Basic global relative invariants for nonlinear differential equations:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Providence, RI
American Mathematical Society
2007
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| Series: | Memoirs of the American Mathematical Society
888 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Volume 190, number 888 (first of three numbers.) |
| Physical Description: | XII, 365 S. |
| ISBN: | 9780821839911 |
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MARC
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| 100 | 1 | |a Chalkley, Roger |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Basic global relative invariants for nonlinear differential equations |c Roger Chalkley |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2007 | |
| 300 | |a XII, 365 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Memoirs of the American Mathematical Society |v 888 | |
| 500 | |a Volume 190, number 888 (first of three numbers.) | ||
| 650 | 4 | |a Invariants | |
| 650 | 7 | |a Invariants |2 ram | |
| 650 | 4 | |a Équations différentielles non linéaires | |
| 650 | 7 | |a Équations différentielles non linéaires |2 ram | |
| 650 | 0 | 7 | |a Nichtlineare Differentialgleichung |0 (DE-588)4205536-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Relative Invariante |0 (DE-588)4177681-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nichtlineare Differentialgleichung |0 (DE-588)4205536-2 |D s |
| 689 | 0 | 1 | |a Relative Invariante |0 (DE-588)4177681-1 |D s |
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| 830 | 0 | |a Memoirs of the American Mathematical Society |v 888 |w (DE-604)BV008000141 |9 888 | |
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Record in the Search Index
| _version_ | 1804137189148196864 |
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| adam_text | Contents
Preface ix
Part 1. Foundations for a General Theory 1
Chapter 1. Introduction 3
1.1. Historical motivation 3
1.2. Context and definitions used throughout Chapters 1 20 4
1.3. Main Theorem 8
1.4. Notational abbreviations employed in Chapters 7 10 10
1.5. Illustrations for the use of formulas (1.18) (1.28) when m 2 10
1.6. Completion of Paul AppelPs research in [7] about Q2 = 0 13
1.7. Inclusion of homogeneous linear differential equations 17
1.8. Order of presentation 20
Chapter 2. The Coefficients c^z) of (1.3) 23
Chapter 3. The Coefficients c**{Q of (1.5) 29
Chapter 4. Isolated Results Needed for Completeness 35
4.1. Nonsolutions of nontrivial equations 35
4.2. Semi invariants of the second kind are isobaric 37
4.3. Substitutions in regard to the derivation for «Sm 38
4.4. All of the relative invariants for (1.1) when m = 1 39
4.5. Isobaric semi invariants of weight 2 when m 2 43
4.6. Further semi invariants of the second kind when m 2 45
Chapter 5. Composite Transformations and Reductions 47
5.1. The composite of substitutions (1.2) and (1.4) 47
5.2. The condition d0,i(C) = do,2(C) = 0 for (5.1) when m 2 49
5.3. Laguerre Forsyth canonical forms for linear equations 50
5.4. A Laguerre Forsyth canonical form for (1.1) when m 2 51
5.5. There are no relative invariants in Q{u;o, 1,1^0,2} 52
Chapter 6. Related Laguerre Forsyth Canonical Forms 53
6.1. Two Laguerre Forsyth forms related by a transformation 54
6.2. Identities for the coefficients of related canonical forms 58
Part 2. The Basic Relative Invariants for Qm = 0 when m 2 67
Chapter 7. Formulas That Involve Litj (z) 69
7.1. The coefficients of (5.1) when do,i«) = doAO = ° 69
iii
iv CONTENTS
7.2. Derivatives of the coefficients in (5.1) when rfo,i(C) = ^o,2(C) = 0 73
7.3. Special combinations of the coefficients for (5.1) 75
Chapter 8. Basic Semi Invariants of the First Kind for m 2 87
Chapter 9. Formulas That Involve Viti ,(z) 93
9.1. The coefficients of (5.1) when do,i(C) = do,2(C) = 0 93
9.2. Derivatives of the coefficients in (5.1) when do,i(C) = ^0,2 (C) =0 97
9.3. Special combinations of the coefficients for (5.1) 99
Chapter 10. Basic Semi Invariants of the Second Kind for m 2 111
Chapter 11. The Existence of Basic Relative Invariants 119
Chapter 12. The Uniqueness of Basic Relative Invariants 121
12.1. Some polynomials that are not relative invariants 121
