| Preface |
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Introduction to Structural Approach --- Overview of the Book |
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1 | (30) |
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Structural Approach to Index of DAE |
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1 | (9) |
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Index of Differential-algebraic Equations |
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1 | (2) |
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Graph-theoretic Structural Approach |
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3 | (4) |
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An Embarrassing Phenomenon |
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7 | (3) |
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What Is Combinatorial Structure? |
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10 | (10) |
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11 | (4) |
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Descriptor Form Rather than Standard Form |
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15 | (2) |
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17 | (3) |
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Mathematics on Mixed Polynomial Matrices |
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20 | (11) |
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20 | (1) |
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Resolution of the Index Problem |
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21 | (5) |
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Block-triangular Decomposition |
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26 | (5) |
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Matrix, Graph, and Matroid |
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31 | (76) |
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31 | (12) |
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Polynomial and Algebraic Independence |
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31 | (2) |
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33 | (3) |
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Rank, Term-rank and Generic-rank |
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36 | (4) |
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40 | (3) |
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43 | (28) |
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Directed Graph and Bipartite Graph |
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43 | (5) |
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Jordan-Holder-type Theorem for Submodular Functions |
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48 | (7) |
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Dulmage-Mendelsohn Decomposition |
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55 | (10) |
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Maximum Flow and Menger-type Linking |
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65 | (2) |
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Minimum Cost Flow and Weighted Matching |
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67 | (4) |
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71 | (36) |
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71 | (2) |
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73 | (4) |
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77 | (1) |
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Basis Exchange Properties |
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78 | (6) |
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Independent Matching Problem |
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84 | (9) |
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93 | (4) |
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Bimatroid (Linking System) |
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97 | (10) |
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Physical Observations for Mixed Matrix Formulation |
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107 | (24) |
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Mixed Matrix for Modeling Two Kinds of Numbers |
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107 | (13) |
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107 | (9) |
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Mixed Matrix and Mixed Polynomial Matrix |
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116 | (4) |
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Algebraic Implication of Dimensional Consistency |
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120 | (6) |
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120 | (1) |
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121 | (2) |
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Total Unimodularity of a Dimensioned Matrix |
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123 | (3) |
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126 | (5) |
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126 | (2) |
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Physical Matrices in a Dynamical System |
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128 | (3) |
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Theory and Application of Mixed Matrices |
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131 | (140) |
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Mixed Matrix and Layered Mixed Matrix |
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131 | (3) |
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134 | (19) |
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Rank Identities for LM-matrices |
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135 | (4) |
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Rank Identities for Mixed Matrices |
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139 | (3) |
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Reduction to Independent Matching Problems |
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142 | (3) |
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145 | (8) |
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Structural Solvability of Systems of Equations |
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153 | (14) |
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Formulation of Structural Solvability |
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153 | (3) |
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Graphical Conditions for Structural Solvability |
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156 | (4) |
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Matroidal Conditions for Structural Solvability |
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160 | (7) |
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Combinatorial Canonical Form of LM-matrices |
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167 | (35) |
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167 | (5) |
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172 | (3) |
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175 | (6) |
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181 | (6) |
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Decomposition of Systems of Equations by CCF |
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187 | (4) |
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191 | (8) |
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199 | (3) |
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Irreducibility of LM-matrices |
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202 | (9) |
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Theorems on LM-irreducibility |
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202 | (3) |
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Proof of the Irreducibility of Determinant |
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205 | (6) |
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Decomposition of Mixed Matrices |
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211 | (10) |
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LU-decomposition of Invertible Mixed Matrices |
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212 | (3) |
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Block-triangularization of General Mixed Matrices |
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215 | (6) |
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221 | (9) |
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Decomposition as Matroid Union |
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221 | (4) |
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225 | (3) |
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Electrical Network with Admittance Expression |
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228 | (2) |
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230 | (20) |
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231 | (4) |
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Existence of Proper Block-triangularization |
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235 | (3) |
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Partial Order Among Blocks |
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238 | (2) |
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Generic Partitioned Matrix |
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240 | (10) |
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Principal Structures of LM-matrices |
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250 | (21) |
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250 | (2) |
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Principal Structure of Submodular Systems |
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252 | (2) |
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Principal Structure of Generic Matrices |
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254 | (3) |
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Vertical Principal Structure of LM-matrices |
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257 | (4) |
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Horizontal Principal Structure of LM-matrices |
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261 | (10) |
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Polynomial Matrix and Valuated Matroid |
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271 | (60) |
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Polynomial/Rational Matrix |
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271 | (9) |
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Polynomial Matrix and Smith Form |
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271 | (1) |
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Rational Matrix and Smith-McMillan Form at Infinity |
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272 | (3) |
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Matrix Pencil and Kronecker Form |
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275 | (5) |
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280 | (51) |
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280 | (1) |
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281 | (1) |
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282 | (3) |
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285 | (2) |
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287 | (3) |
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Induction Through Bipartite Graphs |
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290 | (5) |
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295 | (5) |
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Further Exchange Properties |
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300 | (6) |
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Valuated Independent Assignment Problem |
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306 | (2) |
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308 | (8) |
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Application to Triple Matrix Product |
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316 | (1) |
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Cycle-canceling Algorithms |
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317 | (8) |
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325 | (6) |
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Theory and Application of Mixed Polynomial Matrices |
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331 | (72) |
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Descriptions of Dynamical Systems |
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331 | (4) |
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Mixed Polynomial Matrix Descriptions |
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331 | (1) |
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Relationship to Other Descriptions |
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332 | (3) |
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Degree of Determinant of Mixed Polynomial Matrices |
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335 | (20) |
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335 | (1) |
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336 | (1) |
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337 | (3) |
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Reduction to Valuated Independent Assignment |
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340 | (3) |
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343 | (5) |
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348 | (7) |
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Smith Form of Mixed Polynomial Matrices |
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355 | (9) |
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Expression of Invariant Factors |
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355 | (8) |
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363 | (1) |
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Controllability of Dynamical Systems |
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364 | (20) |
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364 | (1) |
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Structural Controllability |
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365 | (7) |
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Mixed Polynomial Matrix Formulation |
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372 | (3) |
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375 | (4) |
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379 | (5) |
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Fixed Modes of Decentralized Systems |
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384 | (19) |
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384 | (3) |
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387 | (3) |
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Mixed Polynomial Matrix Formulation |
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390 | (5) |
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395 | (3) |
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398 | (5) |
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403 | (50) |
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Combinatorial Relaxation Algorithm |
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403 | (15) |
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403 | (4) |
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407 | (6) |
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Transformation Towards Upper-tightness |
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413 | (4) |
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417 | (1) |
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Combinatorial System Theory |
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418 | (13) |
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Definition of Combinatorial Dynamical Systems |
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419 | (1) |
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420 | (2) |
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Eigensets and Recurrent Sets |
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422 | (4) |
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Controllability of Combinatorial Dynamical Systems |
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426 | (5) |
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Mixed Skew-symmetric Matrix |
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431 | (22) |
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431 | (2) |
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433 | (5) |
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438 | (6) |
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Rank of Mixed Skew-symmetric Matrices |
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444 | (2) |
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Electrical Network Containing Gyrators |
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446 | (7) |
| References |
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453 | (16) |
| Notation Table |
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469 | (10) |
| Index |
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479 | |
