November 14, 2018 Forthcoming
Reference - 312 Pages - 115 B/W Illustrations
ISBN 9781138070820 - CAT# K33657
Series: Discrete Mathematics and Its Applications
Provides a broad introduction for researchers interested in the subject of chip-firing
The text provides historical and current perspectives
Exercises included at the end of each chapter
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.
Introduction
A brief introduction. Origins/History.
Chip-firing on Finite Graphs
The chip-firing process. Confluence. Stabilization. Toppling time. Stabilization with a sink. Long-term stable configurations. The sandpile Markov chain.
Spanning Trees
Spanning trees. Statistics on Trees. Merino’s Theorem. Cori-Le Borgne bijection. Acyclic orientations. Parking functions. Dominoes. Avalanche polynomials.
Sandpile Groups
Toppling dynamics. Group of chip-firing equivalence. Identity. Combinatorial invariance. Sandpile groups and invariant factors. Discriminant groups. Sandpile torsors.
Pattern Formation
Compelling visualizations. Infinite graphs. The one-dimensional grid. Labeled chip-firing. Two and more dimensional grids. Other lattices. The identity element.
Avalanche Finite Systems
M-matrices. Chip-firing on M-matrices. Stability. Burning. Directed graphs. Cartan matrices as M-matrices. M-pairings.
Higher Dimensions
An illustrative example. Cell complexes. Combinatorial Laplacians. Chip-firing in higher dimensions. The sandpile group. Higher-dimensional trees. Sandpile groups. Cuts and flows. Stability.
Divisors
Divisors on curves. The Picard group and Abel-Jacobi theory. Riemann-Roch Theorems. Torelli’s Theorem. The Pic^g (G) torus. Metric graphs and tropical geometry. Arithmetic geometry. Arithmetical graphs. Riemann-Roch for lattices. Two variable zeta-functions. Enumerating arithmetical structures.
Ideals
Ideals. Toppling ideals. Tree ideals. Resolutions. Critical ideals. Riemann-Roch for monomial ideals.
Hardback
January 9, 2019 Forthcoming
Textbook - 611 Pages
ISBN 9781138369917
Series: Chapman & Hall/CRC Texts in Statistical Science
Explains concepts in terms of an interpretation rather than through brute force calculations
Explains probability to people who want to learn statistics
Includes important concepts in stochastic processes including Markov chains, martingales, and Poisson processes
Explains how to run simulations, make visualizations, and do statistical calculations with R
Interesting modern applications such as Google PageRank, applications of MCMC, a few recent legal cases where probability played a role, some finance examples, an example based on Efron and Thisted's work on determining whether a newly discovered sonnet was written by Shakespeare, and various Bayesian statistics examples
Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory.
The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces.
The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment.
The second edition adds many new examples, exercises, and explanations, to deepen understanding of the ideas, clarify subtle concepts, and respond to feedback from many students and readers. New supplementary online resources have been developed, including animations and interactive visualizations, and the book has been updated to dovetail with these resources.
Unsolved Problems and Results
Together with University of Science & Technology
Key problems and conjectures have played an important role in promoting the
development of Ramsey theory, a field where great progress has been made
during the past two decades, with some old problems solved and many new
problems proposed. The present book will be helpful to readers who wish to
learn about interesting problems in Ramsey theory, to see how they are
interconnected, and then to study them in depth. This book is the first problem
book of such scope in Ramsey theory. Many unsolved problems, conjectures
and related partial results in Ramsey theory are presented, in areas such as
extremal graph theory, additive number theory, discrete geometry, functional
analysis, algorithm design, and in other areas. Most presented problems are easy
to understand, but they may be difficult to solve. They can be appreciated on
many levels and by a wide readership, ranging from undergraduate students
majoring in mathematics to research mathematicians. This collection is an
essential reference for mathematicians working in combinatorics and number
theory, as well as for computer scientists studying algorithms.
Some definitions and notations
Ramsey theory
Bi-color diagonal classical Ramsey numbers
Paley graphs and lower bounds for R(k, k)
Bi-color off-diagonal classical Ramsey numbers
Multicolor classical Ramsey numbers
Generalized Ramsey numbers
Folkman numbers
The Erdos-Hajnal conjecture
Other Ramsey-type problems in graph theory
On van der Waerden numbers and Szemeredi’s theorem
More problems of Ramsey type in additive number theory
Sidon-Ramsey numbers
xii, 178 pages, 10 Figures (bw),
20 Schedule (bw)
Hardcover:
ISBN 978-3-11-057651-1
Date of Publication: August 2018
Language of Publication: English
Subjects:
Algebra and Number Theory
Of interest to: Researchers and graduate students in mathematics and computer science.
This comprehensive two-volume work is devoted to the most general beginnings
of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927),
where set theory and the theory of functions were expounded as the fundamental
parts of mathematics in such a way that there was no need for references to other
sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and
integration theory is also included here as the third fundamental part of
contemporary mathematics.The material about sets and numbers is placed in
Volume 1 and the material about functions and measures is placed in Volume 2.
Fundamentals of the theory of classes, sets, and numbers
Characterization of all natural models of Neumann ? Bernays ? Godel and
Zermelo ? Fraenkel set theories
Local theory of sets as a foundation for category theory and its connection with
the Zermelo - Fraenkel set theory
Compactness theorem for generalized second-order language
* A comprehensive, in-depth presentation of the fundamental parts of
mathematics.
* This first volume covers logic, set theory and number systems in detail.
* Of interest to graduate students and researchers in mathematics.
De Gruyter Studies in Mathematics 68/1+2
448 pages
Hardcover:
ISBN 978-3-11-061034-5
Date of Publication: August 2018
Language of Publication: English
Subjects: Analysis
Logic and Set Theory
Of interest to: Graduate students and researchers in mathematics.
This book provides the first systematic treatment of modules over discrete
valuation domains, which play an important role in various areas of algebra,
especially in commutative algebra. Many important results representing the state
of the art are presented in the text along with interesting open problems. This
updated edition presents new approaches on p -adic integers and modules, and on
the determinability of a module by its automorphism group.
* An important contribution to commutative algebra.
* Considers all aspects of modules over discrete valuations domains.
* Updated edition with new approaches on p -adic integers and modules, and the
determinability of a module by its automorphism group.
De Gruyter Expositions in Mathematics 43
2nd, rev. and ext., vi, 331 pages
Hardcover:
ISBN 978-3-11-060977-6
Date of Publication: September 2018
Language of Publication: English
Subjects: Algebra and Number Theory
Of interest to: Researchers and graduate students in mathematics.
EMS Tracts in Mathematics Vol. 30
ISBN print 978-3-03719-190-3, ISBN online 978-3-03719-690-8
DOI 10.4171/190
September 2018, 441 pages, hardcover, 17 x 24 cm.
The present book is a detailed exposition of the author and his collaborators’ work on boundedness, continuity, and differentiability properties of solutions to elliptic equations in general domains, that is, in domains that are not a priori restricted by assumptions such as “piecewise smoothness” or being a “Lipschitz graph”. The description of the boundary behavior of such solutions is one of the most difficult problems in the theory of partial differential equations. After the famous Wiener test, the main contributions to this area were made by the author. In particular, necessary and sufficient conditions for the validity of imbedding theorems are given, which provide criteria for the unique solvability of boundary value problems of second and higher order elliptic equations. Another striking result is a test for the regularity of a boundary point for polyharmonic equations.
The book will be interesting and useful for a wide audience. It is intended for specialists and graduate students working in the theory of partial differential equations.
Keywords: Wiener test, higher order elliptic equations, elasticity systems, Zaremba problem, weighted positivity, capacity
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Editions
Paperback 2019 24.95 20.00
ISBN 9780691182766
272 pp. 0 x 0 25 color + 57 b/w illus. 2 tables.
forthcoming December 2018
The year’s finest mathematical writing from around the world
This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2018 makes available to a wide audience many pieces not easily found anywhere else?and you don’t need to be a mathematician to enjoy them. These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice?and taking readers behind the scenes of today’s hottest mathematical debates.
James Grime shows how to build subtly mischievous dice for playing slightly unfair games, David Rowe investigates the many different meanings and pedigrees of mathematical models, and Michael Barany traces how our appreciation of the societal importance of mathematics has developed since World War II. In other essays, Francis Su extolls the inherent values of learning, doing, and sharing mathematics, and Margaret Wertheim takes us on a mathematical exploration of the mind and the world?with glimpses at science, philosophy, music, art, and even crocheting. And there’s much, much more.
In addition to presenting the year’s most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable pieces on mathematics.
This is a must-read for anyone interested in where math has taken us?and where it is headed.
Mircea Pitici teaches advanced calculus at Syracuse University. He has a PhD in mathematics education from Cornell University and is working on a master’s degree in library and information science at Syracuse’s iSchool. He has edited The Best Writing on Mathematics since 2010 and lives in Ithaca, New York.
“A variety of thoroughly accessible works that tie abstract math to the real world. . . . [G]ives readers an entertaining look at the odd, the amusing, and the utilitarian without requiring any more than a readerly curiosity.”?Publishers Weekly
“Wonderful…. [C]annot be recommended highly enough!”?Robert Schaefer, New York Journal of Books