Implicit Solutions of PDE's

Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a single unknown $\phi$ are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables, whose coefficients are functions of $\phi$ is discussed, and it is shown that if the discriminant of the quadratic vanishes, then an implicit solution of the so-called Universal Field Equation is obtained. The relation to the general solution is discussed.Comment: 11 pages LaTeX2

Study of Quommutators of Quantum Variables and Generalized Derivatives

A general deformation of the Heisenberg algebra is introduced with two deformed operators instead of just one. This is generalised to many variables, and permits the simultaneous existence of coherent states, and the transposition of creation operators.Comment: 17 pages (Previous version was truncated in transmission

Moyal Brackets in M-Theory

The infinite limit of Matrix Theory in 4 and 10 dimensions is described in terms of Moyal Brackets. In those dimensions there exists a Bogomol'nyi bound to the Euclideanized version of these equations, which guarantees that solutions of the first order equations also solve the second order Matrix Theory equations. A general construction of such solutions in terms of a representation of the target space co-ordinates as non-local spinor bilinears, which are generalisations of the standard Wigner functions on phase space, is given.Comment: 10 pages, Latex, no figures. References altered, typos correcte

A Model for Classical Space-time Co-ordinates

Field equations with general covariance are interpreted as equations for a target space describing physical space time co-ordinates, in terms of an underlying base space with conformal invariance. These equations admit an infinite number of inequivalent Lagrangian descriptions. A model for reparametrisation invariant membranes is obtained by reversing the roles of base and target space variables in these considerations.Comment: 9 pages, Latex. This was the basis of a talk given at the Argonne National Laboratory 1996 Summer Institute : Topics on Non-Abelian Duality June 27-July 1