Lnear systems theory and introductory algebraic geometry

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Cover Title page Preface Chapter I General Ideas of Linear Systems Theory 1. Introduction 2. A Definition of a "System" in Terms of Differential Equations 3. Equivalent and Linearly Equivalent Linear Time-Invariant Systems 4. Impulse Response and Transfer Functionsj Observability and Controllability 5. Completely Controllable and Observable Systems Chapter II Ideas of Algebraic Geometry 1. Introduction 2. Polynomials 3. Affine Algebraic Varieties 4. Projective Algebraic Varieties 5. Homogeneous and Inhomogeneous Coordinates 6. The Riemann Sphere and the Complex Projective Line 7. Linear Fractional and Projective Transformations Chapter III The Algebra of Polynomials (blank page) 1. Introduction 2. Polynomials with Coefficients in a Ring 3. Integral Domains and Quotlent Fields 4. The Algebra of Polynomials Over a Field 5. The Resultant of Two Polynomials 6. Resultant System of More Than Two Polynomials Chapter IV General Properties of Affine and Projective Varieties 1. Introduction (missing) 2. Irreducible Varieties and Rational Functions 3. Dimension of Affine Varieties 4. The Zariski Topology 5. The Projective Embedding of the Grassman Space Chapter V Linear Systems with Scalar Inputs and Outputs 1. Introductlon (missing) 2. Hankel Maps and Matrices 3. Ihe Algebraic Structure of finitely Generated Modules Over K[s] 4. Cyclic Vectors and Complete Controllability tor Scalar Input-Output systems 5. Reallzation of Scalar Ratlonal Functions as Frequency Response Functions Chapter VI Rational Maps Between Vector Spaces and Linear Systems Defined Over General Flelds 1. Introduction 2. Quotients of Modules Over Integral Domains 3. Rational Maps Between Vector Spaces 4. Linear Systems from the Standpoint of Rational Maps 5. Some Remarks on the Methodology of Generalization Chapter VII The Realization of Rational Maps as Frequency Responses of Linear Systems 1. Introduction 2. Uniqueness of Realizations as Completely Controllable and Observable Systems 3. Existence of Realizations of Frequency Response Maps 4. Computation of the Dimension of the State Space of a Completely Controllable and Observable Realization Chapter VIII Algebraic Equivalence Relations and Orbit Spaces of Transformation Groups 1. Introduction 2. General Remarks About the "Structure" of Orbit Spaces 3. The Algebraic Equivalence Structure for Linear Systems 4. The Orbit Space for Systems with Scalar Inputs and Outputs Bibliography