12.2. The uniqueness of basic relative invariants 130
12.3. Algebraic independence 131
Chapter 13. Real Valued Functions of a Real Variable 135
13.1. A suitable context for the evaluations when m 2 135
13.2. Appropriate hypotheses when m = 1 140
Part 3. Supplementary Results 141
Chapter 14. Relative Invariants via Basic Ones for m 2 143
14.1. Relative invariants in terms of basic ones and am^ 143
14.2. Combinations of invariants that yield other invariants 146
14.3. The relative invariants of weight 9 for the equations Q2 — 0 153
Chapter 15. Results about Qm as a Quadratic Form 157
15.1. For Qm to have a nontrivial factorization 157
15.2. Relative invariants defined by determinants 162
Chapter 16. Machine Computations 167
16.1. Expansion of D2 in terms of am^, 1m, ,i, 2 m,i,2, and XTO2,2 167
16.2. The expansions for E6 and E7 in (1.81) and (1.82) 168
16.3. A comprehensive check on the consistency of (1.14) (1.38) 169
16.4. The relative invariants of weight 9 for the equations Q2 = 0 172
16.5. Entry of keyboard instructions 177
Chapter 17. The Simplest of the Fano Type Problems for (1.1) 179
17.1. Historical motivation 179
17.2. Results for (1.1) analogous to those in Example 17.1 for (17.1) 180
17.3. An equation (1.1) that has special polynomial solutions 183
Chapter 18. Paul Appell s Condition of Solvability for Qm =0 185
18.1. Context and historical motivation 185
18.2. The equivalent condition of Theorem 18.1 186
18.3. Solutions of Qm = 0 when Appell s condition is satisfied 189
Chapter 19. Appell s Condition for Q2 = 0 and Related Topics 193
CONTENTS v
19.1. Consequences of Chapter 18 for the equations Q2 = 0 193
19.2. An improvement for Proposition 19.6 197
19.3. An example to illustrate Theorem 19.7 198
19.4. Other forms for the nonsingular solutions in Theorem 19.7 199
19.5. Conditions of the type {ui{z))2 Auo(z) u2(z) ^ 0 201
19.6. Equations constructed to have given nonsingular solutions 203
19.7. Absence of movable branch points 206
19.8. Two results for third order linear equations 208
19.9. Extensions to linear equations of higher order 210
19.10. Linear substitutions in binary forms 213
19.11. Properties of the polynomial Tn in (19.120) 216
Chapter 20. Rational Semi Invariants and Relative Invariants 219
20.1. Terminology for an extended context 219
20.2. The integer s in Definition 20.2 220
20.3. A context for the remainder of this chapter 223
20.4. A technical construction needed for Section 20.5 225
20.5. Rational semi invariants of the first kind 230
20.6. A technical construction needed for Section 20.7 234
20.7. Rational semi invariants of the second kind 240
20.8. The structure of rational relative invariants 243
20.9. Substitutions into rational functions of Qm 244
Part 4. Generalizations for Hmn = 0 247
Chapter 21. Introduction to the Equations Hmn = 0 249
21.1. Transformations produced by changing the variables in Hmn = 0 249
21.2. Context and definitions 251
21.3. A summary of results and a derivation for Sm,n 253
21.4. Inclusion of relative invariants for Hm,q = 0 when 1 q n 253
21.5. Nonsolutions of nontrivial equations 255
Chapter 22. Basic Relative Invariants for H ,n = 0 when n 2 257
22.1. Existence 257
22.2. Some polynomials that are not relative invariants 260
22.3. Uniqueness of B2, ..., Bn as basic relative invariants 264
Chapter 23. Laguerre Forsyth Forms for Hm,n = 0 when m 2 265
23.1. The composite of substitutions (21.2) and (21.4) 265
23.2. Laguerre Forsyth reductions when m 2 267
23.3. Related Laguerre Forsyth canonical forms 268
23.4. Identities for the coefficients of related canonical forms 271
23.5. Use of computer algebra to check Theorem 23.8 272
23.6. Properties of am,2 in (23.14) and 6m,2 in (23.16) 273
Chapter 24. Formulas for Basic Relative Invariants when m 2 275
24.1. Formulas for (21.1) that are analogous to (1.18) (1.38) 275
24.2. Notational abbreviations employed in Chapters 25 26 277
24.3. The special situations when n = 2 and when n = 1 277
24.4. Plan to evaluate Laguerre Forsyth sums 278
vi CONTENTS
Chapter 25. Extensions of Chapter 7 to Hm,n = 0, when m 2 279
25.1. The coefficients of (23.1) when cfo,...,o,i(C) = do,...,o,2(C) = 0 279
25.2. Evaluation of Laguerre Forsyth sums 280
Chapter 26. Extensions of Chapter 9 to Hm^n = 0, when m 2 293
