Packing ellipsoids with overlap

The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computational experience is described and illustrated. The motivating application - chromosome organization in the human cell nucleus - is discussed briefly, and some illustrative results are presented

Density-functional study of defects in two-dimensional circular nematic nanocavities

We use density--functional theory to study the structure of two-dimensional defects inside a circular nematic nanocavity. The density, nematic order parameter, and director fields, as well as the defect core energy and core radius, are obtained in a thermodynamically consistent way for defects with topological charge $k=+1$ (with radial and tangential symmetries) and $k=+1/2$. An independent calculation of the fluid elastic constants, within the same theory, allows us to connect with the local free--energy density predicted by elastic theory, which in turn provides a criterion to define a defect core boundary and a defect core free energy for the two types of defects. The radial and tangential defects turn out to have very different properties, a feature that a previous Maier--Saupe theory could not account for due to the simplified nature of the interactions --which caused all elastic constants to be equal. In the case with two $k=+1/2$ defects in the cavity, the elastic r\'egime cannot be reached due to the small radii of the cavities considered, but some trends can already be obtained.Comment: 9 figures. Accepted for publication in liquid crystal

A Sangaku Revived

In this paper we give an account on our mathematical and visual explorations inspired by a sangaku. First we introduce sangakus – traditional Japanese mathematical tablets. Then we give four examples of our static contemporary variants. Finally, we discuss in detail how a fifth sangaku led us to simulate the growth of water lilies, as a means of visualizing the problem. This approach lead to the mathematical field of circle packing, and made it possible to experience the visually intriguing process with different settings of the algorithm