Zeta Functions Of Graphs

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Zeta Functions of Graphs

Audrey Terras
Zeta Functions of Graphs Book PDF

Author : Audrey Terras
Publisher : Cambridge University Press
Page : 253 pages
File Size : 40,5 Mb
Release : 2010-11-18
Category : Mathematics
ISBN : 9781139491785

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Zeta Functions of Graphs by Audrey Terras Pdf

Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

Zeta Functions of Graphs

Audrey Terras
Zeta Functions of Graphs Book PDF

Author : Audrey Terras
Publisher : Unknown
Page : 252 pages
File Size : 44,7 Mb
Release : 2010
Category : Electronic book
ISBN : 0511913117

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Zeta Functions of Graphs by Audrey Terras Pdf

Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

Automorphic Forms and Geometry of Arithmetic Varieties

K. Hashimoto,Y. Namikawa
Automorphic Forms and Geometry of Arithmetic Varieties Book PDF

Author : K. Hashimoto,Y. Namikawa
Publisher : Academic Press
Page : 570 pages
File Size : 55,9 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781483218076

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Automorphic Forms and Geometry of Arithmetic Varieties by K. Hashimoto,Y. Namikawa Pdf

Automorphic Forms and Geometry of Arithmetic Varieties deals with the dimension formulas of various automorphic forms and the geometry of arithmetic varieties. The relation between two fundamental methods of obtaining dimension formulas (for cusp forms), the Selberg trace formula and the index theorem (Riemann-Roch's theorem and the Lefschetz fixed point formula), is examined. Comprised of 18 sections, this volume begins by discussing zeta functions associated with cones and their special values, followed by an analysis of cusps on Hilbert modular varieties and values of L-functions. The reader is then introduced to the dimension formula of Siegel modular forms; the graded rings of modular forms in several variables; and Selberg-Ihara's zeta function for p-adic discrete groups. Subsequent chapters focus on zeta functions of finite graphs and representations of p-adic groups; invariants and Hodge cycles; T-complexes and Ogata's zeta zero values; and the structure of the icosahedral modular group. This book will be a useful resource for mathematicians and students of mathematics.

Zeta and L -functions in Number Theory and Combinatorics

Wen-Ching Winnie Li
Zeta and L functions in Number Theory and Combinatorics Book PDF

Author : Wen-Ching Winnie Li
Publisher : American Mathematical Soc.
Page : 95 pages
File Size : 47,8 Mb
Release : 2019-03-01
Category : Combinatorial number theory
ISBN : 9781470449001

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Zeta and L -functions in Number Theory and Combinatorics by Wen-Ching Winnie Li Pdf

Zeta and L-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and L-functions as a central theme. It provides a systematic and comprehensive account of these functions in a combinatorial setting and establishes, among other things, the combinatorial counterparts of celebrated results in number theory, such as the prime number theorem and the Chebotarev density theorem. The spectral theory for finite graphs and higher dimensional complexes is studied. Of special interest in theory and applications are the spectrally extremal objects, called Ramanujan graphs and Ramanujan complexes, which can be characterized by their associated zeta functions satisfying the Riemann Hypothesis. Explicit constructions of these extremal combinatorial objects, using number-theoretic and combinatorial means, are presented. Research on zeta and L-functions for complexes other than graphs emerged only in recent years. This is the first book for graduate students and researchers offering deep insight into this fascinating and fast developing area.

Prime Numbers and the Riemann Hypothesis

Barry Mazur,William Stein
Prime Numbers and the Riemann Hypothesis Book PDF

Author : Barry Mazur,William Stein
Publisher : Cambridge University Press
Page : 155 pages
File Size : 48,7 Mb
Release : 2016-04-11
Category : Mathematics
ISBN : 9781107101920

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Prime Numbers and the Riemann Hypothesis by Barry Mazur,William Stein Pdf

This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.

Handbook of Mathematical Functions

Milton Abramowitz,Irene A. Stegun
Handbook of Mathematical Functions Book PDF

Author : Milton Abramowitz,Irene A. Stegun
Publisher : Courier Corporation
Page : 1068 pages
File Size : 53,5 Mb
Release : 1965-01-01
Category : Mathematics
ISBN : 0486612724

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Handbook of Mathematical Functions by Milton Abramowitz,Irene A. Stegun Pdf

An extensive summary of mathematical functions that occur in physical and engineering problems

Lectures on the Riemann Zeta Function

H. Iwaniec
Lectures on the Riemann Zeta Function Book PDF

Author : H. Iwaniec
Publisher : American Mathematical Society
Page : 119 pages
File Size : 47,5 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9781470418519

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Lectures on the Riemann Zeta Function by H. Iwaniec Pdf

The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

Series Associated With the Zeta and Related Functions

Hari M. Srivastava,Junesang Choi
Series Associated With the Zeta and Related Functions Book PDF

Author : Hari M. Srivastava,Junesang Choi
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 49,5 Mb
Release : 2001
Category : Mathematics
ISBN : 0792370546

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Series Associated With the Zeta and Related Functions by Hari M. Srivastava,Junesang Choi Pdf

In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s,a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions.

