Following upon a previous paper [1] on the existence of chiral transformations in a foliated version of the Cremmer, Julia and Scherk model, we deduce a couple of interesting properties of the model. These are:

(i) TM4 is isomorphic to a quotient Lie pseudoalgebra on the algebra of basic functions in M11;

(ii) There is a locally trivial fibration which exhibits M11 as M7 × U, U ⊂ W and W is the basic manifold of the foliation [5]

(iii) The chiral group of the model is identified as Clx (L, gL) × Clx (Q, gQ), the factors are respectively the multiplication groups of units in the Clifford algebras Clx (L, gL) and Clx (Q, gQ) and matching of this group with phenomenology is briefly discussed.

The lower part of the spectrum of the Helmholtz equation for a heterogeneous system in a finite region in $d$ dimensions, where the solutions to the corresponding homogeneous system are known, can be systematically approximated by means of iterative methods. These methods only require the specification of an arbitrary ansatz and converge to the desired solution, regardless of the strength of the inhomogeneities, provided the ansatz has a finite overlap with it. In this paper, different boundary conditions at the borders of the domain are assumed, and some applications are used to illustrate the methods.

In this paper we shall derive some asymptotic formulae for spectra of the third boundary value problem in Rn, n = 2 or 3, linked with variation of a positive function entering the boundary conditions. Further results may be obtained.

This paper considers discrete multivariate processes with time-dependent rational spectral density matrices and gives a solution to the spectral factorisation problem. As a result, the corresponding state space representation for the process is obtained. The relationship between multivariate processes with time-dependent rational spectral density matrix functions and multivariate ARMA processes with time-dependent coefficients is discussed. Solutions for the prediction problem are given for the case when only finite data is available and the case when the whole history of the process is known.

In this paper we introduce a set of orthonormal functions, , where ϕn[r] is composed of a sine function and a sigmoidal transformation γr of order r>0. Based on the proposed functions ϕn[r] named by sigmoidal sine functions, we consider a series expansion of a function on the interval [−1,1] and the related convergence analysis. Furthermore, we extend the sigmoidal transformation to the whole real line ℝ and then, by reconstructing the existing sigmoidal cosine functions ψn[r] and the presented functions ϕn[r], we develop two kinds of 2-periodic series expansions on ℝ. Superiority of the presented sigmoidal-type series in approximating a function by the partial sum is demonstrated by numerical examples.

We derive an effective macroscale description for the growth of tissue on a porous scaffold. A multiphase model is employed to describe the tissue dynamics; linearisation to facilitate a multiple-scale homogenisation provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics. In particular, the resulting description admits both interstitial growth and active cell motion. This model comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled with Stokes-type cell problems on the microscale, incorporating dependence on active cell motion and pore scale structure. The cell problems provide the permeability tensors with which the macroscale flow is parameterised. A subset of solutions is illustrated by numerical simulations.

We investigate the behaviour of a reaction described by Michaelis-Menten kinetics in an immobilised enzyme reactor (IER). The IER is treated as a well-stirred flow reactor, with the restriction that bounded and unbounded enzyme species are constrained to remain within the reaction vessel. Our aim is to identify the best operating conditions for the reactor.

The cases in which an iminobilised enzyme reactor is used to either reduce pollutant emissions or to synthesise a product are considered. For the former we deduce that the reactor should be operated using low flow rates whereas for the latter high flow rates are optimal. It is also shown that periodic behaviour is impossible.

A class of partially structured nonlinear programming problems, containing the capacitated nonlinear minimum cost multicommodity flow problem, is considered. Such problems, although large, can often be solved efficiently and with minimal computational storage by gradient projection methods.

In this paper we present an adaptive boundary-element method for a transmission prob-lem for the Laplacian in a two-dimensional Lipschitz domain. We are concerned with an equivalent system of boundary-integral equations of the first kind (on the transmission boundary) involving weakly-singular, singular and hypersingular integral operators. For the h-version boundary-element (Galerkin) discretization we derive an a posteriori error estimate which guarantees a given bound for the error in the energy norm (up to a multiplicative constant). Then, following Eriksson and Johnson this yields an adaptive algorithm steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically good triangulations and are efficient.

Utilising Jones' method associated with the Wiener-Hopf technique, explicit solutions are obtained for the temperature distributions on the surface of a cylindrical rod without an insulated core as well as that inside a cylindrical rod with an insulated inner core when the rod, in either of the two cases, is allowed to enter, with a uniform speed, into two different layers of fluid with different cooling abilities. Simple expressions are derived for the values of the sputtering temperatures of the rod at the points of entry into the respective layers, assuming the upper layer of the fluid to be of finite depth and the lower of infinite extent. Both the problems are solved through a three-part Wiener-Hopf problem of special type and the numerical results under certain special circumstances are obtained and presented in tabular forms.

We generate an algebra on blood phenotypes with multiplication based on the human ABO-blood group inheritance pattern. We assume that gametes are not chosen randomly during meiosis. We investigate some of the properties of this algebra, namely, the set of idempotents, lattice of ideals and the associative enveloping algebra.

