In[]:=
a[mu_,k_,rho_]:=mu*(rho-1)/kvarWaring[a_,k_,rho_]:=a*k*(rho+a-1)*(rho+k-1)/((rho-1)^2*(rho-2))res=Reduce[varWaring[a[mu,k,rho],k,rho]==mu*(1+mu/phi)&&mu>0&&phi>0&&rho>2&&k>0,{rho,k},Reals]
Out[]=
phi>0&&mu>0&&rho+2&&k-||rho>+2&&k-||k+
2mu+phi+muphi+2
2
phi
mu
mu++mu+
2
phi
3
phi
3
phi
4
phi
2
mu
-2mu-phi-muphi+murho
2phi
1
2
4+4muphi+4phi++6mu+-4rho-2muphirho-2phirho-4murho+
2
mu
2
mu
2
phi
2
phi
2
mu
2
phi
2
mu
2
mu
2
phi
2
mu
2
rho
2
phi
2mu+phi+muphi+2
2
phi
mu
mu++mu+
2
phi
3
phi
3
phi
4
phi
2
mu
-2mu-phi-muphi+murho
2phi
1
2
4+4muphi+4phi++6mu+-4rho-2muphirho-2phirho-4murho+
2
mu
2
mu
2
phi
2
phi
2
mu
2
phi
2
mu
2
mu
2
phi
2
mu
2
rho
2
phi
-2mu-phi-muphi+murho
2phi
1
2
4+4muphi+4phi++6mu+-4rho-2muphirho-2phirho-4murho+
2
mu
2
mu
2
phi
2
phi
2
mu
2
phi
2
mu
2
mu
2
phi
2
mu
2
rho
2
phi
In[]:=
rhoBoundRaw=res[[3]][[2]][[1]]rhoBound=Simplify[rhoBoundRaw,Assumptionsrho>2&&mu>0&&phi>0]kvalRaw=res[[3]][[2]][[2]][[1]][[2]]kval=Simplify[kvalRaw,Assumptionsrho>2&&mu>0&&phi>0]
Out[]=
rho>+2
2mu+phi+muphi+2
2
phi
mu
mu++mu+
2
phi
3
phi
3
phi
4
phi
2
mu
Out[]=
rho>2+phi+
phi1+2phi+2
(1+phi)(mu+phi)
mu
Out[]=
-2mu-phi-muphi+murho
2phi
1
2
4+4muphi+4phi++6mu+-4rho-2muphirho-2phirho-4murho+
2
mu
2
mu
2
phi
2
phi
2
mu
2
phi
2
mu
2
mu
2
phi
2
mu
2
rho
2
phi
Out[]=
-
phi+mu(2+phi-rho)++-2muphi(-2-3phi+rho+2phirho)
2
phi
2
mu
2
(2+phi-rho)
2phi
In[]:=
In[]:=
(*Simplify[-2-3phi+2+phi+x+2*phi*(2+phi+x)]*)(*rhocalc[mu_,phi_,rhop_]:=rhop/mu*)(*tauc[mu_,phi_,rho_]=(rho-2-phi)*mu/phi*)tauc[mu_,phi_,rho_]:=-mu*(2+phi-rho)/phikFirst[mu_,phi_,tau_]:=1-tau(*kSecond[mu_,phi_,tau_]:=Sqrt[(phi^2+tau^2-2*mu*phi*(-tau/mu-2*phi+2*phi*(-tau/mu+2+phi)))/(phi^2)]*)(*kSecond[mu_,phi_,tau_]:=Sqrt[1+tau^2-2*mu*(-tau/mu-2+2*(-(tau*phi)/mu+2+phi))]*)kSecond[mu_,phi_,tau_]:=Sqrt[1-4*mu*(1+phi)-(2+4*phi)*tau+tau^2](*kFirst[mu_,phi_,tau_]:=-1+taukSecond[mu_,phi_,tau_]:=Sqrt[1-tau^2-2*(2*mu*phi^2+phi*tau+2*phi*(mu+phi*tau))]*)k[mu_,phi_,tau_]:=-2^-1*(kFirst[mu,phi,tau]+kSecond[mu,phi,tau])tauBound[mu_,phi_,tau_]:=tau>(1+2*phi+2*Sqrt[(1+phi)*(mu+phi)])2+3
Out[]=
2
Null
Out[]=
