Optimization of the branching pattern in coherent phase transitions

Branching can be observed at the austenite-martensite interface of martensitic phase transformations. For a model problem, Kohn and M\"uller studied a branching pattern with optimal scaling of the energy with respect to its parameters. Here, we present finite element simulations that suggest a topologically different class of branching patterns and derive a novel, low dimensional family of patterns. After a geometric optimization within this family, the resulting pattern bears a striking resemblance to our simulation. The novel microstructure admits the same scaling exponents but results in a significantly lower upper energy bound.Comment: 6 pages, 4 figures, 2 tables. correction of minor typesetting error

Leaf-inspired microcontact printing vascular patterns.

The vascularization of tissue grafts is critical for maintaining viability of the cells within a transplanted graft. A number of strategies are currently being investigated including very promising microfluidics systems. Here, we explored the potential for generating a vasculature-patterned endothelial cells that could be integrated into distinct layers between sheets of primary cells. Bioinspired from the leaf veins, we generated a reverse mold with a fractal vascular-branching pattern that models the unique spatial arrangement over multiple length scales that precisely mimic branching vasculature. By coating the reverse mold with 50 μg ml-1 of fibronectin and stamping enabled selective adhesion of the human umbilical vein endothelial cells (HUVECs) to the patterned adhesive matrix, we show that a vascular-branching pattern can be transferred by microcontact printing. Moreover, this pattern can be maintained and transferred to a 3D hydrogel matrix and remains stable for up to 4 d. After 4 d, HUVECs can be observed migrating and sprouting into Matrigel. These printed vascular branching patterns, especially after transfer to 3D hydrogels, provide a viable alternative strategy to the prevascularization of complex tissues

Quasiscarred modes and their branching behavior at an exceptional point

We study quasiscarring phenomenon and mode branching at an exceptional point (EP) in typically deformed microcavities. It is shown that quasiscarred (QS) modes are dominant in some mode group and their pattern can be understood by short-time ray dynamics near the critical line. As cavity deformation increases, high-Q and low-Q QS modes are branching in an opposite way, at an EP, into two robust mode types showing QS and diamond patterns, respectively. Similar branching behavior can be also found at another EP appearing at a higher deformation. This branching behavior of QS modes has its origin on the fact that an EP is a square-root branch point.Comment: 5 pages, 5 figure

Tip Oscillation of Dendritic Patterns in a Phase Field Model

We study dendritic growth numerically with a phase field model. Tip oscillation and regular side-branching are observed in a parameter region where the anisotropies of the surface tension and the kinetic effect compete. The transition from a needle pattern to a dendritic pattern is conjectured to be a supercritical Hopf bifurcation.Comment: 5 figure

<i>Aloe pulcherrima</i> - a beautiful Ethiopian endemic

Aloe pulcherrima is a large-growing, cliff-dwelling species from high altitudes in Ethiopia with a unique stem branching pattern. It is described both in cultivation and in habitat

Interpolation inequalities in pattern formation

We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in the study of branching in superconductors

Reduced branching processes with very heavy tails

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $\alpha\in(0,1]$. We turn to the case of very heavy tailed reproduction distribution $\alpha=0$ assuming Zubkov's regularity condition with parameter $\beta\in(0,\infty)$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.Comment: 15 pages, 1 figur