Inevitable Randomness In Discrete Mathematics

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Inevitable Randomness in Discrete Mathematics

J—zsef Beck
Inevitable Randomness in Discrete Mathematics Book PDF

Author : J—zsef Beck
Publisher : American Mathematical Soc.
Page : 267 pages
File Size : 44,7 Mb
Release : 2009-09-01
Category : Mathematics
ISBN : 9780821847565

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Inevitable Randomness in Discrete Mathematics by J—zsef Beck Pdf

Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the $3n+1$ conjecture, and class field theory. The purpose of this book is to provide examples--and rigorous proofs--of the complexity law: (1) discrete systems are either simple or they exhibit advanced pseudorandomness; (2) a priori probabilities often exist even when there is no intrinsic symmetry. Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics. This book considers randomness and complexity. The traditional approach to complexity--computational complexity theory--is to study very general complexity classes, such as P, NP and PSPACE. What Beck does is very different: he studies interesting concrete systems, which can give new insights into the mystery of complexity. The book is divided into three parts. Part A is mostly an essay on the big picture. Part B is partly new results and partly a survey of real game theory. Part C contains new results about graph games, supporting the main conjecture. To make it accessible to a wide audience, the book is mostly self-contained.

Probabilistic Diophantine Approximation

József Beck
Probabilistic Diophantine Approximation Book PDF

Author : József Beck
Publisher : Springer
Page : 487 pages
File Size : 41,8 Mb
Release : 2014-10-06
Category : Mathematics
ISBN : 9783319107417

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Probabilistic Diophantine Approximation by József Beck Pdf

This book gives a comprehensive treatment of random phenomena and distribution results in diophantine approximation, with a particular emphasis on quadratic irrationals. It covers classical material on the subject as well as many new results developed by the author over the past decade. A range of ideas from other areas of mathematics are brought to bear with surprising connections to topics such as formulae for class numbers, special values of L-functions, and Dedekind sums. Care is taken to elaborate difficult proofs by motivating major steps and accompanying them with background explanations, enabling the reader to learn the theory and relevant techniques. Written by one of the acknowledged experts in the field, Probabilistic Diophantine Approximation is presented in a clear and informal style with sufficient detail to appeal to both advanced students and researchers in number theory.

Positional Games

Dan Hefetz,Michael Krivelevich,Miloš Stojaković,Tibor Szabó
Positional Games Book PDF

Author : Dan Hefetz,Michael Krivelevich,Miloš Stojaković,Tibor Szabó
Publisher : Springer
Page : 146 pages
File Size : 55,5 Mb
Release : 2014-06-13
Category : Mathematics
ISBN : 9783034808255

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Positional Games by Dan Hefetz,Michael Krivelevich,Miloš Stojaković,Tibor Szabó Pdf

This text is based on a lecture course given by the authors in the framework of Oberwolfach Seminars at the Mathematisches Forschungsinstitut Oberwolfach in May, 2013. It is intended to serve as a thorough introduction to the rapidly developing field of positional games. This area constitutes an important branch of combinatorics, whose aim it is to systematically develop an extensive mathematical basis for a variety of two player perfect information games. These ranges from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. These notes cover a variety of topics in positional games, including both classical results and recent important developments. They are presented in an accessible way and are accompanied by exercises of varying difficulty, helping the reader to better understand the theory. The text will benefit both researchers and graduate students in combinatorics and adjacent fields.

LATIN 2012: Theoretical Informatics

David Fernández-Baca
LATIN 2012 Theoretical Informatics Book PDF

Author : David Fernández-Baca
Publisher : Springer
Page : 669 pages
File Size : 46,9 Mb
Release : 2012-04-10
Category : Computers
ISBN : 9783642293443

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LATIN 2012: Theoretical Informatics by David Fernández-Baca Pdf

This book constitutes the proceedings of the 10th Latin American Symposium on Theoretical Informatics, LATIN 2012, held in Arequipa, Peru, in April 2012. The 55 papers presented in this volume were carefully reviewed and selected from 153 submissions. The papers address a variety of topics in theoretical computer science with a certain focus on algorithms, automata theory and formal languages, coding theory and data compression, algorithmic graph theory and combinatorics, complexity theory, computational algebra, computational biology, computational geometry, computational number theory, cryptography, theoretical aspects of databases and information retrieval, data structures, networks, logic in computer science, machine learning, mathematical programming, parallel and distributed computing, pattern matching, quantum computing and random structures.