1-4mu(1+phi)-2(1+2phi)tau+
2
tau
Out[]=
5
In[]:=
FullSimplify[k[mu,phi,tau]]tauBound[mu,phi,tau][[2]]FullSimplify[k[mu,phi,tauBound[mu,phi,tau][[2]]+2/q],Assumptionsq>0]Solve[tau==tauc[mu,phi,rho],{rho},Reals]
Out[]=
1
2
1-4mu(1+phi)+tau(-2-4phi+tau)
Out[]=
1+2phi+2
(1+phi)(mu+phi)
Out[]=
phi+-
(1+phi)(mu+phi)
+1
q
1+2
(1+phi)(mu+phi)
qq
Out[]=
rho-
-2mu-muphi-phitau
mu
In[]:=
Reduce[k[mu,phi,tauc[mu,phi,rho]]==kval&&rho>2&&mu>0&&phi>0&&rhoBound,rho,Reals]
Out[]=
phi>0&&mu>0&&rho>+2
2mu+phi+muphi+2
2
phi
mu
mu++mu+
2
phi
3
phi
3
phi
4
phi
2
mu
In[]:=
Reduce[rhoBound&&tauBound[mu,phi,tauc[mu,phi,rho]]&&mu>0&&phi>0&&rho>2,Reals]===Reduce[rhoBound&&mu>0&&phi>0&&rho>2,Reals]Reduce[rhoBound&&tauBound[mu,phi,tauc[mu,phi,rho]]&&mu>0&&phi>0&&rho>2,Reals]===Reduce[tauBound[mu,phi,tauc[mu,phi,rho]]&&mu>0&&phi>0&&rho>2,Reals]
Out[]=
True
Out[]=
True
In[]:=
FullSimplify[kSecond[mu,phi,(rhoBound[[2]]-2-phi)*mu/phi]]
Out[]=
2
(1+2phi)1+2phi+2
(1+phi)(mu+phi)
In[]:=
In[]:=
Reduce[a[mu,k[mu,phi,rho],rho]==k2[mu,phi,rho]&&mu>0&&phi>0&&rhoBound,{mu,phi,rho},Reals]
Out[]=
Reducek2[mu,phi,rho]&&mu>0&&phi>0&&rho>2+phi+,{mu,phi,rho},
2mu(-1+rho)
1+rho+
1-4mu(1+phi)+(2+4phi)rho+
2
rho
phi1+2phi+2
(1+phi)(mu+phi)
mu
In[]:=
rhoLow[mu_,phi_]:=phi+(phi*(1+2*phi+2*sqrt((1+phi)*(mu+phi))))/murhoLow[a,b]
Out[]=
b+
b(1+2b+2(1+b)(a+b)sqrt)
a
In[]:=
Reduce[a[mu,k[mu,phi,rho],rho]==k[mu,phi2,rho]&&k[mu,phi,rho]==a[mu,k[mu,phi2,rho]]&&mu>0&&phi>0&&rhoBound,{phi2},Reals]Reduce[a[mu,k[mu,phi,rho],rho]==k2[mu,phi2,rho]&&k[mu,phi,rho]==a[mu,k2[mu,phi2,rho]]&&mu>0&&phi>0&&rhoBound,{phi2},Reals]Reduce[a[mu,k2[mu,phi,rho],rho]==k2[mu,phi2,rho]&&k2[mu,phi,rho]==a[mu,k2[mu,phi2,rho]]&&mu>0&&phi>0&&rhoBound,{phi2},Reals]
Out[]=
$Aborted
Out[]=
Reducek2[mu,phi2,rho]&&1+rho+,{phi2},
2mu(-1+rho)
1+rho+
1-4mu(1+phi)+(2+4phi)rho+
2
rho
1
2
1-4mu(1+phi)+(2+4phi)rho+
a[mu,k2[mu,phi2,rho]]&&mu>0&&phi>0&&rho>2+phi+2
rho
phi1+2phi+2
(1+phi)(mu+phi)
mu
Out[]=
Reducek2[mu,phi2,rho]&&k2[mu,phi,rho]a[mu,k2[mu,phi2,rho]]&&mu>0&&phi>0&&rho>2+phi+,{phi2},
mu(-1+rho)
k2[mu,phi,rho]
phi1+2phi+2
(1+phi)(mu+phi)
mu