Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free integro-differential algebras and Gr\"obner-Shirshov bases   The notion of commutative integro-differential algebra was introduced for the
algebraic study of boundary problems for linear ordinary differential
equations. Its noncommutative analog achieves a similar purpose for linear
systems of such equations. In both cases, free objects are crucial for
analyzing the underlying algebraic structures, e.g. of the (matrix) functions.
In this paper we apply the method of Gr\"obner-Shirshov bases to construct the
free (noncommutative) integro-differential algebra on a set. The construction
is from the free Rota-Baxter algebra on the free differential algebra on the
set modulo the differential Rota-Baxter ideal generated by the noncommutative
integration by parts formula. In order to obtain a canonical basis for this
quotient, we first reduce to the case when the set is finite. Then in order to
obtain the monomial order needed for the Composition-Diamond Lemma, we consider
the free Rota-Baxter algebra on the truncated free differential algebra. A
Composition-Diamond Lemma is proved in this context, and a Gr\"obner-Shirshov
basis is found for the corresponding differential Rota-Baxter ideal.

18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: physics
0
The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn