3324.sagews

var('k,y')

(k, y)

xH=1; xL=0.5; phi=2; Lambda=1.5

def q0(x):
    return exp(-Lambda*phi*(xH-x))

def q1(x):
    return Lambda*phi*(xH-x)*exp(-Lambda*phi*(xH-x))

def q2(x):
    return 1/2*Lambda^2*phi^2*(xH-x)^2*exp(-Lambda*phi*(xH-x))

def xaP(k):
    return xH-sqrt(2*k/phi)

def xatilde(k,y):
    return xH-sqrt(2*k/(xH-y))/phi

def xstar(k,y):
    return xH-2*k/phi/(xH-y)

p1=implicit_plot(k==(xH-xL)/2, (k,0,1), (y,0,xL),linestyle = "--")

p2=plot(xL+k-(xH-xaP(k))*(1-q0(xaP(k)))/2/Lambda/q0(xL),(k,0,(xH-xL)/2))

p3=plot(xL-(1-q0(xL)-q1(xL))/Lambda/q0(xL)*(xH-xL)/2,(k,(xH-xL)/2,1))

p4=implicit_plot(2*(xL-y)*(q0(xatilde(k,y))+q1(xatilde(k,y))+q2(xatilde(k,y))-q0(xL)-q1(xL)-q2(xL))/Lambda^2-phi*(xH+y-2*xL)*(xH-xatilde(k,y))*(1-q0(xatilde(k,y))-q1(xatilde(k,y)))/2/Lambda-q0(xatilde(k,y))*k+q0(xL)*(xL+k-y)==0,(k,0,(1-q0(xL))/(1-q0(xL)+q1(xL))*(xH-xL)),(y,0,1))

p5=plot(xL-(1-q0(xL)-q1(xL))/(1-q0(xL)+q1(xL))*(xH-xL),(k,(1-q0(xL))/(1-q0(xL)+q1(xL))*(xH-xL),1))

p6=plot(xL-(xH-xL-k)*(1-q0(xL)-q1(xL))/q1(xL),(k,(xH-xL)/2,(1-q0(xL))/(1-q0(xL)+q1(xL))*(xH-xL)))

p7=implicit_plot(y==xL,(k,0,1),(y,0,1))

p8=implicit_plot(-(2-phi*(xH-y)^2/2/k)*(xH-xstar(k,y))*(1-q0(xstar(k,y))-q1(xstar(k,y)))/2/Lambda-(q0(xstar(k,y))-q0(xL))*k+q1(xstar(k,y))*(xstar(k,y)-y)/Lambda==0,(k,0,(1-q0(xL))/(1-q0(xL)+q1(xL))*(xH-xL)),(y,0,xH))

p9=plot(xH-2*k,(k,0,0.5))

p10=plot(2*xL-xH,(k,0,1))

p11=plot(xH-sqrt(2*k/phi),(k,0,(xH-xL)/2),linestyle = "--",)

p12=plot(xH-sqrt(4*k/phi),(k,0,0.5),linestyle = "--")

p13=implicit_plot(2*k*q0(xL)*(1-2*k/phi/(xH-y)^2)+k*(1-4*k/phi/(xH-y)^2)*(1-q0(xL)-q1(xL))/Lambda,(k,0,(1-q0(xL))/(1-q0(xL)+q1(xL))*(xH-xL)),(y,0,xH))

p14=implicit_plot((xstar(k,y)-y)/(xH-xstar(k,y))*2*2*k*q0(xL)*(1-2*k/phi/(xH-y)^2)+k*(1-4*k/phi/(xH-y)^2)*(1-q0(xL)-q1(xL))/Lambda,(k,0,(1-q0(xL))/(1-q0(xL)+q1(xL))*(xH-xL)),(y,0,xH))

show(p1+p2+p3+p4+p5+p6+p8+p11+p13)