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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" />

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" />

==[[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

==[[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）

respectively  

respectively  

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />  

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />  

For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the

For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the

smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> 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. 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" />

smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> 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.< 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" />

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，这意味着可能存在标度限制。<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配置映射到子网格上的重复配置。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，这意味着可能存在标度限制。<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配置映射到子网格上的重复配置。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" />