Constructing irreducible representations of finitely presented algebras

By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In slightly more detail: Assume that $n$ is a positive integer, that $k$ is a computable field, that $\bar{k}$ denotes the algebraic closure of $k$, and that $M_n(\bar{k})$ denotes the algebra of $n \times n$ matrices with entries in $\bar{k}$. Let $R$ be a finitely presented $k$-algebra. Calculating over $k$, the procedure will (a) decide whether an irreducible representation $R \to M_n(\bar{k})$ exists, and (b) explicitly construct an irreducible representation $R \to M_n(\bar{k})$ if at least one exists. (For (b), it is necessary to assume that $k[x]$ is equipped with a factoring algorithm.) An elementary example is worked through.Comment: 9 pages. Final version. To appear in J. Symbolic Computatio

Some Limited-Interest Problems

The asymptotic probability distribution of identified black-box transfer function models is studied. The main contribution is that we derive variance expressions for the real and imaginary parts of the identified models that are asymptotic in both the number of measurements and the model order. These expressions are considerably simpler than the corresponding ones that hold for fixed model orders, and yet they frequently approximate the true covariance well already with quite modest model orders. We illustrate the relevance of the asymptotic expressions by using them to compute uncertainty regions for the frequency response of an identified model