The steady state bifurcations near a double zero eigenvalue of the reaction diffusion equation associated with a tri-molecular chemical reaction (the Brusselator) are analysed. Special emphasis is put on three degeneracies where previous results of Schaeffer and Golubitsky do not apply. For these degeneracies it is shown by means of a LiapunovSchmidt reduction that the steady state bifurcations are determined by codimension-three normal forms. They are of types (9)31, (8)221 and (6a)ρ,κ in a recent classification of Z(2)-equivariant imperfect bifurcations with corank two. Each normal form couples an ordinary corank-1 bifurcation in the sense of Golubitsky and Schaeffer to a degenerate Z(2)-equivariant corank-1 bifurcation of Golubitsky and Langford in a specific way.

The paper examines a matrix equation given by Ziebur [6] for the growth of a population in which the birth-rate and death-rate are age-dependent. For convenience the population was sub-divided into four age groups, with the same birth-rate and death-rate for individuals in a particular group, and the matrix equation relates the numbers in each sub-division in consecutive years. This avoids delay terms and makes it easier to modify the growth equation but it is shown that the form suggested by Ziebur for the transition matrix leads to some difficulties.

Computing the generalised gradient directly using its standard definition can involve forming the convex hull of a very large number of vectors. Here an alternative concise parametrization is developed for the generalised gradient of the signed rank regression family of objective functions, a class of piecewise linear functions which includes both convex and nonconvex members. The approach uses the geometry of the epigraph explicitly and this suggests extensions to more general functions. A nondegeneracy condition is assumed which is natural in optimization problems.

The paper discusses equilibrium solutions and solutions with period two and period three for the difference equation

where Q and A are real, positive parameters. The equation was used by Bier and Bountis [1] as an example of a difference equation whose iteration diagram can show bubbles of finite length rather than the successive bifurcations usually expected. The paper examines in more detail what kind of solution can occur for given values of Q and A and establishes a series of critical curves which demarcate the regions in the (Q, A) plane where solutions of period two or period three occur and the subregions where these periodic solutions are stable. This makes it easy to see how Q and A can be combined into a one-parameter equation which gives a bubble, or a series of bubbles, in the iteration diagram.

The present paper proves criteria for oscillation of the soultions of functional differential equations of the type

where λ, τ > 0.

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.

Consider the neutral delay differential equation

where p ∈ R, τ ≥ 0, q1 > 0, σ1 ≥ 0, for i = 1, 2, …, k. We prove the following result.

Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation

has no real roots.

The purpose of this work is to begin the development of a theory of generating functions that will not only include the generating functions which are partly bilateral and partly unilateral but also provide a set of expansions, by taking successive partial derivatives with respect to one of the variables of the generating relations. Our starting point is a result of Exton [4] on associated Laguerre polynomials whose application gives certain generating functions of the polynomials of Jacobi and Appell, and functions of n variables of Lauricella.

We consider an optimization problem in which the function being minimized is the sum of the integral functional and the full variation of control. For this problem, we prove the existence theorem, a necessary condition in an integral form and a local necessary condition in the case of monotonic controls.

A class of convex optimal control problems involving linear hereditary systems with linear control constraints and nonlinear terminal constraints is considered. A result on the existence of an optimal control is proved and a necessary condition for optimality is given. An iterative algorithm is presented for solving the optimal control problem under consideration. The convergence property of the algorithm is also investigated. To test the algorithm, an example is solved.

We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.

A mathematical model is presented in which the long Jump is treated as the motion of a projectile under gravity with slight drag. The first two terms of a perturbation solution are obtained and are shown to be more accurate than earlier approximate analytical solutions. Results from the perturbation analysis are just as accurate as results from various numerical schemes, and require far less computer time.

The model is modified to include the observation that a long-jumper's centre of mass is forward of his feet at take-off and behind his feet on landing.

The modified model is used to determine the take-off angle for the current world long jump record, resulting in several interesting observations for athletic coaches.

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.

In this paper, we develop a duality theory for the Apex dual in the case of primal constraints. As suggested by Duffin in [4], the objective function in this framework is a weighted average of the Legendre-Lagrangian function evaluated at key points. We show that whenever this new dual is feasible there is no duality gap for this dual, and moreover, no duality gap for both the Lagrangian and Wolfe duals too. We conclude with an outline of an algorithm to solve constrained minimization problems in the Apex framework.

Solutions are found to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation. The flow in each case is due to a submerged line source or sink, in the presence of a solid horizontal base.