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Some informal background: a Riemannian manifold is a differentiable manifold (where the tangent space at each point has an inner product) with a positivedefinite metric tensor, d (x, y) ≥ 0.

The Lorentzian manifold is a pseudoRiemannian manifold, the generalization of the Riemannian manifold, such that the metric tensor need not be positivedefinite.

The thermal energy due to viscous dissipation appears as a positivedefinite contribution to the various forms of the thermal energy conservation equation.

The aerosolaerosol coagulation scheme is volume and volume concentration conserving, unconditionally stable, and positivedefinite for any time step.

The solution mechanism for simultaneous condensation/dissolution is unconditionally stable, exactly mass conserving, noniterative, and positivedefinite.

Secondary organic condensation is solved simultaneously between the gas phase and among any number of aerosol size bins in any number of size distributions with the same unconditionally stable, noniterative, massconserving, positivedefinite condensation scheme.

The model calculates atmosphereocean exchange for any gas with a noniterative, massconserving, unconditionallystable, positivedefinite scheme that accounts for current ocean composition and pH.

Functions obeying either (a) or (b) are known as positivedefinite functions.

V on which a positivedefinite inner product is defined.

5 Require that all modelers provide studies of the error propagation through their model, of how they handle overflow and underflow conditions, of the exact number of parameters used and their nature, and that they are positivedefinite and that they converge in all conditions.
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