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On the Commutativity of Certain Quasidifferential Expressions I
Author(s) -
Race D.,
Zettl A.
Publication year - 1990
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-42.3.489
Subject(s) - mathematics , equivalence (formal languages) , commutative property , expression (computer science) , differential (mechanical device) , pure mathematics , algebraic number , order (exchange) , commutative ring , set (abstract data type) , algebraic expression , differential algebra , ordinary differential equation , differential equation , mathematical analysis , computer science , physics , finance , economics , thermodynamics , programming language
We consider the question: When do two ordinary, linear, quasi‐differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi‐differential expressions less is known and such an algebraic structure can no longer be exploited. Here, we use the equivalence of matrices which determine the same expression to obtain a complete classification of all real symmetric expressions of both second order and fourth order which commute with any given real symmetric expression of second order. We also classify pairs of two‐term real symmetric expressions of order 2 n which commute. Each of these is a consequence of a corresponding but more general result for commuting expressions having complex coefficients but which are J ‐symmetric.