In two previous papers (1, 2) representations of the unitary groups U4, U2 were found which described some of the properties of nucleons and electrons. In particular, the many electron wave functions were constructed from the irreducible representations of U2 restricted to the proper orthochronous Lorentz group Lp. In this paper the irreducible representations of U4 found in (1) will be shown to be also irreducible representations of the complete homogeneous Lorentz group L0 and the techniques of matrix contraction employed in (2) will be used to find the precise form of the matrices of the infinitesimal ring.

There is developed the theory of the propagation of waves through a system of potential barriers. Many properties are derived and discussed, mainly by the use of matrix theory arising from the description of each barrier by a transmission matrix. The extent to which certain previously used approximate methods are applicable and valid is also examined. Two distinct kinds of resonant phenomena are discussed – first when there is no incident wave and secondly when there is no reflected wave. This latter case of perfect transmission is investigated in detail, an examination being given of both why and when it occurs and the time taken to establish such exceptional fields within the barriers.

The original approach of Blasius to the solution of the differential equation now associated with his name was to develop the unknown function as a power series. Unfortunately, this series has a limited radius of convergence, so that such a representation is not valid over the whole range of interest. It is shown here that, if we work instead with a particular inverse function, this can be expanded as a power series which converges for all relevant values of the independent variable. Moreover, the number associated with the solution which is of principal physical interest can be expressed in terms of the asymptotic properties of the coefficients of this series. Exploiting this relationship, we find upper and lower bounds for this number in terms of the zeros of two particular families of polynomials.

In this paper we study a mixture of a viscous fluid and an elastic solid using a global Clausius–Duhem entropy inequality. Results are similar in character to those obtained by previous writers using a more restricted type of entropy inequality. Special attention is given to the case of a mixture of a Newtonian fluid and an elastic solid.

K. Szymiczek, Grothendieck groups of quadratic forms and G-equivalence of fields

(Proc. Cambridge Phil. Soc.73 (1973), 29–36

Theorem 2 of the paper has been proved under stronger hypotheses than stated explicitly. Denote by g2(k) the subset g(k) = k*/k*2 represented by the form (1, 1) over k and by U2(k) the set of equivalence classes of 2-dimensional universal quadratic forms over k.