Systems Analysis by Graphs and Matroids

1. Preliminaries.- 1. Convention and Notation.- 2. Algebra.- 3. Graph.- 4. Matroid.- 2. Graph-Theoretic Approach to the Solvability of a System of Equations.- 5. Structural Solvability of a System of Equations.- 6. Representation Graph.- 7. Graphical Conditions for Structural Solvability.- 8. Decompositions of a Graph by Menger-type Linkings.- 9. Decompositions and Reductions of a System of Equations.- 10. Application of the Graphical Technique.- 11. Examples.- 3. Graph-Theoretic Approach to the Controllability of a Dynamical System.- 12. Descriptions of a Dynamical System.- 13. Controllability of a Dynamical System.- 14. Graphical Conditions for Structural Controllability.- 15. Discussions.- 4. Physical Observations for Faithful Formulations.- 16. Mixed Matrix for Modeling Two Kinds of Numbers.- 17. Algebraic Implication of Dimensional Consistency.- 18. Physical Matrix.- 5 Matroid-Theoretic Approach to the Solvability of a System of Equations.- 19. Rank of a Mixed Matrix.- 20. Algorithm for Computing the Rank of a Mixed Matrix.- 21. Matroidal Conditions for Structural Solvability.- 22. Combinatorial Canonical Form of a Layered Mixed Matrix.- 23. Relation to Other Decompositions.- 24. Block-Triangularization of a Mixed Matrix.- 25. Decomposition of a System of Equations.- 26. Miscellaneous Notes.- 6. Matroid-Theoretic Approach to the Controllability of a Dynamical System.- 27. Dynamical Degree of a Dynamical System.- 28. Matroidal Conditions for Structural Controllability.- 29. Algorithm for Testing the Structural Controllability.- 30. Examples.- 31. Discussions.- Conclusion.- References.