(* ::Package:: *)

(************************************************************************)\012(* This file was generated automatically by the Mathematica front end.  *)\012(* It contains Initialization cells from a Notebook file, which         *)\012(* typically will have the same name as this file except ending in      *)\012(* ".nb" instead of ".m".                                               *)\012(*                                                                      *)\012(* This file is intended to be loaded into the Mathematica kernel using *)\012(* the package loading commands Get or Needs.  Doing so is equivalent   *)\012(* to using the Evaluate Initialization Cells menu command in the front *)\012(* end.                                                                 *)\012(*                                                                      *)\012(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)\012(* automatically each time the parent Notebook file is saved in the     *)\012(* Mathematica front end.  Any changes you make to this file will be    *)\012(* overwritten.                                                         *)\012(************************************************************************)\012\012

(* ::Input::Initialization:: *)
SetDirectory["C:\\drorbn\\AcademicPensieve\\Talks\\Banff-2607"];
Once[<<KnotTheory`];


(* ::Input::Initialization:: *)
$k=1;


(* ::Input::Initialization:: *)
CCF[\[ScriptCapitalE]_] := ExpandDenominator@ExpandNumerator@Together[\[ScriptCapitalE]]; (*CoefficientCanonical Form *)
CF[\[ScriptCapitalE]_List]:=CF/@\[ScriptCapitalE];
CF[\[ScriptCapitalE]_\[Epsilon]Series]:=CF/@\[ScriptCapitalE];
CF[\[ScriptCapitalE]_] :=Module[{F},Expand@Collect[\[ScriptCapitalE],Subscript[(p|x|\[Pi]|\[Xi]), __],F]/.F->CCF];
(*CF[\[ScriptCapitalE]_] := Module[
{vs=Cases[\[ScriptCapitalE],Subscript[(p|x|\[Pi]|\[Xi]), __],\[Infinity]]},
Total[(CCF[#\[LeftDoubleBracket]2\[RightDoubleBracket]](Times@@(vs^#\[LeftDoubleBracket]1\[RightDoubleBracket])))& /@ CoefficientRules[Expand[\[ScriptCapitalE]], vs]]
];*)
CF[\[ScriptCapitalE]_\[DoubleStruckCapitalE]]:=CF/@\[ScriptCapitalE]; 
CF[Subscript[\[DoubleStruckCapitalE], sp___][\[ScriptCapitalE]s___]]:=CF/@Subscript[\[DoubleStruckCapitalE], sp][\[ScriptCapitalE]s];


(* ::Input::Initialization:: *)
\[Epsilon]Series/: S1_\[Epsilon]Series\[Congruent]S2_\[Epsilon]Series:=Length[S1]==Length[S2]\[And]Inner[CF[#1]==CF[#2]&,S1,S2,And];
\[Epsilon]Series[]:=\[Epsilon]Series@@Table[0,$k+1];
\[Epsilon]Series/: Plus[ss___\[Epsilon]Series]:=Module[{l=Min[Length/@{ss}]},
\[Epsilon]Series@@Total[Take[List@@#,l]&/@{ss}]
];
(*\[Epsilon]Series/: S1_\[Epsilon]Series+S2_\[Epsilon]Series := \[Epsilon]Series@@Table[S1\[LeftDoubleBracket]k\[RightDoubleBracket]+S2\[LeftDoubleBracket]k\[RightDoubleBracket],{k,Min[Length@S1,Length@S2]}];*)
\[Epsilon]Series/: S1_\[Epsilon]Series*S2_\[Epsilon]Series := \[Epsilon]Series@@Table[Sum[S1[[j+1]]*S2[[k-j+1]], {j,0,k}],{k,0,Min[Length@S1,Length@S2]-1}];
\[Epsilon]Series /:c_*S_\[Epsilon]Series:=(c #)& /@ S;
\[Epsilon]Series/:\!\(
\*SubscriptBox[\(\[PartialD]\), \(vs___\)]S_\[Epsilon]Series\):=(s|->\!\(
\*SubscriptBox[\(\[PartialD]\), \(vs\)]s\))/@S;


(* ::Input::Initialization:: *)
{SuperStar[p],SuperStar[x],SuperStar[\[Pi]],SuperStar[\[Xi]]}={\[Pi],\[Xi],p,x}; 
SuperStar[(vs_List)]:=(v|->SuperStar[v])/@vs;
SuperStar[(Subscript[u_, \[Nu]i__])]:=Subscript[(SuperStar[u]), \[Nu]i];


