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INFINITE ELECTRICAL NETWORKS |
| TABLE OF CONTENTS Preface Chapter 1. Introduction 1.1 Notations and Terminology 1.2 Countable Graphs 1.3 0-Graphs 1.4 Electrical Networks 1.5 Kirchhoff's Laws 1.6 Curiouser and Curiouser 1.7 Electrical Analogs for Some Differential Equations 1.8 The Transient Behavior of Linear RLC Networks Chapter 2. Infinite-Power Regimes 2.1 An Example 2.2 The Chainlike Structure 2.3 Halin's Result for Locally Finite Graphs 2.4 An Extension to Countably Infinite Graphs 2.5 Limbs 2.6 Current Regimes Satisfying Kirchhoff's Current Law 2.7 Joints and Chords 2.8 The Equations for a Limb Analysis 2.9 Chord Dominance 2.10 Limb Analysis, Summarized 2.11 Nonlinear Networks 2.12 A Contraction Mapping Result 2.13 A More General Fixed Point Theorem Chapter 3. Finite-Power Regimes: The Linear Case 3.1 Flander's Theorems 3.2 Connections at Infinity: 1-Graphs 3.3 Existence and Uniqueness 3.4 The Validity of Kirchhoff's Laws 3.5 A Dual Analysis 3.6 Transferring Pure Sources 3.7 Networks with Pure Sources 3.8 Thomson's Least Power Principle 3.9 The Concavity of Driving-Point Resistances 3.10 Nondisconnectable 0-Tips 3.11 Effectively Shorted 0-Tips Chapter 4. Finite-Power Regimes: The Nonlinear Case 4.1 Regular Networks 4.2 The Fundamental Operator 4.3 The Modular Sequence Space l_M 4.4 The Space l_M^# 4.5 The Space c_M 4.6 Some Uniform-Variation Conditions 4.7 Current Regimes in l_M 4.8 A Minimization Principle Chapter 5. Transfinite Electrical Networks 5.1 p-Graphs 5.2 omega-Graphs 5.3 Graphs of Still Higher Ranks 5.4 (k,q)-Paths and Terminal Behavior 5.5 k-Networks Chapter 6. Cascades 6.1 Linear Uniform Cascades 6.2 Nonlinear Uniform Cascades 6.3 Backward Mappings of the Axes 6.4 Trajectories near the Origin 6.5 Characteristic Immittances for Nonlinear Uniform Cascades 6.6 Nonlinear Uniform Lattice Cascades 6.7 Nonuniform Cascades: Infinity Imperceptible 6.8 Nonuniform Cascades: Infinity Perceptible 6.9 Input-Output Mappings 6.10 omega^p-Cascades 6.10 Loaded Cascades Chapter 7. Grids 7.1 Laurent Operators and the Fourier-Series Transformation 7.2 n-Dimensional Rectangular Grids 7.3 A Nodal Analysis for Uniform Grids 7.4 One-Dimensional Nonuniformity 7.5 Solving Grounded Semi-Infinite Grids: Infinity Imperceptible 7.6 Solving Ungrounded Semi-Infinite Grids: Infinity Imperceptible 7.7 Semi-Infinite grids: Infinity Perceptible 7.8 Forward and Backward Mappings 7.9 Solving Semi-Infinite Grids: Infinity Perceptible 7.10 Transfinite Grids 7.11 Grids with Two-Dimensionally Transfinite Nonuniformities Chapter 8. Applications 8.1 Surface Operators 8.2 Domain Contractions 8.3 Random Walks 8.4 Operator Networks Bibliography Index of Symbols Index |
