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  • The Research about Invariants of Ordinary Differential Equations

    Document thumbnail: The Research about Invariants of Ordinary Differential Equations
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    Document
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    Abstract. Several basic relative invariants for homogeneous linear differential equations were discovered during the years shortly after 1878. Also, a basic relative invariant was found by Paul Appell in 1889 for a type of nonlinear differential equation. There was little progress during the years 1892--1988 as researchers who worked with homogeneous linear differential equations were unknowingly handicapped by the standard practice of introducing binomial coefficients in the writing of their equations. They thereby failed to develop adequate formulas for the coefficients of equations resulting from a change of the independent variable. Consequently, for relative invariants as the most important kind of invariant, progress was stymied. The notation was simplified in 1989, adequate transformation formulas were developed, and explicit expressions were deduced in 2002 for all of the basic relative invariants of homogeneous linear differential equations. In 2007, explicit formulas were obtained for all of the basic relative invariants of a type of ordinary differential equation involving two parameters m and n that represent positive integers. When n = 1 and m >= 3, the formulas specialize to provide all of the basic relative invariants for homogeneous linear differential equations of order m; and, when m = n = 2, they yield all three of the basic relative invariants for the equations of Paul Appell. A general method developed in 2014 combines two relative invariants of weights p and q for the same type of equation to explicitly obtain a relative invariant of weight p+q +r, for any r >= 0. With that, the principal problems about relative invariants have now been solved. This monograph provides clear perspective about the reformulation begun after 1988 and recently completed. Chapters 15 and 18 show how the major difficulties confronting earlier researchers have been overcome.
    作者:
    Chalkley, Roger
    提交者:
    Roger Chalkley
    上传日期:
    03/21/2019
    更改日期:
    03/21/2019
    创建:
    2018-10-19
    证书:
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