Sandpile groups of Cayley graphs of r 2 Gao, Jiyang; Marx-Kuo, Jared; McDonald, Vaughan ...
Communications in algebra,
10/2/2024, 2024-10-02, Volume:
52, Issue:
10
Journal Article
Peer reviewed
Open access
The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric ...descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of
F
2
r
, focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n-1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries.
A well-balanced second-order finite volume scheme is proposed and analyzed for a 2×2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a ...vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to revert to a well-balanced first-order scheme if it fails to capture or preserve the discrete steady state solution during its time evolution. Further, the scheme maintains the second-order accuracy away from the steady state. The well-balance property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations and provides better resolutions of the physical properties of the state variables, as compared to the existing first-order schemes of the literature.
•Analytically proved that the standard second-order scheme for the Hadeler and Kuttler (HK) model equations is not well-balanced in general.•An adaptive second-order scheme for the HK model is developed and analytically proved that the proposed scheme is well-balanced.•Significance of the proposed scheme is demonstrated through its ability to tackle the challenges faced by conventional second-order scheme.
Particle motion in granular systems is common in nature and industrial processes. Along with particle size and density, the shape of particles also plays a crucial role in the dynamic behavior of ...such systems. To capture the irregular morphology of particles, we have proposed an image-based discrete element method in which the effect of fluid is also considered. We have considered four cases to study how the particle shape and ambient fluid simultaneously affect the properties of sandpiles, that is dry and water-immersed sandpiles, for both irregular and spherical particles. We found that the repose angle of sandpiles is increased in the irregular models while decreased by having ambient water in the system. The contact force chains in irregular particles are also stronger than that of the spherical sandpiles. Meanwhile, the contact force chains are weakened by the ambient water. Furthermore, the largest particle displacement occurs in the spherical sandpile immersed in water.
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•An image-based technique for modeling angular particles is presented.•This method is coupled with fluid.•The developed method is applied to a sandpile problem for dry and wet conditions.•The results of angular modeling are compared with the spherical particles.
A physical study of the LLL algorithm Ding, Jintai; Kim, Seungki; Takagi, Tsuyoshi ...
Journal of number theory,
March 2023, 2023-03-00, Volume:
244
Journal Article
Peer reviewed
Open access
This paper presents a study of the LLL algorithm from the perspective of statistical physics. Based on our experimental and theoretical results, we suggest that interpreting LLL as a sandpile model ...may help understand much of its mysterious behavior. In the language of physics, our work presents evidence that LLL and certain 1-d sandpile models with simpler toppling rules belong to the same universality class.
This paper consists of three parts. First, we introduce sandpile models whose statistics imitate those of LLL with compelling accuracy, which leads to the idea that there must exist a meaningful connection between the two. Indeed, on those sandpile models, we are able to prove the analogues of some of the most desired statements for LLL, such as the existence of the gap between the theoretical and the experimental RHF bounds. Furthermore, we test the formulas from finite-size scaling theory (FSS) against the LLL algorithm itself, and find that they are in excellent agreement. This in particular explains and refines the geometric series assumption (GSA), and allows one to extrapolate various quantities of interest to the dimension limit. In particular, we obtain the estimate that the empirical average RHF converges to ≈1.02265 as the dimension goes to infinity.
Tropical sandpile model (or linearized sandpile model) is the only known
continuous geometric model exhibiting self-organised criticality. This model
represents the scaling limit behavior of a small ...perturbation of the maximal
stable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ of
points in a compact convex domain $\Omega\subset \mathbb R^2$ this linearized
model produces a tropical polynomial $G_P{\bf 0}_\Omega$.
Here we present some quantitative statistical characteristics of this model
and some speculative explanations. Namely, we study the dependence between the
number $n$ of randomly dropped points
$P=\{p_1,\dots,p_n\}\subset0,1^2=\Omega$ and the degree of the tropical
polynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of the
coefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. This
paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf
0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of the
standard square grid lines. This explains a previously known fact that most of
the edges of the tropical curve $C(G_{P}{\bf 0}_\Omega)$ are of directions
$(1,0),(0,1),(1,1),(-1,1)$.
The main theoretical result is that $C(G_{P}{\bf 0}_\Omega)\setminus (P\cap
\partial\Omega)$, i.e. the tropical curve in $\Omega^\circ$ with marked points
$P$ removed, is a tree.
In the real world, the construction of infrastructure networks is influenced by spatial factors. For such complex networks, We refer to them as spatial networks. This study investigates, based on the ...Bak–Tang–Wiesenfeld sandpile model, the impact of spatial factors on sandpile cascading avalanches. Through simulations, we discover that spatial factors not only facilitate the occurrence of moderate-scale avalanches in the system but also have a suppressing effect on the occurrence of the largest cascading avalanches. Using a multiplicative branching process, we analyze the influence of triangular structures composed of nodes on sandpile avalanches and find that nodes within triangular structures are relatively resistant to topples. Because of the relatively stable nature within triangular structures, more load tends to stay within this kind of structures. Once a cascading avalanche occurs in the system, the avalanche scale is often larger than that in the non-spatial counterpart. This increase in avalanche scale results in more load being released by the system, making the spatial networks more relaxed. And their probability of very-large-scale cascading avalanches is lower than that of non-spatial networks.
•We study the sandpile model in spatial networks.•We find that the spatial factors do affect the sandpile dynamics.•We find spatial factors facilitate moderate avalanches but suppress the largest ones.•We focus on the triangle structure in spatial networks to explain the above results.
The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph G. By viewing this chain as a (nonreversible) random walk on an ...abelian group, we give a formula for its eigenvalues and eigenvectors in terms of ``multiplicative harmonic functions'' on the vertices of G. We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on G: If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where G is the complete graph on n vertices, we show that the sandpile chain exhibits cutoff at time \frac {1}{4\pi ^{2}}n^{3}\log n.
This paper investigates the critical group of a faithful representation of a finite group. It computes the order of the critical group in terms of the character values, and gives some restrictions on ...its subgroup structure. It also computes the precise structure of the critical group both for the regular representation of any finite group, and for the reflection representation of Sn.
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