The monotone iterative technique is applied to a system of ordinary differential equations with a singular matrix. The existence of extremal solutions is proved.

This paper deals with the question of the existence of a solution to the stationary-point problem corresponding to a given nonlinear nondifferentiable program. An existence theorem for the stationary-point problem is presented under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the nonlinear program.

In this paper we prove several growth theorems for second-order difference equations.

In this paper we describe an elementary method for calculating the matrix exponential on an arbitrary time scale. An example is also given to illustrate the result.

For the heat equation in two space dimensions we consider semidiscrete and totally discrete variants of the lumped mass modification of the standard Galerkin method, using piecewise linear approximating functions, and demonstrate error estimates of optimal order in L2 and of almost optimal order in L∞.

The two-dimensional Helmholtz equation is studied for an infinite region with two semi-infinite plates extending to infinity in opposite directions and a finite duct in the overlapping region. The solution technique leads to coupled Wiener-Hopf equations, and subsequently to an infinite set of simultaneous linear equations. As an example, an asymptotic expansion is calculated and graphed for the case when the duct length divided by duct width is large enough to ensure damping of all but the zero mode wave in the duct.

When studying deep convection in a compressible medium the effects of viscous dissipation can become important and must be taken into account in any realistic model. But even in shallow convection, for which the Boussinesq approximation is valid, the viscous dissipation effects will become important at high Rayleigh numbers. These effects are estimated with the help of asymptotic methods and the results are compared with those obtained by numerical integration.

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.

Iterative methods for solving a square system of nonlinear equations g(x) = 0 often require that the sum of squares residual γ (x) ≡ ½∥g(x)∥2 be reduced at each step. Since the gradient of γ depends on the Jacobian ∇g, this stabilization strategy is not easily implemented if only approximations Bk to ∇g are available. Therefore most quasi-Newton algorithms either include special updating steps or reset Bk to a divided difference estimate of ∇g whenever no satisfactory progress is made. Here the need for such back-up devices is avoided by a derivative-free line search in the range of g. Assuming that the Bk are generated from an rbitrary B0 by fixed scale updates, we establish superlinear convergence from within any compact level set of γ on which g has a differentiable inverse function g−1.

Conditions are given for a Ck map T to be a Newton map, that is, the map associated with a differentiable real-valued function via Newton's method. For finitely differentiable maps and functions, these conditions are only necessary, but in the smooth case, that is, for k = ∞, they are also sufficient. The characterisation rests upon the structure of the fixed point set of T and the value of the derivative T′ there, and it is best possible as is demonstrated through examples.

A recent paper by Leipnik and Pearce introduced the gamma Weibull distribution. One of the main properties given is its characteristic function, which is expressed as an infinite sum. In this note, we provide a simpler representation in terms of the well-known hypergeometric functions in some special cases. We also derive expressions for moments of the distribution.

The effects of iron on the uniformity of the field produced by a current-carrying axisymmetric conductor are considered. Using a perturbation analysis a simple analytic expression is obtained which describes the field close to the axis of symmetry. A Fourier series approach is also used to provide an analytical solution to the problem and the accuracy of the perturbation method is estimated by comparing results.

A theorem is derived for the hydrodynanuc image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symmetry, a flow past the rigid or shear-free inverse of that surface or sphere.

The theorems are used to simplify the proofs of a number of known results for images of point singularities in plane and spherical rigid and free boundaries, and for a pair of bubbles rising steadily in line in a viscous fluid. They also give for the first time accurate numerical solutions for the velocities of each of a larger number of spherical bubbles rising quasi-steadily in line. These enable one to assess the accuracy of simple approximations to those velocities.

Polynomial identities satisfied by the generators of the Lie groups O(n) and U(n) are rederived. Using these identities the reduced matrix elements of the Lie groups U(n) and O(n) are evaluated as rational functions of the IR labels occurring in the canonical chains

This method does not require an explicit realization of the Lie algebras and their representations using bosons. Finally, trace formulae encountered previously by several authors for finite dimensional irreducible representations are shown to hold on arbitrary representations admitting an infinitesimal character.

We present analytical methods to investigate the Cauchy problem for the complex Ginzburg-Landau equation u1 = (v + iα)Δu − (κ + iβ) |u|2qu + γu in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, κ ≥ 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when .

Tensor identities for finite dimensional representations of arbitrary semi-simple Lie algebras are derived and are applied to the construction of left-projection operators which project out the shift components of tensor operators from the left. The corresponding adjoint identities are also derived and are used for the construction of right-projection operators. It is also shown that, on a finite dimensional irreducible representation, these identities may be considerably reduced. Commutation relations between the shift tensors of a tensor operator are also computed in terms of the roots appearing in the tensor identities.