Therefore a pattern is not an isolated solution but rather generates a manifold of solutions under the action of translations, which is called an orbit under the action of the group of translations.
The problem exhibits manifold optimal solutions (convexity), some variables have a range of non allowed values (solutions space not simply connected) and others are integers.
If a Gauß Newton iteration is used to solve a system of equations that has a manifold of solutions, then the iteration does not produce the minimal norm solution.
In particular, we will describe the manifold that solutions will be following on their way to ∞.
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.
Theorem 3.2 Let, X, Y be Banach spaces and F : X × R → Y a C 2 -map such that F ( x, 0 ) = 0 has a C 2, d-dimensional, manifold of solutions M = { x = ξ ∣ μ ∈ R d }.
First, we assume that T − ′ ( 0 ) ≠ 0. Then a C 2, ( n − 2 ) -dimensional submanifold S of (an open neighborhood of α = 0 in) R n − 1 exists such that T − = T − 0 for any α ∈ S. So, for ε = 0, F ( T +, T −, τ, α, 0 ) = 0 has the ( n − 1 ) -dimensional manifold of solutions ( T +, T −, τ, α ) = ξ : = ( T + , T − 0, τ, α ), ∈ S × R. So, we are in position to apply Theorem 3.2.
Variables such as sonication time, nature and concentration of the acid solution used as leaching solution, leaching temperature, flow-rate of the continuous manifold, leaching solution volume and hair particle size were simultaneously studied by applying a Plackett Burman design approach.
While the MTrL requires the linearized solution manifold to transversally intersect the nonlinear solution manifold at a chosen set of points across the axis of the independent variable, the essential difference of the present MTrL method from its previous version [Roy, D., Kumar, R., 2005. A multistep transversal linearization (MTL) method in nonlinear structural dynamics. J. Sound Vib.
If one is interested in the combinatorial manifold of potential solutions, one would have to enumerate all possible exclusive choices in advance.