(* ::Input::Initialization:: *)
\[DoubleStruckCapitalE]/:\[DoubleStruckCapitalE][\[Omega]1_,Q1_,P1_]\[Congruent]\[DoubleStruckCapitalE][\[Omega]2_,Q2_,P2_]:=CF[\[Omega]1==\[Omega]2]\[And]CF[Q1==Q2]\[And](P1\[Congruent]P2);
\[DoubleStruckCapitalE]/:\[DoubleStruckCapitalE][\[Omega]1_,Q1_,P1_]\[DoubleStruckCapitalE][\[Omega]2_,Q2_,P2_]:=\[DoubleStruckCapitalE][\[Omega]1 \[Omega]2,Q1+Q2,P1+P2];
Subscript[\[DoubleStruckCapitalE], d1_->r1_][\[ScriptCapitalE]1s___]\[Congruent]Subscript[\[DoubleStruckCapitalE], d2_->r2_][\[ScriptCapitalE]2s___] ^:= (d1==d2)\[And](r1==r2)\[And](\[DoubleStruckCapitalE][\[ScriptCapitalE]1s]\[Congruent]\[DoubleStruckCapitalE][\[ScriptCapitalE]2s]);
Subscript[\[DoubleStruckCapitalE], d1_->r1_][\[ScriptCapitalE]1s___]Subscript[\[DoubleStruckCapitalE], d2_->r2_][\[ScriptCapitalE]2s___] ^:= Subscript[\[DoubleStruckCapitalE], (d1\[Union]d2)->(r1\[Union]r2)]@@(\[DoubleStruckCapitalE][\[ScriptCapitalE]1s]\[DoubleStruckCapitalE][\[ScriptCapitalE]2s]);
Subscript[Subscript[\[DoubleStruckCapitalE], dr_][\[ScriptCapitalE]s___], $k_]:=Subscript[\[DoubleStruckCapitalE], dr]@@Subscript[\[DoubleStruckCapitalE][\[ScriptCapitalE]s], $k];


(* ::Input::Initialization:: *)
Subscript[\[DoubleStruckCapitalE], d1_->r1_][\[ScriptCapitalE]1s___]//Subscript[\[DoubleStruckCapitalE], d2_->r2_][\[ScriptCapitalE]2s___] :=Module[{is=r1\[Intersection]d2,lvs},
lvs=Flatten@Table[{Subscript[x, \[Nu],$@i],Subscript[p, \[Nu],$@i]},{\[Nu],3},{i,is}];
Subscript[\[DoubleStruckCapitalE], (d1\[Union]Complement[d2,is])->(r2\[Union]Complement[r1,is])]@@(Subscript[Zip, lvs\[Union]SuperStar[lvs]][SuperStar[lvs] . lvs,Times[
\[DoubleStruckCapitalE][\[ScriptCapitalE]1s]/.Table[Subscript[(u:x|p), \[Nu]_,i]->Subscript[u, \[Nu],$@i],{i,is}],
\[DoubleStruckCapitalE][\[ScriptCapitalE]2s]/.Table[Subscript[(u:\[Xi]|\[Pi]), \[Nu]_,i]->Subscript[u, \[Nu],$@i],{i,is}]
]])
]


(* ::Input::Initialization:: *)
Subscript[Zip, vs_][\[ScriptCapitalF]_,\[ScriptCapitalE]_]:=((*EchoLabel["EZip called with {vs,\[ScriptCapitalF],\[ScriptCapitalE]} = "][{vs,\[ScriptCapitalF],\[ScriptCapitalE]}];*) \[LeftAngleBracket]\[ScriptCapitalF],\[ScriptCapitalE]\[RightAngleBracket]//Subscript[Zip1, vs]//Subscript[EZip23, vs]);
Subscript[Zip, vs_][\[ScriptCapitalF]_,\[ScriptCapitalE]_]:=((*EchoLabel["EZip called with {vs,\[ScriptCapitalF],\[ScriptCapitalE]} = "][{vs,\[ScriptCapitalF],\[ScriptCapitalE]}];*)\[LeftAngleBracket]\[ScriptCapitalF],\[ScriptCapitalE]\[RightAngleBracket]//Subscript[Zip1, vs]//Subscript[Zip2, vs]//Subscript[Zip3, vs]);


