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Recent technology involves large-scale physical or engineering systems consisting of thousands of interconnected elementary units. This monograph illustrates how engineering problems can be solved using the recent results of combinatorial mathematics through appropriate mathematical modeling. The structural solvability of a system of linear or nonlinear equations as well as the structural controllability of a linear time-invariant dynamical system are treated by means of graphs and matroids. Special emphasis is laid on the importance of relevant physical observations to successful mathematical modelings. The reader will become acquainted with the concepts of matroid theory and its corresponding matroid theoretical approach. This book is of interest to graduate students and researchers.
Sommario
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.
