bio | website | mit.edu/~shopkins |
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location | Cambridge, MA | |
age | 23 | |
visits | member for | 2 years, 3 months |
seen | 7 hours ago | |
stats | profile views | 949 |
Oct 5 |
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An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
Well, the definitive source explaining the combinatorics behind the positive Grassmannian is of course arxiv.org/abs/math/0609764. |
Oct 5 |
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An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
It seems that there are already some very similar questions on MO: mathoverflow.net/questions/142841/… mathoverflow.net/questions/143339/what-is-the-amplituhedron |
Aug 18 |
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Notion of infinity in categories
I think the OP is asking about what conditions on a category we can impose to force the transitivity property to hold, not what conditions on objects we can impose. |
Aug 16 |
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Duration and critical groups order in sandpile models and chip firing games
@FelixGoldberg: that's true. But keep in mind there is the subtle distinction between configurations (distributions of sand on the nonsink vertices) and divisors (distributions of sand on all the vertices). So above I am wrong to say the stable divisors form a monoid; it is the stable configurations which do. |
Aug 16 |
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Duration and critical groups order in sandpile models and chip firing games
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Aug 16 |
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Duration and critical groups order in sandpile models and chip firing games
Note that this example also runs exactly opposite to the intuition mentioned by the OP: for $P_n$ we have $2^{n-2}$ stable states, while for $S_n$ we have $n-1$ stable states. So $P_n$ has more states to terminate into. However, the (worst-case) stopping time is much longer for $P_n$ than $S_n$. |
Aug 16 |
revised |
Duration and critical groups order in sandpile models and chip firing games
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Aug 16 |
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Duration and critical groups order in sandpile models and chip firing games
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Aug 15 |
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Embed one Coxeter System into another
@DavidSpeyer: I think that manuscript of Stembridge only addresses embeddings of crystollagraphic root systems in simply-laced ones, in which case there is an easy way to see the embedding via "folding" which arises from the fixed-point set of an automorphism of the Coxeter graph. But the embeddings discussed by the OP (of $H_3$ into $D_6$ or $H_4$ into $E_8$) are different and more complicated than this. |
Aug 15 |
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Duration and critical groups order in sandpile models and chip firing games
By the way, as to the OP's intuition that a larger sandpile group gives "more opportunities to terminate": what is really at issue with such an intuition is not the sandpile group (whose elements are the recurrent configurations) but rather the monoid of stable divisors because we play the game until we reach a state that is stable. The size of this monoid is just the product of all the degrees of the vertices in $G$. |
Aug 15 |
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Duration and critical groups order in sandpile models and chip firing games
However, the quantity $(n-1)(2n\tau)^{1/(n-1)} - 2m$ is negative for many graphs $G$ (such as e.g. a tree or cycle) so the bound on $\lambda_{n-1}$ above is trivial in these cases and we cannot apply it to say anything meaningful about $\ell(N)$. |
Aug 15 |
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Duration and critical groups order in sandpile models and chip firing games
On the other hand, using Lemma 3.6 of this paper (link.springer.com/article/10.1007/BF02669571) we can get an inequality $\lambda_{n-1} \geq (n-1)(2n\tau)^{1/(n-1)} - 2m$, where $m$ is the number of edges of $G$ and $\tau$ is the number of spanning trees (i.e., order of the critical group). So we could combine those inequalities and get our bound on the length in terms of the number of spanning trees and number of edges of $G$. |
Aug 15 |
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Duration and critical groups order in sandpile models and chip firing games
I tried turning these observations into an answer but got stuck. Say $G$ is a graph on $n$ vertices. Let $\ell(N)$ (for small $N$) denote the maximum length of a game on $G$ that terminates and involves $N$ chips. In this paper (cs.elte.hu/~lovasz/morepapers/chips.pdf) the authors show that $\ell(N) \leq 2nN/\lambda_{n-1}$, where $\lambda_{n-1}$ is the smallest positive eigenvalue of the Laplacian of $G$. |
Aug 15 |
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Duration and critical groups order in sandpile models and chip firing games
For definitions of sandpile terms, see arxiv.org/abs/1112.6163. At any rate, the order of the critical group is the number of spanning trees of $G$, which is the determinant of the reduced Laplacian. BLS give a bound on the length of the game in terms of the smallest nonzero eigenvalue of the Laplacian. |
Aug 15 |
answered | Duration and critical groups order in sandpile models and chip firing games |
Aug 14 |
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The ten martini problem - reason for name
Here Terry Tao says it is because Kac offered ten martinis for the solution: terrytao.wordpress.com/2014/08/12/… |
Aug 13 |
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What are the reasons for considering rings without identity?
Another argument in favor of requiring rings to have identity: www-math.mit.edu/~poonen/papers/ring.pdf |
Aug 10 |
awarded | Nice Answer |
Aug 9 |
answered | Examples of unexpected mathematical images |
Aug 9 |
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Examples of unexpected mathematical images
The appearance of Apollonian circle packing in questions related to the scaling limit of the abelian sandpile model and integer superharmonic functions was quite unexpected. See the papers arxiv.org/abs/1208.4839 and arxiv.org/abs/1309.3267. As I understand it, this observation was made by computing some explicit examples and noticing the fractal pattern. |