26.1. The coefficients of (23.1) when do,...,o,i(C) = do,...,oAO = 0 293
26.2. Evaluation of Laguerre Forsyth sums 294
Chapter 27. Basic Relative Invariants for Hm,n — 0 when m 2 307
27.1. Preliminary results 307
27.2. Principal results 314
27.3. Some polynomials that are not relative invariants 316
27.4. Uniqueness of basic relative invariants 316
27.5. The basic relative invariant of index (Zi, ..., ln) when ln = 1 317
27.6. The number of basic relative invariants 318
27.7. Relative invariants via basic ones for m 2 320
27.8. Rational semi invariants for various classes of equations 321
Part 5. Additional Classes of Equations 323
Chapter 28. The Class of Equations Specified by y (z) y (z) 325
28.1. Notation and terminology 325
28.2. Principal formulas 326
28.3. The relative invariants of weight 9 for the equations (28.1) 328
28.4. Computational procedure for Section 28.3 330
28.5. Laguerre Forsyth reductions for the equations (28.1) 331
28.6. All of the relative invariants for the equations (28.1) 332
Chapter 29. Formulations of Greater Generality 335
29.1. Equations characterized by a single monic term 335
29.2. Relative invariants for some nonhomogeneous equations 337
Chapter 30. Invariants for Simple Equations unlike (29.1) 347
30.1. Equations without a dominant term 347
30.2. Another class of homogeneous quadratic equations 353
30.3. The character of Xq as a polynomial absolute invariant 356
Bibliography 357
Index 359
|
| adam_txt |
Contents
Preface ix
Part 1. Foundations for a General Theory 1
Chapter 1. Introduction 3
1.1. Historical motivation 3
1.2. Context and definitions used throughout Chapters 1 20 4
1.3. Main Theorem 8
1.4. Notational abbreviations employed in Chapters 7 10 10
1.5. Illustrations for the use of formulas (1.18) (1.28) when m 2 10
1.6. Completion of Paul AppelPs research in [7] about Q2 = 0 13
1.7. Inclusion of homogeneous linear differential equations 17
1.8. Order of presentation 20
Chapter 2. The Coefficients c^z) of (1.3) 23
Chapter 3. The Coefficients c**{Q of (1.5) 29
Chapter 4. Isolated Results Needed for Completeness 35
4.1. Nonsolutions of nontrivial equations 35
4.2. Semi invariants of the second kind are isobaric 37
4.3. Substitutions in regard to the derivation ' for «Sm 38
4.4. All of the relative invariants for (1.1) when m = 1 39
4.5. Isobaric semi invariants of weight 2 when m 2 43
4.6. Further semi invariants of the second kind when m 2 45
Chapter 5. Composite Transformations and Reductions 47
5.1. The composite of substitutions (1.2) and (1.4) 47
5.2. The condition d0,i(C) = do,2(C) = 0 for (5.1) when m 2 49
5.3. Laguerre Forsyth canonical forms for linear equations 50
5.4. A Laguerre Forsyth canonical form for (1.1) when m 2 51
5.5. There are no relative invariants in Q{u;o, 1,1^0,2} 52
Chapter 6. Related Laguerre Forsyth Canonical Forms 53
6.1. Two Laguerre Forsyth forms related by a transformation 54
6.2. Identities for the coefficients of related canonical forms 58
Part 2. The Basic Relative Invariants for Qm = 0 when m 2 67
Chapter 7. Formulas That Involve Litj (z) 69
7.1. The coefficients of (5.1) when do,i«) = doAO = ° 69
iii
iv CONTENTS
7.2. Derivatives of the coefficients in (5.1) when rfo,i(C) = ^o,2(C) = 0 73
7.3. Special combinations of the coefficients for (5.1) 75
Chapter 8. Basic Semi Invariants of the First Kind for m 2 87
Chapter 9. Formulas That Involve Viti ,(z) 93
9.1. The coefficients of (5.1) when do,i(C) = do,2(C) = 0 93
9.2. Derivatives of the coefficients in (5.1) when do,i(C) = ^0,2 (C) =0 97
9.3. Special combinations of the coefficients for (5.1) 99
Chapter 10. Basic Semi Invariants of the Second Kind for m 2 111
Chapter 11. The Existence of Basic Relative Invariants 119
Chapter 12. The Uniqueness of Basic Relative Invariants 121
12.1. Some polynomials that are not relative invariants 121
12.2. The uniqueness of basic relative invariants 130
12.3. Algebraic independence 131
Chapter 13. Real Valued Functions of a Real Variable 135
13.1. A suitable context for the evaluations when m 2 135
13.2. Appropriate hypotheses when m = 1 140
Part 3. Supplementary Results 141
Chapter 14. Relative Invariants via Basic Ones for m 2 143
14.1. Relative invariants in terms of basic ones and am^ 143
14.2. Combinations of invariants that yield other invariants 146
