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Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />

此外，扩展沙堆模型的循环配置形成了一个阿贝尔群，称为“扩展沙堆群”，其中通常的沙堆群是[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。由于它仅通过将沙粒添加到网格的边界<math>\partial\Gamma</math>而生成，因此扩展的沙堆组还具有维度<math>\partial\Gamma</math>的[[拓扑群|拓扑]]的[[环]]和由通常沙堆组的顺序给出的体积<ref name="Lang2019" />

 

 

  ==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid翻译存疑。a volume given by the order of the usual sandpile group.翻译存疑。==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

  ==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid翻译存疑。a volume given by the order of the usual sandpile group.翻译存疑。==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续[[测地线]]动态变化的问题。这个问题引出了沙堆动力学的定义

特别令人感兴趣的问题是，在这个圆环体通过恒等的连续[[测地线]]s上，循环构型是如何动态变化的。这个问题引出了沙堆动力学的定义

 

  ==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

  ==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）

:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）

This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid.<font color="#ff8000"> Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" /></font>

This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid.<font color="#ff8000"> Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" /></font>

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在缩放极限。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复构型映射到子网格上的重复构型。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在缩放极限。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复构型映射到子网格上的重复构型。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />