There is increasing interest in the cost of railway infrastructure access as a variety of private companies operate trains with different lengths over a common rail network. It is important to have a method for evaluating the cost of adding an additional train to a railway timetable. This is particularly so for single line track with occasional passing loops. The concept of a fixed loop capacity is extended to one that is dependent on the trains. We develop a model for scheduling a heterogeneous set of trains on single line systems with loops. Our method minimizes the total weighted delay. A Lagrangian relaxation technique is used that relaxes the capacity constraints for track segments and super segments. We measure the delay for each train and the total weighted delay for the heterogeneous set of trains. Our model allows us to investigate the robustness of the weighted delay to variation in the departure time of individual trains. The paper demonstrates that a Lagrangian relaxation heuristic provides optimal train schedules for instances of small heterogeneous train sets. The method is used primarily to check the effectiveness of heuristic algorithms commonly used to find schedules for practical problems.

A predator-prey model in which the prey population is subdivided into nine genotypes corresponding to a two-locus, two-allele problem is considered. Sufficient conditions are given which lead to extinction of all the prey allele types except one, as well as conditions which guarantee the persistence of all the allele types.

We study the problem of choosing the best subset of $p$ features in linear regression, given $n$ observations. This problem naturally contains two objective functions including minimizing the amount of bias and minimizing the number of predictors. The existing approaches transform the problem into a single-objective optimization problem. We explain the main weaknesses of existing approaches and, to overcome their drawbacks, we propose a bi-objective mixed integer linear programming approach. A computational study shows the efficacy of the proposed approach.

Knowledge about the foliage angle density g(α) of the leaves in the canopy of trees is crucial in foresty mangement, modelling canopy reflectance, and environmental monitoring. It is usually determined from observations of the contact frequency f(β) by solving a version of the first kind Fredholm integral equation derived by Reeve (Appendix in Warren Wilson [22]). However, for inference purposes, the practitioner uses functionals defined on g(α), such as the leaf area index F, rather than g(α) itself. Miller [12] has shown that F can be computed directly from f(β) without solving the integral equation. In this paper, we show that his result is a special case of a general transformation for linear functionals defined on g(α). The key is the existence of an alternative inversion formula for the integral equation to that derived by Miller [11].

In order to use the method of asymptotic matching for low frequencies, the equations of plane elastostatics are reformulated in terms of the two scalar potentials commonly used in plane elastodynamics. It is shown that the resulting equations of plane elastostatics can be reduced to those first obtained by Muskhelishvili. The use of the formulation is illustrated by considering the case of the plane diffraction of a P wave by a circular, cylindrical cavity of small radius. The results agree with those obtained from the exact solution of the problem.

This paper is concerned with surface water waves produced by small oscillations of a thin vertical plate submerged in deep water. Green's integral theorem in the fluid region is used in a suitable manner to obtain the amplitude for the radiated waves at infinity. Particular results for roll and sway of the plate, and for a line source in the presence of a fixed vertical plate, are deduced.

In this article we study the dilation equation f(x) = ∑h ch f (2x − h) in ℒ2(R) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(R) of much lower resolution. This simpler equation is then “wavelet transformed” to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same.

In this paper we shall develop existence-uniqueness as well as constructive theory for the solutions of systems of nonlinear boundary value problems when only approximations of the fundamental matrix of the associated homogeneous linear differential systems are known. To make the analysis widely applicable, all the results are proved component-wise. An illustration which dwells upon the sharpness as well as the importance of the obtained results is also presented.

We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

The hydrodynamic pressure forces acting upon a slender fish are derived for the case of a fish swimming in a non-uniform velocity field. Possible applications are the effects on fish propulsion of swimming in waves, in turbulent eddies, and in the presence of other fish or a moving ship. The fish is assumed to be a slender body, with no vorticity shed into the fluid except at a single abrupt trailing edge located at the posterior end of the fish, and to be performing small lateral swimming undulations of its body. The non-uniform field through which the fish swims is assumed to be irrotational, and this field as well as the body undulations must be slowly-varying on the length-scale of the lateral fish dimensions. Expressions are derived for the local force and the time-averaged total thrust force. These are applied to the study of steady-state bow-riding and wave-riding of porpoises.