Some swimming microorganisms are sensitive to light, and this can affect the way in which they negotiate their environment. In particular, photophobic cells are repelled from unfavourable light conditions, and in a quiescent fluid environment this can be observed as elevated cell levels in regions away from these light conditions. This photoresponsive effect is of interest due to its potential technological applications. For example, the use of light to focus and direct cells could be used as a convenient means to separate out the algae used in biofuel production (for example, hydrogen), or exploited within devices for biodetection of environmental contaminants. However, in these types of situations the swimming cells will usually be suspended in a flow with shear. In this environment, it has previously been shown that cells can become hydrodynamically trapped in regions of high fluid shear, and so the extent to which photofocusing can occur under these conditions is not immediately clear. Moreover, in applications where the light must pass through appreciable volumes of the suspension, cells will typically absorb light and so shade each other from the illumination. As such, the intensity at any point in the flow is dependent upon the global cell concentration. Hence, in this study we model the coupled influence of fluid shear and cell photosensitivity on a suspension of swimming microorganisms, and ask under what circumstances a suspension of photophobic cells might be focused into high concentration regions.

This paper presents a conformal mapping solution of Laplace's equation in the two dimensional region exterior to two rectangular plates or electrodes at different potentials. Plates with finite and semi-infinite lengths are considered separately and particular emphasis is placed upon the case when the separation between the plates is small. The key results of the paper are expressions for the integral of the square of the normal field along the mid-line between the plates. This integral is of importance in certain gaseous conductor experiments that are sufficiently accurate for a consideration of end effects to be necessary. For small gaps, the dominant end correction to the integral is linear with the gap width. It is also shown that, for small gaps, the simplified (semi-infinite) geometry gives essentially the same value for the integral as the full (finite plate length) geometry.

The problem of surface water wave scattering by two thin nearly vertical barriers submerged in deep water from the same depth below the mean free surface and extending infinitely downwards is investigated here assuming linear theory, where configurations of the two barriers are described by the same shape function. By employing a simplified perturbational analysis together with appropriate applications of Green's integral theorem, first-order corrections to the reflection and transmission coefficients are obtained. As in the case of a single nearly vertical barrier, the first-order correction to the transmission coefficient is found to vanish identically, while the correction for the reflection coefficient is obtained in terms of a number of definite integrals involving the shape function describing the two barriers. The result for a single barrier is recovered when two barriers are merged into a single barrier.

In this paper, we consider the hyperbolic partial differential equation wrr = wrr + 1/r wr − ν2 /r2w, where v ≥ 1/2 or ν = 0 is aprameter, with the Dirichlet, Neumann and mixed boundary conditions. The boundary controllability for such problems is investigated. The main resutl is that all “finite energy” intial states can be steered to the zero state in time T, using a control f ∈ L2 (0, T), provided T > 2. Furthermore, necessary conditions for controllability are also presented.

This paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps.

This paper studies the problem of delay-dependent robust H∞ control for singular systems with multiple delays. Based on a Lyapunov–Krasovskii functional approach, an improved delay-dependent bounded real lemma (BRL) for singular time-delay systems is established without using any of the model transformations and bounding techniques on the cross product terms. Then, by applying the obtained BRL, a delay-dependent condition for the existence of a robust state feedback controller, which guarantees that the closed-loop system is regular, impulse free, robustly stable and satisfies a prescribed H∞ performance index, is proposed in terms of a nonlinear matrix inequality. The explicit expression for the H∞ controller is designed by using linear matrix inequalities and the cone complementarity iterative linearization algorithm. Numerical examples are also given to illustrate the effectiveness of the proposed method.

Sending microwaves through bauxite ore allows almost continuous measurement of moisture content during offload by conveyor belt from a ship. Data and results from a microwave analyser were brought to a European Study Group with Industry at the University of Limerick, with the over-arching question of whether the results are accurate enough. The analyser equipment uses linear regression against phase shifts and signal attenuation to infer moisture content in real time. Simple initial modelling conducted during the Study Group supports this use of linear regression for phase shift data. However, that work also revealed striking and puzzling differences between model and attenuation data.

We present an improved model that allows for multiple reflections of travelling microwaves within the bauxite and in the air above it. Our new model uses four differential equations to describe how electric fields change with distance in each of four layers. By solving these equations and taking reflections into account, we can accurately predict what the receiving antenna will pick up.

Our new solution provides much-improved matches to data from the microwave analyser, and indicates the deleterious effects of reflections. Modelled signal strength behaviour features a highly undesirable noninvertible dependence on bauxite mixture permittivity.

Practical measures that might be expected to reduce the effects of microwave reflections and improve the accuracy of microwave analyser results are suggested based on our improved model solution. This modelling approach and these results are anticipated to extend to the analysis of moisture content during transport on conveyor belts of other ores, slurries, coal, grains and pharmaceutical powders, especially when the depth of the conveyed material is variable.

Four different kinds of positive asymptotic series are identified by the limiting ratio of successive terms. When the limiting ratio is 1 the series is unsummable. When the ratio tends rapidly to a constant, whether greater or less than 1, the series is easily summed. When the ratio tends slowly to a constant not equal to 1 the series is compared with a binomial model which is then used to speed the convergence. When the ratio increases linearly, a limiting binomial and an exponential integral model are both used to speed convergence. The two resulting model sums are consistent and in this case are complex numbers. Truncation at the smallest term is found to be unreliable in the second case, invalid in the third case, and the exponential integral is used to produce a significantly improved truncation in the third case. A divergent series from quantum mechanics is also examined.

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.