Matrices and Matroids for Systems Analysis

Transcription

1 Matrices and Matroids for Systems Analysis Bearbeitet von Kazuo Murota Taschenbuch. xii, 483 S. Paperback ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 1520 g Weitere Fachgebiete > Mathematik > Mathematik Allgemein > Diskrete Mathematik, Kombinatorik Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 Contents Preface... V 1. Introduction to Structural Approach Overview of the Book Structural Approach to Index of DAE Index of Differential-algebraic Equations Graph-theoretic Structural Approach An Embarrassing Phenomenon What Is Combinatorial Structure? Two Kinds of Numbers Descriptor Form Rather than Standard Form Dimensional Analysis Mathematics on Mixed Polynomial Matrices Formal Definitions Resolution of the Index Problem Block-triangular Decomposition Matrix, Graph, and Matroid Matrix Polynomial and Algebraic Independence Determinant Rank, Term-rank and Generic-rank Block-triangular Forms Graph Directed Graph and Bipartite Graph Jordan Hölder-type Theorem for Submodular Functions Dulmage Mendelsohn Decomposition Maximum Flow and Menger-type Linking Minimum Cost Flow and Weighted Matching Matroid From Matrix to Matroid Basic Concepts Examples Basis Exchange Properties

3 X Contents Independent Matching Problem Union Bimatroid (Linking System) Physical Observations for Mixed Matrix Formulation Mixed Matrix for Modeling Two Kinds of Numbers Two Kinds of Numbers Mixed Matrix and Mixed Polynomial Matrix Algebraic Implication of Dimensional Consistency Introductory Comments Dimensioned Matrix Total Unimodularity of a Dimensioned Matrix PhysicalMatrix Physical Matrix Physical Matrices in a Dynamical System Theory and Application of Mixed Matrices Mixed Matrix and Layered Mixed Matrix RankofMixedMatrices Rank Identities for LM-matrices Rank Identities for Mixed Matrices Reduction to Independent Matching Problems Algorithms for the Rank Structural Solvability of Systems of Equations Formulation of Structural Solvability Graphical Conditions for Structural Solvability Matroidal Conditions for Structural Solvability Combinatorial Canonical Form of LM-matrices LM-equivalence Theorem of CCF Construction of CCF Algorithm for CCF Decomposition of Systems of Equations by CCF Application of CCF CCF over Rings Irreducibility of LM-matrices Theorems on LM-irreducibility Proof of the Irreducibility of Determinant Decomposition of Mixed Matrices LU-decomposition of Invertible Mixed Matrices Block-triangularization of General Mixed Matrices Related Decompositions Decomposition as Matroid Union Multilayered Matrix Electrical Network with Admittance Expression

4 Contents XI 4.8 Partitioned Matrix Definitions Existence of Proper Block-triangularization Partial Order Among Blocks Generic Partitioned Matrix Principal Structures of LM-matrices Motivations Principal Structure of Submodular Systems Principal Structure of Generic Matrices Vertical Principal Structure of LM-matrices Horizontal Principal Structure of LM-matrices Polynomial Matrix and Valuated Matroid Polynomial/Rational Matrix Polynomial Matrix and Smith Form Rational Matrix and Smith McMillan Form at Infinity Matrix Pencil and Kronecker Form ValuatedMatroid Introduction Examples Basic Operations Greedy Algorithms Valuated Bimatroid Induction Through Bipartite Graphs Characterizations Further Exchange Properties Valuated Independent Assignment Problem Optimality Criteria Application to Triple Matrix Product Cycle-canceling Algorithms Augmenting Algorithms Theory and Application of Mixed Polynomial Matrices Descriptions of Dynamical Systems Mixed Polynomial Matrix Descriptions Relationship to Other Descriptions Degree of Determinant of Mixed Polynomial Matrices Introduction Graph-theoretic Method Basic Identities Reduction to Valuated Independent Assignment Duality Theorems Algorithm Smith Form of Mixed Polynomial Matrices Expression of Invariant Factors

5 XII Contents Proofs Controllability of Dynamical Systems Controllability Structural Controllability Mixed Polynomial Matrix Formulation Algorithm Examples Fixed Modes of Decentralized Systems Fixed Modes Structurally Fixed Modes Mixed Polynomial Matrix Formulation Algorithm Examples Further Topics Combinatorial Relaxation Algorithm Outline of the Algorithm Test for Upper-tightness Transformation Towards Upper-tightness Algorithm Description Combinatorial System Theory Definition of Combinatorial Dynamical Systems Power Products Eigensets and Recurrent Sets Controllability of Combinatorial Dynamical Systems Mixed Skew-symmetric Matrix Introduction Skew-symmetric Matrix Delta-matroid Rank of Mixed Skew-symmetric Matrices Electrical Network Containing Gyrators References Notation Table Index

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