By Kupiainen M., Sjogreen B.
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Additional info for A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations
T 1/ . 7) (this is the second property). 1/ D Hence, i . N / . N / . N / . 1/ . N / .
Sufficient conditions for a connected natural state space are given in Lemma 12. Connectedness and reachability of the natural state space may also be properties of interest in modeling. We gave conditions in  for the map from system to natural state space to be well defined. It is also pointed out that various differentiability properties of the natural state and natural state trajectory are inherited from the input–output system. These results are differentiability upgrades to the continuity results presented in .
That is, the state vectors are of minimum dimension (or n is minimized). In  a differential equation representation of minimal dimension is determined for a system that is already in state variable or differential form but not necessarily minimal. , it is not necessarily linear, locally Euclidean, or a manifold). As mentioned in the introduction, the natural state space has the essential state properties. The natural state space is minimal as set. This is different than minimizing the dimension of the state vector.
A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations by Kupiainen M., Sjogreen B.