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Algebras and differential equations

  • Helmut Röhrl (a1)

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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[X 1, …, Xn ] of associating but non-commuting variables X 1, …, 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 R Diff m whose objects are precisely the differential equations of arity m and in which the composition of the morphisms is the usual matrix multiplication.

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References

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[1] Bourbaki, N.: Algebra I. Addison-Wesley Publ. 1974.
[2] Coleman, C.: Growth and Decay Estimates near Non-elementary Stationary Points. Can. J. Math. XXII (1970), 11561167.
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Algebras and differential equations

  • Helmut Röhrl (a1)

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