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Combinatorial Matrix Theory

Combinatorial Matrix Theory Author Richard A. Brualdi
ISBN-10 0521322650
Release 1991-07-26
Pages 367
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The book deals with the many connections between matrices, graphs, diagraphs and bipartite graphs. The basic theory of network flows is developed in order to obtain existence theorems for matrices with prescribed combinatorical properties and to obtain various matrix decomposition theorems. Other chapters cover the permanent of a matrix and Latin squares. The book ends by considering algebraic characterizations of combinatorical properties and the use of combinatorial arguments in proving classical algebraic theorems, including the Cayley-Hamilton Theorem and the Jorda Canonical Form.



Combinatorial Matrix Classes

Combinatorial Matrix Classes Author Richard A. Brualdi
ISBN-10 9780521865654
Release 2006-08-10
Pages 544
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A natural sequel to the author's previous book Combinatorial Matrix Theory written with H. J. Ryser, this is the first book devoted exclusively to existence questions, constructive algorithms, enumeration questions, and other properties concerning classes of matrices of combinatorial significance. Several classes of matrices are thoroughly developed including the classes of matrices of 0's and 1's with a specified number of 1's in each row and column (equivalently, bipartite graphs with a specified degree sequence), symmetric matrices in such classes (equivalently, graphs with a specified degree sequence), tournament matrices with a specified number of 1's in each row (equivalently, tournaments with a specified score sequence), nonnegative matrices with specified row and column sums, and doubly stochastic matrices. Most of this material is presented for the first time in book format and the chapter on doubly stochastic matrices provides the most complete development of the topic to date.



A Combinatorial Approach to Matrix Theory and Its Applications

A Combinatorial Approach to Matrix Theory and Its Applications Author Richard A. Brualdi
ISBN-10 1420082248
Release 2008-08-06
Pages 288
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Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on the subject by exploring the connections of these tools to matrices. After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the König digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices that are part of the Perron–Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry. Using combinatorial and graph-theoretical tools, this book enables a solid understanding of the fundamentals of matrix theory and its application to scientific areas.



Nonnegative Matrices and Applications

Nonnegative Matrices and Applications Author R. B. Bapat
ISBN-10 0521571677
Release 1997-03-28
Pages 336
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This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics. The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation. Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely. These results and applications will be of great interest to researchers in linear programming, statistics and operations research. The minimal prerequisites also make the book accessible to first-year graduate students.



Combinatorics Automata and Number Theory

Combinatorics  Automata and Number Theory Author Valérie Berthé
ISBN-10 9780521515979
Release 2010-08-12
Pages 615
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This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and its Applications cover their subjects comprehensively. Less important results may be summarised as exercises at the ends of chapters, For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects.



Permanents

Permanents Author Henryk Minc
ISBN-10 0521302269
Release 1984-12-28
Pages 224
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The purpose of this book, which was first published in 1978, is to give a complete account of the theory of permanents, their history and applications. This volume was the first complete account of the theory of permanents, covering virtually the whole of the subject, a feature that no simple survey of the theory of matrices can even attempt. The work also contains many results stated without formal proofs. This book can be used as a textbook at the advanced undergraduate or graduate level. The only prerequisites are a standard undergraduate course in the theory of matrices and a measure of mathematical maturity.



Matrices and Matroids for Systems Analysis

Matrices and Matroids for Systems Analysis Author Kazuo Murota
ISBN-10 9783642039942
Release 2009-10-27
Pages 483
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A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the 1990's. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. From the reviews: "...The book has been prepared very carefully, contains a lot of interesting results and is highly recommended for graduate and postgraduate students." András Recski, Mathematical Reviews Clippings 2000m:93006



Eigenspaces of Graphs

Eigenspaces of Graphs Author Dragoš M. Cvetković
ISBN-10 0521573521
Release 1997-01-09
Pages 258
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This book describes the spectral theory of finite graphs.



Handbook of Linear Algebra

Handbook of Linear Algebra Author Leslie Hogben
ISBN-10 9781420010572
Release 2006-11-02
Pages 1400
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The Handbook of Linear Algebra provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use handbook format. The esteemed international contributors guide you from the very elementary aspects of the subject to the frontiers of current research. The book features an accessible layout of parts, chapters, and sections, with each section containing definition, fact, and example segments. The five main parts of the book encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, applications of linear algebra to various mathematical and nonmathematical disciplines, and software packages for linear algebra computations. Within each section, the facts (or theorems) are presented in a list format and include references for each fact to encourage further reading, while the examples illustrate both the definitions and the facts. Linearization often enables difficult problems to be estimated by more manageable linear ones, making the Handbook of Linear Algebra essential reading for professionals who deal with an assortment of mathematical problems.



Model Theory

Model Theory Author Wilfrid Hodges
ISBN-10 0521304423
Release 1993-03-11
Pages 772
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Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide range of other areas such as set theory, geometry, algebra and computer science. This book provides an integrated introduction to model theory for graduate students.



Matrix Theory

Matrix Theory Author Fuzhen Zhang
ISBN-10 9781461410997
Release 2011-08-28
Pages 399
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The aim of this book is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. The book contains ten chapters covering various topics ranging from similarity and special types of matrices to Schur complements and matrix normality. This book can be used as a textbook or a supplement for a linear algebra and matrix theory class or a seminar for senior undergraduate or graduate students. The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields. Major changes in this revised and expanded second edition: -Expansion of topics such as matrix functions, nonnegative matrices, and (unitarily invariant) matrix norms -A new chapter, Chapter 4, with updated material on numerical ranges and radii, matrix norms, and special operations such as the Kronecker and Hadamard products and compound matrices -A new chapter, Chapter 10, on matrix inequalities, which presents a variety of inequalities on the eigenvalues and singular values of matrices and unitarily invariant norms.



Matrices and Graphs in Geometry

Matrices and Graphs in Geometry Author Miroslav Fiedler
ISBN-10 9780521461931
Release 2011-02-03
Pages 197
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This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. Each book contains an extensive bibliography. Thus the volumes are encyclopedic references or manageable guides to major subjects.



Special Functions

Special Functions Author George E. Andrews
ISBN-10 0521789885
Release 1999
Pages 664
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An overview of special functions, focusing on the hypergeometric functions and the associated hypergeometric series.



The Theory of Partitions

The Theory of Partitions Author George E. Andrews
ISBN-10 052163766X
Release 1998-07-28
Pages 255
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Discusses mathematics related to partitions of numbers into sums of positive integers.



Applications of Unitary Symmetry and Combinatorics

Applications of Unitary Symmetry and Combinatorics Author James D Louck
ISBN-10 9789814458733
Release 2011-05-11
Pages 380
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This monograph is a synthesis of the theory of the pairwise coupling of the angular momenta of arbitrarily many independent systems to the total angular momentum in which the universal role of doubly stochastic matrices and their quantum-mechanical probabilistic interpretation is a major theme. A uniform viewpoint is presented based on the structure of binary trees. This includes a systematic method for the evaluation of all 3n–j coefficients and their relationship to cubic graphs. A number of topical subjects that emerge naturally are also developed, such as the algebra of permutation matrices, the properties of magic squares and an associated generalized Regge form, the Zeilberger counting formula for alternating sign matrices, and the Heisenberg ring problem, viewed as a composite system in which the total angular momentum is conserved. The readership is intended to be advanced graduate students and researchers interested in learning about the relationship between unitary symmetry and combinatorics and challenging unsolved problems. The many examples serve partially as exercises, but this monograph is not a textbook. It is hoped that the topics presented promote further and more rigorous developments that lead to a deeper understanding of the angular momentum properties of complex systems viewed as composite wholes. Contents:Composite Quantum SystemsAlgebra of Permutation MatricesCoordinates of A in Basis ℙΣn(e,p)Further Applications of Permutation MatricesDoubly Stochastic Matrices in Angular Momentum TheoryMagic SquaresAlternating Sign MatricesThe Heisenberg Magnetic Ring Readership: Graduate students and researchers in physics and mathematics who wish to learn about the relationships between symmetry and combinatorics. Keywords:Angular Momentum Theory;Unitary Symmetry;Combinatorics Binary TreeKey Features:A synthesis of the theory of the coupling of many-body angular momentum systems with application to the universal role of doubly stochastic matrices and their quantum-mechanical probabilistic interpretationAn application of binary coupling theory to the Heisenberg ring Hamiltonian, viewed as a composite system in which the total angular momentum is conserved. The cases for two, three, and four systems are solved exactlyNew formulas for the number of magic squares and a generalization of the Regge form for the coupling of two angular momenta to a general method of realizing all magic squaresNew perspective of the Zeilberger formula for counting the number of alternating sign matrices, based on zerosDevelopment of little-known properties of the algebra of permutation matrices with applications



Tropical and Idempotent Mathematics and Applications

Tropical and Idempotent Mathematics and Applications Author Grigoriĭ Lazarevich Litvinov
ISBN-10 9780821894965
Release 2014
Pages 300
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This volume contains the proceedings of the International Workshop on Tropical and Idempotent Mathematics, held at the Independent University of Moscow, Russia, from August 26-31, 2012. The main purpose of the conference was to bring together and unite researchers and specialists in various areas of tropical and idempotent mathematics and applications. This volume contains articles on algebraic foundations of tropical mathematics as well as articles on applications of tropical mathematics in various fields as diverse as economics, electroenergetic networks, chemical reactions, representation theory, and foundations of classical thermodynamics. This volume is intended for graduate students and researchers interested in tropical and idempotent mathematics or in their applications in other areas of mathematics and in technical sciences.



Convexity and Concentration

Convexity and Concentration Author Eric Carlen
ISBN-10 9781493970056
Release 2017-04-20
Pages 626
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This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute of Mathematics and its Applications during the Spring 2015 where geometric analysis, convex geometry and concentration phenomena were the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The volume is organized into two parts. Part I contains those contributions that focus primarily on problems motivated by probability theory, while Part II contains those contributions that focus primarily on problems motivated by convex geometry and geometric analysis. This book will be of use to those who research convex geometry, geometric analysis and probability directly or apply such methods in other fields.