In[]:=
Out[]=
8.1×
60
10
In[]:=
Solve[Ξκ^(σ/Sqrt[Ξ]),Ξ]
Out[]=
Ξ,Ξ
2
σ
2
Log[κ]
4
2
ProductLog-
1
2
2
σ
1
2
σ
2
Log[κ]
2
Log[κ]
2
σ
2
Log[κ]
4
2
ProductLog
1
2
2
σ
1
2
σ
2
Log[κ]
2
Log[κ]
In[]:=
Ξ/.Last[%]
Out[]=
2
σ
2
Log[κ]
4
2
ProductLog
1
2
2
σ
1
2
σ
2
Log[κ]
2
Log[κ]
In[]:=
FullSimplify[%,{κ>1,σ>0}]
Out[]=
2Wσlog(κ)
1
2
In[]:=
Asymptotic[ProductLog[x],{x,Infinity,2}]
Out[]=
In[]:=
Expand[%]
Out[]=
In[]:=
Exp[2(Log[x/2]-Log[Log[x/2]])]
Out[]=
2Log-LogLog
x
2
x
2
In[]:=
FullSimplify[%,x>0]
Out[]=
2
x
4
2
Log
x
2
In[]:=
Exp[2(Log[x/2])]
Out[]=
2
x
4
In[]:=
Asymptotic[Exp[2ProductLog[x/2]],{x,Infinity,1}]
Out[]=
In[]:=
2ProductLog[x/2]
Out[]=
2ProductLog
x
2
In[]:=
FullSimplify[%,x>0]
Out[]=
2ProductLog
x
2
In[]:=
2
x
4
2
Log
x
2
In[]:=
2
σ
2
Log[κ]
4
2
LogσLog[κ]
1
2
Out[]//TraditionalForm=
2
σ
2
log
4σlog(κ)
2
log
1
2
2Wσlog(κ)
1
2
In[]:=
Exp2ProductLog12Log[κ],/.σ->
2
σ
2
Log[κ]
4
2
LogσLog[κ]
1
2
In[]:=
,,1.625976688086400237032351`2.2330606492521925*^121
2ProductLog[4.032340124650201475155`2.5340906449161738*^60Log[κ]]
1.625976688086400237032351`2.2330606492521925*^121
2
Log[κ]
2
Log[4.032340124650201475155`2.5340906449161738*^60Log[κ]]
2
Log[κ]
Out[]=
,,1.63×
2ProductLog4.0×Log[κ]
60
10
1.63×
121
10
2
Log[κ]
2
Log[4.0×Log[κ]]
60
10
121
10
2
Log[κ]
In[]:=
LogPlot[%,{κ,1,3}]
Out[]=
In[]:=
LogPlot[,{κ,1,3},FrameTrue]
2ProductLog[4.032340124650201475155`2.5340906449161738*^60Log[κ]]
Out[]=
In[]:=
Series[Exp[2ProductLog[1/2σLog[κ]]],{κ,1,2}]
Out[]=
1+σ(κ-1)-σ+
1
2
2
(κ-1)
3
O[κ-1]
In[]:=
fff[κ_]:=Exp2ProductLog12Log[κ]
In[]:=
fff[1.01]
Out[]=
9.5154×
112
10
In[]:=
fff[2]
Out[]=
4.3×
116
10
In[]:=
fff[10]
Out[]=
4.7×
117
10
In[]:=
fff[1.1]
Out[]=
8.43691×
114
10
In[]:=
Out[]=
8.1×
118
10
In[]:=
GridMapText[NumberForm[#,1]]&,"elementary length ()",UnitConvertSqrt[xi],"elementary time ()",UnitConvertSqrt[xi],"elementary energy ()",UnitConvertSqrt[xi],"ElectronVolt","elementary lengths across current universe",2Sqrt[xi],"elements in spatial graph",UnitConvertSqrt[xi]^3,{"elements in branchial graph",xi},"overall updates of universe so far",UnitConvertSqrt[xi],"individual updating events in universe so far",UnitConvertUnitConvertSqrt[xi]^3*Sqrt[xi]/.xi10^116.,{2},FrameAll,FrameStyleLightGray,AlignmentLeft
L
T
E
Out[]=
Max version
Max version
Number of spatial nodes: (tH/T)^3 = (tH/tP)^3 Xi^(3/2)
Xi ~ κ ^ (( (tH/tP)^3 Xi^(3/2)) * (Sqrt[Xi] tH/tP))
Xi = κ ^ ((tH/tP)^4 Xi^2)