We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four classes of trees mentioned above.

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.

These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

The problem of rotating annular disk subjected to a uniformly distributed load is treated in two ways. Stress is divided into a rotating part because of the angular velocity and a bending part due to force loading. New set of equilibrium equations with small deflections is developed. Solutions for radial displacement, deflection, forces and moment resultants, and the rotating and bending stresses of the first-order theory are presented in terms of corresponding quantities of annular disks based on the classical theory. The boundary conditions at the edges of the annular disk are roller supported, clamped or free. Several examples are presented to illustrate the use and accuracy of these relationships. The effects of several parameters on the radial and vertical displacements and rotating and bending stresses are studied. It is observed that the classical theory is sufficient to study the problem of rotating annular disks. However, the inclusion of the effect of shear deformation is necessary to study precisely the curvature of moderately thick annular disks.