Transients and Resonances in Game Theory and General Relativity

Gerald H. Thomas, MSOE

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Abstract

We explore behaviors in differential geometries, with potential applications to decision making (Decision Process Theory) and General Relativity. We find that there is a similarity between behaviors in these theories and behaviors of circuits in electrical engineering. We impose a forcing harmonic. For AC circuits there are a variety of steady state possibilities: transient behaviours, steady state harmonics and resonant harmonics. We suggest this same set of possibilities occurs in decision processes, though in circuits we expect to solve coupled linear equations whereas in more complex geometries, we solve coupled non-linear PDEs. Resonances occur in physical media such as musical instruments, bells and bridges. When such systems are driven with a harmonic force, they exhibit a large response to what may seem like a small driving force. Not surprisingly perhaps, this property is shared by non-linear theories based on differential geometry. In such systems, the harmonic driving force can by introduced through the initial conditions. We apply this to game theory and general relativity.

LRC Circuit

Transient Behaviors

Solution--No Sources

iInput=Quantity[4UnitStep[t],"amps"] 
r=Quantity[2,"ohm"]
c=Quantity[0.5,"Farad"]
L=Quantity[1,"Henry"]

e1=QuantityMagnitude[iInput]==QuantityMagnitude[UnitConvert[((Quantity[v[t],"volt"])/r)+c Quantity[v'[t],"volt"]/Quantity[1,"second"]+Quantity[i[t],"amp"],"amp"]]
e2=v[t]==QuantityMagnitude[UnitConvert[L Quantity[i'[t],"amp"]/Quantity[1,"second"],"volt"]]

sol=First[DSolve[{e1,e2,v[0]==0,i[0]==1},{v,i},{t,0,10}]]
iNum[t_]=Chop[ComplexExpand[Re[i[t]/.sol]]]
vNum[t_]=Chop[ComplexExpand[Re[v[t]/.sol]]]

Plot[{vNum[t],iNum[t]},{t,0,10},PlotStyle->{Blue,Red}]

α=UnitConvert[(1/(2 r c)),"Hertz"]
τ=UnitConvert[(1/α),"second"]
(Subscript[ω, 0])=UnitConvert[(1/(Sqrt[L c])),"Hertz"]
β=UnitConvert[(Sqrt[(α^2)-((Subscript[ω, 0])^2)]),"Hertz"]
f=Chop[((-I β)/(2Pi))]
T=UnitConvert[(1/f),"second"]

Plot[{vNum[t],iNum[t]},{t,0,10},PlotStyle->{Blue,Red}]

Sources--Harmonics with Transients

Plot[(6/26)Sin[2t]+(9/26)Cos[2t]+(17/26)Exp[-t]Cos[3 t]-(1/3)(47/26)Exp[- t]Sin[3 t],{t,0,10},PlotStyle->Thick]

Differential Geometry Flow Conservation

(Subscript[g, ab]) ((dV^b)/dt)=(Subscript[f, ab]) (V^b)+(Subscript[T, a])

No Sources--Resonant Harmonics

Sources--Transient and Forced Harmonics

Steady State Behaviors

Toy Model--Sine Gordon Equation

cycles=5
mesh = 16
ω=0
init=1.7
solutiong[z_]=First[NDSolve[{g''[z]+(ω^2)g[z]+ Sin[ g[z]]==init Sum[(( Sin[m z])/m),{m,1,mesh}],g'[-Pi]==g'[Pi],g[Pi]==g[-Pi]},{g[z],g'[z]},{z,-cycles Pi,cycles Pi},MaxSteps->∞]]
gNum[z_]=g[z]/.solutiong[z]
gNumPrime[z_]=g'[z]/.solutiong[z]
Plot[Sum[(( Sin[m z])/m),{m,1,mesh}],{z,- cycles Pi,cycles Pi},PlotStyle->Green,PlotLabel->"Driver"]
Plot[{gNum[z],gNumPrime[z]},{z,-cycles Pi,cycles Pi},PlotStyle->{Blue,Red},PlotLabel->"n cycles of NDSolve g[z] (blue) and g'[z] (red)"]
ParametricPlot[{gNum[z],gNumPrime[z]},{z,-cycles Pi,cycles Pi},AspectRatio->1,AxesLabel->{"g[z]","g'[z]"},PlotStyle->Purple]

