Dror Bar-Natan: Talks: Fields-0911:

Dror Bar-Natan: Academic Pensieve: 2009-11:

Hilbert's 13th Problem

Pensieve Header: Hilbert 13th, step 3.

SetDirectory[
$MachineName /. {
"dror-x61" -> "C:/drorbn/AcademicPensieve/2009-11"
} /. $MachineName -> "."
];
<< Hilbert13th-Program.m

Do[
phi2[i]= Phi[Identity, 2, 0.02, 0.95, i/5];
g2[i]=G[f, phi2[i]],
{i,0,4}
];
Do[
phi3[i]= Phi[Identity, 3, 0.01, 0.95, i/5];
g3[i]=G[f, phi3[i]],
{i,0,4}
]

Plot[Table[phi3[i][x], {i, 0, 4}] , {x,0,1}]

Plot[
Table[g3[i][z], {i, 0, 4}], {z,0,1+λ},
PlotPoints-> 200
]

Rasterize[
Plot3D[phi3[0][x]+λ*phi3[0][y], {x,0,1}, {y,0,1},
Mesh -> 11,ViewPoint -> {-2,-2,1}, AxesLabel ->Automatic, NormalsFunction -> None, ColorFunction -> (Hue[g3[0][#3]]&)
]
]

Rasterize[
DensityPlot[
g3[0][phi3[0][x]+λ*phi3[0][y]],
{x,0,1}, {y,0,1}
]
]

Step3FiveCascades = Rasterize[
GraphicsRow[
Table[
Plot3D[phi3[i][x]+λ*phi3[i][y], {x,0,1}, {y,0,1},
Mesh -> None, Boxed -> False,ViewPoint -> {-2,-2,1}, Axes -> None, AxesLabel ->None, Ticks -> None, NormalsFunction -> None, ColorFunction -> (Hue[g3[i][#3]]&)
], {i,0,4}
], Spacings -> 0
], ImageSize -> 900
]

Export[
"Step3FiveCascades.png",
ImageCrop[Step3FiveCascades]
]

Rasterize[
DensityPlot[
(1/3)*Sum[g3[k][phi3[k][x]+λ*phi3[k][y]],  {k ,0, 4}],
{x,0,1}, {y,0,1}, Frame -> False
]
]

Rasterize[
DensityPlot[
(1/5)*Sum[g3[k][phi3[k][x]+λ*phi3[k][y]],  {k ,0, 4}],
{x,0,1}, {y,0,1}
]
]

Rasterize[
Plot3D[
(1/5)*Sum[g3[k][phi3[k][x]+λ*phi3[k][y]],  {k ,0, 4}],
{x,0,1}, {y,0,1}
]
]

Step3Density = Rasterize[
DensityPlot[
(1/5)*Sum[g2[k][phi2[k][x]+λ*phi2[k][y]],  {k ,0, 4}],
{x,0,1}, {y,0,1}, PlotPoints -> 150, Frame -> False
]
]

Export[
"Step3Density.png",
Step3Density
]

Step3Trenches=Rasterize[
Plot3D[
(1/5)*Sum[g2[k][phi2[k][x]+λ*phi2[k][y]],  {k ,0, 4}],
{x,0,1}, {y,0,1}, PlotPoints -> 50, Axes ->None, Boxed -> False
]
]

Export[
"Step3Trenches.png",
Step3Trenches
]