Introduction of loose ribbons in geographic information system

886-898In a geographic information system, we use principally many models, such as points, polylines and regions to represent spatial objects. But, usually, lines represent linear objects that have a width, whereas from a mathematical point of view, lines have no width. To solve this paradox, in previous papers, the notion of rectilinear lines was replaced by rectangular ribbons. The rectangular ribbon was used to represent longish objects such as streets, roads and rivers. However, the problems come from their mathematical modeling because in reality, rivers and roads can have irregular widths and measurement errors must be taken into account. So, not all longish objects have rectangular shapes, but they can have loose ones. To solve this problem, the concept of a loose ribbon need be developed. In this paper, we address the eventual mutation of the topological relations between loose ribbons into other topological relations, according to certain criteria, when downscaling

Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity

Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory, in which the pullback of the curvature to the boundary is self-dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all $SU(2)$ Chern-Simon theories defined by all choices of punctures and representations on the spatial boundary $\cal S$. The integer level $k$ of Chern-Simons theory is found to be given by $k= 6\pi /G^2 \Lambda + \alpha$, where $\Lambda$ is the cosmological constant and $\alpha$ is a $CP$ breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. The Beckenstein bound and 't Hooft-Susskind holographic hypothesis are confirmed, (in the limit of large area and small cosmological constant) in the sense that once the two metric of the boundary has been measured, the subspace of the physical state space that describes the further information that the observer on the boundary may obtain about the interior has finite dimension equal to the exponent of the area of the boundary, in Planck units, times a fixed constant. Finally,the construction of the state space for quantum gravity in a region from that of all Chern-Simon theories defined on its boundary confirms the categorical-theoretic ``ladder of dimensions picture" of Crane.Comment: TEX File, Minor Changes Made, 59 page

A New Euler's Formula for DNA Polyhedra

DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components , of crossings , and of Seifert circles are related by a simple and elegant formula: . This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler's formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus

The theory and applications of writhing

Writhe measures the extent to which a curve is kinked and coiled about itself in space. It has generally been expressed as a double integral. This measure can be interpreted as the average number of signed crossings seen by each viewer, over all possible viewpoints of the curve. This simple geometrical interpretation is used to describe the established properties of the writhe, as applied to closed spacecurves. These descriptions differ from previous work as they do not require the construction of an artificial ribbon structure. A major feature of this thesis concerns the evaluation of the writhe along a preferred direction. A directional measure termed the polar writhe will be developed which can be applied to generic curves (open or closed) . This single integral expression is shown to be equivalent to the double integral writhe measure for closed curves. However for open curves the two measures are shown to differ. Further, it is shown that the polar writhe has distinct advantages when analysing curves with a strong directional bias. The thesis then discusses in detail the properties of both the writhe and the polar writhe measures for open curves. The use of artificial closures for both measures is examined. In the case of the writhe a new closure is defined that allows the evaluation of the writhe using single integral expression via the theorems of Fuller. This closure is unique in that it can be applied to open curves whose end points are in general position. A simple expression for calculating the non-local polar writhe is derived which generalises a closed curve expression defined in (Berger Prior J. Phys. A: Math. Gen. 39, 8321-8348, (2006)). A quantitative study on the effect of the choice of evaluation direction of the polar writhe is conducted. The polar writhe formulation is applied to a simple linear force-free magnetic field model where the field lines form loops above a boundary plane. Loops with a sufficient amount of kinking are generally seen to form S or inverse S (Z) shaped structures. Such field lines structures are commonly observed in the Sun’s corona. A popular measure of the field line morphology is the magnetic helicity. We use the polar writhe, the correct form for the writhe helicity in the coronal region, to challenge some popular assumptions of the field. Firstly, the writhe of field lines of significant aspect ratio (the apex height divided by the foot point width) can often have the opposite sign to that assumed in a recent review paper by Green et al (Solar Phys., 365-391, (2007)). Secondly, we demonstrate the possibility of field lines forming apparent Z shaped structures which are in fact constructed from a pair of S shapes and have a writhe sign expected of an S shaped structure. Such field lines could be misinterpreted without full knowledge of the line’s three dimensional structure. Thirdly, we show that much of the interesting morphological behaviour occurs for field lines located next to separatrices

Kwinking as the plastic forming mechanism of B19' NiTi martensite

Irreversible plastic forming of B19$^\prime$ martensite of the NiTi shape memory alloy is discussed within the framework of continuum mechanics. It is suggested that the main mechanism arises from coupling between martensite reorientation and coordinated $[100](001)_{\rm M}$ dislocation slip. A heuristic model is proposed, showing that the ${(20\bar{1})_{\rm M}}$ deformation-twin bands, commonly observed in experiments, can be interpreted as a combination of dislocation-mediated kink bands, appearing due to strong plastic anisotropy, and reversible twinning of martensite. We introduce a term 'kwinking' for this combination of reversible twinning and irreversible plastic kinking. The model is subsequently formulated using the tools of nonlinear elasticity theory of martensite and crystal plasticity, introducing 'kwink interfaces' as planar, kinematically compatible interfaces between two differently plastically slipped variants of martensite. It is shown that the ${(20\bar{1})_{\rm M}}$ kwink bands may be understood as resultsing from energy minimization, and that their nucleation and growth and their pairing with $(100)_{\rm M}$ twins into specific patterns enables low-energy plastic forming of NiTi martensite. We conclude that kwinking makes plastic deformation of B19$^\prime$ martensite in polycrystalline NiTi possible despite only one slip system being available.Comment: Revised version of the manuscript submitted to the International Journal of Plasticit

Feature-Based Uncertainty Visualization

While uncertainty in scientific data attracts an increasing research interest in the visualization community, two critical issues remain insufficiently studied: (1) visualizing the impact of the uncertainty of a data set on its features and (2) interactively exploring 3D or large 2D data sets with uncertainties. In this study, a suite of feature-based techniques is developed to address these issues. First, a framework of feature-level uncertainty visualization is presented to study the uncertainty of the features in scalar and vector data. The uncertainty in the number and locations of features such as sinks or sources of vector fields are referred to as feature-level uncertainty while the uncertainty in the numerical values of the data is referred to as data-level uncertainty. The features of different ensemble members are indentified and correlated. The feature-level uncertainties are expressed as the transitions between corresponding features through new elliptical glyphs. Second, an interactive visualization tool for exploring scalar data with data-level and two types of feature-level uncertainties — contour-level and topology-level uncertainties — is developed. To avoid visual cluttering and occlusion, the uncertainty information is attached to a contour tree instead of being integrated with the visualization of the data. An efficient contour tree-based interface is designed to reduce users’ workload in viewing and analyzing complicated data with uncertainties and to facilitate a quick and accurate selection of prominent contours. This thesis advances the current uncertainty studies with an in-depth investigation of the feature-level uncertainties and an exploration of topology tools for effective and interactive uncertainty visualizations. With quantified representation and interactive capability, feature-based visualization helps people gain new insights into the uncertainties of their data, especially the uncertainties of extracted features which otherwise would remain unknown with the visualization of only data-level uncertainties

Statistical Computing on Non-Linear Spaces for Computational Anatomy

International audienceComputational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings