Basic global relative invariants for homogeneous linear differential equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
AMS
2002
|
| Series: | Memoirs of the American Mathematical Society
744 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | "Volume 156, number 744 (end of volume)." |
| Physical Description: | XI, 204 S. |
| ISBN: | 0821827812 |
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MARC
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| 100 | 1 | |a Chalkley, Roger |d 1931- |e Verfasser |0 (DE-588)134257227 |4 aut | |
| 245 | 1 | 0 | |a Basic global relative invariants for homogeneous linear differential equations |c Roger Chalkley |
| 264 | 1 | |a Providence, RI |b AMS |c 2002 | |
| 300 | |a XI, 204 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Memoirs of the American Mathematical Society |v 744 | |
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| 650 | 7 | |a Differentiaalvergelijkingen |2 gtt | |
| 650 | 7 | |a Invarianten |2 gtt | |
| 650 | 4 | |a Differential equations, Linear | |
| 650 | 4 | |a Invariants | |
| 830 | 0 | |a Memoirs of the American Mathematical Society |v 744 |w (DE-604)BV008000141 |9 744 | |
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Record in the Search Index
| _version_ | 1804129106894258176 |
|---|---|
| adam_text | Contents
Chapter 1. Introduction 1
1.1. General summary 1
1.2. The ring 7£m of polynomials to be employed throughout 2
1.3. Terminology used throughout (except as modified in Chapter 12) 3
1.4. Principal results 5
1.5. Symbolism for polynomials in Hm 7
1.6. Miscellaneous observations 8
1.7. Order of presentation 9
Chapter 2. Some Problems of Historical Importance 12
2.1. The older semi invariants 12
2.2. A challenging problem posed by J. Liouville in 1839 13
2.3. The first construction of a decisive set for any m 3 16
2.4. Decisive sets of semi invariants 18
2.5. Awkward formulations involving the older semi invariants 19
Chapter 3. Illustrations for Some Results in Chapters 1 and 2 21
3.1. {G2, ¦ ¦ •, Gm} and {G3, ..., Gm} are not decisive sets 21
3.2. A simple check on the consistency of (1.20) (1.27) 22
3.3. A simple check on the consistency of (1.12)—(1.15) 23
3.4. Two types of symbolic sums and their evaluation 24
3.5. Computations when m is a symbol for an integer 3 27
3.6. A comprehensive check on the consistency of (1.8) (1.27) 28
Chapter 4. Ln and /„,, as Semi Invariants of the First Kind 31
Chapter 5. Vn and Jn i as Semi Invariants of the Second Kind 34
Chapter 6. The Coefficients of Transformed Equations 39
6.1. Alternative formulas for c**{C) in (1.5) 39
6.2. The coefficients of a composite transformation 40
6.3. Several examples 44
6.4. Proof of an old observation 45
6.5. Conditions for transformed equations 46
6.6. Formulas for later reference 49
Chapter 7. Formulas That Involve Ln{z) or In,n(z) 50
7.1. The coefficients of (6.8) when di(C) s d2(£) = 0 50
7.2. Derivatives for the coefficients of (6.8) when di(Q = d2«) =0 53
7.3. Identities for the coefficients of (6.8) when d «) = d2(C) =0 55
vii
viii CONTENTS
Chapter 8. Formulas That Involve Vn{z) or Jn n(z) 61
8.1. The coefficients of (6.8) when di{Q = d2(C) =0 61
8.2. Derivatives for the coefficients of (6.8) when di(Q = d2(C) = 0 64
8.3. Identities for the coefficients of (6.8) when d^Q = d2(C) =0 66
Chapter 9. Verification of /„,„ = Jn%n and Various Observations 71
9.1. Proof for the first part of the Main Theorem in Chapter 1 71
9.2. Global sets 73
9.3. A fourth type of invariant: an absolute invariant 74
9.4. Laguerre Forsyth canonical forms 75
Chapter 10. The Local Constructions of Earlier Research 77
10.1. Standard techniques 77
10.2. An improved computational procedure 78
10.3. Hindrances to earlier research 79
Chapter 11. Relations for Gu Ht, and Lt That Yield
Equivalent Formulas for Basic Relative Invariants 81
11.1. The identity Ht = G; A.m i 81
11.2. Formulas for Lo, ..., Lm in terms of G?o, ¦ ¦ ¦, Gm 83
11.3. Formulas for L3, ..., Lm in terms of H3, ..., Hm 87
11.4. Formulas for Go, ¦¦¦, Gm in terms of Lo, ..., Lm 90
11.5. Formulas for H3, ..., Hm in terms of L3, ..., Lm 93
Chapter 12. Real Valued Functions of a Real Variable 94
12.1. A context suitable for the required evaluations 94
12.2. Xm 3, ..., 1m,m are relative invariants of global character 95
12.3. Decisive sets for Context 12.1 96
12.4. Global sets 100
Chapter 13. A Constructive Method for Imposing Conditions
on Laguerre Forsyth Canonical Forms 104
13.1. Conditions imposed directly on the coefficients of (1.1) 104
13.2. Illustrations of the technique 106
Chapter 14. Additional Formulas for Kitj, Uij, Ai%j, Ditj, ... 108
14.1. Alternative formulas for Kitj in (1.12) (1.14) 108
14.2. Alternative formulas for U j in (1.21) (1.23) 109
14.3. Alternative formulas for Aid in (2.19) (2.21) 110
14.4. Alternative formulas for D{j in (2.42) (2.44) 110
14.5. Alternative formulas for A j(z) in (6.2) (6.3) 111
14.6. Alternative formulas for f i,j(z) in (6.10) (6.11) 112
14.7. Polynomials Qitj and Ri for use in Section 15.2 112
Chapter 15. Three Canonical Forms Are Now Available 114
15.1. Reduction of (1.1) to a Halphen canonical form 114
