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Matrix Inequalities for Iterative Systems

Author: Hanjo Taubig
Publisher: Portland CRC Press 2016
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Acknowledgments -- Preface -- Table of Contents -- Symbol Description -- I: INTRODUCTION -- 1: Notation and Basic Facts -- 1.1 Matrices and Vectors -- 1.1.1 Matrices -- 1.1.2 Vectors -- 1.1.3 Matrix Classes -- 1.2 Graphs -- 1.2.1 Degrees -- 1.2.2 Adjacency Matrix -- 1.3 Number of Walks in Graphs -- 1.3.1 Walks -- 1.3.2 Number of Walks -- 1.4 Entry
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Material Type: Document, Internet resource
Document Type: Book, Computer File, Internet Resource
All Authors / Contributors: Hanjo Taubig
ISBN: 9781351679091 1351679090 9781498777773 1498777775
OCLC Number: 1020857858
Description: 1 Online-Ressource (219 Seiten)
Contents: INTRODUCTIONNotation and Basic Facts Matrices and VectorsGraphs Number of Walks in Graphs Entry Sums and Matrix Powers Subsets, Submatrices, and Weighted Entry Sums Elementary Inequalities for Vectors and Sequences Motivation Simple Combinatorial Problems Automata and Formal Languages Graph Density, Maximum Clique and Densest k-Subgraph The Number of Paths and Extremal Graph Theory Means, Variances, and Irregularity Random Walks and Markov Chains Largest Eigenvalue Population Genetics and Evolutionary Theory Theoretical Chemistry Iterated Line Digraphs Other Applications Diagonalization and Spectral Decomposition Relevant Literature Similar and Diagonalizable Matrices Hermitian and Real Symmetric Matrices Adjacency Matrices and Number of Walks in Graphs Quadratic Forms for Diagonalizable Matrices UNDIRECTED GRAPHS / HERMITIAN MATRICES General Results Related Work for Products of Quadratic FormsGeneralizations for Products of Quadratic Related Work for Powers of Quadratic Forms Generalizations for Powers of Quadratic Forms The Number of Walks and Degree Powers Invalid InequalitiesRelaxed Inequalities Using Geometric Means Restricted Graph Classes Counterexamples for Special Cases Semiregular Graphs Trees Subdivision Graphs DIRECTED GRAPHS / NONSYMMETRIC MATRICES Walks and Alternating Walks in Directed Graphs Chebyshev's Sum Inequality Relaxed Inequalities Using Geometric Means Alternating Matrix Powers and Alternating Walks Powers of Row and Column SumsDegree Powers and Walks in Directed GraphsRow and Column Sums in Nonnegative MatricesOther Inequalities for the Number of WalksGeometric MeansAPPLICATIONS Bounds for the Largest Eigenvalue Matrix FoundationsGraphs and Adjacency Matrices Eigenvalue Moments and EnergyIterated Kernels Related WorkOur Results Sidorenko's Conjecture
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Abstract:

Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Acknowledgments -- Preface -- Table of Contents -- Symbol Description -- I: INTRODUCTION -- 1: Notation and Basic Facts -- 1.1 Matrices and Vectors -- 1.1.1 Matrices -- 1.1.2 Vectors -- 1.1.3 Matrix Classes -- 1.2 Graphs -- 1.2.1 Degrees -- 1.2.2 Adjacency Matrix -- 1.3 Number of Walks in Graphs -- 1.3.1 Walks -- 1.3.2 Number of Walks -- 1.4 Entry Sums and Matrix Powers -- 1.4.1 Entry Sums of Matrix Powers -- 1.4.2 Number of Walks and Matrix Powers -- 1.4.3 Local Decomposition -- 1.5 Subsets, Submatrices, and Weighted Entry Sums -- 1.5.1 Selected Walks and Submatrices -- 1.5.2 Weighted Entry Sums -- 1.5.3 Quadratic Forms and Hermitian Matrices -- 1.5.4 Global Decomposition -- 1.6 Elementary Inequalities for Vectors and Sequences -- 1.6.1 The Inequalities of Cauchy, Bunyakovsky, and Schwarz -- 1.6.2 The Inequalities of Rogers and Hölder -- 1.6.3 Chebyshev's Sum Inequality -- 1.6.4 Inequalities for Arithmetic, Geometric, Harmonic, and Power Means -- 2: Motivation -- 2.1 Simple Combinatorial Problems -- 2.2 Automata and Formal Languages -- 2.3 Graph Density, Maximum Clique and Densest k-Subgraph -- 2.4 The Number of Paths and Extremal Graph Theory -- 2.5 Means, Variances, and Irregularity -- 2.6 Random Walks and Markov Chains -- 2.7 Largest Eigenvalue -- 2.8 Population Genetics and Evolutionary Theory -- 2.9 Theoretical Chemistry -- 2.10 Iterated Line Digraphs -- 2.11 Other Applications -- 3: Diagonalization and Spectral Decomposition -- 3.1 Relevant Literature -- 3.2 Similar and Diagonalizable Matrices -- 3.3 Hermitian and Real Symmetric Matrices -- 3.4 Adjacency Matrices and Number of Walks in Graphs -- 3.5 Quadratic Forms for Diagonalizable Matrices -- II: UNDIRECTED GRAPHS / HERMITIAN MATRICES -- 4: General Results -- 4.1 Related Work for Products of Quadratic Forms

4.1.1 The Number of Walks in Graphs -- 4.1.1.1 The Results of Lagarias, Mazo, Shepp, and McKay -- 4.1.1.2 The Results of Dress and Gutman -- 4.1.1.3 Further Results -- 4.1.2 Real Symmetric, Positive-Semidefinite, and Hermitian Matrices -- 4.1.2.1 The Results of Marcus and Newman -- 4.1.2.2 The Results of Lagarias, Mazo, Shepp, and McKay -- 4.2 Generalizations for Products of Quadratic Forms -- 4.2.1 A Generalization of the Inequality by Dress and Gutman -- 4.2.2 The Method of Expectation of Random Variables -- 4.2.3 The Complex-Weighted Sandwich Theorem -- 4.2.3.1 A Related Theorem by Taussky and Marcus -- 4.2.4 Positive-Semidefinite Matrices -- 4.2.5 Negative-Semidefinite Matrices -- 4.2.6 The Density Implication -- 4.3 Related Work for Powers of Quadratic Forms -- 4.3.1 The Results of Erdős, Simonovits, and Godsil -- 4.3.2 The Results of Ilić and Stevanović -- 4.3.3 The Results of Scheuer and Mandel, Mulholland and Smith, Blakley and Roy, Blakley and Dixon, and London -- 4.3.4 The Results of Hyyrö, Merikoski, and Virtanen -- 4.4 Generalizations for Powers of Quadratic Forms -- 4.4.1 More General Results for Walk Numbers -- 4.4.2 A Special Case for the Average Number of Walks -- 4.4.3 The Density Implication -- 4.4.4 Other Theorems Involving Powers -- 4.4.5 The Method of Expectation of Random Variables -- 4.5 The Number of Walks and Degree Powers -- 4.5.1 Related Work -- 4.5.2 Refined Inequalities for Powers of Degrees or Row Sums -- 4.5.3 Other Degree-Related Inequalities -- 4.6 Invalid Inequalities -- 4.6.1 Known Results -- 4.6.2 Extended Results -- 4.7 Relaxed Inequalities Using Geometric Means -- 4.7.1 Remarks on Another Relaxed Inequality -- 4.7.2 An Improved Inequality Implied by Logarithmic Means -- 4.7.3 An Arithmetic-Geometric-Harmonic Mean Inequality -- 5: Restricted Graph Classes -- 5.1 Counterexamples for Special Cases

5.1.1 Bipartite Graphs -- 5.1.2 Forests and Trees -- 5.1.3 Construction of Worst Case Trees -- 5.2 Semiregular Graphs -- 5.3 Trees -- 5.3.1 Stars and Paths -- 5.3.2 Walks of Length 2 and 3 in Trees -- 5.3.3 Walks of Length 4 and 5 in Trees -- 5.3.3.1 (1, n1, n2)-Barbell Graphs (Trees with Diameter 3) -- 5.3.3.2 (2, n1, n2)-Barbell Graphs -- 5.3.3.3 Proof for the w5-Inequality for Trees -- 5.3.4 A Conjecture for Trees -- 5.4 Subdivision Graphs -- 5.4.1 Walks of Length 2 and 3 -- 5.4.2 Walks of Length 4 and 5 -- III: DIRECTED GRAPHS / NONSYMMETRIC MATRICES -- 6: Walks and Alternating Walks in Directed Graphs -- 6.1 Chebyshev's Sum Inequality -- 6.2 Relaxed Inequalities Using Geometric Means -- 6.3 Alternating Matrix Powers and Alternating Walks -- 6.3.1 Definitions and Interpretations -- 6.3.2 A Simple Generalization -- 6.3.3 Related Work -- 6.3.4 Transferring Hermitian Matrix Results to General Matrices -- 6.3.5 Generalizations for Nonnegative Matrices -- 6.3.5.1 Implications for Hermitian Matrices -- 6.3.6 Generalizations for Complex Matrices -- 6.3.6.1 Implications for Hermitian Matrices -- 7: Powers of Row and Column Sums -- 7.1 Degree Powers and Walks in Directed Graphs -- 7.1.1 Related Work -- 7.1.2 Extended Results -- 7.2 Row and Column Sums in Nonnegative Matrices -- 7.3 Other Inequalities for the Number of Walks -- 7.4 Geometric Means -- IV: APPLICATIONS -- 8: Bounds for the Largest Eigenvalue -- 8.1 Matrix Foundations -- 8.1.1 Complex Matrices -- 8.1.2 Normal Matrices -- 8.1.3 Real Symmetric and Hermitian Matrices -- 8.1.4 Nonnegative Matrices -- 8.1.5 Symmetric Nonnegative Matrices -- 8.2 Graphs and Adjacency Matrices -- 8.2.1 Directed Graphs -- 8.2.2 Directed Graphs with Normal Adjacency Matrix -- 8.2.3 Undirected Graphs -- 8.2.4 Other Bounds -- 8.3 Eigenvalue Moments and Energy -- 8.3.1 Energy Bounds -- 8.3.2 Closed Walks Bounds

9: Iterated Kernels -- 9.1 Related Work -- 9.2 Our Results -- 9.2.1 Hermitian Kernels -- 9.2.2 Nonnegative Symmetric Kernels -- 9.2.3 Nonnegative Kernels -- 9.3 Sidorenko's Conjecture -- Conclusion -- Bibliography -- Index

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and Paths -- 5.3.2 Walks of Length 2 and 3 in Trees -- 5.3.3 Walks of Length 4 and 5 in Trees -- 5.3.3.1 (1, n1, n2)-Barbell Graphs (Trees with Diameter 3) -- 5.3.3.2 (2, n1, n2)-Barbell Graphs -- 5.3.3.3 Proof for the w5-Inequality for Trees -- 5.3.4 A Conjecture for Trees -- 5.4 Subdivision Graphs -- 5.4.1 Walks of Length 2 and 3 -- 5.4.2 Walks of Length 4 and 5 -- III: DIRECTED GRAPHS \/ NONSYMMETRIC MATRICES -- 6: Walks and Alternating Walks in Directed Graphs -- 6.1 Chebyshev\'s Sum Inequality -- 6.2 Relaxed Inequalities Using Geometric Means -- 6.3 Alternating Matrix Powers and Alternating Walks -- 6.3.1 Definitions and Interpretations -- 6.3.2 A Simple Generalization -- 6.3.3 Related Work -- 6.3.4 Transferring Hermitian Matrix Results to General Matrices -- 6.3.5 Generalizations for Nonnegative Matrices -- 6.3.5.1 Implications for Hermitian Matrices -- 6.3.6 Generalizations for Complex Matrices -- 6.3.6.1 Implications for Hermitian Matrices -- 7: Powers of Row and Column Sums -- 7.1 Degree Powers and Walks in Directed Graphs -- 7.1.1 Related Work -- 7.1.2 Extended Results -- 7.2 Row and Column Sums in Nonnegative Matrices -- 7.3 Other Inequalities for the Number of Walks -- 7.4 Geometric Means -- IV: APPLICATIONS -- 8: Bounds for the Largest Eigenvalue -- 8.1 Matrix Foundations -- 8.1.1 Complex Matrices -- 8.1.2 Normal Matrices -- 8.1.3 Real Symmetric and Hermitian Matrices -- 8.1.4 Nonnegative Matrices -- 8.1.5 Symmetric Nonnegative Matrices -- 8.2 Graphs and Adjacency Matrices -- 8.2.1 Directed Graphs -- 8.2.2 Directed Graphs with Normal Adjacency Matrix -- 8.2.3 Undirected Graphs -- 8.2.4 Other Bounds -- 8.3 Eigenvalue Moments and Energy -- 8.3.1 Energy Bounds -- 8.3.2 Closed Walks Bounds<\/span>\" ;\u00A0\u00A0\u00A0\nschema:description<\/a> \"4.1.1 The Number of Walks in Graphs -- 4.1.1.1 The Results of Lagarias, Mazo, Shepp, and McKay -- 4.1.1.2 The Results of Dress and Gutman -- 4.1.1.3 Further Results -- 4.1.2 Real Symmetric, Positive-Semidefinite, and Hermitian Matrices -- 4.1.2.1 The Results of Marcus and Newman -- 4.1.2.2 The Results of Lagarias, Mazo, Shepp, and McKay -- 4.2 Generalizations for Products of Quadratic Forms -- 4.2.1 A Generalization of the Inequality by Dress and Gutman -- 4.2.2 The Method of Expectation of Random Variables -- 4.2.3 The Complex-Weighted Sandwich Theorem -- 4.2.3.1 A Related Theorem by Taussky and Marcus -- 4.2.4 Positive-Semidefinite Matrices -- 4.2.5 Negative-Semidefinite Matrices -- 4.2.6 The Density Implication -- 4.3 Related Work for Powers of Quadratic Forms -- 4.3.1 The Results of Erd\u0151s, Simonovits, and Godsil -- 4.3.2 The Results of Ili\u0107 and Stevanovi\u0107 -- 4.3.3 The Results of Scheuer and Mandel, Mulholland and Smith, Blakley and Roy, Blakley and Dixon, and London -- 4.3.4 The Results of Hyyr\u00F6, Merikoski, and Virtanen -- 4.4 Generalizations for Powers of Quadratic Forms -- 4.4.1 More General Results for Walk Numbers -- 4.4.2 A Special Case for the Average Number of Walks 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