ParametricPlot[{gNum[z],gNumPrime[z]},{z,-cycles Pi,cycles Pi},AspectRatio->1,AxesLabel->{"g[z]","g'[z]"},PlotStyle->Purple]

Elasticity (WTC2013)

The system is deterministic and occasionally chaotic: it is also elastic. Neighbor events impact each other and are not independent! We will investigate decision process theory that has all of these characteristics and roughly can be thought of as producing behavior of the following equations which are similar to the motion of charged particles in an electromagnetic field

Attack defense model--Decision Process Model

The general form of a dynamic game with two players is the following for player 1, where each player has two strategies, one internal self strategy and two valuations:

MatrixForm[payoff[n1_,n2_,n3_,n4_,s1_,s2_]={{0,s2,-n1,-n3},{-s2,0,-n2,-n4},{n1,n2,0,s1},{n3,n4,-s1,0}}]
payoffValue[v1_,v2_,v3_,v4_]={v3,v4,-v1,-v2}
valuePotential[y1_,y2_,y3_,y4_,v1_,v2_,v3_,v4_]=payoffValue[v1,v2,v3,v4].{y1,y2,y3,y4}
tmin=-300
tmax=300
e1z=(1/2);e2z=(3/5)
initialSpaceFlow[r_,u_,σ1_,σ2_]={((r-u+(Sqrt[2])σ2)/2),((r-u-(Sqrt[2])σ2)/2),((r+u+(Sqrt[2])σ1)/2),((r+u-(Sqrt[2])σ1)/2)}
initialSpaceFlow[1,0,-(1/(Sqrt[2])),-(1/(Sqrt[2]))]
maxInitialTimeFlow=20
initialTimeFlow=4
size[z_,L_]=(L/Pi)Sin[((Pi z)/L)]
ϕ[w_,tL_,α_,β_,γ_,δ_]=(1/2)γ (δ^2)  Log[(w^2)+(δ^2)]+β w Sin[α tL]
MatrixForm[gab=DiagonalMatrix[{-1,-1,-1,-1}]]
g00[w_,tL_,α_,β_,γ_,δ_]=Exp[2ϕ[w,tL,α,β,γ,δ]]
MatrixForm[inverseg=Inverse[gab]]
gradϕ[w_]=Simplify[D[ϕ[w,tL,α,β,γ,δ],w]]
gradtL[w_,tL_,α_,β_,γ_,δ_]=Simplify[D[ϕ[w,tL,α,β,γ,δ],tL]]
gradVector[w_,tL_,α_,β_,γ_,δ_]=gradϕ[w]{-1/2,-1/2,1/2,1/2}
gradForce[w_,tL_,α_,β_,γ_,δ_]=inverseg.gradVector[w,tL,α,β,γ,δ]g00[w,tL,α,β,γ,δ]

Add the characteristic payoff to the mix, and don’t forget to come back and add the value terms as a potential. But first the characteristic payoff:

characteristicPotential[y1_,y2_,y3_,y4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_]=((-1)/2)size[payoff[cn1,cn2,cn3,cn4,cs1,cs2].{y1,y2,y3,y4},αα]
tmp12=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[2]],y1]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[1]],y2]
tmp13=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[3]],y1]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[1]],y3]
tmp14=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[4]],y1]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[1]],y4]
tmp23=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[3]],y2]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[2]],y3]
tmp24=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[4]],y2]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[2]],y4]
tmp34=D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[4]],y3]-D[characteristicPotential[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2][[3]],y4]
MatrixForm[characteristicPayoff[y1_,y2_,y3_,y4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_]={{0,tmp12,tmp13,tmp14},{-tmp12,0,tmp23,tmp24},{-tmp13,-tmp23,0,tmp34},{-tmp14,-tmp24,-tmp34,0}}]

MatrixForm[payoff[n1,n2,n3,n4,s1,s2]]

We have also set some of the numerical parameters to be used later. We will solve the dynamic equations using parametric solution of the differential equations. As a starting values we choose the values for the prisoner’s dilemma.

inertiaPD=initialTimeFlow
MatrixForm[payoffPD1={{0,0,-(9/10),-1},{0,0,0,-(1/10)},{(9/10),0,0,0},{1,(1/10),0,0}}]
MatrixForm[payoffPD2={{0,0,(9/10),0},{0,0,1,(1/10)},{-(9/10),-1,0,0},{0,-(1/10),0,0}}]
MatrixForm[payoffTotal[n1_,n2_,n3_,n4_,s1_,s2_,y1_,y2_,y3_,y4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,e0_,e1_,e2_]=e1 payoffPD1+e2 payoffPD2+payoff[n1,n2,n3,n4,s1,s2]+e0  characteristicPayoff[y1,y2,y3,y4,cn1,cn2,cn3,cn4,cs1,cs2,αα]]
n1Z=0;n2Z=0;n3Z=0;n4Z=0;s1Z=0;s2Z=0;v1Z=0;v2Z=0;v3Z=0;v4Z=0;rz=1;uz=0;σ1z=-(1/(Sqrt[2]));σ2z=-(1/(Sqrt[2]));αz=1;βz=0;γz=1;δz=1;tLowz=tmin/6;tHighz=tmax/6;cn1Z=0;cn2Z=0;cn3Z=0;cn4Z=0;cs1Z=0;cs2Z=0;ααZ=1;ββZ=1;

valuePotentialPD1=valuePotential[y1,y2,y3,y4,(1/(10inertiaPD)),(1/(10inertiaPD)),(1/(10inertiaPD)),(1/(10inertiaPD))]
valuePotentialPD2=valuePotential[y1,y2,y3,y4,((-1)/(10inertiaPD)),((-1)/(10inertiaPD)),((-1)/(10inertiaPD)),((-1)/(10inertiaPD))]
(*valuePotentialdelta=valuePotential[size[y1,ββ],size[y2,ββ],size[y3,ββ],size[y4,ββ],v1,v2,v3,v4]*)
valuePotentialdelta=size[valuePotential[y1,y2,y3,y4,v1,v2,v3,v4],ββ]
valuePotentialTotal[y1_,y2_,y3_,y4_,v1_,v2_,v3_,v4_]=e1 valuePotentialPD1+e2 valuePotentialPD2+valuePotentialdelta
ValueFieldTotal[y1_,y2_,y3_,y4_,v1_,v2_,v3_,v4_,ββ_]={D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y1],D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y2],D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y3],D[valuePotentialTotal[y1,y2,y3,y4,v1,v2,v3,v4],y4]}

The model will be a dynamic game model with two sets of variables: the flows and the preferences. The payoff matrix and inertia term of the initial flow vector are part of the parameter set. The equation we will solve is the coupled set of equations for the flow vector (x^a)=(V^a): (Subscript[g, ab])((dV^b)/(dτ))=(Subscript[eF, ab])(V^b). Here (Subscript[g, ab]) is a diagonal matrix whose fifth component has the opposite sign of the other components (which are -1) and (Subscript[F, ab]) is the payoff matrix for player 1. All quantities are from that player’s perspective. The preferences are (y^a) and are determined from the flow: ((dy^a)/(dτ))=(V^a). The “time” parameter is the proper time and represents the distance along the path, τ. We start the preference at 0.

