Boundedness, compactness and Schatten-class membership of weighted composition operators

The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces $\mathcal{A}^2 \alpha(\mathbb{D})$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.Comment: 12 page

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

Boundedness of weighted composition operators W u,φ acting on the classical Dirichlet space D as W u,φ f=u(f∘φ) is studied in terms of the multiplier space associated to the symbol φ , i.e., M(φ)={u∈D:W u,φ is bounded on D} . A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of W u,φ in D whenever φ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen et al. (J. Funct. Anal. 265:1749–1777, 2013) to the context of the Dirichlet space

A Mathematical Model of Liver Cell Aggregation In Vitro

The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work

Biodegradable synthetic polymers for tissue engineering

This paper reviews biodegradable synthetic polymers focusing on their potential in tissue engineering applications. The major classes of polymers are briefly discussed with regard to synthesis, properties and biodegradability, and known degradation modes and products are indicated based on studies reported in the literature. A vast majority of biodegradable polymers studied belongs to the polyester family, which includes polyglycolides and polylactides. Some disadvantages of these polymers in tissue engineering applications are their poor biocompatibility, release of acidic degradation products, poor processability and loss of mechanical properties very early during degradation. Other degradable polymers such as polyorthoesters, polyanhydrides, polyphosphazenes, and polyurethanes are also discussed and their advantages and disadvantages summarised. With advancements in tissue engineering it has become necessary to develop polymers that meet more demanding requirements. Recent work has focused on developing injectable polymer compositions based on poly (propylene fumarate) and poly (anhydrides) to meet these requirements in orthopaedic tissue engineering. Polyurethanes have received recent attention for development of degradable polymers because of their great potential in tailoring polymer structure to achieve mechanical properties and biodegradability to suit a variety of applications