15.2. A neglected canonical form for any (1.1) having m 2 116
15.3. Semi invariants R2, ..., Rm, analogous to G2, ..., Gm 119
15.4. {JK2, •.., Rm} and {#3, ..., Rm} are not decisive sets 121
15.5. Semi invariants 53, ..., Sm analogous to H3, ..., Hm 121
15.6. The identity Si = Ri Ti,m i 122 J
CONTENTS ix
15.7. Solving (14.28) for Wi in terms of Rq, ..., Rm 125
Chapter 16. Interesting Problems that Require Further Study 130
16.1. Constructive characterizations for Fano s research 130
16.2. Generalizations based on Lie groups 132
Appendix A. Results Needed for Self Containment 133
A.I. The coefficients c|(z) for (1.3) 134
A.2. The coefficients cf*(C) for (1.5) 135
A.3. Semi invariants of the second kind given by S* 1 + kb S 138
A.4. Non solutions for non zero equations 139
A.5. Semi invariants of the second kind are isobaric 140
A.6. Supplementary observations about invariants 142
A.7. Identities relating the coefficients of (1.3) or (1.5) to (1.1) 146
A.8. Pencil and paper computation for J33 148
A.9. Machine computations for the coefficients Cj(z) of (15.32) 149
Appendix B. Related Studies for a Class of Nonlinear Equations 151
B.I. P. AppelPs influence on our solution of J. Liouville s problem 151
B.2. Deduction of (B.16) from [12, page 139, Theorem 4.1] 154
Appendix C. Polynomials That Are Linear in a Key Variable 156
C.I. Some polynomials that are not relative invariants 156
C.2. The uniqueness of basic relative invariants and a proof
for the second part of the Main Theorem in Chapter 1 163
Appendix D. Rational Semi Invariants and Relative Invariants 165
D.I. Introduction 165
D.2. Definitions of rational semi invariants and relative invariants 166
D.3. The integer s in Definition D.2 166
D.4. A context for the remainder of this appendix 169
D.5. A technical construction needed for Section D.6 172
D.6. Rational semi invariants of the first kind 176
D.7. A technical construction needed for Section D.8 180
D.8. Rational semi invariants of the second kind 185
D.9. The structure of rational relative invariants 188
D.10. The structure of absolute invariants 188
D.ll. Substitutions into fractions of Qm 189
Appendix E. Generating Additional Relative Invariants 192
E.I. Two constructions due to G. H. Halphen and A. R. Forsyth 192
E.2. Examples for the constructions 194
E.3. Historical observations 195
E.4. A challenging problem for further research 195
Bibliography 197
Index 200
|
| any_adam_object | 1 |
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| discipline | Mathematik |
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| id | DE-604.BV014220716 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:59:52Z |
| institution | BVB |
| isbn | 0821827812 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009750696 |
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| physical | XI, 204 S. |
| publishDate | 2002 |
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| publishDateSort | 2002 |
| publisher | AMS |
| record_format | marc |
| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spelling | Chalkley, Roger 1931- Verfasser (DE-588)134257227 aut Basic global relative invariants for homogeneous linear differential equations Roger Chalkley Providence, RI AMS 2002 XI, 204 S. txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society 744 "Volume 156, number 744 (end of volume)." Differentiaalvergelijkingen gtt Invarianten gtt Differential equations, Linear Invariants Memoirs of the American Mathematical Society 744 (DE-604)BV008000141 744 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009750696&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chalkley, Roger 1931- Basic global relative invariants for homogeneous linear differential equations Memoirs of the American Mathematical Society Differentiaalvergelijkingen gtt Invarianten gtt Differential equations, Linear Invariants |
| title | Basic global relative invariants for homogeneous linear differential equations |
| title_auth | Basic global relative invariants for homogeneous linear differential equations |
| title_exact_search | Basic global relative invariants for homogeneous linear differential equations |
| title_full | Basic global relative invariants for homogeneous linear differential equations Roger Chalkley |
| title_fullStr | Basic global relative invariants for homogeneous linear differential equations Roger Chalkley |
| title_full_unstemmed | Basic global relative invariants for homogeneous linear differential equations Roger Chalkley |
| title_short | Basic global relative invariants for homogeneous linear differential equations |
| title_sort | basic global relative invariants for homogeneous linear differential equations |
| topic | Differentiaalvergelijkingen gtt Invarianten gtt Differential equations, Linear Invariants |
| topic_facet | Differentiaalvergelijkingen Invarianten Differential equations, Linear Invariants |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009750696&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
| work_keys_str_mv | AT chalkleyroger basicglobalrelativeinvariantsforhomogeneouslineardifferentialequations |