Integral equations on the half line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by a positive number β. A novel technique is used here to rederive a number of classical results on the existence and uniqueness of the solution of the Wiener-Hopf and related equations, and is then extended to obtain existence, uniqueness and convergence results for the corresponding finite-section equations. Unlike the methods used in the recent work of Anselone and Sloan, the present methods are constructive, and result in explicit asymptotic bounds for the error introduced by the finite-section approximation.

Recently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems. The relationship between invexity and other generalizations of convexity is illustrated. Finally, it is shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification.

The known linear model reference adaptive control (MRAC) technique is extended to cover nonlinear and nonlinearizable systems (several equilibria, etc) and used to stabilize the system about a model. The method proposed applies the same Liapunov Design Technique but avoids the classical error equation. Instead it operates in the product of the state spaces of plant and model, aiming at convergence to a diagonal set. Control program, Liapunov functions and adaptive law are specified. The case is illustrated on a two-degrees of freedom robotic manipulator.

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.

Recently, Hanson and Mond formulated a type of generalized convexity and used it to establish duality between the nonlinear programming problem and the Wolfe dual. Elsewhere, Mond and Weir gave an alternate dual, different from the Wolfe dual, that allowed the weakening of the convexity requirements. Here we establish duality between the nonlinear programming problem and the Mond-Weir dual using Hanson-Mond generalized convexity conditions.

A model for thermal ignition by intense light is studied. The governing non-linear parabolic equation is linearized in a two-step manner with the aid of a non-linear ordinary differential equation which captures the salient features of the non-linear parabolic equation. The critical parameters are computed from the steady-state solution of the ordinary differential equation, which can be obtained without actually solving the equation. Comparison with available data shows that the present method yields good results.

In this paper the relations between semi-infinite programs and optimisation problems with finitely many variables and constraints are reviewed. Two classes of convex semi-infinite programs are defined, one based on the fact that a convex set may be represented as the intersection of closed halfspaces, while the other class is defined using the representation of the elements of a convex set as convex combinations of points and directions. Extension to nonconvex problems is given. A common technique of solving a semi-infinite program computationally is to derive necessary conditions for optimality in the form of a nonlinear system of equations with finitely many equations and unknowns. In the three-phase algorithm, this system is constructed from the optimal solution of a discretised version of the given semi-infinite program. i.e. a problem with finitely many variables and constraints. The system is solved numerically, often by means of some linearisation method. One option is to use a direct analog of the familiar SOLVER method.

We show the strongly stable convergence of some non-collectively-compact approximations of compact operators. Special attention is devoted to Anselone's singularity subtraction discretization of certain singular integral operators. Numerical experiments are provided.

A new method is described which allows an exact solution in a closed form to the following non-axisymmetric mixed boundary-value problem for a charged sphere: arbitrary potential values are given at the surface of a spherical segment while an arbitrary charge distribution is prescribed on the rest of the sphere. The method is founded on a new integral representation of the kernel of the governing integral equation. Several examples are considered. All the results are expressed in elementary functions. Some further applications of the method are discussed. No similar result seems to have been published previously.

The problem of passing from an L∞ function to a Wiener-Hopf factorization is considered. It is shown that a small L∞ perturbation which does not change the factorization indices will lead to small Lp (1 < p < ∞) perturbations in the Wiener-Hopf factors, but can lead to large L∞ perturbations, unless the derivatives are controlled during the perturbation.

We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Fréchet). When the range space is a countably order-complete Banach lattice, these conditions are also necessary. In particular, every order-bounded convex operator from an Asplund space into such a lattice is generically Fréchet differentiable, if and only if the lattice has weakly-compact order intervals, if and only if the lattice has strongly-exposed order intervals. Applications are given which indicate how such results relate to optimization theory.

A ratio game approach to the generalized fractional programming problem is presented and duality relations established. This approach suggests certain solution procedures for solving fractional programs involving several ratios in the objective function.

Under suitable hypotheses on the function f, the two constrained minimization problems:

are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to:

in which T and S are convex cones, S* is the dual cone of S, and ∂y denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This duality result confirms a conjecture by Chandra, which had previously been proved only in special cases.

Sparse matrix factorizations of transfer matrices for the interactions round a face model are reviewed. The sparse factors of a more general Ising model containing first, second and third nearest neighbour interactions are also presented. For both models the factorizations are achieved by considering the required auxiliary spin sets as a hierarchy of interacting spins.

A continuous-time model of a two-sector trading economy with a finitely-saturating production function and constant population is constructed. To apply it to the farm-economy problem of optimal trading of cereal for chemical nitrogenous fertilizer and of optimal allocation of land for green manure the special assumptions are made of similar technologies in both sectors and of a nitrogen-capital loss proportional to the cereal production. Optimal-control theory is applied to get the pair of controls that maximizes an infinite-horizon integral of utility of consumption. The analysis of the model is shown to reduce to that of the Ramsey-Koopmans one-sector neoclassical model of optimal growth. Calculations of the feedback control and optimal time-paths for the standardized dimensionless model are tabulated for a range of 7 utility functions of constant elasticity of marginal utility of magnitude 1 to 4 and of production functions of degree 2 to 4. Two particular analytic solutions are given. The application of the results both to the farm-economy model and to the Ramsey-Koopmans model is illustrated.

Polynomial identities for the generators of a simple basic classical Lie superalgebra are derived in arbitrary representations generated by a maximal (or minimal) weight vector. The infinitesimal characters occurring in the tensor product of two finite dimensional irreducible representations are also determined.

A heuristic methodology for the identification of a circuit passing through all the vertices only once in a graph is presented. The procedure is based upon defining a normal form of a matrix and then transforming the adjacency matrix into its normal form. For a class of graphs known to be Hamiltonian, it is conjectured that this methodology will identify circuits in a small number of steps and in many cases merely by observation.

We consider the Volterra-Lotka equations for two competing species in which the right-hand sides are periodic in time. Using topological degree, we show that conditions recently given by K. Gopalsamy, which imply the existence of a periodic solution with positive components, also imply the uniqueness and asymptotic stability of the solution. We also give optimal upper and lower bounds for the components of the solution.

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:

In summary, we prove that, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible set M[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible set M[ H, G] is stable (perturbations of H and G produce homeomorphic feasible sets) if and only if MFCQ holds; under a stability condition, two lower level sets of f with a Kuhn-Tucker point between them are homotopically related by attachment of a k-cell (k being the stationary index in the sense of Kojima).

The explicit solitary Rossby wave solutions found by Larichev, Reznik and Berestov are shown to be unique for the model equations considered, in the sense that there are no other antisymmetric wave solutions which are not of these forms. This is done by adapting arguments used by Amick and Fraenkel to show the uniqueness of the Hill's vortex solution. It is based on the maximum principle and the domain folding method of Gidas, Ni and Nirenberg, and involves showing that the function ψ/y is radially symmetric, where ψ is the streamfunction of a solitary wave and y the horizontal cartesian coordinate perpendicular to the x-axis, along which the waves move at steady positive speed. This argument is also used to show the uniqueness of the well-known explicit solutions for cylindrical vortices. The result does not apply directly to rider solutions of Flierl et al., which are not antisymmetric, but it does restrict the possible rider solutions that can form because of their association with particular antisymmetric solutions.