var={(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]),(Subscript[x, 4]),(Subscript[x, 5]),(Subscript[y, 1]),(Subscript[y, 2]),(Subscript[y, 3]),(Subscript[y, 4]),(Subscript[y, 5])}
flow[t_]={(Subscript[x, 1])[t],(Subscript[x, 2])[t],(Subscript[x, 3])[t],(Subscript[x, 4])[t]}
vBarupperZ[t_]=(((Subscript[x, 5])[t]-valuePotentialTotal[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],v1,v2,v3,v4])/(g00[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]));
vUpper[t_]=vBarupperZ[t]-characteristicPotential[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2].flow[t];
preference[t_]={(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t]}
parameterList={{n1,-2,2},{n2,-2,2},{n3,-2,2},{n4,-2,2},{s1,-2,2},{s2,-2,2},{v1,-2,2},{v2,-2,2},{v3,-2,2},{v4,-2,2},{cn1,-2,2},{cn2,-2,2},{cn3,-2,2},{cn4,-2,2},{cs1,-2,2},{cs2,-2,2},{αα,1/10,10},{ββ,1/10,10},{inertia,2,maxInitialTimeFlow},{e1,-2,2},{e2,-2,2},{r,-2,2},{u,-2,2},{σ1,-2,2},{σ2,-2,2},{α,0,2Pi},{β,-1,1},{γ,0,2},{δ,0.1,5},{tLow,tmin,0},{tHigh,0,tmax}}
force[t_]=Table[(inverseg.payoffTotal[n1,n2,n3,n4,s1,s2,(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2,αα,(Subscript[x, 5])[t],e1,e2].flow[t])[[k]],{k,4}];
gradTimeForce1[α_,β_,γ_,δ_,v1_,v2_,v3_,v4_]=gradForce[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]((vBarupperZ[t])^2)-inverseg.characteristicPotential[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2]gradtL[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]g00[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ]((vUpper[t])^2);
gradTimeForce2[α_,β_,γ_,δ_,v1_,v2_,v3_,v4_]=g00[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ](((vBarupperZ[t])^2)-(((flow[t].characteristicPotential[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],cn1,cn2,cn3,cn4,cs1,cs2]))^2))gradtL[(((Subscript[y, 3])[t]+(Subscript[y, 4])[t]-(Subscript[y, 1])[t]-(Subscript[y, 2])[t])/2),(Subscript[y, 5])[t],α,β,γ,δ];
gradTimeForce3[α_,β_,γ_,δ_,v1_,v2_,v3_,v4_]=inverseg.ValueFieldTotal[(Subscript[y, 1])[t],(Subscript[y, 2])[t],(Subscript[y, 3])[t],(Subscript[y, 4])[t],v1,v2,v3,v4,ββ]vBarupperZ[t];

eqns[t_]=Join[Table[flow'[t][[k]]==force[t][[k]]+gradTimeForce1[α,β,γ,δ,v1,v2,v3,v4][[k]]+gradTimeForce3[α,β,γ,δ,v1,v2,v3,v4][[k]],{k,4}],{(Subscript[x, 5])'[t]==gradTimeForce2[α,β,γ,δ,v1,v2,v3,v4]},Table[preference'[t][[k]]==flow[t][[k]],{k,4}],{(Subscript[y, 5])'[t]==vUpper[t]}];
vBarupperZerotmp=((inertia-valuePotentialTotal[0,0,0,0,v1,v2,v3,v4])/(g00[0,0,α,β,γ,δ]));

initialFlow={initialSpaceFlow[r,u,σ1,σ2][[1]],initialSpaceFlow[r,u,σ1,σ2][[2]],initialSpaceFlow[r,u,σ1,σ2][[3]],initialSpaceFlow[r,u,σ1,σ2][[4]]};
normalizedFlow=initialFlow/(Sqrt[Simplify[initialFlow.gab.initialFlow+g00[0,0,α,β,γ,δ](vBarupperZerotmp^2)]])
vBarupperZero=vBarupperZerotmp/(Sqrt[Simplify[initialFlow.gab.initialFlow+g00[0,0,α,β,γ,δ](vBarupperZerotmp^2)]])
vBarlowerZero=g00[0,0,α,β,γ,δ]vBarupperZero+valuePotentialTotal[0,0,0,0,v1,v2,v3,v4];

initEqns=Simplify[Join[Table[(Subscript[x, k])[0]==normalizedFlow[[k]],{k,4}],Table[(Subscript[y, k])[0]==0,{k,5}],{(Subscript[x, 5])[0]==vBarlowerZero}]];

