Pristine Transfinite Graphs and Permissive Electrical Networks

Pristine Transfinite Graphs and Permissive Electrical Networks by Armen H. Zemanian.

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Bibliographic Details
Main Author: Zemanian, Armen H. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2001.
Edition:1st ed. 2001.
Series:Springer eBook Collection.
Subjects:
Discrete mathematics.
Electrical engineering.
Computational complexity.
Control engineering.
Robotics.
Mechatronics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1 Introduction
  • 1.1 Notations and Terminology
  • 1.2 Transfinite Nodes and Graphs
  • 1.3 A Need for Transfiniteness
  • 1.4 Pristine Graphs
  • 2 Pristine Transfinite Graphs
  • 2.1 0-Graphs and 1-Graphs
  • 2.2 ?-Graphs and (? + 1)-Graphs
  • 2.3 $$ \mathop{\omega }\limitŝ{ \to } $$-Graphs and ?-Graphs
  • 2.4 Transfinite Graphs of Higher Ranks
  • 3 Some Transfinite Graph Theory
  • 3.1 Nondisconnectable Tips and Connectedness
  • 3.2 Sections
  • 3.3 Transfinite Versions of König’s Lemma
  • 3.4 Countable Graphs
  • 3.5 Locally Finite Graphs
  • 3.6 Transfinite Ends
  • 4 Permissive Transfinite Networks
  • 4.1 Linear Electrical Networks
  • 4.2 Permissive 1-Networks
  • 4.3 The 1-Metric
  • 4.4 The Recursive Assumptions
  • 4.5 Permissive (? + l)-Networks
  • 4.6 Permissive Networks of Ranks $$ \mathop{\omega }\limitŝ{ \to } $$, ?, and Higher
  • 5 Linear Networks; Tellegen Regimes
  • 5.1 A Tellegen-Type Fundamental Theorem
  • 5.2 Node Voltages
  • 5.3 Transfinite Current Flows—Some Ideas
  • 5.4 Current Flows at Natural-Number Ranks
  • 5.5 Current Flows at the Rank ?
  • 6 Monotone Networks; Kirchhoff Regimes
  • 6.1 Some Assumptions
  • 6.2 Minty’s Colored-Graph Theorem
  • 6.3 Wolaver’s No-Gain Property
  • 6.4 Duffin’s Theorem on Operating Points
  • 6.5 The Minty-Calvert Theorem
  • 6.6 Potentials and Branch Voltages
  • 6.7 Existence of a Potential
  • 6.8 Existence of an Operating Point
  • 6.9 Uniqueness of an Operating Point
  • 6.10 Monotones ?-Networks
  • 6.11 Reconciling Two Theories
  • 7 Some Maximum Principles
  • 7.1 Input Resistance Matrices
  • 7.2 Some Maximum Principles for Node Voltages
  • 8 Transfinite Random Walks
  • 8.1 The Nash-Williams Rule
  • 8.2 Transfinite Walks
  • 8.3 Transfiniteness for Random Walks
  • 8.4 Reaching a Bordering Node
  • 8.5 Leaving a Bordering Node
  • 8.6 Transitions for Adjacent Bordering Nodes
  • 8.7 Wandering on a v-Network
  • References
  • Index of Symbols.

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