A Panorama of Discrepancy Theory

William Chen,Anand Srivastav,Giancarlo Travaglini
A Panorama of Discrepancy Theory Book PDF

Author : William Chen,Anand Srivastav,Giancarlo Travaglini
Publisher : Springer
Page : 695 pages
File Size : 54,5 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9783319046969

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A Panorama of Discrepancy Theory by William Chen,Anand Srivastav,Giancarlo Travaglini Pdf

This is the first work on Discrepancy Theory to show the present variety of points of view and applications covering the areas Classical and Geometric Discrepancy Theory, Combinatorial Discrepancy Theory and Applications and Constructions. It consists of several chapters, written by experts in their respective fields and focusing on the different aspects of the theory. Discrepancy theory concerns the problem of replacing a continuous object with a discrete sampling and is currently located at the crossroads of number theory, combinatorics, Fourier analysis, algorithms and complexity, probability theory and numerical analysis. This book presents an invitation to researchers and students to explore the different methods and is meant to motivate interdisciplinary research.

Random Trees

Michael Drmota
Random Trees Book PDF

Author : Michael Drmota
Publisher : Springer Science & Business Media
Page : 466 pages
File Size : 50,7 Mb
Release : 2009-04-16
Category : Mathematics
ISBN : 9783211753576

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Random Trees by Michael Drmota Pdf

The aim of this book is to provide a thorough introduction to various aspects of trees in random settings and a systematic treatment of the mathematical analysis techniques involved. It should serve as a reference book as well as a basis for future research.

Real Solutions to Equations from Geometry

Frank Sottile
Real Solutions to Equations from Geometry Book PDF

Author : Frank Sottile
Publisher : American Mathematical Soc.
Page : 214 pages
File Size : 45,7 Mb
Release : 2011-08-31
Category : Mathematics
ISBN : 9780821853313

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Real Solutions to Equations from Geometry by Frank Sottile Pdf

Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a difficult problem with many applications outside of mathematics. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure often coming from geometry. This book focuses on equations from toric varieties and Grassmannians. Not only is much known about these, but such equations are common in applications. There are three main themes: upper bounds on the number of real solutions, lower bounds on the number of real solutions, and geometric problems that can have all solutions be real. The book begins with an overview, giving background on real solutions to univariate polynomials and the geometry of sparse polynomial systems. The first half of the book concludes with fewnomial upper bounds and with lower bounds to sparse polynomial systems. The second half of the book begins by sampling some geometric problems for which all solutions can be real, before devoting the last five chapters to the Shapiro Conjecture, in which the relevant polynomial systems have only real solutions.

Koszul Cohomology and Algebraic Geometry

Marian Aprodu,Jan Nagel
Koszul Cohomology and Algebraic Geometry Book PDF

Author : Marian Aprodu,Jan Nagel
Publisher : American Mathematical Soc.
Page : 138 pages
File Size : 47,5 Mb
Release : 2010
Category : Geometry, Algebraic
ISBN : 9780821849644

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Koszul Cohomology and Algebraic Geometry by Marian Aprodu,Jan Nagel Pdf

The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

Combinatorial Convexity

Imre Bárány
Combinatorial Convexity Book PDF

Author : Imre Bárány
Publisher : American Mathematical Soc.
Page : 148 pages
File Size : 43,9 Mb
Release : 2021-11-04
Category : Education
ISBN : 9781470467098

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Combinatorial Convexity by Imre Bárány Pdf

This book is about the combinatorial properties of convex sets, families of convex sets in finite dimensional Euclidean spaces, and finite points sets related to convexity. This area is classic, with theorems of Helly, Carathéodory, and Radon that go back more than a hundred years. At the same time, it is a modern and active field of research with recent results like Tverberg's theorem, the colourful versions of Helly and Carathéodory, and the (p,q) (p,q) theorem of Alon and Kleitman. As the title indicates, the topic is convexity and geometry, and is close to discrete mathematics. The questions considered are frequently of a combinatorial nature, and the proofs use ideas from geometry and are often combined with graph and hypergraph theory. The book is intended for students (graduate and undergraduate alike), but postdocs and research mathematicians will also find it useful. It can be used as a textbook with short chapters, each suitable for a one- or two-hour lecture. Not much background is needed: basic linear algebra and elements of (hyper)graph theory as well as some mathematical maturity should suffice.

