Pensieve Header: Wheeled Semi-Symmetrized calculus in the 2D quotient: Solving for the “conj” coefficients, III.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\w-Computations"];
<< "Wheeled Semi-Symmetrized 2D Calculus.m"

conj2-3 are designed to work only in the feedback case.

conj2[y_,x_][μ_] := Module[
{γ, ξc},
ξc = D[μ, h[x]] /. t[s_] :> c[s];
γ = Coefficient[μ, ar[y,x]];
μCollect[μ + W[γ, ξc]]
];

μ3 =(Subscript[α, 1]) ar[1,1] + (Subscript[α, 3]) ar[1,2] + (Subscript[α, 4]) ar[1,3];
μ3 =(Subscript[α, 1]) ar[1,1] + (Subscript[α, 2]) ar[2,1] + (Subscript[α, 3]) ar[1,2] + (Subscript[α, 4]) ar[1,3];
Riffle[
ComposeList[
ops={conj2[1,3], conj2[1,2], hm[2,3,2]},
μ3 =(Subscript[α, 1]) ar[1,1] + 0(Subscript[α, 2]) ar[2,1] + (Subscript[α, 3]) ar[1,2] + (Subscript[α, 4]) ar[1,3]
]// μForm,
ops
]

Riffle[
ComposeList[
ops={hm[2,3,2], conj2[1,2]},
μ3 
]// μForm,
ops
]

{c2l = μ3 // conj2[1,2] // conj2[1,3] // hm[2,3,2], c2r = μ3 //  hm[2,3,2] // conj2[1,2]}

c2l-c2r // FullSimplify

sols=SolveAlways[c2l==c2r, {t[1], t[2], h[1], h[2], (Subscript[α, 1]), (Subscript[α, 2]), (Subscript[α, 3]), (Subscript[α, 4]), c[1], W}]

W(a1+a2 c[y])γ/(1+(a3+a4 c[y])γ)

(W(a1+a2 c[y])γ/(1+(a3+a4 c[y]γ))) /. sols

conj3

conj3[y_,x_][μ_] := Module[
{γ, ξc},
γ = Coefficient[μ, ar[y,x]];
μCollect[μ + W Log[c[y]γ+1]]
];

μ3 =(Subscript[α, 1]) ar[1,1] + (Subscript[α, 3]) ar[1,2] + (Subscript[α, 4]) ar[1,3];
μ3 =(Subscript[α, 1]) ar[1,1] + (Subscript[α, 2]) ar[2,1] + (Subscript[α, 3]) ar[1,2] + (Subscript[α, 4]) ar[1,3];
Riffle[
ComposeList[
ops={conj3[1,3], conj3[1,2], hm[2,3,2]},
μ3 =(Subscript[α, 1]) ar[1,1] + 0(Subscript[α, 2]) ar[2,1] + (Subscript[α, 3]) ar[1,2] + (Subscript[α, 4]) ar[1,3]
]// μForm,
ops
]

Riffle[
ComposeList[
ops={hm[2,3,2], conj3[1,2]},
μ3 
]// μForm,
ops
]

Simplify[
(W (Log[1+c[1] (Subscript[α, 3])]+Log[1+c[1] (Subscript[α, 4])]) == W Log[(1+c[1] (Subscript[α, 3])) (1+c[1] (Subscript[α, 4]))]) /.
Log[a_] + Log[b_] :> Log[a b]
]