(* ::Input::Initialization:: *)
\!\(\*SubscriptBox[\(Zip1\), \({}\)]\)=Identity;
Subscript[Zip1, vs_]@\[LeftAngleBracket]\[ScriptCapitalF]_,\[DoubleStruckCapitalE][\[Omega]_,Q_,P_]\[RightAngleBracket] := Module[{\[ScriptCapitalI],F,G,u,v},
(*EchoLabel["EZip1 called with {vs,\[ScriptCapitalF],\[Omega],Q,P} = "][{vs,\[ScriptCapitalF],\[Omega],Q,P}];*)
\[ScriptCapitalI]=IdentityMatrix@Length@vs;
F=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(u, v\)]\[ScriptCapitalF]\),{u,SuperStar[vs]},{v,SuperStar[vs]}];
G=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(u, v\)]Q\),{u,vs},{v,vs}];
\[LeftAngleBracket]CF[SuperStar[vs] . F . Inverse[\[ScriptCapitalI]-G . F] . SuperStar[vs]/2], \[DoubleStruckCapitalE][CF@PowerExpand@Factor[\[Omega]  Det[\[ScriptCapitalI]- G . F]^(-1/2)],CF[Q-vs . G . vs/2],P]\[RightAngleBracket]
]


(* ::Input::Initialization:: *)
\!\(\*SubscriptBox[\(Zip2\), \({}\)]\)=Identity;
Subscript[Zip2, vs_]@\[LeftAngleBracket]\[ScriptCapitalF]_,\[DoubleStruckCapitalE][\[Omega]_,Q_,P_]\[RightAngleBracket] := Module[{F,Y,u,v},
(*EchoLabel["EZip2 called with {vs,\[ScriptCapitalF],\[Omega],Q,P} = "][{vs,\[ScriptCapitalF],\[Omega],Q,P}];*)
F=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(u, v\)]\[ScriptCapitalF]\),{u,SuperStar[vs]},{v,SuperStar[vs]}];
Y=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(v\)]Q\),{v,vs}];
CF/@\[LeftAngleBracket]\[ScriptCapitalF], \[DoubleStruckCapitalE][\[Omega],Q-Y . vs+Y . F . Y/2, P/.Thread[vs->vs+F . Y]]\[RightAngleBracket]
]


(* ::Input::Initialization:: *)
Subscript[Zip3, vs_]@\[LeftAngleBracket]\[ScriptCapitalF]_,\[DoubleStruckCapitalE][\[Omega]_,Q_,P_]\[RightAngleBracket] := Module[{Z,u,v,m,j},
(*EchoLabel["EZip3 called with {vs,\[ScriptCapitalF],\[Omega],Q,P} = "][{vs,\[ScriptCapitalF],\[Omega],Q,P}];*)
Z[0]=P;
For[m=0,m<2$k,++m,
Z[m+1]=CF[1/(2(m+1)) Sum[\!\(
\*SubscriptBox[\(\[PartialD]\), \(
\*SuperscriptBox[\(u\), \(*\)], 
\*SuperscriptBox[\(v\), \(*\)]\)]\[ScriptCapitalF]\)(\!\(
\*SubscriptBox[\(\[PartialD]\), \(u, v\)]\(Z[m]\)\)+Sum[(\!\(
\*SubscriptBox[\(\[PartialD]\), \(u\)]\(Z[j]\)\))(\!\(
\*SubscriptBox[\(\[PartialD]\), \(v\)]\(Z[m - j]\)\)),{j,0,m}]),{u,vs},{v,vs}]]
];
\[DoubleStruckCapitalE][\[Omega],Q,CF[Sum[Z[m],{m,0,2$k}]/.Table[v->0,{v,vs}]]]
]


(* ::Input::Initialization:: *)
Subscript[EZip23, vs_]@\[LeftAngleBracket]\[ScriptCapitalF]_,\[DoubleStruckCapitalE][\[Omega]_,Q_,P_]\[RightAngleBracket] := Module[
{nP,n\[ScriptCapitalF],nQ,j=0,ps,c,t,rr={(*release rules*)}},
nP=Total[
CoefficientRules[#, vs] /. (ps_->c_) :> (AppendTo[rr,t[++j]->CF@c]; t[j](Times@@(vs^ps)))
]& /@ P;
nQ=Total[CoefficientRules[Q, vs] /. (ps_->c_) :> (AppendTo[rr,t[++j]->CF@c]; t[j](Times@@(vs^ps)))];
n\[ScriptCapitalF]=Total[CoefficientRules[\[ScriptCapitalF], SuperStar[vs]] /. (ps_->c_) :> (AppendTo[rr,t[++j]->CF@c]; t[j](Times@@((SuperStar[vs])^ps)))];
CF[Expand[\[LeftAngleBracket]n\[ScriptCapitalF],\[DoubleStruckCapitalE][\[Omega],nQ,nP]\[RightAngleBracket]//Subscript[Zip2, vs]//Subscript[Zip3, vs]]/.rr]
]


