For classical electromagnetism, differential forms are natural objects to model field quantities on manifolds.
Start with a three-manifold Σ equipped with quantities which, when Σ is embedded as a spacelike submanifold of spacetime, become initial data for the vacuum field equations.
The actuators can affect the internal states (operational parameters) of diesel engine such as inlet manifold pressure, EGR rate, quantity and timing of pilot and main injection which in turn will influence the engine emissions and performance.
Recently, the authors Wang et al. [9] introduced an algorithm of computing the singular point quantities on center manifold, which misses the above tedious course.
In Section 2, we give some preliminaries about Liapunov constants, the focal values and singular point quantities on center manifold for a three-dimensional system (1).
Particularly, if for each 1 ≤ k ≤ m − 1, v2k+1(2π) = 0 holds, we can get v2(2π) = v4(2π) = ··· = v2m(2π) = 0. Here, we recall the definition of the singular point quantities on center manifold.
The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law.
PEC utilizes a Pareto efficiency, and introduces a manifold drift term as a measure for determining the adequacy of using a certain combustion-manifold model to predict selected quantities of interest.
We have seen in this approach we need to subtract the geometrical quantities belong to two different manifolds.
The explicit algebraic equivalent relation between singular point quantities and Liapunov constants on center manifold is rigorously proved.
