Parametric Lie group actions on global generalised solutions of nonlinear PDEs: including a solution to Hilbert's Fifth Problem
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1998
|
| Schriftenreihe: | Mathematics and its applications
452 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008447032&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XVII, 234 S. graph. Darst. |
| ISBN: | 0792352327 |
Internformat
MARC
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| 100 | 1 | |a Rosinger, Elemer E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Parametric Lie group actions on global generalised solutions of nonlinear PDEs |b including a solution to Hilbert's Fifth Problem |c by Elemér E. Rosinger |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 1998 | |
| 300 | |a XVII, 234 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 452 | |
| 650 | 4 | |a Differential equations, Nonlinear |x Numerical solutions | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Table of Contents
Preface xiii
1. Introduction 1
The Main Results 1
Parametrisation : Two Advantages 2
Functorial Advantage 2
Calculus Advantage 2
Parametrisations 3
Classical Lie Group Actions 4
Lie Group Actions on Generalised Functions 5
Answer to Hilbert s Fifth Problem . 5
Why Use the Nowhere Dense Differential Algebras ? 5
Open Problems 8
Towards a General Relativistic Quantum Theory ? 8
Acknowledgements 10
2. Actions on functions, difficulties 13
A Simple, Basic Observation 16
3. Parametric representation of functions 17
Need for Global Approach 17
Parametric Representation 18
Canonical Parametrisation 18
Classes of Parametrisations 19
Comparing Parametrisations 19
Basic Properties 21
Staying with Usual Functions 21
Equivalent Parametrisations 22
Parametric Functions and a Two Track Approach 23
4. Action on parametric representations 25
Natural Definition 25
Properties 25
vii
viii
Comments 30
Semigroup Actions 31
5. Parametric functions as solutions 33
Detour to the Results in the Appendix 33
Back to Usual, Nonparametric Functions 34
Advantages of Parametrisation 36
Singularity Points 37
Nonsingular Points 38
Derivatives of Parametric Functions 39
Parametric Functions as Solutions 41
Definition of Parametric Solution 41
Note on the Domain Dy 43
Equivalent Definition of Parametric Solution 44
Symmetries of Parametric Solutions 44
The Nature of the Domain Condition (5.74) 49
Contact Transformations 50
6. Rarefaction waves and Riemann solvers of the nonlinear shock
wave equation 53
7. Arbitrary nonlinear Lie group actions on generalised functions 61
7.1. Nowhere dense algebras 62
7.2. Lie group actions on sequences of functions 65
7.3. Lie group actions on generalised functions, preliminaries 68
7.4. Lie group actions on generalised functions, definition 72
7.4.1. Family of commutative diagrams associated to a
function 73
7.4.2. Uniform family of commutative diagrams associated
to a sequence of functions 75
7.4.3. The domain of definition of S 76
7.4.4. The definition of S 80
7.4.5. Constructing arbitrary Lie group actions on
generalised functions 85
7.4.6. The particular case of projectable Lie group actions 86
ix
7.4.7. The computation of S 88
7.4.8. Extension of the family giving the values of S 92
7.5. Two basic applications 95
7.5.1. Arbitrary Lie group actions on C°° smooth
functions 96
The Classical Case 98
7.5.2. Arbitrary Lie group actions on generalised functions 99
Global Cauchy Kovalevskaia Theorem 100
The Class Snd{ty of Generalised Functions 101
The Second Test 104
A General Method for Lie Group Actions on Snd(Q) 107
7.5.3. Two examples 109
7.6. The spaces C[nj and £ [n] and their extensions 119
Forward to the Second Track, Parametric Approach 122
7.7. The Closure property of the mapping 5 123
8. Nonprojectable Lie group symmetries of rarefaction waves
and Riemann solvers 125
The First Track, Usual, Nonparametric Approach 126
Generalised Solutions Go Into Generalised Solutions 127
8.1. Rarefaction waves 127
8.2. Riemann solvers 131
9. General parametric approach to Lie symmetries 141
9.1. Symmetries of parametric generalised solutions 142
Parametric Generalised Solutions 142
Note on the Concept of Solution 146
Invariance of Parametric Generalised Solutions 146
Four Concepts of Solution 148
9.2. Symmetries of generalised solutions in An 151
9.3. Revisiting the symmetries of shock waves 156
9.4. A general approach to the quotient spaces (C£°(M))N/ mnd 158
9.5. Conclusions 160
X
10. Projectable Lie group actions and Hilbert s fifth problem 163
11. Nonprojectable Lie group actions and an answer to Hilbert s
fifth problem 169
12. Singularities and the nowhere dense algebras of generalised
functions 173
Generalised Functions Means Dealing with
Singularities 174
How to Introduce Large Classes of Singularities
without Growth Conditions 176
The Naturaleness of the Nowhere Dense Ideals 180
Flabbiness and the Largest Class of Singularities 180
How about the Distributions ? 181
A Dichotomy 183
Nowhere Dense Algebras Applied in Abstract
Diferential Geometry and General Relativity 185
13. Lie semigroup actions and semisymmetries 189
13.1. Examples of semisymmetries of solutions of
nonlinear PDEs 191
13.2. How to generate genuine Lie semigroup actions 193
Lie Groups, Actions, Infinitesimal Generators,
ODEs, Flows and Evolution Operators 194
Finding Infinitesimal Generators for Genuine Lie
Semigroups 194
A Simple Example of Singular Nonautonomous ODE and
its Solutions 196
A Class of more General Nonautonomous Singular ODE
Systems and Solutions 198
Nonremovable Singularities 200
13.3. Standard reduction of nonautonomous ODEs to autonomous
ODEs 201
13.4. Applications to earlier examples of actions 206
13.5. Continuity, smoothness and domain of action 209
13.6. Remark on singularities 210
13.7. Evolution PDEs and genuine Lie semigroups 211
xi
13.8. On other instances of semigroups of actions 213
Appendix 215
Bibliography 225
Index 233
|
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| id | DE-604.BV012447805 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:30:27Z |
| institution | BVB |
| isbn | 0792352327 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008447032 |
| oclc_num | 39498668 |
| open_access_boolean | |
| owner | DE-824 DE-703 DE-11 |
| owner_facet | DE-824 DE-703 DE-11 |
| physical | XVII, 234 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Kluwer Acad. Publ. |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spellingShingle | Rosinger, Elemer E. Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem Mathematics and its applications Differential equations, Nonlinear Numerical solutions Lie groups Lie-Gruppe (DE-588)4035695-4 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4128900-6 |
| title | Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem |
| title_auth | Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem |
| title_exact_search | Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem |
| title_full | Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem by Elemér E. Rosinger |
| title_fullStr | Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem by Elemér E. Rosinger |
| title_full_unstemmed | Parametric Lie group actions on global generalised solutions of nonlinear PDEs including a solution to Hilbert's Fifth Problem by Elemér E. Rosinger |
| title_short | Parametric Lie group actions on global generalised solutions of nonlinear PDEs |
| title_sort | parametric lie group actions on global generalised solutions of nonlinear pdes including a solution to hilbert s fifth problem |
| title_sub | including a solution to Hilbert's Fifth Problem |
| topic | Differential equations, Nonlinear Numerical solutions Lie groups Lie-Gruppe (DE-588)4035695-4 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Differential equations, Nonlinear Numerical solutions Lie groups Lie-Gruppe Nichtlineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008447032&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT rosingerelemere parametricliegroupactionsonglobalgeneralisedsolutionsofnonlinearpdesincludingasolutiontohilbertsfifthproblem |