(* ::Input::Initialization:: *)
Subscript[m, i_,j_->k_]:=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({i, j} \[Rule] {k}\)]\)[1,\!\(
\*SubsuperscriptBox[\(\[Sum]\), \(\[Nu] = 1\), \(3\)]\((\(-
\*SubscriptBox[\(\[Xi]\), \(\[Nu], i\)]\) 
\*SubscriptBox[\(\[Pi]\), \(\[Nu], j\)] + \((
\*SubscriptBox[\(\[Pi]\), \(\[Nu], i\)] + 
\*SubscriptBox[\(\[Pi]\), \(\[Nu], j\)])\) 
\*SubscriptBox[\(p\), \(\[Nu], k\)] + \((
\*SubscriptBox[\(\[Xi]\), \(\[Nu], i\)] + 
\*SubscriptBox[\(\[Xi]\), \(\[Nu], j\)])\) 
\*SubscriptBox[\(x\), \(\[Nu], k\)])\)\),\[Epsilon]Series[]]


(* ::Input::Initialization:: *)
Subscript[Kink, i_]:=Subscript[CC, 3] Subscript[R, 1,2]//Subscript[m, 2,3->2]//Subscript[m, 2,1->i];
Subscript[\!\(\*OverscriptBox[\(Kink\), \(_\)]\), i_]:=Subscript[CC, 3] Subscript[\!\(\*OverscriptBox[\(R\), \(_\)]\), 1,2]//Subscript[m, 1,3->1]//Subscript[m, 1,2->i]


(* ::Input::Initialization:: *)
RVK[pd_PD]:=Module[{n,xs,x,rots,front={0},k},
n=Length@pd;rots=Table[0,{2n}];
xs=Cases[pd,x_X:>\[Piecewise]{
 {Xp[x[[4]],x[[1]]], PositiveQ@x},
 {Xm[x[[2]],x[[1]]], True}
}];
For[k=0,k<2n,++k,If[k==0\[Or]FreeQ[front,-k],
front=Flatten[front/.k->(xs/.{
Xp[k+1,l_]|Xm[l_,k+1]:>{l,k+1,1-l},
Xp[l_,k+1]|Xm[k+1,l_]:>(++rots[[l]];{1-l,k+1,l})
})],
Cases[front,k|-k]/.{k,-k}:>--rots[[k+1]];
]];
RVK[xs,rots] ];
RVK[K_]:=RVK[PD[K]];


(* ::Input::Initialization:: *)
rot[i_,0]:=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({} \[Rule] {i}\)]\)[1,0,\[Epsilon]Series[]];
rot[i_,n_]:=Module[{j},If[n>0,rot[i,n-1]Subscript[CC, j],rot[i,n+1]Subscript[\!\(\*OverscriptBox[\(CC\), \(_\)]\), j]]//Subscript[m, i,j->i]];


