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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: http://dx.doi.org/10.1090/memo/0888
MathSciNet review: 2356350
MSC (2000): Primary 34A34; Secondary 12H20, 34A25, 34C14, 34M15

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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)

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