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# Basic global relative invariants for nonlinear differential equations

### About this Title

**Roger Chalkley**

Publication: Memoirs of the American Mathematical Society

Publication Year:
2007; Volume 190, Number 888

ISBNs: 978-0-8218-3991-1 (print); 978-1-4704-0494-9 (online)

DOI: https://doi.org/10.1090/memo/0888

MathSciNet review: 2356350

MSC: Primary 34A34; Secondary 12H20, 34A25, 34C14, 34M15

### Table of Contents

**Chapters**

- Part 1. Foundations for a general theory
- 1. Introduction
- 2. The coefficients $c^*_{i,j}(z)$ of (1.3)
- 3. The coefficients $c^{**}_{i,j}(\zeta )$ of (1.5)
- 4. Isolated results needed for completeness
- 5. Composite transformations and reductions
- 6. Related Laguerre-Forsyth canonical forms
- Part 2. The basic relative invariants for $Q_m=0$ when $m\ge 2$
- 7. Formulas that involve $L_{i,j}(z)$
- 8. Basic semi-invariants of the first kind for $m \geq 2$
- 9. Formulas that involve $V_{i,j}(z)$
- 10. Basic semi-invariants of the second kind for $m \geq 2$
- 11. The existence of basic relative invariants
- 12. The uniqueness of basic relative invariants
- 13. Real-valued functions of a real variable
- Part 3. Supplementary results
- 14. Relative invariants via basic ones for $m \geq 2$
- 15. Results about $Q_m$ as a quadratic form
- 16. Machine computations
- 17. The simplest of the Fano-type problems for (1.1)
- 18. Paul Appell’s condition of solvability for $Q_m = 0$
- 19. Appell’s condition for $Q_2 = 0$ and related topics
- 20. Rational semi-invariants and relative invariants
- Part 4. Generalizations for $H_{m,n}=0$
- 21. Introduction to the equations $H_{m,n} = 0$
- 22. Basic relative invariants for $H_{1,n} = 0$ when $n \geq 2$
- 23. Laguerre-Forsyth forms for $H_{m,n} = 0$ when $m \geq 2$
- 24. Formulas for basic relative invariants when $m \geq 2$
- 25. Extensions of Chapter 7 to $H_{m,n} = 0$, when $m \geq 2$
- 26. Extensions of Chapter 9 to $H_{m,n} = 0$, when $m \geq 2$
- 27. Basic relative invariants for $H_{m,n} = 0$ when $m \geq 2$
- Part 5. Additional classes of equations
- 28. The class of equations specified by $y''(z) y’(z)$
- 29. Formulations of greater generality
- 30. Invariants for simple equations unlike (29.1)