The Riemann Zeta-Function

Anatoly A. Karatsuba,S. M. Voronin
The Riemann Zeta Function Book PDF

Author : Anatoly A. Karatsuba,S. M. Voronin
Publisher : Walter de Gruyter
Page : 409 pages
File Size : 41,5 Mb
Release : 2011-05-03
Category : Mathematics
ISBN : 9783110886146

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The Riemann Zeta-Function by Anatoly A. Karatsuba,S. M. Voronin Pdf

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Emerging Applications of Number Theory

Dennis A. Hejhal,Joel Friedman,Martin C. Gutzwiller,Andrew M. Odlyzko
Emerging Applications of Number Theory Book PDF

Author : Dennis A. Hejhal,Joel Friedman,Martin C. Gutzwiller,Andrew M. Odlyzko
Publisher : Springer Science & Business Media
Page : 693 pages
File Size : 42,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461215448

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Emerging Applications of Number Theory by Dennis A. Hejhal,Joel Friedman,Martin C. Gutzwiller,Andrew M. Odlyzko Pdf

Most people tend to view number theory as the very paradigm of pure mathematics. With the advent of computers, however, number theory has been finding an increasing number of applications in practical settings, such as in cryptography, random number generation, coding theory, and even concert hall acoustics. Yet other applications are still emerging - providing number theorists with some major new areas of opportunity. The 1996 IMA summer program on Emerging Applications of Number Theory was aimed at stimulating further work with some of these newest (and most attractive) applications. Concentration was on number theory's recent links with: (a) wave phenomena in quantum mechanics (more specifically, quantum chaos); and (b) graph theory (especially expander graphs and related spectral theory). This volume contains the contributed papers from that meeting and will be of interest to anyone intrigued by novel applications of modern number-theoretical techniques.

Fourier Analysis on Finite Groups and Applications

Audrey Terras
Fourier Analysis on Finite Groups and Applications Book PDF

Author : Audrey Terras
Publisher : Cambridge University Press
Page : 456 pages
File Size : 40,7 Mb
Release : 1999-03-28
Category : Mathematics
ISBN : 0521457181

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Fourier Analysis on Finite Groups and Applications by Audrey Terras Pdf

It examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research.

The Mathematics of Chip-Firing

Caroline J. Klivans
The Mathematics of Chip Firing Book PDF

Author : Caroline J. Klivans
Publisher : CRC Press
Page : 296 pages
File Size : 40,7 Mb
Release : 2018-11-15
Category : Computers
ISBN : 9781351800990

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The Mathematics of Chip-Firing by Caroline J. Klivans Pdf

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.

The Riemann Hypothesis

Peter B. Borwein
The Riemann Hypothesis Book PDF

Author : Peter B. Borwein
Publisher : Springer Science & Business Media
Page : 543 pages
File Size : 52,7 Mb
Release : 2008
Category : Mathematics
ISBN : 9780387721255

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The Riemann Hypothesis by Peter B. Borwein Pdf

The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors." The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

Topological Crystallography

Toshikazu Sunada
Topological Crystallography Book PDF

Author : Toshikazu Sunada
Publisher : Springer Science & Business Media
Page : 229 pages
File Size : 53,7 Mb
Release : 2012-12-23
Category : Mathematics
ISBN : 9784431541776

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Topological Crystallography by Toshikazu Sunada Pdf

Geometry in ancient Greece is said to have originated in the curiosity of mathematicians about the shapes of crystals, with that curiosity culminating in the classification of regular convex polyhedra addressed in the final volume of Euclid’s Elements. Since then, geometry has taken its own path and the study of crystals has not been a central theme in mathematics, with the exception of Kepler’s work on snowflakes. Only in the nineteenth century did mathematics begin to play a role in crystallography as group theory came to be applied to the morphology of crystals. This monograph follows the Greek tradition in seeking beautiful shapes such as regular convex polyhedra. The primary aim is to convey to the reader how algebraic topology is effectively used to explore the rich world of crystal structures. Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal. For that reason the title Topological Crystallography has been chosen. Topological crystals can be described as “living in the logical world, not in space,” leading to the question of how to place or realize them “canonically” in space. Proposed here is the notion of standard realizations of topological crystals in space, including as typical examples the crystal structures of diamond and lonsdaleite. A mathematical view of the standard realizations is also provided by relating them to asymptotic behaviors of random walks and harmonic maps. Furthermore, it can be seen that a discrete analogue of algebraic geometry is linked to the standard realizations. Applications of the discussions in this volume include not only a systematic enumeration of crystal structures, an area of considerable scientific interest for many years, but also the architectural design of lightweight rigid structures. The reader therefore can see the agreement of theory and practice.

Introduction to Quantum Graphs

Gregory Berkolaiko,Peter Kuchment
Introduction to Quantum Graphs Book PDF

Author : Gregory Berkolaiko,Peter Kuchment
Publisher : American Mathematical Soc.
Page : 291 pages
File Size : 46,5 Mb
Release : 2013
Category : Mathematics
ISBN : 9780821892114

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Introduction to Quantum Graphs by Gregory Berkolaiko,Peter Kuchment Pdf

A ``quantum graph'' is a graph considered as a one-dimensional complex and equipped with a differential operator (``Hamiltonian''). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., ``meso-'' or ``nano-scale'') system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician's heart. Work on quantum graphs has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications.