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Divisors and Sandpiles: An Introduction to Chip-Firing

Author: Scott Corry

Publisher: American Mathematical Soc.

Published: 2018-07-23

Total Pages: 325

ISBN-13: 1470442183

Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces. Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book. Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, M-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.

The Mathematics of Chip-Firing

Author: Caroline J. Klivans

Publisher: CRC Press

Published: 2018-11-15

Total Pages: 296

ISBN-13: 135180099X

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The Mathematics of Chip-firing is a solid introduction and overview of the growing field of chip-firing. It offers an appreciation for the richness and diversity of the subject. Chip-firing refers to a discrete dynamical system — a commodity is exchanged between sites of a network according to very simple local rules. Although governed by local rules, the long-term global behavior of the system reveals fascinating properties. The Fundamental properties of chip-firing are covered from a variety of perspectives. This gives the reader both a broad context of the field and concrete entry points from different backgrounds. Broken into two sections, the first examines the fundamentals of chip-firing, while the second half presents more general frameworks for chip-firing. Instructors and students will discover that this book provides a comprehensive background to approaching original sources. Features: Provides a broad introduction for researchers interested in the subject of chip-firing The text includes historical and current perspectives Exercises included at the end of each chapter About the Author: Caroline J. Klivans received a BA degree in mathematics from Cornell University and a PhD in applied mathematics from MIT. Currently, she is an Associate Professor in the Division of Applied Mathematics at Brown University. She is also an Associate Director of ICERM (Institute for Computational and Experimental Research in Mathematics). Before coming to Brown she held positions at MSRI, Cornell and the University of Chicago. Her research is in algebraic, geometric and topological combinatorics.

A Project-Based Guide to Undergraduate Research in Mathematics

Author: Pamela E. Harris

Publisher: Springer Nature

Published: 2020-04-17

Total Pages: 324

ISBN-13: 3030378535

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This volume provides accessible and self-contained research problems designed for undergraduate student projects, and simultaneously promotes the development of sustainable undergraduate research programs. The chapters in this work span a variety of topical areas of pure and applied mathematics and mathematics education. Each chapter gives a self-contained introduction on a research topic with an emphasis on the specific tools and knowledge needed to create and maintain fruitful research programs for undergraduates. Some of the topics discussed include:• Disease modeling• Tropical curves and surfaces• Numerical semigroups• Mathematics EducationThis volume will primarily appeal to undergraduate students interested in pursuing research projects and faculty members seeking to mentor them. It may also aid students and faculty participating in independent studies and capstone projects.

SOFSEM 2019: Theory and Practice of Computer Science

Author: Barbara Catania

Publisher: Springer

Published: 2019-01-10

Total Pages: 548

ISBN-13: 3030108015

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This book constitutes the refereed proceedings of the 45th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2019, held in Nový Smokovec, Slovakia, in January 2019. The 34 full papers presented together with 6 invited talks were carefully reviewed and selected from 92 submissions. They presented new research results in the theory and practice of computer science in the each sub-area of SOFSEM 2019: Foundations of theoretical Computer Science, foundations of data science and engineering, and foundations of software engineering.

The Mathematics of Chip-Firing

Author: Caroline J. Klivans

Publisher: CRC Press

Published: 2018-11-15

Total Pages: 203

ISBN-13: 1351801007

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Tropical and Non-Archimedean Geometry

Author: Omid Amini

Publisher: American Mathematical Soc.

Published: 2014-12-26

Total Pages: 274

ISBN-13: 1470410214

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Over the past decade, it has become apparent that tropical geometry and non-Archimedean geometry should be studied in tandem; each subject has a great deal to say about the other. This volume is a collection of articles dedicated to one or both of these disciplines. Some of the articles are based, at least in part, on the authors' lectures at the 2011 Bellairs Workshop in Number Theory, held from May 6-13, 2011, at the Bellairs Research Institute, Holetown, Barbados. Lecture topics covered in this volume include polyhedral structures on tropical varieties, the structure theory of non-Archimedean curves (algebraic, analytic, tropical, and formal), uniformisation theory for non-Archimedean curves and abelian varieties, and applications to Diophantine geometry. Additional articles selected for inclusion in this volume represent other facets of current research and illuminate connections between tropical geometry, non-Archimedean geometry, toric geometry, algebraic graph theory, and algorithmic aspects of systems of polynomial equations.

Deterministic Abelian Sandpile Models and Patterns

Author: Guglielmo Paoletti

Publisher: Springer Science & Business Media

Published: 2013-09-16

Total Pages: 171

ISBN-13: 3319012045

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The model investigated in this work, a particular cellular automaton with stochastic evolution, was introduced as the simplest case of self-organized-criticality, that is, a dynamical system which shows algebraic long-range correlations without any tuning of parameters. The author derives exact results which are potentially also interesting outside the area of critical phenomena. Exact means also site-by-site and not only ensemble average or coarse graining. Very complex and amazingly beautiful periodic patterns are often generated by the dynamics involved, especially in deterministic protocols in which the sand is added at chosen sites. For example, the author studies the appearance of allometric structures, that is, patterns which grow in the same way in their whole body, and not only near their boundaries, as commonly occurs. The local conservation laws which govern the evolution of these patterns are also presented. This work has already attracted interest, not only in non-equilibrium statistical mechanics, but also in mathematics, both in probability and in combinatorics. There are also interesting connections with number theory. Lastly, it also poses new questions about an old subject. As such, it will be of interest to computer practitioners, demonstrating the simplicity with which charming patterns can be obtained, as well as to researchers working in many other areas.

Computing and Combinatorics

Author: Donghyun Kim

Publisher: Springer Nature

Published: 2020-08-27

Total Pages: 678

ISBN-13: 3030581500

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This book constitutes the proceedings of the 26th International Conference on Computing and Combinatorics, COCOON 2020, held in Atlanta, GA, USA, in August 2020. Due to the COVID-19 pandemic COCOON 2020 was organized as a fully online conference. The 54 papers presented in this volume were carefully reviewed and selected from 126 submissions. The papers cover various topics, including algorithm design, approximation algorithm, graph theory, complexity theory, problem solving, optimization, computational biology, computational learning, communication network, logic, and game theory.

Leavitt Path Algebras

Author: Gene Abrams

Publisher: Springer

Published: 2017-11-30

Total Pages: 289

ISBN-13: 1447173449

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This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and symbolic dynamics. With its descriptive writing style, this book is highly accessible.

The Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me About Ultimate Reality, The Meaning of Life, And How to Be Happy

Author: Rudy Rucker

Publisher:

Published: 2016-10-31

Total Pages: 470

ISBN-13: 9781940948256

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A playful and profound survey of the concept of computation across the entire spectrum of human thought-written by a mathematician novelist who spent twenty years as a Silicon Valley computer scientist. The logic is correct, and the conclusions are startling. Simple rules can generate gnarly patterns. Physics obeys laws, but the outcomes aren't predictable. Free will is real. The mind is like a quantum computer. Social strata are skewed by universal scaling laws. And there can never be a simple trick for answering all possible questions about our world's natural processes. We live amid splendor beyond our control.