博客來
本書的第一部分介紹了Chip-firing的基本原理。第一章以對Chip-firing的簡單介紹開始，包括基本動力學的一個擴展的例子。第二章展示了Chip-firing動力學的細節，比如阿貝爾性質、穩定性與臨界性。第三章與組合學有很大的聯繫，其結果是不同的長期穩定構形的數量等於圖的生成樹的數量。此外，我們提出了美利奴(Merino)定理，該定理改進了這個計算，並且由於它與面數的聯繫從而引起了人們對Chip-firing組合學的極大興趣。
第四章和第五章繼續介紹Chip-firing的早期觀點。第四章處理了沙堆群——一個自然與碎片配置相關聯的有限阿貝爾群。第五章討論了模式的形成，包括沙堆群的單位元。正是在這裡，人們發現了Chip-firing模型迷人的分形行為。
本書的第二部分展示了更通用的Chip-firing的一般框架。對Chip-firing是一種由圖拉普拉斯(Laplace)算子控制的離散擴散形式的觀察構成了第六章的基礎內容。將圖拉普拉斯算子適當地推廣到其他算子中會產生新的系統，但它們都具有類似的良好屬性。在這種情況下，Chip-firing可以被視為一種能量最小化系統。
第七章介紹了高維的Chip-firing。高維模型不是圖頂點上的碎片，而是由拓撲複形單元上的流組成的。更高維的Chip-firing帶來了胞腔理論生成的樹和組合拉普拉斯算子。
第八章介紹了一個由代數幾何驅動的方向。將圖形解釋為代數曲線的組合類比，碎片構形可以被認為是曲線上的除數。第八章還包括了從算術幾何角度出發的Chip-firing以及與二變數zeta函數的連接。
最後，第九章考慮了組合交換代數角度的Chip-firing。Chip-firing的動作在被稱為傾覆理想的二項式理想中被編碼。這裡還介紹了單項式初始理想，即圖的樹理想。我們將看到樹理想的標準單項式與Chip-firing系統的長期穩定構形是雙射的。
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Chip-firing中的數學(英文)的圖書 |
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Chip-firing中的數學(英文) 作者：（美）卡洛琳·J.克利文 出版社：哈爾濱工業大學出版社 出版日期：2024-04-01 語言：簡體中文 規格：平裝 / 303頁 / 19 x 26 x 1 cm / 普通級/ 1-1 |
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圖書介紹 - 資料來源：博客來 評分：
圖書名稱：Chip-firing中的數學(英文)
內容簡介
目錄
Preface
Ⅰ Fundamentals
1 Introduction
1.1 A brief introduction
1.2 Origins and History
1.2.1 The abelian sandpile model
1.2.2 A combinatorial game
1.2.3 Abstract rewriting systems
2 Chip-firing on Finite Graphs
2.1 The chip-firing process
2.1.1 The graph Laplacian
2.1.2 Cluster-fires
2.2 Confluence
2.3 Stabilization
2.4 Toppling time
2.5 Stabilization with a sink
2.6 Long-term stable configurations
2.6.1 Criticality
2.6.2 Firing equivalence
2.6.3 Superstability
2.6.4 Energy minimization
2.6.5 Duality
2.6.6 Structure
2.6.7 Burning
2.7 The sandpile Markov chain
2.7.1 Avalanche operators
2.8 Exercises
3 Spanning Trees
3.1 Spanning trees
3.2 Statistics on trees
3.2.1 Level
3.2.2 Activity
3.2.3 The Tutte polynomial
3.3 Merino‘s theorem
3.3.1 The O-conjecture
3.4 Cori Le Bcrgne bijection
3.5 Acyclic orientations
3.5.1 Hyperplane arrangements
3.6 Parking functions
3.7 Dominoes
3.8 Avalanche polynomials
3.8.1 Avalanche polynomials of trees
3.9 Exercises
4 Sandpile Groups
4.1 Toppling dynamics
4.2 Group of chip-firing equivalence
4.3 Identity
4.4 Combinatorial invariance
4.5 Sandpile groups and invariant factors
4.5.1 Explicit forms of the sandpile group
4.5.2 Sandpile groups of random graphs
4.6 Discriminant groups
4.7 Sandpile tcrsors
4.7.1 Rotor-routing
4.7.2 Bernardi process
4.7.3 Cycle-cocycle reversal
4.8 Exercises
5 Pattern Formation
5.1 Compelling visualizations
5.2 Infinite graphs
5.3 The one-dimensional grid
5.4 Labeled chip-firing
5.5 Two and more dimensional grids
5.5.1 Odometer
5.5.2 Support
5.5.3 Backgrounds
5.5.3.1 Higher dimensions
5.5.4 Scaling limits
5.6 Other lattices
5.7 Tile identity element
5.8 Exercises
Ⅱ Extensions
6 Avalanche Finite Systems
6.1 M-matrices
6.2 Chip-firing on M-matrices
6.3 Stability
6.3.1 Superstability
6.3.2 Criticality
6.3.3 Energy minimization
6.3.4 Uniqueness
6.4 Burning
6.5 Directed graphs
6.5.1 Digraphs
6.5.2 Stabilization
6.5.3 Toppling time
6.5.4 Oriented spanning trees
6.6 Cartan matrices as M-matrices
6.7 M-pairings
6.8 Exercises
7 Higher Dimensions
7.1 Illustrative examples
7.2 Cell complexes
7.3 Combinatorial Laplacians
7.4 Chip-firing in higher dimensions
7.5 The sandpile group
7.6 Higher-dimensional trees
7.6.1 Enumeration of trees
7.7 Sandpile groups
7.7.1 Precise forms of sandpile groups
7.8 Cuts and flows
7.9 Stability
7.9.1 M-pairings
7.10 Exercises
8 Divisors
8.1 Divisors on curves
