So, with some help from Ember, I figured out how to convert data from Pari-GP into mathematica so I can make 3d graphs now. Everything is behaving exactly as I expected.
Here is a graph of the surface \((\varphi_1,\varphi_2,\varphi_3\) in the equation:
\[
3 <0.5>_{\varphi_1} \left(3<1.5>_{\varphi_2} 3\right) = 3 <1.5>_{\varphi_3} 4\\
\]
This is done over the box \(-0.5 \le \varphi_1,\varphi_2 \le 0.5\). The surface is almost planar it's fascinating. So there is a single value on this surface that we want.
I'm hoping to make a widget so that we can observe the evolution of this surface as we move \(s\) in \(([0,1]\).
Here is a graph of the surface:
\[
3 <0.3>_{\varphi_1} \left(3<1.3>_{\varphi_2} 3\right) = 3 <1.3>_{\varphi_3} 4\\
\]
Very little changes, and again, it's very planar:
You can expect the evolution to be fairly static and planar. This is very good news!!! It means we not only have locality, it means we'll have good regular global structure. I'm going to do more experimenting, I'll try bigger values and check the evolution more!
Regards, James
Also another interesting tidbit. If we only move \(y\), and write \(\varphi_3(y) = \varphi_2(y+1)\), then we actually get a First Order Difference Equation which looks pretty solvable:
\[
\varphi_2(y+1) = \log_{(y+1)^{1/(y+1)}}^{\circ s+1}\left(x <s>_{\varphi_1} (x <s+1>_{\varphi_2(y)} y)\right) - y - 1 - \log^{\circ s+1}_{(y+1)^{1/(y+1)}}(x)\\
\]
This is effectively the first restriction; which then lowers the dimension by \(1\); and we only have to worry about \(\varphi_1\). Funny how everything comes full circle, I love me some first order difference equations!
Here is a graph of the surface \((\varphi_1,\varphi_2,\varphi_3\) in the equation:
\[
3 <0.5>_{\varphi_1} \left(3<1.5>_{\varphi_2} 3\right) = 3 <1.5>_{\varphi_3} 4\\
\]
This is done over the box \(-0.5 \le \varphi_1,\varphi_2 \le 0.5\). The surface is almost planar it's fascinating. So there is a single value on this surface that we want.
I'm hoping to make a widget so that we can observe the evolution of this surface as we move \(s\) in \(([0,1]\).
Here is a graph of the surface:
\[
3 <0.3>_{\varphi_1} \left(3<1.3>_{\varphi_2} 3\right) = 3 <1.3>_{\varphi_3} 4\\
\]
Very little changes, and again, it's very planar:
You can expect the evolution to be fairly static and planar. This is very good news!!! It means we not only have locality, it means we'll have good regular global structure. I'm going to do more experimenting, I'll try bigger values and check the evolution more!
Regards, James
Also another interesting tidbit. If we only move \(y\), and write \(\varphi_3(y) = \varphi_2(y+1)\), then we actually get a First Order Difference Equation which looks pretty solvable:
\[
\varphi_2(y+1) = \log_{(y+1)^{1/(y+1)}}^{\circ s+1}\left(x <s>_{\varphi_1} (x <s+1>_{\varphi_2(y)} y)\right) - y - 1 - \log^{\circ s+1}_{(y+1)^{1/(y+1)}}(x)\\
\]
This is effectively the first restriction; which then lowers the dimension by \(1\); and we only have to worry about \(\varphi_1\). Funny how everything comes full circle, I love me some first order difference equations!