In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression. - Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering
Author(s): Robert Hermann (Eds.)
Series: Mathematics in Science and Engineering 49
Publisher: Elsevier Science
Year: 1968
Language: English
Pages: iii-vii, 3-440
Content:
Edited by
Page iii
Copyright page
Page v
Preface
Pages v-vii
R. Hermann
1 Introduction
Pages 3-5
2 Tangent Vector-Vector Field Formalism
Pages 6-10
3 Differential Forms
Pages 11-20
4 Specialization to Euclidean Spaces: Differential Manifolds
Pages 21-27
5 Mappings, Submanifolds, and the Implicit Function Theorem
Pages 28-33
6 The Jacobi Bracket and the Lie Theory of Ordinary Differential Equations
Pages 34-45
Jacobi Bracket
7 Lie Derivation and Exterior Derivative; Integration on Manifolds
Pages 46-62
8 The Frobenius Complete Integrability Theorem
Pages 63-72
9 Reduction of Dimension when a Lie Algebra of Vector Fields Leaves a Vector-Field Invariant
Pages 73-80
10 Lie Groups
Pages 81-97
11 Classical Mechanics of Particles and Continua
Pages 98-109
12 Differential Forms and Variational Problems
Pages 113-121
13 Hamilton-Jacobi Theory
Pages 122-141
14 Extremal Fields and Sufficient Conditions for a Minimum
Pages 142-151
15 The Ordinary Problems of the Calculus of Variations
Pages 152-169
16 Groups of Symmetries of Variational Problems: Applications to Mechanics
Pages 170-231
17 Elliptic Functions
Pages 232-240
18 Accessibility Problems for Path Systems
Pages 241-257
19 Affine Connections on Differential Manifolds
Pages 261-271
20 The Riemannian Affine Connection and the First Variation Formula
Pages 272-283
21 The Hopf-Rinow Theorem Applications to the Theory of Covering Spaces
Pages 284-290
22 The Second Variation Formula and Jacobi Vector Fields
Pages 291-299,301
23 Sectional Curvature and the Elementary Comparison Theorems
Pages 302-317
24 Submanifolds of Riemannian Manifolds
Pages 318-341
25 Groups of Isometries
Pages 342-361
26 Deformation of Submanifolds in Riemannian Spaces
Pages 362-369
27 First-Order Invariants of Submanifolds and Convexity for Affinely Connected Manifolds
Pages 373-377
28 Affine Groups of Automorphisms. Induced Connections on Submanifolds. Projective Changes of Connection
Pages 378-385
29 The Laplace-Beltrami Operator
Pages 386-393
30 Characteristics and Shock Waves
Pages 394-400
31 The Morse Index Theorem
Pages 401-419
32 Complex Manifolds and Their Submanifolds
Pages 420-426
33 Mechanics on Riemannian Manifolds
Pages 427-430
Bibliography
Pages 431-433
Subject Index
Pages 435-440
