Bessmertnyi Realizations, Representations, and Related Problems in Multiphase Composites

We discuss multivariate functions that can be represented as the Schur complement of a linear (matrix-, tensor-, or operator-valued) pencil, i.e., the class of Bessmertnyi realizable functions. We motivate this by showing that for multiphase composites, both the effective operator in the theory of composites as well as the DtN map for an electrical network are in this class in which the associated linear pencil is of positive semidefinite type. This naturally leads to multivariate (matrix-, tensor-, or operator-valued) Herglotz-Nevanlinna functions in this class and several open problems in realizability theory. Next, we present Bessmertnyi realizations as the ``universal" state-space model/realization [compared to others common in electrical engineering (e.g., Kalman-type, Fornasini-Marchesini, Givone-Roesser) that are just a special cases of the Bessmertnyi realization]. Then we discuss our recent work (see doi:10.1007/s11785-021-01150-2 and doi:10.1016/j.laa.2021.06.007) on extensions of the Bessmertnyi realization theorem for multivariate rational functions, Schur complement algebra and operations, and their application in symmetric determinantal representations of polynomials. Finally, we will show how the latter relates to the open realization problems above. This is based on joint work with Anthony Stefan (Florida Institute of Technology).