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Basic Global Relative Invariants for Nonlinear Differential Equations

Roger Chalkley University of Cincinnati, Cincinnati, OH
Available Formats:
Electronic ISBN: 978-1-4704-0494-9
Product Code: MEMO/190/888.E
List Price: $108.00 MAA Member Price:$97.20
AMS Member Price: $64.80 Click above image for expanded view Basic Global Relative Invariants for Nonlinear Differential Equations Roger Chalkley University of Cincinnati, Cincinnati, OH Available Formats:  Electronic ISBN: 978-1-4704-0494-9 Product Code: MEMO/190/888.E  List Price:$108.00 MAA Member Price: $97.20 AMS Member Price:$64.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 1902007; 365 pp
MSC: Primary 34;

The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order $m$ was initiated in 1879 with Edmund Laguerre's success for the special case $m = 3$. It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any $m \geq3$, each of the $m - 2$ basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations.

With respect to any fixed integer $\,m \geq 1$, the author begins by explicitly specifying the basic relative invariants for the class $\,\mathcal{C}_{m,2}$ that contains equations like $Q_{m} = 0$ in which $Q_{m}$ is a quadratic form in $y(z), \, \dots, \, y^{(m)}(z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y^{(m)}(z) \bigr)^{2}$ is $1$. Then, in terms of any fixed positive integers $m$ and $n$, the author explicitly specifies the basic relative invariants for the class $\,\mathcal{C}_{m,n}$ that contains equations like $H_{m,n} = 0$ in which $H_{m,n}$ is an $n$th-degree form in $y(z), \, \dots, \, y^{(m)}(z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y^{(m)}(z) \bigr)^{n}$ is $1$. These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equations.

• 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)
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 1902007; 365 pp
MSC: Primary 34;

The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order $m$ was initiated in 1879 with Edmund Laguerre's success for the special case $m = 3$. It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any $m \geq3$, each of the $m - 2$ basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations.

With respect to any fixed integer $\,m \geq 1$, the author begins by explicitly specifying the basic relative invariants for the class $\,\mathcal{C}_{m,2}$ that contains equations like $Q_{m} = 0$ in which $Q_{m}$ is a quadratic form in $y(z), \, \dots, \, y^{(m)}(z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y^{(m)}(z) \bigr)^{2}$ is $1$. Then, in terms of any fixed positive integers $m$ and $n$, the author explicitly specifies the basic relative invariants for the class $\,\mathcal{C}_{m,n}$ that contains equations like $H_{m,n} = 0$ in which $H_{m,n}$ is an $n$th-degree form in $y(z), \, \dots, \, y^{(m)}(z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y^{(m)}(z) \bigr)^{n}$ is $1$. These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equations.

• 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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