Differential Equations and parametric solution (WTC2013)

We use ParametricNDSolve to obtain the solution:

solution=ParametricNDSolve[Join[eqns[t],initEqns],var,{t,tLow,tHigh},parameterList,MaxSteps->∞]
flowNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=Table[(Subscript[x, k])[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]/.solution,{k,4}];
preferenceNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=Table[(Subscript[y, k])[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]/.solution,{k,5}];
flowNum5[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=(Subscript[x, 5])[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]/.solution;

g00Num[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=g00[(1/2)(preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]]+preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]]),preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[5]],α,β,γ,δ];
valuePotentialNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]= valuePotentialTotal[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],v1,v2,v3,v4];
vBarupperZNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]= (flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]-valuePotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t])/g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

characteristicPotentialNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=characteristicPotential[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],cn1,cn2,cn3,cn4,cs1,cs2];
flowNumUpper5[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=vBarupperZNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]-characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t].flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

forceNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=inverseg.payoffTotal[n1,n2,n3,n4,s1,s2,preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],cn1,cn2,cn3,cn4,cs1,cs2,αα,flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t],e1,e2].flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];
forceNum5[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=ValueFieldTotal[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],v1,v2,v3,v4,ββ].flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

flowNumComplete[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=Join[flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t],{flowNumUpper5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]}];

aNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=valuePotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];
ϕNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=ϕ[(1/2)(preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]]+preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]]-preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]]),preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[5]],α,β,γ,δ];
conservedNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

pathNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t]((vBarupperZNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t])^2)+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t].gab.flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

transformedFlowNum[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={(1/2)(flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]),(1/2)(-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[2]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]+flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[4]]),(1/(Sqrt[2]))(flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[3]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t] [[4]]),(1/(Sqrt[2]))(flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[1]]-flowNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t] [[2]]),flowNumComplete[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[5]]};(*r,u,s1,s2,t*)
spacePayoff=payoffTotal[n1,n2,n3,n4,s1,s2,preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],cn1,cn2,cn3,cn4,cs1,cs2,αα,flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t],e1,e2];
timePayoff=ValueFieldTotal[preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[1]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[2]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[3]],preferenceNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t][[4]],v1,v2,v3,v4,ββ];
combinedPayoff[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={{spacePayoff[[1]][[1]],spacePayoff[[1]][[2]],spacePayoff[[1]][[3]],spacePayoff[[1]][[4]],timePayoff[[1]]},{spacePayoff[[2]][[1]],spacePayoff[[2]][[2]],spacePayoff[[2]][[3]],spacePayoff[[2]][[4]],timePayoff[[2]]},{spacePayoff[[3]][[1]],spacePayoff[[3]][[2]],spacePayoff[[3]][[3]],spacePayoff[[3]][[4]],timePayoff[[3]]},{spacePayoff[[4]][[1]],spacePayoff[[4]][[2]],spacePayoff[[4]][[3]],spacePayoff[[4]][[4]],timePayoff[[4]]},{-timePayoff[[1]],-timePayoff[[2]],-timePayoff[[3]],-timePayoff[[4]],0}};

inverseg00=1/g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ]+characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ].inverseg.characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ];
inversega0=-inverseg.characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ];
inversegab[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={{inverseg[[1]][[1]],inverseg[[1]][[2]],inverseg[[1]][[3]],inverseg[[1]][[4]],inversega0[[1]]},{inverseg[[2]][[1]],inverseg[[2]][[2]],inverseg[[2]][[3]],inverseg[[2]][[4]],inversega0[[2]]},{inverseg[[3]][[1]],inverseg[[3]][[2]],inverseg[[3]][[3]],inverseg[[3]][[4]],inversega0[[3]]},{inverseg[[4]][[1]],inverseg[[4]][[2]],inverseg[[4]][[3]],inverseg[[4]][[4]],inversega0[[4]]},{inversega0[[1]],inversega0[[2]],inversega0[[3]],inversega0[[4]],inverseg00}};
normalg0a=g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh] [t]characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ];
normalgmn=Table[gab[[j]][[k]]+g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh] [t]characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[j]]characteristicPotentialNum[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ][[k]],{j,4},{k,4}];
normalgab[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]={{normalgmn[[1]][[1]],normalgmn[[1]][[2]],normalgmn[[1]][[3]],normalgmn[[1]][[4]],normalg0a[[1]]},{normalgmn[[2]][[1]],normalgmn[[2]][[2]],normalgmn[[2]][[3]],normalgmn[[2]][[4]],normalg0a[[2]]},{normalgmn[[3]][[1]],normalgmn[[3]][[2]],normalgmn[[3]][[3]],normalgmn[[3]][[4]],normalg0a[[3]]},{normalgmn[[4]][[1]],normalgmn[[4]][[2]],normalgmn[[4]][[3]],normalgmn[[4]][[4]],normalg0a[[4]]},{normalg0a[[1]],normalg0a[[2]],normalg0a[[3]],normalg0a[[4]],g00Num[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t] }};
deltaab[n1_,n2_,n3_,n4_,s1_,s2_,v1_,v2_,v3_,v4_,cn1_,cn2_,cn3_,cn4_,cs1_,cs2_,αα_,ββ_,inertia_,e1_,e2_,r_,u_,σ1_,σ2_,α_,β_,γ_,δ_,tLow_,tHigh_][t_]=normalgab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t].inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t];