The Invariant Theory of Matrices

Corrado De Concini,Claudio Procesi
The Invariant Theory of Matrices Book PDF

Author : Corrado De Concini,Claudio Procesi
Publisher : American Mathematical Soc.
Page : 153 pages
File Size : 42,6 Mb
Release : 2017-11-16
Category : Invariants
ISBN : 9781470441876

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The Invariant Theory of Matrices by Corrado De Concini,Claudio Procesi Pdf

This book gives a unified, complete, and self-contained exposition of the main algebraic theorems of invariant theory for matrices in a characteristic free approach. More precisely, it contains the description of polynomial functions in several variables on the set of matrices with coefficients in an infinite field or even the ring of integers, invariant under simultaneous conjugation. Following Hermann Weyl's classical approach, the ring of invariants is described by formulating and proving (1) the first fundamental theorem that describes a set of generators in the ring of invariants, and (2) the second fundamental theorem that describes relations between these generators. The authors study both the case of matrices over a field of characteristic 0 and the case of matrices over a field of positive characteristic. While the case of characteristic 0 can be treated following a classical approach, the case of positive characteristic (developed by Donkin and Zubkov) is much harder. A presentation of this case requires the development of a collection of tools. These tools and their application to the study of invariants are exlained in an elementary, self-contained way in the book.

Generalized Ricci Flow

Mario Garcia-Fernandez,Jeffrey Streets
Generalized Ricci Flow Book PDF

Author : Mario Garcia-Fernandez,Jeffrey Streets
Publisher : American Mathematical Soc.
Page : 248 pages
File Size : 49,5 Mb
Release : 2021-04-06
Category : Education
ISBN : 9781470462581

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Generalized Ricci Flow by Mario Garcia-Fernandez,Jeffrey Streets Pdf

The generalized Ricci flow is a geometric evolution equation which has recently emerged from investigations into mathematical physics, Hitchin's generalized geometry program, and complex geometry. This book gives an introduction to this new area, discusses recent developments, and formulates open questions and conjectures for future study. The text begins with an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as ‘canonical metrics’ in generalized Riemannian and complex geometry. The book then introduces generalized Ricci flow as a tool for constructing such metrics and proves extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kähler-Ricci flow, leading to global convergence results and applications to complex geometry. Finally, the book gives a purely mathematical introduction to the physical idea of T-duality and discusses its relationship to generalized Ricci flow. The book is suitable for graduate students and researchers with a background in Riemannian and complex geometry who are interested in the theory of geometric evolution equations.

Noncommutative Motives

Gonçalo Tabuada
Noncommutative Motives Book PDF

Author : Gonçalo Tabuada
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 40,7 Mb
Release : 2015-09-21
Category : Algebraic varieties
ISBN : 9781470423971

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Noncommutative Motives by Gonçalo Tabuada Pdf

The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a "universal cohomology theory of algebraic varieties". The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a "universal invariant of noncommutative algebraic varieties". This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.

Polynomial Methods in Combinatorics

Larry Guth
Polynomial Methods in Combinatorics Book PDF

Author : Larry Guth
Publisher : American Mathematical Soc.
Page : 273 pages
File Size : 45,9 Mb
Release : 2016-06-10
Category : Combinatorial analysis
ISBN : 9781470428907

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Polynomial Methods in Combinatorics by Larry Guth Pdf

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

Robert Steinberg

Robert Steinberg
Robert Steinberg Book PDF

Author : Robert Steinberg
Publisher : American Mathematical Soc.
Page : 160 pages
File Size : 47,7 Mb
Release : 2016-12-22
Category : Algebraic geometry -- Algebraic groups -- Algebraic groups
ISBN : 9781470431051

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Robert Steinberg by Robert Steinberg Pdf

Robert Steinberg's Lectures on Chevalley Groups were delivered and written during the author's sabbatical visit to Yale University in the 1967–1968 academic year. The work presents the status of the theory of Chevalley groups as it was in the mid-1960s. Much of this material was instrumental in many areas of mathematics, in particular in the theory of algebraic groups and in the subsequent classification of finite groups. This posthumous edition incorporates additions and corrections prepared by the author during his retirement, including a new introductory chapter. A bibliography and editorial notes have also been added.

Function Theory and ℓp Spaces

Raymond Cheng,Javad Mashreghi,William T. Ross
Function Theory and p Spaces Book PDF

Author : Raymond Cheng,Javad Mashreghi,William T. Ross
Publisher : American Mathematical Soc.
Page : 219 pages
File Size : 55,5 Mb
Release : 2020-05-28
Category : Education
ISBN : 9781470455934

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Function Theory and ℓp Spaces by Raymond Cheng,Javad Mashreghi,William T. Ross Pdf

The classical ℓp sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces ℓpA of analytic functions whose Taylor coefficients belong to ℓp. Relations between the Banach space ℓp and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of ℓpA and a discussion of the Wiener algebra ℓ1A. Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is self-contained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.