Basic Global Relative Invariants for Nonlinear Differential Equations (Memoirs of the American Mathematical Society)

Basic Global Relative Invariants for Nonlinear Differential Equations (Memoirs of the American Mathematical Society)

By: Roger Chalkley (author)Paperback

Up to 1 WeekUsually despatched within 1 week

Description

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.

Contents

Part 1. Foundations for a General Theory: Introduction The coefficients $c {i,j}^{*}(z)$ of (1.3) The coefficients $c {i,j}^{**}(\zeta)$ of (1.5) Isolated results needed for completeness Composite transformations and reductions Related Laguerre-Forsyth canonical forms Part 2. The Basic Relative Invariants for $Q {m} = 0$ when $m\geq 2$: Formulas that involve $L {i,j}(z)$ Basic semi-invariants of the first kind for $m \geq 2$ Formulas that involve $V {i,j}(z)$ Basic semi-invariants of the second kind for $m \geq 2$ The existence of basic relative invariants The uniqueness of basic relative invariants Real-valued functions of a real variable Part 3. Supplementary Results: Relative invariants via basic ones for $m \geq 2$ Results about $Q {m}$ as a quadratic form Machine computations The simplest of the Fano-type problems for (1.1) Paul Appell's condition of solvability for $Q {m} = 0$ Appell's condition for $Q {2} = 0$ and related topics Rational semi-invariants and relative invariants Part 4. Generalization for $H {m, n} = 0$: Introduction to the equations $H {m, n} = 0$ Basic relative invariants for $H {1,n} = 0$ when $n \geq 2$ Laguerre-Forsyth forms for $H {m, n} = 0$ when $m \geq 2$ Formulas for basic relative invariants when $m \geq 2$ Extensions of Chapter 7 to $H {m,n} = 0$, when $m \geq 2$ Extensions of Chapter 9 to $H {m,n} = 0$, when $m \geq 2$ Basic relative invariants for $H {m, n} = 0$ when $m \geq2$ Additional Classes of Equations: The class of equations specified by $y""(z)$$y'(z)$ Formulations of greater generality Invariants for simple equations unlike (29.1) Bibliography Index.

Product Details

  • ISBN13: 9780821839911
  • Format: Paperback
  • Number Of Pages: 365
  • ID: 9780821839911
  • ISBN10: 0821839918

Delivery Information

  • Saver Delivery: Yes
  • 1st Class Delivery: Yes
  • Courier Delivery: Yes
  • Store Delivery: Yes

Prices are for internet purchases only. Prices and availability in WHSmith Stores may vary significantly

Close