| Description |
1 online resource : illustrations |
| Series |
New mathematical methos, systems and applications set |
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New mathematical methos, systems and applications set
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| Contents |
Intro; Title page; Table of Contents; Copyright; Preface; Errata for Volume 1; List of Notation; Chapter 1: Field Extensions and Differential Field extensions; Chapter 2: General Topology; Chapter 3: Topological Vector Spaces; Chapter 4: Measure, Integration, Function spaces; Chapter 5: Sheaves; 1: Field Extensions and Differential Field Extensions; Abstract; 1.1 Galois theory; 1.2 Transcendental extensions; 1.3 Differential Galois theory; 1.4 Differentially transcendental extensions; 2: General Topology; Abstract; 2.1 Introduction to general topology; 2.2 Filters and nets |
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2.3 Topological structures2.4 Uniform structures; 2.5 Bornologies; 2.6 Baire spaces, Polish spaces, Suslin spaces, and LindelÃœf spaces; 2.7 Uniform function spaces; 2.8 Topological algebra; 3: Topological Vector Spaces; Abstract; 3.1 Introduction; 3.2 General topological vector spaces; 3.3 Locally convex spaces; 3.4 Important types of locally convex spaces; 3.5 Weak topologies; 3.6 Dual of a locally convex space; 3.7 Bidual and reflexivity; 3.8 Additional notes about â#x84;± and â#x84;±S-spaces and their duals; 3.9 Continuous multilinear mappings; 3.10 Hilbert spaces; 3.11 Nuclear spaces |
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4: Measure and Integration, Function SpacesAbstract; 4.1 Measure and integration; 4.2 Functions in a single complex variable; 4.3 Function spaces; 4.4 Generalized function spaces; 5: Sheaves; Abstract; 5.1 Introduction; 5.2 General results about sheaves; 5.3 Sheaves of Modules; 5.4 Cohomology of sheaves; Bibliography; Cited Authors; Index |
| Summary |
The three volumes of this series of books, of which this is the second, put forward the mathematical elements that make up the foundations of a number of contemporary scientific methods: modern theory on systems, physics and engineering. Whereas the first volume focused on the formal conditions for systems of linear equations (in particular of linear differential equations) to have solutions, this book presents the approaches to finding solutions to polynomial equations and to systems of linear differential equations with varying coefficients. Fundamentals of Advanced Mathematics, Volume 2: Field Extensions, Topology and Topological Vector Spaces, Functional Spaces, and Sheaves begins with the classical Galois theory and the theory of transcendental field extensions. Next, the differential side of these theories is treated, including the differential Galois theory (Picard-Vessiot theory of systems of linear differential equations with time-varying coefficients) and differentially transcendental field extensions. The treatment of analysis includes topology (using both filters and nets), topological vector spaces (using the notion of disked space, which simplifies the theory of duality), and the radon measure (assuming that the usual theory of measure and integration is known). In addition, the theory of sheaves is developed with application to the theory of distributions and the theory of hyperfunctions (assuming that the usual theory of functions of the complex variable is known). This volume is the prerequisite to the study of linear systems with time-varying coefficients from the point-of-view of algebraic analysis and the algebraic theory of nonlinear systems |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from READ title page (OverDrive, viewed February 05, 2018) |
| Subject |
Mathematics.
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Field extensions (Mathematics)
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Topology.
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Linear topological spaces.
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Vector fields.
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Sheaf theory.
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Mathematics
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MATHEMATICS -- Essays.
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MATHEMATICS -- Pre-Calculus.
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MATHEMATICS -- Reference.
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Field extensions (Mathematics)
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Linear topological spaces
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Mathematics
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Sheaf theory
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Topology
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Vector fields
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| Form |
Electronic book
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| ISBN |
9780081023853 |
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0081023855 |
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