Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Basic Global Relative Invariants for Nonlinear Differential Equations
 
Roger Chalkley University of Cincinnati, Cincinnati, OH
Basic Global Relative Invariants for Nonlinear Differential Equations
eBook 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
Basic Global Relative Invariants for Nonlinear Differential Equations
Click above image for expanded view
Basic Global Relative Invariants for Nonlinear Differential Equations
Roger Chalkley University of Cincinnati, Cincinnati, OH
eBook 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.

  • 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)
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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)
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.