14.3. The relative invariants of weight 9 for the equations Q2 — 0 153
Chapter 15. Results about Qm as a Quadratic Form 157
15.1. For Qm to have a nontrivial factorization 157
15.2. Relative invariants defined by determinants 162
Chapter 16. Machine Computations 167
16.1. Expansion of D2 in terms of am^, 1m,\,i, 2"m,i,2, and XTO2,2 167
16.2. The expansions for E6 and E7 in (1.81) and (1.82) 168
16.3. A comprehensive check on the consistency of (1.14) (1.38) 169
16.4. The relative invariants of weight 9 for the equations Q2 = 0 172
16.5. Entry of keyboard instructions 177
Chapter 17. The Simplest of the Fano Type Problems for (1.1) 179
17.1. Historical motivation 179
17.2. Results for (1.1) analogous to those in Example 17.1 for (17.1) 180
17.3. An equation (1.1) that has special polynomial solutions 183
Chapter 18. Paul Appell's Condition of Solvability for Qm =0 185
18.1. Context and historical motivation 185
18.2. The equivalent condition of Theorem 18.1 186
18.3. Solutions of Qm = 0 when Appell's condition is satisfied 189
Chapter 19. Appell's Condition for Q2 = 0 and Related Topics 193
CONTENTS v
19.1. Consequences of Chapter 18 for the equations Q2 = 0 193
19.2. An improvement for Proposition 19.6 197
19.3. An example to illustrate Theorem 19.7 198
19.4. Other forms for the nonsingular solutions in Theorem 19.7 199
19.5. Conditions of the type {ui{z))2 Auo(z) u2(z) ^ 0 201
19.6. Equations constructed to have given nonsingular solutions 203
19.7. Absence of movable branch points 206
19.8. Two results for third order linear equations 208
19.9. Extensions to linear equations of higher order 210
19.10. Linear substitutions in binary forms 213
19.11. Properties of the polynomial Tn in (19.120) 216
Chapter 20. Rational Semi Invariants and Relative Invariants 219
20.1. Terminology for an extended context 219
20.2. The integer s in Definition 20.2 220
20.3. A context for the remainder of this chapter 223
20.4. A technical construction needed for Section 20.5 225
20.5. Rational semi invariants of the first kind 230
20.6. A technical construction needed for Section 20.7 234
20.7. Rational semi invariants of the second kind 240
20.8. The structure of rational relative invariants 243
20.9. Substitutions into rational functions of Qm 244
Part 4. Generalizations for Hmn = 0 247
Chapter 21. Introduction to the Equations Hmn = 0 249
21.1. Transformations produced by changing the variables in Hmn = 0 249
21.2. Context and definitions 251
21.3. A summary of results and a derivation ' for Sm,n 253
21.4. Inclusion of relative invariants for Hm,q = 0 when 1 q n 253
21.5. Nonsolutions of nontrivial equations 255
Chapter 22. Basic Relative Invariants for H\,n = 0 when n 2 257
22.1. Existence 257
22.2. Some polynomials that are not relative invariants 260
22.3. Uniqueness of B2, ., Bn as basic relative invariants 264
Chapter 23. Laguerre Forsyth Forms for Hm,n = 0 when m 2 265
23.1. The composite of substitutions (21.2) and (21.4) 265
23.2. Laguerre Forsyth reductions when m 2 267
23.3. Related Laguerre Forsyth canonical forms 268
23.4. Identities for the coefficients of related canonical forms 271
23.5. Use of computer algebra to check Theorem 23.8 272
23.6. Properties of am,2 in (23.14) and 6m,2 in (23.16) 273
Chapter 24. Formulas for Basic Relative Invariants when m 2 275
24.1. Formulas for (21.1) that are analogous to (1.18) (1.38) 275
24.2. Notational abbreviations employed in Chapters 25 26 277
24.3. The special situations when n = 2 and when n = 1 277
24.4. Plan to evaluate Laguerre Forsyth sums 278
vi CONTENTS
Chapter 25. Extensions of Chapter 7 to Hm,n = 0, when m 2 279
25.1. The coefficients of (23.1) when cfo,.,o,i(C) = do,.,o,2(C) = 0 279
25.2. Evaluation of Laguerre Forsyth sums 280
Chapter 26. Extensions of Chapter 9 to Hm^n = 0, when m 2 293
26.1. The coefficients of (23.1) when do,.,o,i(C) = do,.,oAO = 0 293
26.2. Evaluation of Laguerre Forsyth sums 294
Chapter 27. Basic Relative Invariants for Hm,n — 0 when m 2 307
27.1. Preliminary results 307
27.2. Principal results 314
27.3. Some polynomials that are not relative invariants 316
27.4. Uniqueness of basic relative invariants 316
27.5. The basic relative invariant of index (Zi, ., ln) when ln = 1 317
