We give here some extensions of inequalities of Popoviciu and Rado. The idea is to use an inequality [C. P. Niculescu and L. E. Persson, Convex functions. Basic theory and applications (Universitaria Press, Craiova, 2003), Page 4] which gives an approximation of the arithmetic mean of n values of a given convex function in terms of the value at the arithmetic mean of the arguments. We also give more general forms of this inequality by replacing the arithmetic mean with others. Finally we use these inequalities to establish similar inequalities of Popoviciu and Rado type.

An analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

The current geothermal and volcanic activity in the North Island of New Zealand is explained as a consequence of Pacific and Australian plate interactions over the last 20 million years. The primary hypothesis is that the Kermadec subduction zone has for the last 20 million years or more been retreating in a south-easterly direction at about five centimetres per year. It is surmised that this motion and interaction with another subduction zone almost at right angles to it under the North Island resulted in plate tearing due to the incompatibility of the plate geometry where these subduction zones interacted. The nature and consequences of this plate tearing are partially revealed in published maps of the plate currently under the North Island. If the subducted parts of this plate, as shown in Eiby’s maps, [G. A. Eiby, “The New Zealand sub-crustal rift”, New Zeal. J. Geol. Geophy.7 (1964) 109–133] are straightened, then the plate edge lies on a curve giving a rough picture of their position before being torn and subducted by the Kermadec trench motion. This map of the tear suggests the shape of the edge of a missing plate segment torn from the plate, and implies a rotation of the upper North Island, clockwise approximately 20 degrees, about a point just south of the Thames estuary. A consequence of this plate tearing is that the solid retreating crustal wave generating magma pressure beneath the crest of the solid wave has the potential to inject significant basaltic magma into the crust through the tears. These intrusive magma fluxes have the ability to generate geothermal fields and rhyolitic lavas from crustal melts. This could explain the geothermal activity along the Coromandel peninsula five to seven million years ago, the ignimbrite outcrops about Lake Taupo and the current geothermal and volcanic activity stretching from Taupo to Rotorua.

Mathematical modelling of Wairakei geothermal field is reviewed, both lumped-parameter and distributed-parameter models. In both cases it is found that reliable predictions require five to ten years of history for calibration. With such calibration distributed-parameter models are now used for field management. A prudent model of Wairakei, constructed without such historical data, would underestimate field capacity and provide only general projections of the type of changes in surface activity and subsidence.

There are advantages in viewing orthogonal functions as functions generated by a random variable from a basis set of functions. Let Y be a random variable distributed uniformly on [0,1]. We give two ways of generating the Zernike radial polynomials with parameter l, {Zll+2n(x), n≥0}. The first is using the standard basis {xn,n≥0} and the random variable Y1/(l+1). The second is using the nonstandard basis {xl+2n,n≥0} and the random variable Y1/2. Zernike polynomials are important in the removal of lens aberrations, in characterizing video images with a small number of numbers, and in automatic aircraft identification.