If we are predicting t steps ahead , we need to do t^2 operations
In[]:=
ArrayPlot[CellularAutomaton[30,{{1},0},5],Mesh->True]
Out[]=
In[]:=
ArrayPlot[ReplacePart[CellularAutomaton[30,{{1},0},5],{4,6}->-1],ColorRules->{-1->Red}]
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In[]:=
Module[{x=1,y=x+1},{x,y}]
Out[]=
{1,1+x}
In[]:=
modca[rn_,init_,t_,{tm_,xm_}]:=Module[{u=CellularAutomaton[rn,init,{t,All}],v},v=CellularAutomaton[rn,MapAt[1-#&,u[[tm]],xm],t-tm+1];Join[Take[u,tm-1],MapThread[If[#1==#2,#1,#1-2]&,{Drop[u,tm-1],v},2]]]
In[]:=
modcaplot[rn_,init_,t_,{tm_,xm_},opts___]:=ArrayPlot[modca[rn,init,t,{tm,xm}],ColorRules->{0->White,1->Black,-1->Red,-2->Purple},opts]
In[]:=
modcaplot[30,{{1},0},5,{4,6},Mesh->True]
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In[]:=
modcaplot[30,{{1},0},50,{30,50}]
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In[]:=
modcaplot[30,{{1},0},50,{30,70}]
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Region that is relevant is the intersection of the forward and past light cone
Area is (maybe): 2 dx dt
In[]:=
Table[Length[Flatten[Trace[ArrayPlot[With[{c=CellularAutomaton[30,{{1},0},t]},Table[If[#1!=#2,1,0]&[c[[-1,t+1]],CellularAutomaton[30,MapAt[1-#&,c[[j]],i],t+1-j][[-1,t+1]]],{j,t+1},{i,2t+1}]],Mesh->True,ColorRules->{1->Lighter[Red,.2],-1->Lighter[Blue,.2],0->LightGray}],_CellularAutomaton]]],{t,10}]
Out[]=
{7,16,29,46,67,92,121,154,191,232}
In[]:=
FindSequenceFunction[%,t]
Out[]=
2+3t+2
2
t
In[]:=
#[[1,-1]]&/@With[{t=2},Flatten[Trace[ArrayPlot[With[{c=CellularAutomaton[30,{{1},0},t]},Table[If[#1!=#2,1,0]&[c[[-1,t+1]],CellularAutomaton[30,MapAt[1-#&,c[[j]],i],t+1-j][[-1,t+1]]],{j,t+1},{i,2t+1}]],Mesh->True,ColorRules->{1->Lighter[Red,.2],-1->Lighter[Blue,.2],0->LightGray}],_CellularAutomaton]]]
Out[]=
{2,2,2,2,2,2,1,1,1,1,1,0,0,0,0,0}
In[]:=
Table[Total[#[[1,-1]]^2&/@Flatten[Trace[ArrayPlot[With[{c=CellularAutomaton[30,{{1},0},t]},Table[If[#1!=#2,1,0]&[c[[-1,t+1]],CellularAutomaton[30,MapAt[1-#&,c[[j]],i],t+1-j][[-1,t+1]]],{j,t+1},{i,2t+1}]],Mesh->True,ColorRules->{1->Lighter[Red,.2],-1->Lighter[Blue,.2],0->LightGray}],_CellularAutomaton]]],{t,10}]
Out[]=
{4,29,107,286,630,1219,2149,3532,5496,8185}
In[]:=
FindSequenceFunction[%,t]
Out[]=
1
6
(t+11
2
t
+8
3
t
+4
4
t
)
Out[]=
w
1
ϕ[
w
2
ϕ[x
w
3
]+
w
4
ϕ[x
w
5
]+
w
6
ϕ[x
w
7
]]+
w
8
ϕ[
w
9
ϕ[x
w
10
]+
w
11
ϕ[x
w
12
]+
w
13
ϕ[x
w
14
]]+
w
15
ϕ[
w
16
ϕ[x
w
17
]+
w
18
ϕ[x
w
19
]+
w
20
ϕ[x
w
21
]]
Out[]=
In[]:=
∂
x
d[c[b[a[x]]]]
Out[]=
′
a
[x]
′
b
[a[x]]
′
c
[b[a[x]]]
′
d
[c[b[a[x]]]]
In[]:=
∂
x
c[b[a[x]]]
Out[]=
′
a
[x]
′
b
[a[x]]
′
c
[b[a[x]]]
In[]:=
Series[d[c[b[a[x+ϵ]]]],{ϵ,0,2}]
Out[]=
d[c[b[a[x]]]]+
′
a
[x]
′
b
[a[x]]
′
c
[b[a[x]]]
′
d
[c[b[a[x]]]]ϵ+
′
d
[c[b[a[x]]]]
1
2
′
c
[b[a[x]]]
′
b
[a[x]]
′′
a
[x]+
2
′
a
[x]
′′
b
[a[x]]+
1
2
2
′
a
[x]
2
′
b
[a[x]]
′′
c
[b[a[x]]]+
1
2
2
′
a
[x]
2
′
b
[a[x]]
2
′
c
[b[a[x]]]
′′
d
[c[b[a[x]]]]
2
ϵ
+
3
O[ϵ]
To find what the effect on the bottom center cell is .... Say w1[t,x] is the pattern from a single cell initial condition.... e[t,x] (which depends on tf, xf which is the point whose change you’re looking at