In this paper, we consider a linear program with only equality constraints but containing interval and random coefficients. We first address the linear program with interval coefficients, and establish some structural properties of this linear program. On this basis, a solution method is proposed. We then move on to consider the linear program with random coefficients. Using the chance constraint approach and a new approach, the satisfaction degree approach, we obtain the two respective deterministic equivalent formulations. Then the results and the numerical solution methods obtained for these two linear models are applied to the original linear problem which contains both interval and random coefficients. By way of illustration, we consider a practical problem, where the optimal mixing proportions need to be determined for the mix slurry in the production process of aluminium with sintering. This gives rise to a linear program with interval and random coefficients. Its deterministic equivalent formulations are presented. Preliminary numerical examples show that the proposed models and the solution methods are promising.

The steady, axisymmetric flow induced by a point sink (or source) submerged in an unbounded inviscid fluid is computed. The resulting deformation of the free surface is obtained, and a limit of steady solutions is found that is quite different to those obtained in past work. More accurate solutions indicate that the old limiting flow rate was too high and, in fact, the breakdown of steady solutions at a lower flow rate is characterized by the appearance of spurious wavelets at the free surface.

In this paper, we show that the cost of an optimal train journey on level track over a fixed distance is a strictly decreasing and strictly convex function of journey time. The precise structure of the cost–time curves for individual trains is an important consideration in the design of energy-efficient timetables on complex rail networks. The development of optimal timetables for busy metropolitan lines can be considered as a two-stage process. The first stage seeks to find optimal transit times for each individual journey segment subject to the usual trip-time, dwell-time, headway and connection constraints in such a way that the total energy consumption over all proposed journeys is minimized. The second stage adjusts the arrival and departure times for each journey while preserving the individual segment times and the overall journey times, in order to best synchronize the collective movement of trains through the network and thereby maximize recovery of energy from regenerative braking. The precise nature of the cost–time curve is a critical component in the first stage of the optimization.

The existence and selection of steady-state travelling planar fronts in a set of typical phase field equations for solidification are investigated by a combination of numerical and analytical methods. Such solutions are conjectured to exist only for a unique velocity of propagation and to be unique except for translation. This behaviour is in marked contrast to the situation in conventional Stefan models in which travelling fronts exist for all velocities. The value of the steady-state velocity depends upon the various material parameters which enter the phase field equations. Numerical and, in certain tractable limits, analytical results for the velocity are presented for a number of physical situations.

The leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.

The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

We study certain types of composite nonsmooth minimization problems by introducing a general smooth approximation method. Under various conditions we derive bounds on error estimates of the functional values of original objective function at an approximate optimal solution and at the optimal solution. Finally, we obtain second-order necessary optimality conditions for the smooth approximation prob lems using a recently introduced generalized second-order directional derivative.

An improved model of water cresting towards horizontal wells is presented, using a hodograph solution to the lateral edge drive model with a constant potential boundary at a finite outer radius. In this model, the water crest tends to a horizontal asymptote far from the well, correcting previous approaches which led to the unphysical result of a crest which never levels off but, rather, tends to a parabolic curve. The hodograph solution yields the shape of the free water-oil interface. It also yields integral representations for the lengths of boundary segments, and these have enabled the derivation of an explicit expression for the critical rate in terms of the outer radius. The critical rates calculated using the improved model do not differ significantly from those calculated using previous approaches. The main advantage of the model, therefore, is not a correction to the quantitative predictions of the critical rate, but the removal of physical inconsistencies in the underlying theory.

The paper contains mainly three theorems involving generating functions expressed in terms of single and double Laplace and beta integrals. The theorems, in turn, yield, as special cases, a number of bilinear and bilateral generating functions of generalized functions particularly general double and triple hypergeometric series. One variable special cases of the generalized functions are important in several applied problems.

The generalized diffusion equation with a nonlinear source term which encompasses the Fisher, Newell-Whitehead and Fitzhugh-Nagumo equations as particular forms and appears in a wide variety of physical and engineering applications has been analysed for its generalized symmetries (isovectors) via the isovector approach. This yields a new and exact solution to the generalized diffusion equation. Further applications of group theoretic techniques on the travelling wave reductions of the Fisher, Newell-Whitehead and Fitzhugh-Nagumo equations result in integrability conditions and Lie vector fields for these equations. The Lie group of transformations obtained from the exponential vector fields reduces these equations in generalized form to a standard second-order differential equation of nonlinear type, which for particular cases become the Weierstrass and Jacobi elliptic equations. A particular solution to the generalized case yields the exact solutions that have been obtained through different techniques. The group-theoretic integrability relations of the Fisher and Newell-Whitehead equations have been cross-checked through Painlevé analysis, which yields a new solution to the Fisher equation in a complex-valued function form.

This paper considers a single-product industries, fixed capital model discussed briefly by Sraffa in Chaapter X if his book. The analysis is elementary, being based on the direct calculation of a certain martrix inverse. This generalises the approach adopted for circulating capital models. It also demonstrates that common economic and mathematical frameworks exist for both circulating and fixed capital models.

