Journal of Physics A: Mathematical, Nuclear and General

With the aid of a single operator identity, the derivation of the Mori generalized linear Langevin equation is simplified and a new generalized nonlinear Langevin equation is obtained. The flexibility of the method is stressed which allows the derivation of various generalized nonlinear Langevin equations that can be used as bases for devising approximation schemes such as the mode coupling scheme.

Graphs associated with the Gel'fand-Yaglom equations for arbitrary spin are used in a simple method to find the number of conditions which a good quantizable free-field theory must satisfy. The number of conditions is seen to be simply related to the line independence numbers of the s block subgraphs of the theory. This provides a rough guide as to whether a particular theory is possibly quantizable. Some examples are given. Also a general result is given which enables quick elimination of a range of 'bad' theories by visual examination of their graphs.

A new generalization of Pade approximants to power series in two (or more) variables is given, along with an example of its application.

Estimates are presented for the magnetic specific heat of an infinite chain of spins with isotropic nearest-neighbour exchange. The calculations are performed for S=¹/₂, 1, ³/₂, 2 and ⁵/₂. The results are presented graphically. A simple expression is used to represent the results for ferromagnetic as well as for antiferromagnetic interaction.

Evidence is presented that the arrival direction distribution of cosmic rays with energies above 10¹⁹ eV is not isotropic. Harmonic analyses indicate that there is, prima facie, a first harmonic (amplitude=44%, chance probability=2.6%, phase (in right ascension)=13.3+or-1.5 h) at declinations above 0 degrees , and that there is a significant second harmonic (amplitude=101%, chance probability=0.2%, phase 7.8, 19.8+or-1 h) at declinations below -30 degrees . Supporting evidence from low energy data is also discussed.

Graphs associated with the Gel'fand-Yaglom equations for arbitrary spin are used in a simple method to find the number of conditions which a good quantizable free-field theory must satisfy. The number of conditions is seen to be simply related to the line independence numbers of the s block subgraphs of the theory. This provides a rough guide as to whether a particular theory is possibly quantizable. Some examples are given. Also a general result is given which enables quick elimination of a range of 'bad' theories by visual examination of their graphs.

By extensive use of methods developed by Rieckers it is shown that power series of the free field, while not strictly localizable in general, are essentially local with respect to all those Gel'fand spaces S^alpha on which they can be defined at all.