A numerical procedure for calculating the evolution of a periodic interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell is described. The motion of the interface is determined in a stepwise manner with its new velocity at exach time step being derived as a numerical solution of a boundary integral equation. Attention is focused on the case of unstable displacement charaterised physically by the “fingering” of the interface and computationally by the growth of numerical errors regardless of the numerical method employed. Here the growth of such error is reduced and the usable part of the calculation extended to finite amplitudes. Numerical results are compared with an exact “finger” solution and the calculated behaviour of an initial sinusoidal displacement, as a function of interfacial tension, initial amplitude and wavelength, is discussed.

An analysis is made of quadrature viatwo-point formulae when the integrand is Lipschitz or of bounded variation. The error estimates are shown to be as good as those found in recent studies using Simpson (three-point) formulae.

It is proved that the Neumann boundary value problem, which Mays and Norbury have recently connected with a certain fluid dynamics equation, has a positive solution for any positive value of a particular parameter. Uniform bounds for the solutions and symmetry on a given range of the parameter are also introduced. The proofs include Krasnoselskii's classical fixed-point theorem on cones of a Banach space and basic comparison techniques.

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

A stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.

The bi-objective Cost-time Trade-off Three Axial Sums' Transportation Problem is shown to be equivalent to a single-objective standard Three Axial Sums' problem, which can be solved easily by the existing efficient methods. The equivalence is established for some specially defined solutions termed as Lexicographic optimal solutions with minimum pipe-line.

A model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

The unsteady Hele-Shaw problem is a model nonlinear system that, for a certain parameter ranger, exhibits the phenomenon known as viscous fingering. While not directly applicable to multiphase porous-media flow, it does prove to be an adequate mathematical model for unstable dieplacement in laboratory parallel-plate devices. We seek here to determine, by use of an accurate boundary-integral frount-tracking scheme, the extent to which the simplified system captures the canonical nonlinear behavior of displacement flows and, in particular, to ascertain the role of noise in such systems. We choose to study a particular pattern of injection and production “wells.” The pattern chosen is the isolated “five-spot,” that is a single source surrounded by four symmetrically-placed sinks in an infinite two-dimensional “reservoir.” In cases where the “pusher” fluid has negligible viscosity, sweep efficiency is calculated for a range of values of the single dimensionless parameter τ, an inverse capillary number. As this parameter is reduced, corresponding to increased flow rate or reduced interfacial tension, this efficiency decreases continuously. For small values of τ, these stable displacements change abruptly to a regime characterized by unstable competing fingers and a significant reduction in sweep efficiency. A simple stability argument appears to correctly predict the noise level required to transit from the stable to the competing-finger regimes. Published compilations of experimental results for sweep efficiency as a function of viscosity ratio showed an unexplained divergence when the pusher fluid is less viscous. Our simulations produce a similar divergence when, for a given viscosity ratio, the parameter τ is varied.

In a recent paper the authors give upper and lower bounds for the motion of the moving boundary for the classical Stefan problem for plane, cylindrical and spherical geometries. On comparison with the exact Neumann solution for the plane geometry and no surface radiation the bounds obtained are seen to be quite adequate for practical purposes except for the lower bound at small Stefan numbers. Here improved lower bounds are obtained which in some measure remove this inadequacy. Time dependent surface conditions are also examined and the new lower bounds obtained for the classical problem are illustrated numerically.

We consider a hybrid model, created by coupling a continuum and an agent-based model of infectious disease. The framework of the hybrid model provides a mechanism to study the spread of infection at both the individual and population levels. This approach captures the stochastic spatial heterogeneity at the individual level, which is directly related to deterministic population level properties. This facilitates the study of spatial aspects of the epidemic process. A spatial analysis, involving counting the number of infectious agents in equally sized bins, reveals when the spatial domain is nonhomogeneous.