The group law on the tropical Hesse pencil

We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions through the ultradiscretization procedure. A tropical analogue of the Hessian group, the group of linear automorphisms acting on the Hesse pencil, is also investigated; it is shown that the dihedral group of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.Comment: 17 pages, 1 figure, submitted to Special Issue of the Journal Mathematics and Computers in Simulation on "Nonlinear Waves: Computation and Theory

On the geometry of C^3/D_27 and del Pezzo surfaces

We clarify some aspects of the geometry of a resolution of the orbifold X = C3/D_27, the noncompact complex manifold underlying the brane quiver standard model recently proposed by Verlinde and Wijnholt. We explicitly realize a map between X and the total space of the canonical bundle over a degree 1 quasi del Pezzo surface, thus defining a desingularization of X. Our analysis relys essentially on the relationship existing between the normalizer group of D_27 and the Hessian group and on the study of the behaviour of the Hesse pencil of plane cubic curves under the quotient.Comment: 23 pages, 5 figures, 2 tables. JHEP style. Added references. Corrected typos. Revised introduction, results unchanged

Complexity in biomaterials for tissue engineering

The molecular and physical information coded within the extracellular milieu is informing the development of a new generation of biomaterials for tissue engineering. Several powerful extracellular influences have already found their way into cell-instructive scaffolds, while others remain largely unexplored. Yet for commercial success tissue engineering products must be not only efficacious but also cost-effective, introducing a potential dichotomy between the need for sophistication and ease of production. This is spurring interest in recreating extracellular influences in simplified forms, from the reduction of biopolymers into short functional domains, to the use of basic chemistries to manipulate cell fate. In the future these exciting developments are likely to help reconcile the clinical and commercial pressures on tissue engineering