Rational conchoid and offset constructions: algorithms and implementation

This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages

Quantificação das estruturas de fluxo sin-magmáticas do Plutão de Vila Pouca de Aguiar: uma ferramenta para a quantificação estrutural e da qualidade da rocha

Mineral distribution pattern of Variscan post-tectonic granites from Vila Pouca de Aguiar Pluton (NE Portugal) were analyzed with methods partially based on fractal geometry and, with respect to rock inhomogeneity and anisotropy. The result of the analysis provides information about magmatic flux and mineral equilibrium processes in a crystallizing magma chamber. In addition, the used methods may also provide important information for the ornamental rock industry, because they allow fast and automatic evaluation of economic rock parameters.Os padrões de distribuição mineral dos granitos póstectónicos do plutão de Vila Pouca de Aguiar foram analisados com métodos parcialmente baseados na geometria fractal, atendendo à homogeneidade e anisotropia da rocha. O resultado desta análise forneceu informação acerca do fluxo magmático e dos processos de equilíbrio mineral na cristalização no interior de uma câmara magmática. Adicionalmente, os métodos utilizados ainda disponibilizaram informação importante para a indústria de pedra ornamental, pois permitem uma avaliação rápida e automática dos parâmetros que valorizam economicamente a rocha.(undefined

On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

The influence of strain rate and presence of dispersed second phases on the deformation behaviour of polycrystalline D₂O ice

This contribution discusses results obtained from 3-D neutron diffraction and 2-D fabric analyser in situ deformation experiments on laboratory-prepared polycrystalline deuterated ice and ice containing a second phase. The two-phase samples used in the experiments are composed of an ice matrix with (1) air bubbles, (2) rigid, rhombohedral-shaped calcite and (3) rheologically soft, platy graphite. Samples were tested at 10°C below the melting point of deuterated ice at ambient pressures, and two strain rates of 1 × 10−5 s−1 (fast) and 2.5 × 10−6 s−1 (medium). Nature and distribution of the second phase controlled the rheological behaviour of the ice by pinning grain boundary migration. Peak stresses increased with the presence of second-phase particles and during fast strain rate cycles. Ice-only samples exhibit well-developed crystallographic preferred orientations (CPOs) and dynamically recrystallized microstructures, typifying deformation via dislocation creep, where the CPO intensity is influenced in part by the strain rate. CPOs are accompanied by a concentration of [c]-axes in cones about the compression axis, coinciding with increasing activity of prismatic- slip activity. Ice with second phases, deformed in a relatively slower strain rate regime, exhibit greater grain boundary migration and stronger CPO intensities than samples deformed at higher strain rates or strain rate cycles

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.Comment: 72 pages, 6 figures. A shortened version of this paper has appeared in Publications math\'ematiques de l'IH\'ES. The final publication is available at http://www.springerlink.co

On quadratic two-parameter families of spheres and their envelopes

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R3. The four dimensional cyclographic model of the set of spheres in R3 is an appropriate framework to show that a quadratic triangular Bézier patch in R4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties

Surfaces of general type with pg=1, q=0, K2=6 and grassmannians

We construct examples of surfaces of general type with = 1, = 0 and 2 = 6. We use as key varieties Fano fourfolds and Calabi–Yau threefolds that are zero section of some special homogeneous vector bundle on Grassmannians. We link as well our construction to a classical Campedelli surface, using the Pfaffian–Grassmannian correspondence

Hough Transform and Laguerre Geometry for the Recognition and Reconstruction of Special 3D Shapes

We put the Hough transform, a method from Image Processing, into relation to Laguerre geometry, a concept of classical geometry, and study both concepts in the 3D case. It is shown how Laguerre geometry, which works in the set of oriented planes, is used in the detection of special shapes such as planes, spheres, rotational cones and cylinders, general cones and cylinders, and general developable surfaces. We perform shape recognition tasks by principal component analysis on a set of points in the so-called Blaschke model of Laguerre geometry. These points are Blaschke image points of estimated tangent planes at the given data points. Finally we present examples and show how the implementation also takes advantage of mathematical morphology on images, which are defined on meshes

Influence of pre-existing microstructure on mechanical properties of marine ice during compression experiments

Marine ice is an important component of ice shelves in Antarctica. It accretes in substantial amounts at weak points and below ice shelves. It is likely to exhibit peculiar rheological properties, which are crucial to understanding its potential role in stabilizing ice-shelf flow. Due to its location and consolidation processes, marine ice can present a variety of textures which are likely to influence its rheological properties. We present a new dataset of unconfined uniaxial compression experiments on folded marine ice samples that have been cut at various angles to the folds. Texture and fabric analyses are described 'before' and 'after' the deformation experiment. It is shown that, in the given stress configuration, the geometry of the anisotropy controls the rheological behaviour of the marine ice. During secondary creep, folded marine ice is harder to deform than weakly textured ice when compressed parallel or perpendicular to the folds' hinges, while the reverse is true for ice compressed at 45 degrees. The observed range of values for the n exponent in Glen's flow law is between 2.1 and 4.1. Surprisingly, we see that tertiary creep tends to develop at a higher total strain than for randomly oriented impurity-free meteoric ice