33 research outputs found
de Branges-Rovnyak spaces: basics and theory
For a contractive analytic operator-valued function on the unit disk
, de Branges and Rovnyak associate a Hilbert space of analytic
functions and related extension space
consisting of pairs of analytic functions on the unit disk . This
survey describes three equivalent formulations (the original geometric de
Branges-Rovnyak definition, the Toeplitz operator characterization, and the
characterization as a reproducing kernel Hilbert space) of the de
Branges-Rovnyak space , as well as its role as the underlying
Hilbert space for the modeling of completely non-isometric Hilbert-space
contraction operators. Also examined is the extension of these ideas to handle
the modeling of the more general class of completely nonunitary contraction
operators, where the more general two-component de Branges-Rovnyak model space
and associated overlapping spaces play key roles. Connections
with other function theory problems and applications are also discussed. More
recent applications to a variety of subsequent applications are given in a
companion survey article
Structured models of cell migration incorporating molecular binding processes
The dynamic interplay between collective cell movement and the various
molecules involved in the accompanying cell signalling mechanisms plays a
crucial role in many biological processes including normal tissue development
and pathological scenarios such as wound healing and cancer. Information about
the various structures embedded within these processes allows a detailed
exploration of the binding of molecular species to cell-surface receptors
within the evolving cell population. In this paper we establish a general
spatio-temporal-structural framework that enables the description of molecular
binding to cell membranes coupled with the cell population dynamics. We first
provide a general theoretical description for this approach and then illustrate
it with two examples arising from cancer invasion