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Algebras and differential equations

Published online by Cambridge University Press:  22 January 2016

Helmut Röhrl
Helmut Röhrl*
Affiliation:
University of California at San Diego and Nagoya University
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One purpose of this paper is a purely algebraic study of (systems of) ordinary differential equations of the type

where the coefficients are taken from a fixed associative, commutative, unital ring R, such as the field R of real or C of complex numbers or a commutative, unital Banach algebra. The right hand sides of D are considered to be elements in the polynomial ring R[X1, …, Xn] of associating but non-commuting variables X1, …, Xn. An algebraic study calls for maps between such differential equations and, in fact, morphisms are defined between differential equations having the same arity m but not necessarily the same dimension n. These morphisms are rectangular matrices with entries in R which satisfy certain relations. This leads to a category RDiffm whose objects are precisely the differential equations of arity m and in which the composition of the morphisms is the usual matrix multiplication.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 68 , November 1977 , pp. 59 - 122
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1977

References

[1] Bourbaki, N.: Algebra I. Addison-Wesley Publ. 1974.Google Scholar
[2] Coleman, C.: Growth and Decay Estimates near Non-elementary Stationary Points. Can. J. Math. XXII (1970), 1156–1167.Google Scholar
[3] Courant, R. and Hilbert, D.: Methods of Mathematical Physics, Vol. II. Interscience Publ. 1962.Google Scholar
[4] Koecher, M.: Über Abbildungen, die Lösungen von gewissen gewöhnlichen Differ entialgleichungen erhalten. Ms. Google Scholar
[5] Kreisel, G. and Krivine, : Modelltheorie. Springer Verlag 1972.Google Scholar
[6] Lefschetz, S.: Differential Equations: Geometric Theory, 2nd ed. Inter science Publ. 1963.Google Scholar
[7] Levin, J. J. and Shatz, S.: On the Matrix Riccati Equation. Proc. AMS 10 (1959), 519–524.Google Scholar
[8] Levin, J. J. and Shatz, S.: Riccati Algebras. Duke Math. J. 30 (1963), 579–594.Google Scholar
[9] Markus, L.: Quadratic Differential Equations and Non-associative Algebras. Contributions to the Theory of Non-linear Oscillations, Vol. V, 185–213. Princeton University Press 1960.CrossRefGoogle Scholar
[10] Pareigis, B.: Categories and Functors. Academic Press 1970.Google Scholar
[11] Röhrl, H.: Algebren und Riccati Gleichungen. Sitz. ber. d. Berliner Math. Ges. (1972/4).Google Scholar
[12] Röhrl, H. and Wischnewsky, M. B.: Subalgebras that are Cyclic as Submodules. Manuscripta Math. 9 (1976), 195–209.Google Scholar
[13] Röhrl, H.: A theorem on non-associated algebras and its applications to differential equations. Manuscripta Math. 21 (1977), 181–187.Google Scholar