(* ::Input::Initialization:: *)
Z[K_] := Z[RVK@K];
Z[rvk_RVK] := (*Z[rvk] =*)Module[{todo,n,rots,\[Zeta],done,st,cx, \[Zeta]1, i,j,k,k1,k2,k3},
{todo,rots}=List@@rvk;
AppendTo[rots,0];
n=Length[todo];
\[Zeta]=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({} \[Rule] {0}\)]\)[1,0,\[Epsilon]Series[]];
done={0};
st=Range[0,2n+1];
While[{}=!=($M=todo),
{cx}=MaximalBy[todo, Length[done\[Intersection]{#[[1]], #[[2]], #[[1]]-1,#[[2]]-1}]&,1];
{i,j}=List@@cx;
\[Zeta]1=Switch[Head[cx],
 Xp, (Subscript[R, i,j] Subscript[\!\(\*OverscriptBox[\(Kink\), \(_\)]\), k])//Subscript[m, j,k->j],
 Xm, (Subscript[\!\(\*OverscriptBox[\(R\), \(_\)]\), i,j] Subscript[Kink, k])//Subscript[m, j,k->j]
];
\[Zeta]1=(rot[k,rots[[i]]]\[Zeta]1)//Subscript[m, k,i->i]; rots[[i]]=0;
\[Zeta]1=(\[Zeta]1 rot[k, rots[[i+1]]])//Subscript[m, i,k->i]; rots[[i+1]]=0;
\[Zeta]1=(rot[k,rots[[j]]]\[Zeta]1)//Subscript[m, k,j->j]; rots[[j]]=0;
\[Zeta]1=(\[Zeta]1 rot[k, rots[[j+1]]])//Subscript[m, j,k->j]; rots[[j+1]]=0;
\[Zeta]*=\[Zeta]1;
If[MemberQ[done, i], \[Zeta]=\[Zeta]//Subscript[m, i,i+1->i]; st=st/.st[[i+2]]->st[[i+1]]];
If[MemberQ[done, i-1],  \[Zeta]=\[Zeta]//Subscript[m, st[[i]],i->st[[i]]]; st=st/.st[[i+1]]->st[[i]]]; 
If[MemberQ[done, j],  \[Zeta]=\[Zeta]//Subscript[m, j,j+1->j]; st=st/.st[[j+2]]->st[[j+1]]]; 
If[MemberQ[done, j-1],  \[Zeta]=\[Zeta]//Subscript[m, st[[j]],j->st[[j]]]; st=st/.st[[j+1]]->st[[j]]];
done=done\[Union]{i-1,i,j-1,j};
todo=DeleteCases[todo, cx]
];
CF/@ (\[Zeta] (*/. {Subscript[x, 0]\[Rule]x,Subscript[y, 0]\[Rule]y,Subscript[a, 0]\[Rule]a}*))
]


(* ::Input::Initialization:: *)
Subscript[R, i_,j_]:=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({} \[Rule] {i, j}\)]\)[1,(-1+Subscript[T, 1]) Subscript[x, 1,j] (Subscript[p, 1,i]-Subscript[p, 1,j]),\[Epsilon]Series[0,1/2 (-1+Subscript[T, 1]) \!\(\*SubsuperscriptBox[\(x\), \(1, j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(p\), \(1, i\), \(2\)]\)+Subscript[x, 1,i] Subscript[x, 1,j] Subscript[p, 1,i] Subscript[p, 1,j]+1/2 (1-3 Subscript[T, 1]) \!\(\*SubsuperscriptBox[\(x\), \(1, j\), \(2\)]\) Subscript[p, 1,i] Subscript[p, 1,j]]];
Subscript[\!\(\*OverscriptBox[\(R\), \(_\)]\), i_,j_]:=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({} \[Rule] {i, j}\)]\)[1,(-1+1/Subscript[T, 1]) Subscript[x, 1,j] (Subscript[p, 1,i]-Subscript[p, 1,j]),\[Epsilon]Series[0,-(((-1+Subscript[T, 1]) Subscript[x, 1,i] Subscript[x, 1,j] \!\(\*SubsuperscriptBox[\(p\), \(1, i\), \(2\)]\))/\!\(\*SubsuperscriptBox[\(T\), \(1\), \(2\)]\))-((1-Subscript[T, 1]) \!\(\*SubsuperscriptBox[\(x\), \(1, j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(p\), \(1, i\), \(2\)]\))/(2 \!\(\*SubsuperscriptBox[\(T\), \(1\), \(3\)]\))-(Subscript[x, 1,i] Subscript[x, 1,j] Subscript[p, 1,i] Subscript[p, 1,j])/\!\(\*SubsuperscriptBox[\(T\), \(1\), \(2\)]\)-((-1-Subscript[T, 1]) \!\(\*SubsuperscriptBox[\(x\), \(1, j\), \(2\)]\) Subscript[p, 1,i] Subscript[p, 1,j])/(2 \!\(\*SubsuperscriptBox[\(T\), \(1\), \(3\)]\))]];
Subscript[CC, i_]:=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({} \[Rule] {i}\)]\)[Sqrt[Subscript[T, 1]],0,\[Epsilon]Series[0,-((Subscript[x, 1,i] Subscript[p, 1,i])/Subscript[T, 1])]];
Subscript[\!\(\*OverscriptBox[\(CC\), \(_\)]\), i_]:=\!\(\*SubscriptBox[\(\[DoubleStruckCapitalE]\), \({} \[Rule] {i}\)]\)[1/Sqrt[Subscript[T, 1]],0,\[Epsilon]Series[0,(Subscript[x, 1,i] Subscript[p, 1,i])/Subscript[T, 1]]];