An analytical solution of a two-dimensional bow and stern flow model based on a flat ship theory is presented for the first time. The flat ship theory is a counterpart to Michell's thin ship theory and leads to a mixed initial-boundary value problem, which is usually difficult to solve analytically. Starting from the transient problem, we shall first show that a steady state is attainable at the large time limit. Then the steady problem is solved in detail by means of the Wiener-Hopf technique and closed-form far-field results are obtained for an arbitrary hull shape. Apart from providing a better understanding of the underlying physics, the newly found analytical solution has shed some light on solving a longtime outstanding problem in the engineering practice of ship building, the optimisation of hull shape.

The BFGS formula is arguably the most well known and widely used update method for quasi-Newton algorithms. Some authors have claimed that updating approximate Hessian information via the BFGS formula with a Cholesky factorisation offers greater numerical stability than the more straightforward approach of performing the update directly. Other authors have claimed that no such advantage exists and that any such improvement is probably due to early implementations of the DFP formula in conjunction with low accuracy line searches.

This paper supports the claim that there is no discernible advantage in choosing factorised implementations (over non-factorised implementations) of BFGS methods when approximate Hessian information is available to full machine precision. However the results presented in this paper show that a factorisation strategy has clear advantages when approximate Hessian information is available only to limited precision. These results show that a conjugate directions factorisation outperforms the other methods considered in this paper (including Cholesky factorisation).

In the framework of Mond-Weir duality a new equivalence between nonlinear programming and a matrix game is given. Finally, certain conclusions about convex programming with nested maxima and matrix games are also included.

We consider a generalisation of the stochastic formulation of smoothing splines, and discuss the smoothness properties of the resulting conditional expectation (generalised smoothing spline), and the sensitivity of the numerical algorithms. One application is to the calculation of smoothing splines with less than the usual order of continuity at the data points.

We consider the optimal proportional reinsurance from an insurer’s point of view to maximize the expected utility and minimize the value at risk. Under the general premium principle, we prove the existence and uniqueness of the optimal strategies and Pareto optimal solution, and give the relationship between the optimal strategies. Furthermore, we study the optimization problem with the variance premium principle. When the total claim sizes are normally distributed, explicit expressions for the optimal strategies and Pareto optimal solution are obtained. Finally, some numerical examples are presented to show the impact of the major model parameters on the optimal results.

An inequality involving the logarithmic mean is established. Specifically, we show that

where . Then several generalizations are given.

Online retailers are increasingly adding buy-online and pick-up-in-store (BOPS) modes to order fulfilment. In this paper, we study a system of BOPS by developing a stochastic Nash equilibrium model with incentive compatibility constraints, where the online retailer seeks optimal online sale prices and an optimal delivery schedule in an order cycle, and the offline retailer pursues a maximal rate of sharing the profit owing to the consignment from the online retailer. By an expectation method and optimality conditions, the equilibrium model is first transformed into a system of constrained nonlinear equations. Then, by a case study and sensitivity analysis, the model is validated and the following practical insights are revealed. (I) Our method can reliably provide an equilibrium strategy for the online and offline retailers under BOPS mode, including the optimal online selling price, the optimal delivery schedule, the optimal inventory and the optimal allocation of profits. (II) Different model parameters, such as operational cost, price sensitivity coefficient, cross-sale factor, opportunity loss ratio and loss ratio of unsold goods, generate distinct impacts on the equilibrium solution and the profits of the BOPS system. (III) Optimization of the delivery schedule can generate greater consumer surplus, and makes the offline retailer share less sale profit from the online retailer, even if the total profit of the BOPS system becomes higher. (IV) Inventory subsidy is an indispensable factor to improve the applicability of the game model in BOPS mode.

We propose a new nonrigid registration algorithm which is based on the optimal control approach. In our previously proposed methods, the Jacobian determinant and the curl vector were used as control functions. In this algorithm, we use a new set of control functions. A main advantage of using the new controls is that the positivity and normalization of the Jacobian determinant are satisfied automatically. Numerical results on large deformation brain images are provided to show the accuracy and efficiency of the algorithm.

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic solid with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and mixed displacement-traction boundary conditions yield standard boundary-value problems of potential theory for which uniqueness and existence of solutions are well established. However, the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry. Uniqueness and existence of solutions, within arbitrary rigid motions, are now established for the traction problem for general domains.

The modulation of short gravity waves by long waves or currents is described for the situation when the flow is irrotational and when the short waves are described by linearised equations. Two cases are distinguished depending on whether the basic flow can be characterised as a deep-water current, or a shallow-water current. In both cases the basic flow has a current which has finite amplitude, while in the first case the free surface slope of the basic flow can be finite, but in the second case is small. The modulation equations are the local dispersion relation of the short waves, the kinematic equation for conservation of wave crests and the wave action equation. The results incorporate and extend the earlier work of Longuet-Higgins and Stewart [10, 11].