ON TWO-DIMENSIONAL MANIFOLDS WITH CONSTANT GAUSSIAN CURVATURE AND THEIR ASSOCIATED EQUATIONS
The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Backlund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the equations. As a consequence of studying the prolongation structure of each equation, a Lax pair and Backlund transformation are obtained.
The asymptotic properties of the machine interference model were described in this paper. When the properties of the system were appropriately normalized and represented in terms of the variable w (=nu/v), the size dependencies of these properties were of order 1/n in regions of the most interest, that is, w < 1. Furthermore, the model displayed a functionally different behavior in the region w ≥ 1. The region around w = 1 was described as an unstable region in which an equivalent simulation model could be expected to stabilize slowly and probably give results different from the mathematical model. The asymptotic formulas are simple in form and quite accurate, and they allow the performance of queuing analysis with very little effort. In the last section of the paper, the solution and the properties of the generalized machine interference model were given.