Partitions of Minimal Length on Manifolds

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a $\Gamma$-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation

Steiner minimum trees for equidistant points on two sides of an angle

In this paper we deal with the Steiner minimum tree problem for a special type of point sets. These sets consist of the vertex of an angle 2a and equidistant points lying on the two sides of this angle

Applications to Biological Networks of Adaptive Hagen-Poiseuille Flow on Graphs

Physarum polycephalum is a single-celled, multi-nucleated slime mold whose body constitutes a network of veins. As it explores its environment, it adapts and optimizes its network to external stimuli. It has been shown to exhibit complex behavior, like solving mazes, finding the shortest path, and creating cost-efficient and robust networks. Several models have been developed to attempt to mimic its network's adaptation in order to try to understand the mechanisms behind its behavior as well as to be able to create efficient networks. This thesis aims to study a recently developed, physically-consistent model based on adaptive Hagen-Poiseuille flows on graphs, determining the properties of the trees it creates and probing them to understand if they are realistic and consistent with experiment. It also intends to use said model to produce short and efficient networks, applying it to a real-life transport network example. We have found that the model is able to create networks that are consistent with biological networks: they follow Murray's law at steady state, exhibit structures similar to Physarum's networks, and even present peristalsis (oscillations of the vein radii) and shuttle streaming (the back-and-forth movement of cytoplasm inside Physarum's veins) in some parts of the networks. We have also used the model paired with different stochastic algorithms to produce efficient, short, and cost-efficient networks; when compared to a real transport network, mainland Portugal's railway system, all algorithms proved to be more efficient and some proved to be more cost-efficient.Comment: 106 pages, 59 figure

Calibrations for minimal networks in a covering space setting

In this paper we define a notion of calibration for an equivalent approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon