Pristine Transfinite Graphs and Permissive Electrical Networks

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 $$ \mathop{\omega }\limits^{ \to } $$-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 $$ \mathop{\omega }\limits^{ \to } $$, ?, 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.