#  ## Multivariable linear systems and projective algebraic geometry

Peter Falb

This monograph is an introduction to the ideas of algebraic geometry written for graduate students in systems, control, and applied mathematics. An extension of an earlier volume, this self-contained work has an applied flavor in its presentation of the core ideas in the algebro-geometric treatment of scalar linear system theory with the emphasis on constructive methods rather than on abstraction. Exercises, which are an integral part of the exposition throughout, five appendices containing supplementary material, and extensive bibliography of related literature make this a valuable classroom tool or good self-study resource.

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[目次]

• Scalar input or scalar output systems
• two or three input, two output systems - some examples
• the transfer and Hankel matrices
• polynomial matrices
• projective space
• projective algebraic geometry I - basic concepts
• projective algebraic geometry II - regular functions, local rings, morphisms
• exterior algebra and grassmannians
• the Laurent isomorphism theorem I
• projective algebraic geometric III - products, projections, degree
• the Laurent isomorphism theorem II
• projective algebraic geometry IV - families, projections, degree
• the state space - realizations, controllability, observability, equivalence
• projective algebraic geometry V - fibres of morphisms
• projective algebraic geometry VI - tangents, differentials, simple subvarieties
• the geometry quotient theorem
• projective algebraic geometry VII -divisors
• projective algebraic geometry VIII - intersections
• state feedback
• output feedback
• formal power series, completions, regular local rings, and Hilbert polynomials
• specialization, generic points and spectra
• differentials
• the space nm
• review of affine algebraic geometry.

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[目次]

• 1 Scalar Input or Scalar Output Systems.- 2 Two or Three Input, Two Output Systems: Some Examples.- 3 The Transfer and Hankel Matrices.- 4 Polynomial Matrices.- 5 Projective Space.- 6 Projective Algebraic Geometry I: Basic Concepts.- 7 Projective Algebraic Geometry II: Regular Functions, Local Rings, Morphisms.- 8 Exterior Algebra and Grassmannians.- 9 The Laurent Isomorphism Theorem: I.- 10 Projective Algebraic Geometry III: Products, Graphs, Projections.- 11 The Laurent Isomorphism Theorem: II.- 12 Projective Algebraic Geometry IV: Families, Projections, Degree.- 13 The State Space: Realizations, Controllability, Observability, Equivalence.- 14 Projective Algebraic Geometry V: Fibers of Morphisms.- 15 Projective Algebraic Geometry VI: Tangents, Differentials, Simple Subvarieties.- 16 The Geometric Quotient Theorem.- 17 Projective Algebraic Geometry VII: Divisors.- 18 Projective Algebraic Geometry VIII: Intersections.- 19 State Feedback.- 20 Output Feedback.- Appendices.- A Formal Power Series, Completions, Regular Local Rings, and Hubert Polynomials.- B Specialization, Generic Points and Spectra.- C Differentials.- D The Space.- E Review of Affine Algebraic Geometry.- References.- Glossary of Notations.

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### 書名 Multivariable linear systems and projective algebraic geometry Falb, P. L Falb P.L. Falb Peter L. Methods of algebraic geometry in control theory Birkhäuser c1999 viii, 390 p. 25 cm 3764341130 0817641130 BA43546054 ※クリックでCiNii Booksを表示 英語 アメリカ合衆国

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