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

Phage engineering: how advances in molecular biology and synthetic biology are being utilized to enhance the therapeutic potential of bacteriophages

Background The therapeutic potential of bacteriophages has been debated since their first isolation and characterisation in the early 20th century. However, a lack of consistency in application and observed efficacy during their early use meant that upon the discovery of antibiotic compounds research in the field of phage therapy quickly slowed. The rise of antibiotic resistance in bacteria and improvements in our abilities to modify and manipulate DNA, especially in the context of small viral genomes, has led to a recent resurgence of interest in utilising phage as antimicrobial therapeutics. Results In this article a number of results from the literature that have aimed to address key issues regarding the utility and efficacy of phage as antimicrobial therapeutics utilising molecular biology and synthetic biology approaches will be introduced and discussed, giving a general view of the recent progress in the field. Conclusions Advances in molecular biology and synthetic biology have enabled rapid progress in the field of phage engineering, with this article highlighting a number of promising strategies developed to optimise phages for the treatment of bacterial disease. Whilst many of the same issues that have historically limited the use of phages as therapeutics still exist, these modifications, or combinations thereof, may form a basis upon which future advances can be built. A focus on rigorous in vivo testing and investment in clinical trials for promising candidate phages may be required for the field to truly mature, but there is renewed hope that the potential benefits of phage therapy may finally be realised