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BL
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| Record Number
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860945
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| Main Entry
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Martínez-Guerra, Rafael
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| Title & Author
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Algebraic and Differential Methods for Nonlinear Control Theory : : Elements of Commutative Algebra and Algebraic Geometry /\ Rafael Martínez-Guerra, Oscar Martínez-Fuentes, Juan Javier Montesinos-García.
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| Publication Statement
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Cham, Switzerland :: Springer Nature,, [2019]
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| Series Statement
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Mathematical and analytical techniques with applications to engineering
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| Page. NO
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1 online resource
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| ISBN
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3030120252
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: 9783030120252
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3030120244
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9783030120245
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Acknowledgements; Contents; Notations and Abbreviations; 1 Mathematical Background; 1.1 Introduction to Set Theory; 1.1.1 Set Operations and Other Properties; 1.2 Equivalence Relations; 1.3 Functions or Maps; 1.3.1 Classification of Functions or Maps; 1.4 Well-Ordering Principle and Mathematical Induction; References; 2 Group Theory; 2.1 Basic Definitions; 2.2 Subgroups; 2.3 Homomorphisms; 2.4 The Isomorphism Theorems; References; 3 Rings; 3.1 Basic Definitions; 3.2 Ideals, Homomorphisms and Rings; 3.3 Isomorphism Theorems in Rings; 3.4 Some Properties of Integers.
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3.4.1 Divisibility3.4.2 Division Algorithm; 3.4.3 Greatest Common Divisor; 3.4.4 Least Common Multiple; 3.5 Polynomials Rings; References; 4 Matrices and Linear Equations Systems; 4.1 Properties of Algebraic Operations with Real Numbers; 4.2 The Set mathbbRn and Linear Operations; 4.2.1 Linear Operations in mathbbRn; 4.3 Background of Matrix Operations; 4.4 Gauss-Jordan Method; 4.5 Definitions; References; 5 Permutations and Determinants; 5.1 Permutations Group; 5.2 Determinants; References; 6 Vector and Euclidean Spaces; 6.1 Vector Spaces and Subspaces; 6.2 Generated Subspace.
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6.3 Linear Dependence and Independence6.4 Bases and Dimension; 6.5 Quotient Space; 6.6 Cayley-Hamilton Theorem; 6.7 Euclidean Spaces; 6.8 GramSchmidt Process; References; 7 Linear Transformations; 7.1 Background; 7.2 Kernel and Image; 7.3 Linear Operators; 7.4 Associate Matrix; References; 8 Matrix Diagonalization and Jordan Canonical Form; 8.1 Matrix Diagonalization; 8.2 Jordan Canonical Form; 8.2.1 Generalized Eigenvectors; 8.2.2 Dot Diagram Method; References; 9 Differential Equations; 9.1 Motivation: Some Physical Origins of Differential Equations; 9.1.1 Free Fall.
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9.1.2 Simple Pendulum Problem9.1.3 Friction Problem; 9.2 Definitions; 9.3 Separable Differential Equations; 9.4 Homogeneous Equations; 9.5 Exact Equations; 9.6 Linear Differential Equations; 9.7 Homogeneous Second Order Linear Differential Equations; 9.8 Variation of Parameters Method; 9.9 Initial Value Problem; 9.10 Indeterminate Coefficients; 9.11 Solution of Differential Equations by Means of Power Series; 9.11.1 Some Criterions of Convergence of Series; 9.11.2 Solution of First and Second Order Differential Equations; 9.12 Picard's Method; 9.13 Convergence of Picard's Iterations.
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| Abstract
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This book is a short primer in engineering mathematics with a view on applications in nonlinear control theory. In particular, it introduces some elementary concepts of commutative algebra and algebraic geometry which offer a set of tools quite different from the traditional approaches to the subject matter. This text begins with the study of elementary set and map theory. Chapters 2 and 3 on group theory and rings, respectively, are included because of their important relation to linear algebra, the group of invertible linear maps (or matrices) and the ring of linear maps of a vector space. Homomorphisms and Ideals are dealt with as well at this stage. Chapter 4 is devoted to the theory of matrices and systems of linear equations. Chapter 5 gives some information on permutations, determinants and the inverse of a matrix. Chapter 6 tackles vector spaces over a field, Chapter 7 treats linear maps resp. linear transformations, and in addition the application in linear control theory of some abstract theorems such as the concept of a kernel, the image and dimension of vector spaces are illustrated. Chapter 8 considers the diagonalization of a matrix and their canonical forms. Chapter 9 provides a brief introduction to elementary methods for solving differential equations and, finally, in Chapter 10, nonlinear control theory is introduced from the point of view of differential algebra.
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| Subject
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Commutative algebra.
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Geometry, Algebraic.
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| Subject
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Nonlinear control theory.
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Commutative algebra.
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Geometry, Algebraic.
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| Subject
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MATHEMATICS-- General.
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Nonlinear control theory.
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| Subject
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TECHNOLOGY ENGINEERING-- Engineering (General)
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| Dewey Classification
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629.836
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| LC Classification
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QA402.35
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| Added Entry
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Martínez-Fuentes, Oscar
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Montesinos-García, Juan Javier
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