This paper organizes in a systematic manner the major features of a general theory of m-tone rows. A special case of this development is the twelve-tone row system of musical composition as introduced by Arnold Schoenberg and his Viennese school. The theory as outlined here applies to tone rows of arbitrary length, and can be applied to microtonal composition for electronic media.

The flow between two eccentric rotating cylinders with a slotted sleeve placed around the inner cylinder is determined numerically using an exponentially fitted finite-volume method. The flow field is determined for various Reynolds numbers, eccentricities and rotational speeds for the cases when the cylinders rotate in the same sense and rotate in opposite senses. The flow field developed when both cylinders rotate in the same sense is characterised, for sufficiently large eccentricity and rotational rate, by two counter-rotating eddies. Only one eddy is observed when the cylinders rotate in opposite senses. The presence of these eddies restricts the flow through the slotted sleeve in the former case but encourages through flow in the latter. For both cases, the eccentricity affects the location of the eddies, while changing the relative rotational rate only affects the eddy location for the case when the cylinders rotate in opposite directions. The change in Reynolds number has little effect on the flow field for the problems considered here. The vorticity generated by the slotted sleeve is convected into the main body of the flow field. No inviscid core within the main body of the flow field is observed for the range of Reynolds number considered.

The fractal kinetics curve derived by Savageau is analysed to show that its parameters are not uniquely determined given four appropriately situated data points. Comparison is made with an alternate fractal Michaelis-Menten equation derived by Lopez-Quintela and Casado.

We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

The laser Lorenz equations are studied by reducing them to a form suitable for application of an extension of a method developed by Kuzmak. The method generates a flow in a Poincaré section from which it is inferred that a certain Hopf bifurcation is always subcritical.

We show existence and uniqueness of solutions to an initial boundary value problem that entails a pantograph type functional partial differential equation with two advanced nonlocal terms. The problem models cell growth and division into two daughter cells of different sizes. There is a paucity of information about the solution to the problem for an arbitrary initial cell distribution.

A mixed boundary-valued problem associated with the diffusion equation, that involves the physical problem of cooling of an infinite slab in a two-fluid medium, is solved completely by using the Wiener-Hopf technique. An analytical solution is derived for the temperature distribution at the quench fronts being created by two different layers of cold fluids having different cooling abilities moving on the upper surface of the slab at constant speed. Simple expressions are derived for the values of the sputtering temperatures of the slab at the points of contact with the respective layers, assuming one layer of the fluid to be of finite extent and the other of infinite extent. The main problem is solved through a three-part Wiener-Hopf problem of a special type, and the numerical results under certain special circumstances are obtained and presented in the form of a table.

Before wastewaters can be released into the environment, they must be treated to reduce the concentration of organic pollutants in the effluent stream. There is a growing concern as to whether wastewater treatment plants are able to effectively reduce the concentration of micropollutants that are also contained in their influent streams. We investigate the removal of micropollutants in treatment plants by analysing a model that includes biodegradation and sorption as the main mechanisms of micropollutant removal. For the latter a linear adsorption model is used in which adsorption only occurs onto particulates.

The steady-state solutions of the model are found and their stability is determined as a function of the residence time. In the limit of infinite residence time, we show that the removal of biodegradable micropollutants is independent of the processes of adsorption and desorption. The limiting concentration can be decreased by increasing the concentration of growth-related macropollutants. Although, in principle, it is possible that the concentration of micropollutants is minimized at a finite value of the residence time, this was found not to be the case for the particular biodegradable micropollutants considered.

For nonbiodegradable pollutants, we show that their removal is always optimized at a finite value of the residence time. For finite values of the residence time, we obtain a simple condition which identifies whether biodegradation is more or less efficient than adsorption as a removal mechanism. Surprisingly, we find that, for the micropollutants considered, adsorption is always more important than biodegradation, even when the micropollutant is classified as being highly biodegradable with low adsorption.

Sufficient conditions for an equilibrium point to be an attractor or a global attractor are derived for a class of first-order difference equations which need not be continuous at the equilibrium point. These conditions involve Lyapunovlike functions which need not be continuous and are applied to the logistic equation with a piecewise continuous control.

Denoising of images corrupted by multiplicative noise is an important task in various applications, such as laser imaging, synthetic aperture radar and ultrasound imaging. We propose a combined first-order and second-order variational model for removal of multiplicative noise. Our model substantially reduces the staircase effects while preserving edges in the restored images, since it combines advantages of the first-order and second-order total variation. The issues of existence and uniqueness of a minimizer for this variational model are analysed. Moreover, a gradient descent method is employed to solve the associated Euler–Lagrange equation, and several numerical experiments are given to show the efficiency of our model. In particular, a comparison with an existing model in terms of peak signal-to-noise ratio and structural similarity index is provided.

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.

In this paper, we consider convex programs with linear constraints where the objective function involves nested maxima of linear functions as well as a convex function. A dual program is constructed which has interpretational significance and may be easier to solve than the primal formulation. A numerical example is given to illustrate the method.

We show that a combination of Taylor series and boundary integral methods can lead to an effective scheme for solving a class of nonlinear partial differential equations. The method is illustrated through its application to an equation from two dimensional fluid mechanics.

A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

In this paper we will study a feature of a localised topographic flow. We will prove existence of an ideal fluid containing a bounded vortex, approaching a uniform flow at infinity and passing over a localised seamount. The domain of the fluid is the upper half-plane and the data prescribed is the rearrangement class of the vorticity field.

The heart of the Lanczos algorithm is the systematic generation of orthonormal bases of invariant subspaces of a perturbed matrix. The perturbations involved are special since they are always rank-1 and are the smallest possible in certain senses. These minimal perturbation properties are extended here to more general cases.

Rank-1 perturbations are also shown to be closely connected to inverse iteration, and thus provide a novel explanation of the global convergence phenomenon of Rayleigh quotient iteration.

Finally, we show that the restriction to a Krylov subspace of a matrix differs from the restriction of its inverse by a rank-1 matrix.

The method of asymptotic matching introduced by Buchwald [I] is adapted to the case of the diffraction of plane longitudinal and shear waves by cylindrical cavities with elliptic cross-sections. It is assumed that the dimensions of the cross section are small compared with the wavelength of the incident waves. Asymptotic formulae for the scattered wave potentials are obtained.

The method is valid when the cavity reduces to a two-dimensional stress free crack whose length is small compared with the wavelength. Formulae for the scattered waves, and for the stress-concentrations at the crack tips are obtained.

The steady, axisymmetric flow induced by a point sink (or source) submerged in an inviscid fluid of infinite depth is computed and the resulting deformation of the free surface is obtained. The effect of surface tension on the free surface is determined and is the new component of this work. The maximum Froude numbers at which steady solutions exist are computed. It is found that the determining factor in reaching the critical flow changes as more surface tension is included. If there is zero or a very small amount of surface tension, the limiting factor appears to be the formation of small wavelets on the free surface; but, as the surface tension increases, this is replaced by a tendency for the lowest point on the free surface to descend sharply as the Froude number is increased.