A linear programming model for optimally assigning diameters to a gas pipeline network is discussed. Computational results for a real life situation are presented, and certain properties that have to be satisfied by an optimal assignment are derived.

Some initial value problems are considered which arise in the treatment of a one-dimensional gas of point particles interacting with a “hard-core” potential.

Two basic types of initial conditions are considered. For the first, one particle is specified to be at the origin with a given velocity. The positions in phase space of the remaining background of particles are represented by continuous distribution functions. The second problem is a periodic analogue of the first.

Exact equations for the delta-function part of the single particle distribution functions are derived for the non-periodic case and approximate equations for the periodic case. These take the form of differential operator equations. The spectral and asymptotic properties of the operators associated with the two cases are examined and compared. The behaviour of the solutions is also considered.

The convergence properties of a very general class of adaptive recursive algorithms for the identification of discrete-time linear signal models are studied for the stochastic case using martingale convergence theorems. The class of algorithms specializes to a number of known output error algorithms (also called model reference adaptive schemes) and equation error schemes including extended (and standard) least squares schemes. They also specialize to novel adaptive Kalman filters, adaptive predictors and adaptive regulator algorithms. An algorithm is derived for identification of uniquely parameterized multivariable linear systems.

A passivity condition (positive real condition in the time invariant model case) emerges as the crucial condition ensuring convergence in the noise-free cases. The passivity condition and persistently exciting conditions on the noise and state estimates are then shown to guarantee almost sure convergence results for the more general adaptive schemes.

Of significance is that, apart from the stability assumptions inherent in the passivity condition, there are no stability assumptions required as in an alternative theory using convergence of ordinary differential equations.

The problem of an infinitely long rigid punch of uniform cross-section moving across a viscoelastic half-space at constant velocity, large enough so that inertial effects cannot be neglected, is examined and solved in various approximations. Frictional shear is assumed to exist between the punch and the half-space. The method, which is an extension of that developed in previous papers [6, 7], is applicable for any form of viscoelastic behaviour in the half-space. For the special case of discrete spectrum behaviour the method is described in detail. For the case where the punch is cylindrical and viscoelastic effects are small compared with elastic effects, explicit expressions are given for all quantities of interest, in particular the coefficient of hysteretic friction. A general Hilbert transform formula is derived in the appendix.

In this note we consider various theoretical aspects of the problem of least-squares approximation subject to constraints on the range of the approximating polynomial. The problem is treated from an optimization theory viewpoint. Rice's parameter space procedure is discussed.

The problem of determining the temperature, displacement and stress fields around a single crack in an anisotropic slab is considered. The problem is reduced to Fredholm integral equations which may be solved numerically.