This paper presents an algorithm to solve the least squares problem when the parameters are restricted to be non-negative. The algorithm is based on the branch and bound method which has been suggested for this problem, and shares with it the property that an unrestricted least squares subproblem is solved at each step. However, improvements have been made to the branching rules by making use of the convexity of the problem, and the Kuhn–Tucker conditions are used to test for optimality. The resulting algorithm becomes essentially iterative in nature, and linearity of the number of subproblems solved can be shown under assumptions which have always been observed in practice.

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

Using Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

This paper establishes an estimate for the asymptotic behaviour of the spectrum of a direct strain feedback (DSF) control system. The results show that the system operator corresponding to the closed loop system cannot have an analytic extension and that the decay rate for the system energy is not proportional to the feedback constant.

In this paper, we consider a class of systems governed by second order linear parabolic delay-partial differential equations with first boundary conditions. Our main results are reported in Theorems 3.1 and 3.2. As in [9, Theorems 4.1 and 4.2], the coefficients and forcing terms of the system considered in Theorem 3.1 are linear in the control variables. On the other hand, the forcing terms of the system considered in Theorem 3.2 are allowed to be nonlinear in the control variables at the expense of dropping the control variables in the cost integrand.

Some multipoint iterative methods without memory, for approximating simple zeros of functions of one variable, are described. For m > 0, n ≧ 0, and k satisfying m + 1 ≧ k > 0, there exist methods which, for each iteration, use one evaluation of f, f′, … f(m) followed by n evaluations of f(k), and have order of convergence m + 2n + 1. In particular, there are methods of order 2(n + 1) which use one function evaluation and n + 1 derivative evaluations per iteration. These methods naturally generalize the known cases n = 0 (Newton's method) and n = 1 (Jarratt's fourth-order method), and are useful if derivative evaluations are less expensive than function evaluations. To establish the order of convergence of the methods we prove some results, which may be of independent interest, on orthogonal and “almost orthogonal” polynomials. Explicit, nonlinear, Runge-Kutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. The theoretical results are illustrated by several numerical examples.

When faced with the task of solving hyperbolic partial differential equations (PDEs), high order, strong stability-preserving (SSP) time integration methods are often needed to ensure preservation of the nonlinear strong stability properties of spatial discretizations. Among such methods, SSP second derivative time-stepping schemes have been recently introduced and used for evolving hyperbolic PDEs. In previous works, coupling of forward Euler and a second derivative formulation led to sufficient conditions for a second derivative general linear method (SGLM), which preserve the strong stability properties of spatial discretizations. However, for such methods, the types of spatial discretizations that can be used are limited. In this paper, we use a formulation based on forward Euler and Taylor series conditions to extend the SSP SGLM framework. We investigate the construction of SSP second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs with order $p=r=s$ and stage order $q=p,p-1$ up to order eight, where r is the number of external stages and s is the number of internal stages of the method. Proposed methods are examined on some one-dimensional linear and nonlinear systems to verify their theoretical order, and show potential of these schemes in preserving some nonlinear stability properties such as positivity and total variation.

We analyse the performance of the additive observable proportional navigation guidance system, which is well-suited for low-cost homing missiles with bearings-only measurements. Closed-form solutions are derived for both manoeuvring and non-manoeuvring targets. Guidelines on how to select the navigation constants of the control law are presented. We show that the additive observable proportional navigation guidance system can cover a larger capture area than can a conventional proportional navigation system.

A delayed predator-prey system with Holling type III functional response is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of suitable Lyapunov functionals, sufficient conditions are derived for the local and global asymptotic stability of a positive equilibrium of the system. Numerical simulations are presented to illustrate the feasibility of our main results.

In this short paper, it is shown that the geodesic deviation equation admits a “constant of the motion” and so can be solved exactly. We also derive an expression for the energy E of relative motion between two freely falling test particles. We can infer that, in general, E will not be a linear superposition of kinetic and potential energies.

A new method of field quantization, in which the number operator has the form a*a, is proposed. Representations of this method are considered, particularly in reference to what conditions unique-vacuum state representations are required to satisfy.

This paper deals with a class of network optimization problems in which the flow is a function of time rather than static as in the classical network flow problem, and storage is permitted at the nodes. A solution method involving discretization will be presented as an application of the ASG algorithm. We furnish a proof that the discretized solution converges to the exact continuous solution. We also apply the method to a water distribution network where we minimize the cost of pumping water to meet supply and demand, subject to both linear and nonlinear constraints.

An improvement is made to two results of Brown and Shepp which are useful in calculations with fractal sets.

A mathematical model is developed for the process of gas exchange in lung capillaries, taking into account the transport mechanisms of molecular diffusion and the facilitated diffusion of the species due to haemoglobin. We have assumed here equilibrium conditions which enable us to neglect advection effects. The nth order one-step kinetics of oxygen uptake by haemoglobin, proposed by Sharan and Singh [8], have been incorporated. The solution of this coupled nonlinear facilitated diffusion-reaction problem together with the physiologically-relevant boundary conditions is obtained in the closed form.

It is found that about 97.15% of total haemoglobin has combined with oxygen and 2.85% free pigment is left, which is present as carbaminohaemoglobin, met haemoglobin, carboxy haemoglobin etc. It is also shown that the percentage of free haemoglobin at a given PO2 and PCO2 is independent of total haemoglobin content present in the blood.

The well-known Hill's empirical relation is deduced from our solution. The results obtained from our model, based on physical formulation, are in good agreement with the documented data [6] and those computed from the Kelman [3] empirical relation.

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.

We consider the effect the competing mechanisms of buoyancy-driven acceleration (arising from heating a surface) and streamline curvature (due to curvature of a surface) have on the stability of boundary-layer flows. We confine our attention to vortex type instabilities (commonly referred to as Görtler vortices) which have been identified as one of the dominant mechanisms of instability in both centrifugally and buoyancy driven boundary layers. The particular model we consider consists of the boundary-layer flow over a heated (or cooled) curved rigid body. In the absence of buoyancy forcing the flow is centrifugally unstable to counter-rotating vortices aligned with the direction of the flow when the curvature is concave (in the fluid domain) and stable otherwise. Heating the rigid plate to a level sufficiently above the fluid's ambient (free-stream) temperature can also serve to render the flow unstable. We determine the level of heating required to render an otherwise centrifugally stable flow unstable and likewise, the level of body cooling that is required to render a centrifugally unstableflow stable.

Solutions of a homogeneous (r + 1)-term linear difference equation are given in two different forms. One involves the elements of a certain matrix, while the other is in terms of certain lower Hessenberg determinants. The results generalize some earlier results of Brown [1] for the solution of a 3-term linear difference equation.

Sufficient conditions are derived for the relative controllability of nonlinear neutral Volterra integrodifferential systems with distributed delays in the control variables. The results are a generalization of previous results and are obtained by using Schauder's fixed-point theorem.

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.

In a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.