often the case with John's work; partially because his material was almost always a departure from conventional wisdom, but also because John's writing could be rather formal and obscure.” The junior author of the 1981 papers now has achieved considerable mastery of the subject area, but couldn't have been expected to then take the helm from the opinionated Mahony who had initiated the study through his successful proposal to the Australian Research Council.

The velocities of Rayleigh surface waves and, when they exist, Stoneley interface waves can be obtained as the roots of two irrational functions. Here previous results are extended by using standard operations related to the Wiener-Hopf technique to provide expressions in quadrature for these roots.

The paper revisits characterizations of strong stability and strong regularity of KarushKuhn-Tucker solutions of nonlinear programs with twice differentiable data. We give a unified framework to handle both concepts simultaneously.

We obtain representations for the Mellin transform of the product of generalized hypergeometric functions0F1[−a2x2]1F2[−b2x2]fora, b > 0. The later transform is a generalization of the discontinuous integral of Weber and Schafheitlin; in addition to reducing to other known integrals (for example, integrals involving products of powers, Bessel and Lommel functions), it contains numerous integrals of interest that are not readily available in the mathematical literature. As a by-product of the present investigation, we deduce the second fundamental relation for3F2[1]. Furthermore, we give the sine and cosine transforms of1F2[−b2x2].