New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
Return to List

Basic Global Relative Invariants for Nonlinear Differential Equations
Roger Chalkley, University of Cincinnati, OH
 SEARCH THIS BOOK:
Memoirs of the American Mathematical Society
2007; 365 pp; softcover
Volume: 190
ISBN-10: 0-8218-3991-8
ISBN-13: 978-0-8218-3991-1
List Price: US$108 Individual Members: US$64.80
Institutional Members: US\$86.40
Order Code: MEMO/190/888

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

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. Generalizations 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
 AMS Home | Comments: webmaster@ams.org © Copyright 2014, American Mathematical Society Privacy Statement