The low-velocity impact of two convex surfaces comprised of identical material, which approach each other along the direction of the normal at first contact, and obey a J2 = k2 plastic yield condition, is shown for very early times to satisfy the following conditions: the interior surface which separates the two bodies is equivalent to either the locus of points formed by the intersecting curves resulting from moving the two bodies towards each other along their normal; or to the locus of points formed from the level surfaces (suitably parametrized) drawn about each body at the time of first contact. This separating surface lies midway between the geometrical overlap of the two approaching surfaces for times sufficiently short for inertial effects not to significantly affect the approaching velocities.

In this paper we consider simplifying a model of the nitrogen cycle in Port Phillip Bay, Victoria, Australia. The approach taken is to aggregate state variables that are linearly related using a projection in state space. The technique involved is a modification of proper orthogonal decomposition and was developed so that a resulting simplified model retains an ecological interpretation. It can be applied automatically, and enables insights into the system to be gained that were not obvious beforehand. In the case of the Port Phillip Bay model, we find that the variables representing water and sand are unaffected by the remaining variables, while only variables on the same trophic level can be grouped together. The validity of the aggregation under several nutrient loads is also discussed.

In 2002 the Mathematics in Industry Study Group (MISG) investigated the question of optimally scheduling cyclic production in a battery charging and finishing facility. The facility produces various types of battery and the scheduling objective is to maximize battery throughout subject to achieving a pre-specified product-mix. In this paper we investigate the robustness of such schedules using simulation experiments that span multiple production cycles. We simulate random variations (delays) in battery charging time and find that an optimal off-line schedule yields higher throughput in comparison to a common on-line dispatching rule. This result has been found to hold for a range of expected charging-time delays and has significant practical implications for scheduling battery charging and finishing facilities.

We present an alternative matrix mnemonic for the basic equations of simple thermodynamics. When normalized, this permits an explicit generalized inverse, allowing inversion of the mechanical and chemical thermodynamic equations. As an application, the natural variables S, V, P and T are derived from the four energies E (internal), F (free), G (Gibbs) and H (enthalpy).

The diffusion equation is used to model and analyze sorption, a process used in the purification or separation of fluids. For the diffusion inside a spherical porous solid immersed in a limited-volume and well-stirred fluid, Ruthven [5], Crank [3] and, for the analogous flow of heat, Carslaw and Jaeger [2] give an eigenfunction expansion solution to the diffusion equation that provides accurate long-time solutions when only a few terms are used. However, to obtain the same accuracy for short-time solutions the number of eigenfunction terms required increases exponentially. An alternative error function solution of Carman and Haul [1] is accurate for sufficiently short times but not for long times. Although their solution is well quoted [3, 4, 6], Carman and Haul do not provide a derivation in their paper. This paper provides a full derivation of the short-time solution of Carman and Haul that uses only the first term of a negative exponential series in the Laplace domain. It is shown that the accuracy and range of the short-time result is improved by the inclusion of additional terms of the negative exponential series. An analysis of short-time and long-time resultsis presented, together with recommendations as to their use.

In this paper, a function involving the divided difference of the psi function is proved to be completely monotonic, a class of inequalities involving sums is found, and an equivalent relation between complete monotonicity and one of the class of inequalities is established.

This paper is concerned with constructing polynomial solutions to ordinary boundary value problems. A semi-analytic technique using two-point Hermite interpolation is compared with conventional methods via a series of examples and is shown to be generally superior, particularly for problems involving nonlinear equations and/or boundary conditions.

In this paper we establish some generalizations of a weighted trapezoidal inequality for monotonic functions and give several applications for the r-moments, the expectation of a continuous random variable and the Beta and Gamma functions.

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.

The state of a patient is an important concept in biomedical sciences. While analytical methods for predicting and exploring treatment strategies of disease dynamics have proven to have useful applications in public health policy and planning, the state of a patient has attracted less attention, at least mathematically. As a result, models constructed in relation to treatment strategies may not be very informative. We derive a patient-dependent parameter from an age-physiology dependent population model, and show that a single treatment strategy is not always optimal. Also, we derive a function which increases with the patient dependence parameter and describes the effort expended to be in good health.