L-systems in Geometric Modeling

We show that parametric context-sensitive L-systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bezier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L-systems, which were limited to subdivision curves.Comment: In Proceedings DCFS 2010, arXiv:1008.127

A complementary view on the growth of directory trees

Trees are a special sub-class of networks with unique properties, such as the level distribution which has often been overlooked. We analyse a general tree growth model proposed by Klemm {\em et. al.} (2005) to explain the growth of user-generated directory structures in computers. The model has a single parameter $q$ which interpolates between preferential attachment and random growth. Our analysis results in three contributions: First, we propose a more efficient estimation method for $q$ based on the degree distribution, which is one specific representation of the model. Next, we introduce the concept of a level distribution and analytically solve the model for this representation. This allows for an alternative and independent measure of $q$. We argue that, to capture real growth processes, the $q$ estimations from the degree and the level distributions should coincide. Thus, we finally apply both representations to validate the model with synthetically generated tree structures, as well as with collected data of user directories. In the case of real directory structures, we show that $q$ measured from the level distribution are incompatible with $q$ measured from the degree distribution. In contrast to this, we find perfect agreement in the case of simulated data. Thus, we conclude that the model is an incomplete description of the growth of real directory structures as it fails to reproduce the level distribution. This insight can be generalised to point out the importance of the level distribution for modeling tree growth.Comment: 16 pages, 7 figure

Basic Understanding of Condensed Phases of Matter via Packing Models

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298

Multiscale Bone Remodelling with Spatial P Systems

Many biological phenomena are inherently multiscale, i.e. they are characterized by interactions involving different spatial and temporal scales simultaneously. Though several approaches have been proposed to provide "multilayer" models, only Complex Automata, derived from Cellular Automata, naturally embed spatial information and realize multiscaling with well-established inter-scale integration schemas. Spatial P systems, a variant of P systems in which a more geometric concept of space has been added, have several characteristics in common with Cellular Automata. We propose such a formalism as a basis to rephrase the Complex Automata multiscaling approach and, in this perspective, provide a 2-scale Spatial P system describing bone remodelling. The proposed model not only results to be highly faithful and expressive in a multiscale scenario, but also highlights the need of a deep and formal expressiveness study involving Complex Automata, Spatial P systems and other promising multiscale approaches, such as our shape-based one already resulted to be highly faithful.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

Natural history of Arabidopsis thaliana and oomycete symbioses

Molecular ecology of plant–microbe interactions has immediate significance for filling a gap in knowledge between the laboratory discipline of molecular biology and the largely theoretical discipline of evolutionary ecology. Somewhere in between lies conservation biology, aimed at protection of habitats and the diversity of species housed within them. A seemingly insignificant wildflower called Arabidopsis thaliana has an important contribution to make in this endeavour. It has already transformed botanical research with deepening understanding of molecular processes within the species and across the Plant Kingdom; and has begun to revolutionize plant breeding by providing an invaluable catalogue of gene sequences that can be used to design the most precise molecular markers attainable for marker-assisted selection of valued traits. This review describes how A. thaliana and two of its natural biotrophic parasites could be seminal as a model for exploring the biogeography and molecular ecology of plant–microbe interactions, and specifically, for testing hypotheses proposed from the geographic mosaic theory of co-evolution

Shall We (Math and) Dance?

Can we use mathematics, and in particular the abstract branch of category theory, to describe some basics of dance, and to highlight structural similarities between music and dance? We first summarize recent studies between mathematics and dance, and between music and categories. Then, we extend this formalism and diagrammatic thinking style to dance.Comment: preprin

An Approach to the Bio-Inspired Control of Self-reconfigurable Robots

Self-reconfigurable robots are robots built by modules which can move in relationship to each other. This ability of changing its physical form provides the robots a high level of adaptability and robustness. Given an initial configuration and a goal configuration of the robot, the problem of self-regulation consists on finding a sequence of module moves that will reconfigure the robot from the initial configuration to the goal configuration. In this paper, we use a bio-inspired method for studying this problem which combines a cluster-flow locomotion based on cellular automata together with a decentralized local representation of the spatial geometry based on membrane computing ideas. A promising 3D software simulation and a 2D hardware experiment are also presented.National Natural Science Foundation of China No. 6167313

Genetic Control of Organ Shape and Tissue Polarity

A combination of experimental analysis and mathematical modelling shows how the genetic control of tissue polarity plays a fundamental role in the development and evolution of form