Manipulate[{MatrixForm[inversegab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tHigh]],MatrixForm[normalgab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tHigh]],MatrixForm[Chop[deltaab[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][tHigh]]],
flowNum5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][0],Plot[flowNumUpper5[n1,n2,n3,n4,s1,s2,v1,v2,v3,v4,cn1,cn2,cn3,cn4,cs1,cs2,αα,ββ,inertia,e1,e2,r,u,σ1,σ2,α,β,γ,δ,tLow,tHigh][t ],{t,tLow,tHigh},ImageSize->500]},{{n1,n1Z,"(Subscript[ΔG, 11])"},parameterList[[1]][[2]],parameterList[[1]][[3]]},{{n2,n2Z,"(Subscript[ΔG, 12])"},parameterList[[2]][[2]],parameterList[[2]][[3]]},{{n3,n3Z,"(Subscript[ΔG, 21])"},parameterList[[3]][[2]],parameterList[[3]][[3]]},{{n4,n4Z,"(Subscript[ΔG, 22])"},parameterList[[4]][[2]],parameterList[[4]][[3]]},{{s1,s1Z,"(Subscript[ΔS1, 12])"},parameterList[[5]][[2]],parameterList[[5]][[3]]},{{s2,s2Z,"(Subscript[ΔS2, 12])"},parameterList[[6]][[2]],parameterList[[6]][[3]]},{{v1,v1Z,"(Subscript[Δv, 1])"},parameterList[[7]][[2]],parameterList[[7]][[3]]},{{v2,v2Z,"(Subscript[Δv, 2])"},parameterList[[8]][[2]],parameterList[[8]][[3]]},{{v3,v3Z,"(Subscript[Δv, 3])"},parameterList[[9]][[2]],parameterList[[9]][[3]]},{{v4,v4Z,"(Subscript[Δv, 4])"},parameterList[[10]][[2]],parameterList[[10]][[3]]},{{cn1,cn1Z,"(Subscript[cpG, 11])"},parameterList[[11]][[2]],parameterList[[11]][[3]]},{{cn2,cn2Z,"(Subscript[cpG, 12])"},parameterList[[12]][[2]],parameterList[[12]][[3]]},{{cn3,cn3Z,"(Subscript[cpG, 21])"},parameterList[[13]][[2]],parameterList[[13]][[3]]},{{cn4,cn4Z,"(Subscript[cpG, 22])"},parameterList[[14]][[2]],parameterList[[14]][[3]]},{{cs1,s1Z,"(Subscript[cpS1, 12])"},parameterList[[15]][[2]],parameterList[[15]][[3]]},{{cs2,s2Z,"(Subscript[cpS2, 12])"},parameterList[[16]][[2]],parameterList[[16]][[3]]},{{αα,ααZ,"cpBoxSize"},parameterList[[17]][[2]],parameterList[[17]][[3]]},{{ββ,ββZ,"valueBoxSize"},parameterList[[18]][[2]],parameterList[[18]][[3]]},{{inertia,inertiaPD},parameterList[[19]][[2]],parameterList[[19]][[3]]},{{e1,e1z,"(Subscript[e, 1])"},parameterList[[20]][[2]],parameterList[[20]][[3]]},{{e2,e2z,"(Subscript[e, 2])"},parameterList[[21]][[2]],parameterList[[21]][[3]]},{{r,rz,"(Subscript[V, r])"},parameterList[[22]][[2]],parameterList[[22]][[3]]},{{u,uz,"(Subscript[V, u])"},parameterList[[23]][[2]],parameterList[[23]][[3]]},{{σ1,σ1z,"(Subscript[V, (Subscript[s, 1])])"},parameterList[[24]][[2]],parameterList[[24]][[3]]},{{σ2,σ2z,"(Subscript[V, (Subscript[s, 2])])"},parameterList[[25]][[2]],parameterList[[25]][[3]]},{{α,αz,"driven frequency α"},parameterList[[26]][[2]],parameterList[[26]][[3]]},{{β,βz,"driven amplitude β"},parameterList[[27]][[2]],parameterList[[27]][[3]]},{{γ,γz,"gravity amplitude γ"},parameterList[[28]][[2]],parameterList[[28]][[3]]},{{δ,δz,"gravity range δ"},parameterList[[29]][[2]],parameterList[[29]][[3]]},{{tLow,tLowz,"(Subscript[t, Low])"},parameterList[[30]][[2]],parameterList[[30]][[3]]},{{tHigh,tHighz,"(Subscript[t, High])"},parameterList[[31]][[2]],parameterList[[31]][[3]]}]

Stock Market Evidence of Causal Decision Processes

Get AAPLX Stock Prices

listdata=FinancialData["APPLX",{{2015,1,1},{2015,10,1}}];
len=Length[listdata]
DateListPlot[listdata]

Recurrence Plot of Stock Market Data

listArray=Table[Abs[listdata[[i]][[2]]-listdata[[j]][[2]]],{i,len},{j,len}];
ListContourPlot[listArray,ColorFunction->Hue,Contours->20,PlotLabel->"AAPLX from January-October 2015"]

Decision Process Theory Example

e.g. ( (Subscript[f, 2])WWW model from vol. 2)

Game Theory viewed as General Relativity

Strategic Decisions occur in a strategic spacetime

Natural Harmonics are Discrete Solutions of (Subscript[g, ab]) ((dV^b)/dt)=(Subscript[f, ab]) (V^b)
Forced Harmonics are Continuum Solutions of (Subscript[g, ab]) ((dV^b)/dt)=(Subscript[f, ab]) (V^b)+(Subscript[T, a])

Decision Process Theory Results--Standing/Traveling Waves

Summary

Details and much more in upcoming book--To be published 2016

Geometry, Language and Strategy--Vol. 2 Dynamics of Decision Processes, World Scientific, 2016
website: http://decisionprocesstheory.com/

Similarity to Circuits: Harmonic Gauge Coordinate Equations are Linear

Phasor circuit equations are coupled linear equation, which become coupled non-linear PDEs

Therefore we get resonant harmonics, forced harmonics and transients

Non-linearity: Steady state geometry is not linear