Steady two-dimensional flows in a domain bounded below by an infinite horizontal wall and above by a semi-infinite horizontal wall, a vertical wall and a free surface are considered. The fluid is assumed to be inviscid and incompressible, and gravity is taken into account. The problem is solved numerically by series truncation. It is shown that for a given length of the vertical wall, there are two families of solutions. One family is characterized by a continuous slope at the separation point and a limiting configuration with a stagnation point and a 120° angle corner at the separation point. The other family is characterized by a stagnation point and a 90° angle corner at the separation point. Flows under a sluice gate with and without a rigid lid approximation upstream are also considered.

The contact problem investigated in this paper may be more fully described as a three dimensional elastic body with a circular hole through it; inside this tunnel is press fitted a solid elastic plug of finite length. Shear stresses are taken to be absent along the contact interface.

An influence coefficient technique is used to model the governing integral equation. For the elastic region the displacement influence coefficients due to bands of constant pressure are determined using a numerical quadrature on Fourier integrals. However, the plug, being of finite length, requires the superposition of two separate solutions to boundary value problems before the displacement influence coefficients can be determined.

Contact pressure distributions are presented for a sample of parameter variations and also for a case where hydrostatic pressure is present in the tunnel in the elastic region. Despite both components being elastic the imposition of a constant interference displacement along the interface still gives rise to the characteristic singularity in contact pressure at the edges of contact due to the strain discontinuity at these points.

Elastic behaviour of a nonhomogeneous transversely isotropic half-space is studied under the action of a smooth rigid axisymmetric indentor. Hankel transforms of different orders have been used. It is observed that in contrast to a homogeneous medium, the pressure distribution in the contact region in a nonhomogeneous medium is not directly available, rather it is obtainable from the solution of a Fredholm integral equation. The integral equation is solved for a flat-ended punch and paraboloidal indentations for various values of the nonhomogeneity parameter, and the effects of nonhomogeneity in elastic behaviour on stresses have been shown graphically. The results of the associated homogeneous case are readily available from the results of the present study.

The large-amplitude oscillations and buckling of an anisotropic cylindrical shell subjected to the initial inplane biaxial normal stresses have been analysed. The concept of anisotropy used by Lekhnitsky has been introduced into the field equations for cylindrical shells of isotropic material deduced by Donnell. The method of Galerkin and the method of successive approximation have been used to obtain the desired approximate solution. The expression for the critical loads for the buckling of anisotropic cylindrical shells has been obtained during intermediate stages of analysis. Some relevant frequency response graphs of the obtained solution are also presented. The minimum critical loads for various classes of anisotropy have also been given at the end of the discussion, to exhibit the effects of large deflections and imperfections on elastic buckling.

The movement of the interface between two immiscible fluids flowing through a porous medium is discussed. It is assumed that one of the fluids, which is a liquid, is much more viscous than the other. The problem is transformed by replacing the pressure with an integral of pressure with respect to time. Singularities of pressure and the transformed variable are seen to be related.

Some two-dimensional problems may be solved by comparing the singularities of certain analytic functions, one of which is derived from the new variable. The implications of the approach of a singularity to the moving boundary are examined.

An exact invariant is found for the one-dimensional oscillator with equation of motion . The method used is that of linear canonical transformations with time-dependent coeffcients. This is a new approach to the problem and has the advantage of simplicity. When f(t) and g(t) are zero, the invariant is related to the well-known Lewis invariant. The significance of extension to higher dimension of these results is indicated, in particular for the existence of non-invariance dynamical symmetry groups.

In this paper we establish the existence of solutions of a more general class of stochastic integral equation of Volterra type. The main tools used here are the measure of noncompactness and the fixed point theorem of Darbo. The results generalize the results of Tsokos and Padgett [9] and Szynal and Wedrychowicz [7]. An application to a stochastic model arising in chemotherapy is discussed.

We consider some general switching inequalities of Brenner and Alzer. It is shown that Brenner's Theorem B below does not hold in general without further conditions. A simple proof is given of Alzer's Corollary D.

Step changes in current through either grounded or ungrounded wires lying on the surface of a uniformly conducting half-space produce image current sources within the surface the conductor. This image current is effectively the only source term for initial changes in ∂1 Bz, Ex and Ey. The general steady state electric and magnetic field components resulting from steady currents flowing through either grounded or ungrounded wires of finite length lying on the surface of a uniform half- space are derived. Then the operators mapping these steady fields into the early values of ∂t Bz, Ex and Ey on or above the conducting half-plane resulting from instantaneously stopping the current flow through the wires are derived.

The problem of principal component analysis of a symmetric matrix (finding a p-dimensional eigenspace associated with the largest p eigenvalues) can be viewed as a smooth optimization problem on a homogeneous space. A solution in terms of the limiting value of a continuous-time dynamical system is presented. A discretization of the dynamical system is proposed that exploits the geometry of the homogeneous space. The relationship between the proposed algorithm and classical methods are investigated.

This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as the uniqueness of a positive solution. In order to obtain these results we employ monotonicity methods, singular spectral theory and a fixed point index.

We consider in this article an evolutionary monotone follower problem in [0,1]. State processes under consideration are controlled diffusion processes , solutions of dyx(t) = g(yx(t), t)dt + σu(yx(t), t) dwt + dυt with yx(0) = x ∈[0, 1], where the control processes υt are increasing, positive, and adapted. The cost functional is of integral type, with certain explicit cost of control action including the cost of jumps. We shall present some analytic results of the value function, mainly its characterisation, by standard dynamic programming arguments.

We study the structure of solutions of an initial value problem arising in the study of steadily rotating spiral waves in the kinematic theory of excitable media. In particular, we prove that under certain conditions there is a unique global positive monotone increasing solution.

For Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.

Brockett has studied the isospectral flow Ḣ = [H, [H, N]], with [A, B] = AB ∔ BA, on spaces of real symmetric matrices. The flow diagonalises real symmetric matrices and can be used to solve linear programming problems with compact convex constraints. We show that the flow converges exponentially fast to the optimal solution of the programming problem and we give explicit estimates for the time needed by the flow to approach an ε-neighbourhood of the optimum. An interior point algorithm for the standard simplex is analysed in detail and a comparison is made with a continuous time version of Karmarkar algorithm.

If the terms of a series behave like n−k where k is an exactly known constant, a formula using two terms transforms the series into a series of terms like n−k −2 provided k ≠ 1. The multiple use of this transformation is demonstrated in summing three series.

Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.