Seiler Involution

The Formal Theory of Differential Equations and its Applications in Computer Algebra
1. Auflage 2009
ISBN: 978-3-642-01287-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 24, 650 Seiten

Reihe: Algorithms and Computation in Mathematics

Autoren/Hrsg.

Seiler, Werner M.

Inhaltsverzeichnis

1;Involution
;5
1.1;1 Introduction;20
1.2;2 Formal Geometry of Differential Equations;28
1.2.1;2.1 A Pedestrian Approach to Jet Bundles;29
1.2.2;2.2 An Intrinsic Approach to Jet Bundles;37
1.2.2.1;Addendum: The Contact Structure à la Gardner–Shadwick;46
1.2.3;2.3 Differential Equations;48
1.2.4;2.4 Some Examples;67
1.2.5;2.5 Notes;77
1.3;3 Involution I: Algebraic Theory;81
1.3.1;3.1 Involutive Divisions;82
1.3.1.1;Addendum: Some Algorithmic Considerations;90
1.3.2;3.2 Polynomial Algebras of Solvable Type;94
1.3.3;3.3 Hilbert's Basis Theorem and Gröbner Bases;104
1.3.3.1;Iterated Polynomial Algebras of Solvable Type;105
1.3.3.2;Polynomial Algebras with Centred Commutation Relations;107
1.3.3.3;Filtered Algebras;109
1.3.3.4;Polynomial Algebras over Fields;110
1.3.4;3.4 Involutive Bases;112
1.3.5;3.5 Notes;118
1.4;4 Completion to Involution;123
1.4.1;4.1 Constructive Divisions;124
1.4.2;4.2 Computation of Involutive Bases;128
1.4.2.1;Addendum: Right and Two-Sided Ideals;136
1.4.3;4.3 Pommaret Bases and -Regularity;140
1.4.4;4.4 Construction of Minimal Bases and Optimisations;150
1.4.5;4.5 Semigroup Orders;159
1.4.6;4.6 Involutive Bases over Rings;174
1.4.7;4.7 Notes;179
1.5;5 Structure Analysis of Polynomial Modules;184
1.5.1;5.1 Combinatorial Decompositions;185
1.5.2;5.2 Dimension and Depth;192
1.5.3;5.3 Noether Normalisation and Primary Decomposition;199
1.5.3.1;Addendum: Standard Pairs;207
1.5.4;5.4 Syzygies and Free Resolutions;210
1.5.4.1;Addendum: Iterated Polynomial Algebras of Solvable Type;224
1.5.5;5.5 Minimal Resolutions and Castelnuovo–Mumford Regularity;227
1.5.6;5.6 Notes;245
1.6;6 Involution II: Homological Theory;252
1.6.1;6.1 Spencer Cohomology and Koszul Homology;253
1.6.2;6.2 Cartan's Test;263
1.6.3;6.3 Pommaret Bases and Homology;271
1.6.4;6.4 Notes;277
1.7;7 Involution III: Differential Theory;280
1.7.1;7.1 (Geometric) Symbol and Principal Symbol;281
1.7.2;7.2 Involutive Differential Equations;298
1.7.3;7.3 Completion of Ordinary Differential Equations;313
1.7.3.1;Addendum: Constrained Hamiltonian Systems;319
1.7.4;7.4 Cartan–Kuranishi Completion;322
1.7.5;7.5 The Principal Symbol Revisited;327
1.7.6;7.6 -Regularity and Extended Principal Symbols;334
1.7.7;7.7 Notes;339
1.8;8 The Size of the Formal Solution Space;345
1.8.1;8.1 General Solutions;346
1.8.2;8.2 Cartan Characters and Hilbert Function;350
1.8.3;8.3 Differential Relations and Gauge Symmetries;359
1.8.3.1;Addendum: Einstein's Strength;368
1.8.4;8.4 Notes;369
1.9;9 Existence and Uniqueness of Solutions;372
1.9.1;9.1 Ordinary Differential Equations;373
1.9.2;9.2 The Cauchy–Kovalevskaya Theorem;385
1.9.3;9.3 Formally Well-Posed Initial Value Problems;389
1.9.4;9.4 The Cartan–Kähler Theorem;399
1.9.5;9.5 The Vessiot Distribution;407
1.9.5.1;Addendum: Generalised Prolongations;419
1.9.5.2;Addendum: Symmetry Theory and the Method of Characteristics;422
1.9.6;9.6 Flat Vessiot Connections;427
1.9.7;9.7 Notes;439
1.10;10 Linear Differential Equations;445
1.10.1;10.1 Elementary Geometric Theory;446
1.10.2;10.2 The Holmgren Theorem;450
1.10.3;10.3 Elliptic Equations;454
1.10.4;10.4 Hyperbolic Equations;463
1.10.5;10.5 Basic Algebraic Analysis;472
1.10.6;10.6 The Inverse Syzygy Problem;480
1.10.6.1;Addendum: Computing Extension Groups;487
1.10.6.2;Addendum: Algebraic Systems Theory;489
1.10.7;10.7 Completion to Involution;494
1.10.8;10.8 Linear Systems of Finite Type with Constant Coefficients;508
1.10.9;10.9 Notes;518
1.11;A Miscellaneous;522
1.11.1;A.1 Multi Indices and Orders;522
1.11.1.1;Addendum: Computing Derivative Trees;528
1.11.2;A.2 Real-Analytic Functions;530
1.11.3;A.3 Elementary Transformations of Differential Equations;532
1.11.3.1;Reduction to first order;532
1.11.3.2;Quasi-Linearisation;534
1.11.3.3;Transformation to one dependent variable;535
1.11.4;A.4 Modified Stirling Numbers;538
1.12;B Algebra;542
1.12.1;B.1 Some Basic Algebraic Structures;543
1.12.2;B.2 Homological Algebra;557
1.12.3;B.3 Coalgebras and Comodules;572
1.12.4;B.4 Gröbner Bases for Polynomial Ideals and Modules;580
1.13;C Differential Geometry;598
1.13.1;C.1 Manifolds;598
1.13.2;C.2 Vector Fields and Differential Forms;605
1.13.3;C.3 Distributions and the Frobenius Theorem;613
1.13.4;C.4 Connections;617
1.13.5;C.5 Lie Groups and Algebras;621
1.13.6;C.6 Symplectic Geometry and Generalisations;623
1.14;References;630
1.15;Glossary;650
1.16;Index;652