Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Transglutaminase activation in neurodegenerative diseases

The following review examines the role of calcium in promoting the in vitro and in vivo activation of transglutaminases in neurodegenerative disorders. Diseases such as Alzheimer's disease, Parkinson's disease and Huntington's disease exhibit increased transglutaminase activity and rises in intracellular calcium concentrations, which may be related. The aberrant activation of transglutaminase by calcium is thought to give rise to a variety of pathological moieties in these diseases, and the inhibition has been shown to have therapeutic benefit in animal and cellular models of neurodegeneration. Given the potential clinical relevance of transglutaminase inhibitors, we have also reviewed the recent development of such compounds