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## Section: Research Program

### Algebraic analysis of functional systems

We want to further develop our expertise in the computational aspects of algebraic analysis by continuing to develop effective versions of results of module theory, homological algebra, category theory and sheaf theory [136] which play important roles in algebraic analysis [45], [103], [104] and in the algorithmic study of linear functional systems. In particular, we shall focus on linear systems of integro-differential-constant/varying/distributed delay equations [124], [126] which play an important role in mathematical systems theory, control theory, and signal processing [124], [131], [125], [128].

The rings of integro-differential operators are highly more complicated than the purely differential case (i.e. Weyl algebras) [12], due to the existence of zero-divisors, or the fact of having a coherent ring instead of a noetherian ring [42]. Therefore, we want to develop an algorithmic study of these rings. Following the direction initiated in [126] for the computation of zero divisors (based on the polynomial null spaces of certain operators), we first want to develop algorithms for the computation of left/right kernels and left/right/generalized inverses of matrices with entries in such rings, and to use these results in module theory (e.g. computation of syzygy modules, (shorter/shortest) free resolutions, split short/long exact sequences). Moreover, Stafford's results [137], algorithmically developed in [12] for rings of partial differential operators (i.e. the Weyl algebras), are known to still hold for rings of integro-differential operators. We shall study their algorithmic extensions. Our corresponding implementation will be extended accordingly.

Finally, within a computer algebra viewpoint, we shall continue to algorithmically study issues on rings of integro-differential-delay operators [124], [125] and their applications to the study of equivalences of differential constant/varying/distributed delay systems (e.g. Artstein's reduction, Fiagbedzi-Pearson's transformation) which play an important role in control theory.