Variational principles for circle patterns

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz' theorem on the geometric realizability of abstract polyhedra. We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and Rivin for circle packings and circle patterns from our variational principles. In the case of Br\"agger's and Rivin's functionals. Leibon's functional for hyperbolic circle patterns cannot be derived directly from our functionals. But we construct yet another functional from which both Leibon's and our functionals can be derived. We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics

Glassy dynamics in granular compaction: sand on random graphs

We discuss the use of a ferromagnetic spin model on a random graph to model granular compaction. A multi-spin interaction is used to capture the competition between local and global satisfaction of constraints characteristic for geometric frustration. We define an athermal dynamics designed to model repeated taps of a given strength. Amplitude cycling and the effect of permanently constraining a subset of the spins at a given amplitude is discussed. Finally we check the validity of Edwards' hypothesis for the athermal tapping dynamics.Comment: 13 pages Revtex, minor changes, to appear in PR

IST Austria Thesis

In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations. The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density, and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics. We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss the implications on persistence for periodic data sets

Topological Progression in Proliferating Epithelia Is Driven by a Unique Variation in Polygon Distribution

Morphogenesis is consequence of lots of small coordinated variations that occur during development. In proliferating stages, tissue growth is coupled to changes in shape and organization. A number of studies have analyzed the topological properties of proliferating epithelia using the Drosophila wing disc as a model. These works are based in the existence of a fixed distribution of these epithelial cells according to their number of sides. Cell division, cell rearrangements or a combination of both mechanisms have been proposed to be responsible for this polygonal assembling. Here, we have used different system biology methods to compare images from two close proliferative stages that present high morphological similarity. This approach enables us to search for traces of epithelial organization. First, we show that geometrical and network characteristics of individual cells are mainly dependent on their number of sides. Second, we find a significant divergence between the distribution of polygons in epithelia from mid-third instar larva versus early prepupa. We show that this alteration propagates into changes in epithelial organization. Remarkably, only the variation in polygon distribution driven by morphogenesis leads to progression in epithelial organization. In addition, we identify the relevant features that characterize these rearrangements. Our results reveal signs of epithelial homogenization during the growing phase, before the planar cell polarity pathway leads to the hexagonal packing of the epithelium during pupal stages.España, Ministero de Ciencia BFU2011-2573

Topological Progression in Proliferating Epithelia Is Driven by a Unique Variation in Polygon Distribution

Morphogenesis is consequence of lots of small coordinated variations that occur during development. In proliferating stages, tissue growth is coupled to changes in shape and organization. A number of studies have analyzed the topological properties of proliferating epithelia using the Drosophila wing disc as a model. These works are based in the existence of a fixed distribution of these epithelial cells according to their number of sides. Cell division, cell rearrangements or a combination of both mechanisms have been proposed to be responsible for this polygonal assembling. Here, we have used different system biology methods to compare images from two close proliferative stages that present high morphological similarity. This approach enables us to search for traces of epithelial organization. First, we show that geometrical and network characteristics of individual cells are mainly dependent on their number of sides. Second, we find a significant divergence between the distribution of polygons in epithelia from mid-third instar larva versus early prepupa. We show that this alteration propagates into changes in epithelial organization. Remarkably, only the variation in polygon distribution driven by morphogenesis leads to progression in epithelial organization. In addition, we identify the relevant features that characterize these rearrangements. Our results reveal signs of epithelial homogenization during the growing phase, before the planar cell polarity pathway leads to the hexagonal packing of the epithelium during pupal stages.LME is supported by the Miguel Servet (Instituto Carlos III) program that also funded the work. LME and DSG are funded by the Spanish Ministry of Science (BFU2011-25734). AS is funded by the Consejería de Innovación, Ciencia y Empresa of the Junta de Andalucía.Peer Reviewe

A geospatiotemporal and causal inference epidemiological exploration of substance and cannabinoid exposure as drivers of rising US pediatric cancer rates

Background: Age-adjusted US total pediatric cancer incidence rates (TPCIR) rose 49% 1975–2015 for unknown reasons. Prenatal cannabis exposure has been linked with several pediatric cancers which together comprise the majority of pediatric cancer types. We investigated whether cannabis use was related spatiotemporally and causally to TPCIR. Methods: State-based age-adjusted TPCIR data was taken from the CDC Surveillance, Epidemiology and End Results cancer database 2003–2017. Drug exposure was taken from the nationally-representative National Survey of Drug Use and Health, response rate 74.1%. Drugs included were: tobacco, alcohol, cannabis, opioid analgesics and cocaine. This was supplemented by cannabinoid concentration data from the Drug Enforcement Agency and ethnicity and median household income data from US Census. Results: TPCIR rose while all drug use nationally fell, except for cannabis which rose. TPCIR in the highest cannabis use quintile was greater than in the lowest (β-estimate = 1.31 (95%C.I. 0.82, 1.80), P = 1.80 × 10− 7) and the time:highest two quintiles interaction was significant (β-estimate = 0.1395 (0.82, 1.80), P = 1.00 × 10− 14). In robust inverse probability weighted additive regression models cannabis was independently associated with TPCIR (β-estimate = 9.55 (3.95, 15.15), P = 0.0016). In interactive geospatiotemporal models including all drug, ethnic and income variables cannabis use was independently significant (β-estimate = 45.67 (18.77, 72.56), P = 0.0009). In geospatial models temporally lagged to 1,2,4 and 6 years interactive terms including cannabis were significant. Cannabis interactive terms at one and two degrees of spatial lagging were significant (from β-estimate = 3954.04 (1565.01, 6343.09), P = 0.0012). The interaction between the cannabinoids THC and cannabigerol was significant at zero, 2 and 6 years lag (from β-estimate = 46.22 (30.06, 62.38), P = 2.10 × 10− 8). Cannabis legalization was associated with higher TPCIR (β-estimate = 1.51 (0.68, 2.35), P = 0.0004) and cannabis-liberal regimes were associated with higher time:TPCIR interaction (β-estimate = 1.87 × 10− 4, (2.9 × 10− 5, 2.45 × 10− 4), P = 0.0208). 33/56 minimum e-Values were \u3e 5 and 6 were infinite. Conclusion: Data confirm a close relationship across space and lagged time between cannabis and TPCIR which was robust to adjustment, supported by inverse probability weighting procedures and accompanied by high e-Values making confounding unlikely and establishing the causal relationship. Cannabis-liberal jurisdictions were associated with higher rates of TPCIR and a faster rate of TPCIR increase. Data inform the broader general consideration of cannabinoid-induced genotoxicity

Studies into the structure and function of various domains of obscurin and titin

Muscles give our bodies the ability to move by stretching and contracting. While contraction is accomplished by the well-known actin-myosin interaction, not much is known about stretch. Two integral muscle proteins involved in stretch are titin and obscurin; both are long rope-like protein molecules that seem to act as molecular springs. Mutations in these two proteins can lead to diseases such as hypertrophic cardiomyopathy and muscular dystrophy, as well as a variety of cancers. In an effort to understand muscle stretch and signaling on a more fundamental level, here we present the high resolution structure of obscurin Ig59, a domain involved in titin/obscurin binding. We also describe how unbound titin moves when stretched. Last, we describe ongoing work in elucidating the high-resolution structures and activation/inhibition mechanisms of obscurin domains Rho-GEF, Rho-GEF-PH, kinase I (KI), and kinase II (KII)