8.2 The Picard group and Abel-Jacobi theory
8.3 Riemann-Roch Theorems
8.3.1 The rank function
8.3.2 Proof
8.4 Torelli’s theorem
8.5 The Picg(G) torus
8.6 Metric graphs and tropical geometry
8.7 Arithmetic geometry
8.8 Riemann-Roch for lattices
8.9 Two-variable zeta-functions
8.10 Enumerating arithmetical structures
8.11 Exercises
9 Ideals
9.1 Toppling ideals
9.2 Tree ideals
9.3 Resolutions
9.3.1 Cellular resolutions
9.3.2 Betti numbers
9.4 Critical ideals
9.5 Riemann Roch for monomial ideals
9.6 Exercises
List of Figures
Bibliography
Index
編輯手記
Ⅰ Fundamentals
1 Introduction
1.1 A brief introduction
1.2 Origins and History
1.2.1 The abelian sandpile model
1.2.2 A combinatorial game
1.2.3 Abstract rewriting systems
2 Chip-firing on Finite Graphs
2.1 The chip-firing process
2.1.1 The graph Laplacian
2.1.2 Cluster-fires
2.2 Confluence
2.3 Stabilization
2.4 Toppling time
2.5 Stabilization with a sink
2.6 Long-term stable configurations
2.6.1 Criticality
2.6.2 Firing equivalence
2.6.3 Superstability
2.6.4 Energy minimization
2.6.5 Duality
2.6.6 Structure
2.6.7 Burning
2.7 The sandpile Markov chain
2.7.1 Avalanche operators
2.8 Exercises
3 Spanning Trees
3.1 Spanning trees
3.2 Statistics on trees
3.2.1 Level
3.2.2 Activity
3.2.3 The Tutte polynomial
3.3 Merino‘s theorem
3.3.1 The O-conjecture
3.4 Cori Le Bcrgne bijection
3.5 Acyclic orientations
3.5.1 Hyperplane arrangements
3.6 Parking functions
3.7 Dominoes
3.8 Avalanche polynomials
3.8.1 Avalanche polynomials of trees
3.9 Exercises
4 Sandpile Groups
4.1 Toppling dynamics
4.2 Group of chip-firing equivalence
4.3 Identity
4.4 Combinatorial invariance
4.5 Sandpile groups and invariant factors
4.5.1 Explicit forms of the sandpile group
4.5.2 Sandpile groups of random graphs
4.6 Discriminant groups
4.7 Sandpile tcrsors
4.7.1 Rotor-routing
4.7.2 Bernardi process
4.7.3 Cycle-cocycle reversal
4.8 Exercises
5 Pattern Formation
5.1 Compelling visualizations
5.2 Infinite graphs
5.3 The one-dimensional grid
5.4 Labeled chip-firing
5.5 Two and more dimensional grids
5.5.1 Odometer
5.5.2 Support
5.5.3 Backgrounds
5.5.3.1 Higher dimensions
5.5.4 Scaling limits
5.6 Other lattices
5.7 Tile identity element
5.8 Exercises
Ⅱ Extensions
6 Avalanche Finite Systems
6.1 M-matrices
6.2 Chip-firing on M-matrices
6.3 Stability
6.3.1 Superstability
6.3.2 Criticality
6.3.3 Energy minimization
6.3.4 Uniqueness
6.4 Burning
6.5 Directed graphs
6.5.1 Digraphs
6.5.2 Stabilization
6.5.3 Toppling time
6.5.4 Oriented spanning trees
6.6 Cartan matrices as M-matrices
6.7 M-pairings
6.8 Exercises
7 Higher Dimensions
7.1 Illustrative examples
7.2 Cell complexes
7.3 Combinatorial Laplacians
7.4 Chip-firing in higher dimensions
7.5 The sandpile group
7.6 Higher-dimensional trees
7.6.1 Enumeration of trees
7.7 Sandpile groups
7.7.1 Precise forms of sandpile groups
7.8 Cuts and flows
7.9 Stability
7.9.1 M-pairings
7.10 Exercises
8 Divisors
8.1 Divisors on curves
8.2 The Picard group and Abel-Jacobi theory
8.3 Riemann-Roch Theorems
8.3.1 The rank function
8.3.2 Proof
8.4 Torelli’s theorem
8.5 The Picg(G) torus
8.6 Metric graphs and tropical geometry
8.7 Arithmetic geometry
8.8 Riemann-Roch for lattices
8.9 Two-variable zeta-functions
8.10 Enumerating arithmetical structures
8.11 Exercises
9 Ideals
9.1 Toppling ideals
9.2 Tree ideals
9.3 Resolutions
9.3.1 Cellular resolutions
9.3.2 Betti numbers
9.4 Critical ideals
9.5 Riemann Roch for monomial ideals
9.6 Exercises
List of Figures
Bibliography
Index
編輯手記
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