(* ::Input::Initialization:: *)
\[ScriptCapitalL][Subscript[X, i_,j_][s_]]:=\!\(
\*SubsuperscriptBox[\(T\), \(3\), \(s\)] \(\[DoubleStruckCapitalE][CF@{\[IndentingNewLine]
\*SubsuperscriptBox[\(\[Sum]\), \(\[Nu] = 1\), \(3\)]\((
\*SubscriptBox[\(x\), \(\[Nu]\[InvisibleComma]i\)] \((
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]
\*SuperscriptBox[\(i\), \(+\)]\)] - 
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]i\)])\) + 
\*SubscriptBox[\(x\), \(\[Nu]\[InvisibleComma]j\)] \((
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]
\*SuperscriptBox[\(j\), \(+\)]\)] - 
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]j\)])\) + \((
\*SubsuperscriptBox[\(T\), \(\[Nu]\), \(s\)] - 1)\) 
\*SubscriptBox[\(x\), \(\[Nu]\[InvisibleComma]i\)] \((
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]
\*SuperscriptBox[\(i\), \(+\)]\)] - 
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]
\*SuperscriptBox[\(j\), \(+\)]\)])\))\), \[IndentingNewLine]\((
\*SubsuperscriptBox[\(T\), \(1\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(1\[InvisibleComma]i\)] \((
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] - 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]j\)])\), \[IndentingNewLine]\[Epsilon]\ s \((
\*SubsuperscriptBox[\(T\), \(3\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(1\[InvisibleComma]j\)] \((
\*SubscriptBox[\(p\), \(2\[InvisibleComma]i\)] - 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)])\) 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)]/\((
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] - 1)\), \[IndentingNewLine]\[Epsilon]\ s \((1/2 + 
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] 
\*SubscriptBox[\(p\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] - 
\*SubscriptBox[\(p\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]j\)] - 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] - \((
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] + \((
\*SubsuperscriptBox[\(T\), \(3\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] + 2\ 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] + 
\*SubscriptBox[\(p\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]j\)] - 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]i\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]j\)] - 
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]j\)] + \((\((
\*SubsuperscriptBox[\(T\), \(1\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(1\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(1\[InvisibleComma]i\)] \((
\*SubsuperscriptBox[\(T\), \(2\), \(2  s\)] 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]i\)] - 
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]j\)] - \((
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] + 1)\) \((
\*SubsuperscriptBox[\(T\), \(3\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] + 
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]j\)])\) + \((
\*SubsuperscriptBox[\(T\), \(3\), \(s\)] - 1)\) 
\*SubscriptBox[\(p\), \(3\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] \((1 - 
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] 
\*SubscriptBox[\(p\), \(1\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(1\[InvisibleComma]i\)] + 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]j\)] + \((
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] - 2)\) 
\*SubscriptBox[\(p\), \(2\[InvisibleComma]j\)] 
\*SubscriptBox[\(x\), \(2\[InvisibleComma]j\)])\))\)/\((
\*SubsuperscriptBox[\(T\), \(2\), \(s\)] - 1)\))\)}]\)\)


(* ::Input::Initialization:: *)
\[ScriptCapitalL][Subscript[C, i_][\[CurlyPhi]_]]:=\!\(
\*SubsuperscriptBox[\(T\), \(3\), \(\[CurlyPhi]\)] \(\[DoubleStruckCapitalE][
\*SubsuperscriptBox[\(\[Sum]\), \(\[Nu] = 1\), \(3\)]
\*SubscriptBox[\(x\), \(\[Nu]\[InvisibleComma]i\)] \((
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]
\*SuperscriptBox[\(i\), \(+\)]\)] - 
\*SubscriptBox[\(p\), \(\[Nu]\[InvisibleComma]i\)])\) + \[Epsilon]\ \[CurlyPhi] \((
\*SubscriptBox[\(p\), \(3\[InvisibleComma]i\)] 
\*SubscriptBox[\(x\), \(3\[InvisibleComma]i\)] - 1/2)\)]\)\)