27.6. The number of basic relative invariants 318
27.7. Relative invariants via basic ones for m 2 320
27.8. Rational semi invariants for various classes of equations 321
Part 5. Additional Classes of Equations 323
Chapter 28. The Class of Equations Specified by y"(z) y'(z) 325
28.1. Notation and terminology 325
28.2. Principal formulas 326
28.3. The relative invariants of weight 9 for the equations (28.1) 328
28.4. Computational procedure for Section 28.3 330
28.5. Laguerre Forsyth reductions for the equations (28.1) 331
28.6. All of the relative invariants for the equations (28.1) 332
Chapter 29. Formulations of Greater Generality 335
29.1. Equations characterized by a single monic term 335
29.2. Relative invariants for some nonhomogeneous equations 337
Chapter 30. Invariants for Simple Equations unlike (29.1) 347
30.1. Equations without a dominant term 347
30.2. Another class of homogeneous quadratic equations 353
30.3. The character of Xq as a polynomial absolute invariant 356
Bibliography 357
Index 359 |
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| id | DE-604.BV022947668 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:00:36Z |
| indexdate | 2024-07-09T21:08:20Z |
| institution | BVB |
| isbn | 9780821839911 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016152202 |
| oclc_num | 184969045 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-29T DE-11 DE-188 DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-11 DE-188 DE-83 |
| physical | XII, 365 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Memoirs of the American Mathematical Society |
| series2 | Memoirs of the American Mathematical Society |
| spelling | Chalkley, Roger Verfasser aut Basic global relative invariants for nonlinear differential equations Roger Chalkley Providence, RI American Mathematical Society 2007 XII, 365 S. txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society 888 Volume 190, number 888 (first of three numbers.) Invariants Invariants ram Équations différentielles non linéaires Équations différentielles non linéaires ram Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd rswk-swf Relative Invariante (DE-588)4177681-1 gnd rswk-swf Nichtlineare Differentialgleichung (DE-588)4205536-2 s Relative Invariante (DE-588)4177681-1 s DE-604 Memoirs of the American Mathematical Society 888 (DE-604)BV008000141 888 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016152202&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chalkley, Roger Basic global relative invariants for nonlinear differential equations Memoirs of the American Mathematical Society Invariants Invariants ram Équations différentielles non linéaires Équations différentielles non linéaires ram Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Relative Invariante (DE-588)4177681-1 gnd |
| subject_GND | (DE-588)4205536-2 (DE-588)4177681-1 |
| title | Basic global relative invariants for nonlinear differential equations |
| title_auth | Basic global relative invariants for nonlinear differential equations |
| title_exact_search | Basic global relative invariants for nonlinear differential equations |
| title_exact_search_txtP | Basic global relative invariants for nonlinear differential equations |
| title_full | Basic global relative invariants for nonlinear differential equations Roger Chalkley |
| title_fullStr | Basic global relative invariants for nonlinear differential equations Roger Chalkley |
| title_full_unstemmed | Basic global relative invariants for nonlinear differential equations Roger Chalkley |
| title_short | Basic global relative invariants for nonlinear differential equations |
| title_sort | basic global relative invariants for nonlinear differential equations |
| topic | Invariants Invariants ram Équations différentielles non linéaires Équations différentielles non linéaires ram Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Relative Invariante (DE-588)4177681-1 gnd |
| topic_facet | Invariants Équations différentielles non linéaires Nichtlineare Differentialgleichung Relative Invariante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016152202&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
| work_keys_str_mv | AT chalkleyroger basicglobalrelativeinvariantsfornonlineardifferentialequations |