Abstract
It has been spent a huge amount of efforts in the study of Abelian Sandpile Model, as prototype of Self Organized Criticality.
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Notes
- 1.
Here and in the following, bold letters \({\varvec{k}},{\varvec{v}},\ldots \) are vectors in \(\mathbb Z ^2\) if not otherwise stated.
- 2.
We use here the notation of Chap. 2
- 3.
The sites we are interested in are the ones on the border of the framing polygons described in Sect. 5.4
- 4.
For more details see appendix D.
- 5.
Each pair also define a parallelogram, whose area is constant, equal to the magnitude of the cross product.
- 6.
In the following we will use \(\varvec{v}_3\) in addition to \(\varvec{v}_1\) and \(\varvec{v}_2\) with the constraint \(\varvec{v}_1+\varvec{v}_2+\varvec{v}_3=0\).
- 7.
The name blue cells comes from the color for sites of height 2 used in our representation.
- 8.
We call tile size the area of the corresponding framing polygon and we denote it by \(|F(\cdot )|\).
- 9.
We want \(|\varvec{v}_1(B)\wedge \varvec{v}_2(B)|=|F(P)|\) and we know \(|F(P)|=|\varvec{k}|^2-|F(B(P))|\), with \(|F(B(P))|=\varvec{w}_1\wedge \varvec{w}_2\). So we have \(|\varvec{v}_1(B)\wedge \varvec{v}_2(B)|=i\varvec{k}\wedge i(\varvec{w}_\beta \pm i\varvec{k})=\varvec{k}\wedge \varvec{w}_\beta \pm \varvec{k}\wedge \varvec{k}=\varvec{k}\wedge \varvec{w}_\beta \pm |\varvec{k}|^2\), the last two summands have to subtract each other, but \(|\varvec{k}|^2\) is always positive, so its sign has to change as the sign of \(\varvec{k}\wedge \varvec{w}_\beta \) changes.
- 10.
\(\varvec{k}\) and \(-\varvec{k}\) indentify the same string.
- 11.
\(\varvec{v}_1(B_1)\wedge \varvec{v}_2(B_1)=i\varvec{k}_0\wedge i(\varvec{p}+i\varvec{k}_0)=(\varvec{p}+\varvec{q})\wedge \varvec{p}+ \varvec{k}_0\wedge \varvec{k}_0=\varvec{v}_1(B_0)\wedge \varvec{v}_2(B_0)+|\varvec{k}_0|^2>0\)
- 12.
Only two regions, if there are two sides, one horizontal and one vertical.
- 13.
Building the triangle with sides of 1, 2 or 3 tiles, each of the tiles of the triangle is in contact with the exterior, and in this condition the burning test works.
- 14.
The sequence of Mersenne numbers http://oeis.org/A000225, are of the form \(M_n\equiv 2^n-1\) where \(n\) is a positive integer. The Mersenne numbers consist of all 1s in base-2, and are therefore binary repunits. Sometimes Mersenne numbers are considered to be only the ones with \(n\) prime.
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Paoletti, G. (2014). Pattern Formation. In: Deterministic Abelian Sandpile Models and Patterns. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01204-9_5
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