Counting Configurations in Designs

AbstractGiven a t-(v, k, λ) design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and λ and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design

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

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences