tropicalsand
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Ci-dessous, les différences entre deux révisions de la page.
| Les deux révisions précédentesRévision précédenteProchaine révision | Révision précédente | ||
| tropicalsand [2015/05/05 01:43] – kalinin0 | tropicalsand [2017/03/10 23:16] (Version actuelle) – kalinin0 | ||
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| Ligne 4: | Ligne 4: | ||
| - | **Tropical curves | + | You can see our two talks on the BIRS-CMO conference Sandpiles Groups |
| + | http:// | ||
| - | Abstract: We consider the sandpile model on a part of the square lattice, bounded by a polygon. We modify the maximal stable state by adding a grain of sand at each of the $n$ fixed points: the consequent relaxation produces pictures where we can see tropical curves. These curves pass through the same $n$ fixed points andÌý | + | http://www.birs.ca/ |
| - | solve a version of the Steiner tree problem: minimization of {\it tropical symplectic area}. In order to show this, we develop several technics to deal with particular integer-valued solutions ofÌý | + | |
| - | certain Dirichlet problems and to study the continuous version of the consideredÌý | + | |
| - | relaxation which reveals an interesting dynamics on polytopes. | + | |
| - | Keywords: | ||
| - | {{:Sand.pdf|}} | + | Our presentations: |
| + | See the simulation on Youtube: https:// | ||
| - | **Tropical series and sandpiles in arbitrary convex domains, Nikita Kalinin, Mikhail Shkolnikov.** | ||
| - | This is a draft of the second part of the article entitled ``Tropical curvesÌý | + | **TROPICAL ANALYTIC CURVES IN 2-DIMENSIONAL SANDPILE MODEL, Nikita Kalinin, Mikhail Shkolnikov.** |
| - | in $2$-dimensional sandpile model'' | + | |
| - | arbitrary boundary. | + | |
| - | {{: | ||
| - | {{::sandcircle1.png?200|}} | + | Abstract: We consider the sandpile model on a convex part of the square lattice. We modify the maximal stable state by adding a grain of sand at each of the $n$ fixed points: the consequent relaxation produces pictures where we can see tropical curves. These curves pass through the same $n$ fixed points and solve a version of the Steiner tree problem: minimization of {\it tropical symplectic area}. In order to show this, we develop several techniques to deal with particular integer-valued solutions ofÌý |
| + | a certain Dirichlet problems. The continuous version of the considered relaxation reveals an interesting dynamics on polytopes. | ||
| - | and in the limit we always obtain something like | + | Keywords: |
| + | Ìý | ||
| + | {{:: | ||
| + | Ìý | ||
| + | https:// | ||
| + | Ìý | ||
| + | https:// | ||
| + | Ìý | ||
| + | Ìý | ||
| + | Ìý | ||
| + | {{:: | ||
| - | {{:: | ||
| - | Soon, we will upload here code and more pictures. | ||
| For more information about sand read http:// | For more information about sand read http:// | ||
| Ligne 41: | Ligne 44: | ||
| or manage to use the package Tkinter by yourself. | or manage to use the package Tkinter by yourself. | ||
| - | 2. Use the file {{: | + | 2. Use the file {{: |
| - | 3. On the top part of the file you can change some parametres, choose the boudary etc. But you can start by clicking and observing what happens. | + | 3. On the top part of the file you can change some parametres, choose the boudary etc. But you can start by pressing F5 (run file), |
| 4. Basically, we start from the grid with 3 grains of sand everywhere. When you left-click on the left half of the screen you add an additional grain of sand. White color means that at this point we have 3 grains of sand. | 4. Basically, we start from the grid with 3 grains of sand everywhere. When you left-click on the left half of the screen you add an additional grain of sand. White color means that at this point we have 3 grains of sand. | ||
| 5. You can rescale the picture if you left-click in the top half of the right half of the screen. You can undo return to the previous picture by clicking in the bottom half of the right half of the screen. | 5. You can rescale the picture if you left-click in the top half of the right half of the screen. You can undo return to the previous picture by clicking in the bottom half of the right half of the screen. | ||
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tropicalsand.1430783008.txt.gz · Dernière modificationÌý: de kalinin0