Existence of piecewise optimal control is proved when the cost function includes one or both of (a) a cost of sudden switching (discontinuity) of control variables, and (b) a cost associated with the maximum rate of variation of the control over segments of the path for which the control is continuous.

We consider a single server queueing system where each customer visits the queue a fixed number of times before departure. A customer on his j th visit to the queue is defined to be a class-j -customer. We obtain the joint probability generating function for the number of class-j-customers and also obtain the Laplace-Stieltjes transform for the total response time of a customer.

The spectral function , where are the eigenvalues of the two-dimensional Laplacian, is studied for a variety of domains. The dependence of θ(t) on the connectivity of a domain and the impedance boundary conditions is analysed. Particular attention is given to a doubly-connected region together with the impedance boundary conditions on its boundaries.

In two dimensions it is found that the most general autonomous Hamiltonian possessing a Laplace-Runge-Lenz vector is The Poisson bracket of the two components of this vector leads to a third first-integral, cubic in the momenta. The Lie algebra of the three integrals under the operation of the Poisson bracket closes, and is shown to be so(3) for negative energy and so(2, 1) for positive energy. In the case of zero energy, the algebra is W(3, 1). The result does not have a three-dimensional analogue, apart from the usual Kepler problem.

Under the assumptions that the spatial variable is one dimensional and the distributed delay kernel is the general Gamma distributed delay kernel, when the average delay is small, the existence of travelling wave solutions for the population genetics model with distributed delay is obtained by using the linear chain trick and geometric singular perturbation theory. On the other hand, for the population genetics model with small discrete delay, the existence of travelling wave solutions is obtained by employing a technique which is based on a result concerning the existence of the inertial manifold for small discrete delay equations.

The hyper-Wiener index of a connected graph G is defined as , where V (G) is the set of all vertices of G and d(u,v) is the distance between the vertices u,v∈V (G). In this paper we find an exact expression for the hyper-Wiener index of TUHC6[2p,q], the zigzag polyhex nanotube.

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 pair of multi-objective programming problems is shown to be symmetric dual by associating a vector-valued infinite game to the given pair. This symmetric dual pair seems to be more general than those studied in the literature.

In this paper a bridging method is introduced for numerical solutions of one-dimensional global optimization problems where a continuously differentiable function is to be minimized over a finite interval which can be given either explicitly or by constraints involving continuously differentiable functions. The concept of a bridged function is introduced. Some properties of the bridged function are given. On this basis, several bridging algorithm are developed for the computation of global optimal solutions. The algorithms are demonstrated by solving several numerical examples.

A class of non-standard optimal control problems is considered. The non-standard feature of these optimal control problems is that they are of neither fixed final time nor of fixed final state. A method of solution is devised which employs a computational algorithm based on control parametrization techniques. The method is applied to the problem of maximizing the range of an aircraft-like gliding projectile with angle of attack control.

The problem of reflection and refraction of elastic waves due to an incident quasi-primary $(qP)$ wave at a plane interface between two dissimilar nematic elastomer half-spaces has been investigated. The expressions for the phase velocities corresponding to primary and secondary waves are given. It is observed that these phase velocities depend on the angle of propagation of the elastic waves. The reflection and refraction coefficients corresponding to the reflected and refracted waves, respectively, are derived by using appropriate boundary conditions. The energy transmission of the reflected and refracted waves is obtained, and the energy ratios and the reflection and refraction coefficients are computed numerically.

The present paper presents a ray analysis for a problem of technical importance in fragmentation studies. The problem is that of suddenly punching a circular hole in either isotropic or transversely isotropic plates subjected to a uniaxial tension field. The ray method, which involves only differentiation, integration, and simple algebra, is shown to be particularly useful in clarifying the propagation process of the resulting unloading waves and obtaining the attendant discontinuities of the various quantities involved. Numerical results obtained from the ray analysis are presented in graphical form and compared with those obtained by more elaborate schemes.

Consider a density-dependent birth-death process XN on a finite state space of size N. Let PN be the law (on D([0, T]) where T > 0 is arbitrary) of the density process XN/N and let πN be the unique stationary distribution (on[0,1]) of XN/N, if it exists. Typically, these distributions converge weakly to a degenerate distribution as N → ∞ so the probability of sets not containing the degenerate point will tend to 0; large deviations is concerned with obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel theory is used to establish the large deviations behaviour (as N → ∞) of PN. In the one-dimensional case, a large deviations principle for the stationary distribution πN is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so πN no longer exists), the same elementary computations are still applicable to the quasi-stationary distribution, and we show that the quasi-stationary distributions obey the same large deviations principle as in the recurrent case. In addition, we address some questions related to the estimated time to absorption and obtain a large deviations principle for the invariant distribution in higher dimensions by studying a quasi-potential.

This is a review of progress made since [R. McKibbin, “An attempt at modelling hydrothermal eruptions”, Proc. 11th New Zealand Geothermal Workshop 1989 (University of Auckland, 1989), 267–273] began development of a mathematical model for progressive hydrothermal eruptions (as distinct from “blasts”). Early work concentrated on modelling the underground process, while lately some attempts have been made to model the eruption jet and the flight and deposit of ejected material. Conceptually, the model is that of a boiling and expanding two-phase fluid rising through porous rock near the ground surface, with a vertical high-speed jet, dominated volumetrically by the gas phase, ejecting rock particles that are then deposited on the ground near the eruption site. Field observations of eruptions in progress and experimental results from a laboratory-sized model have confirmed the conceptual model. The quantitative models for all parts of the process are based on the fundamental conservation equations of motion and thermodynamics, using a continuum approximation for each of the components.

It is well known that the trapezoidal rule of quadrature is exact for linear functions on [0, 1], and easy to see that it is exact for functions of the form f = l+g where l is linear and g is odd about ½. Not so well known is an example of a continuous function for which the trapezoidal rule is exact but which does not have this form. In this paper we show that if the trapezoidal rule is exact for f then f has the form above provided it has either absolutely convergent Fourier series or continuous second derivative. We consider one-sided versions in which the approximate integrals are non-negative, and also characterize those sequences arising as the approximate integrals of a function with absolutely convergent Fourier series.

Certain well known results on linear programming duality and matrix game equivalence are extended to nonlinear and fractional programming problems.

In this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

This paper is concerned with constructing polynomial solutions to ordinary boundary value problems. A semi-analytic technique using two-point Hermite interpolation is compared with conventional methods via a series of examples and is shown to be generally superior, particularly for problems involving nonlinear equations and/or boundary conditions.

In this paper, we consider the system governed via the coefficients of a semilinear elliptic equation and give the necessary conditions for optimal control. Furthermore, we obtain the necessary conditions for an optimal domain in a domain optimization problem.

The combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.