An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

Los arenales costeros del litoral catalán (la bahía de Rosas)

[ES] Se distinguen dos fuentes para los minerales pesados que se encuentran en las playas de la bahía de Rosas: los basaltos de Olot, para la augita, olivino e hiperstena y las rocas metamórficas del macizo de los Alberes y Cabo de Creus de donde proceden la andalucita, silimanita y distena. La distribución de los minerales, se explica por el transporte efectuado por las corrientes de deriva, los temporales y el viento. Las anomaiías en la distribucidn de algunas especies se deben a los accicentes del terreno y a las condiciones dinámicas muy activas de la bahía, que afectan a la seleccidn de minerales.[EN] The heavy minerals of Gulf of Rosas coastal sand have two different sources. The more frequent heavy minerals are augite and olivine which come from Olot basalts. The metamorfic association is presented by andalusite, siliimanite and kyanite and they come from the metamorfic rocks of the Pyrenees and Cap of Creus massif. Homblende can have two origins: the metamorfic and the granitic rocks of river Muga basin.Peer reviewe

The geometry of entanglement: metrics, connections and the geometric phase

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.Comment: 42 page

Pattern formation in directional solidification under shear flow. I: Linear stability analysis and basic patterns

An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near the onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies, described in a companion paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.Comment: 20 pages, Latex, accepted for Physical Review

Spatial Heterogeneity of Seasonal Grazing Pressure Created by Herd Movement Patterns on Hilly Rangelands Using GPS and GIS

The spatial heterogeneity of grazing pressure on extensive rangelands has management implications (Adler et al., 2001) but it has traditionally been difficult to quantify. Combination of technologies based on GPS (Global Positioning System) and GIS (Geographic Information Systems) is a quantum leap in our ability to address this issue. These tools were used to estimate the spatial heterogeneity of grazing pressure at a farm scale, and examine the relation between local landscape features and local grazing pressure

A Derivation of Three-Dimensional Inertial Transformations

The derivation of the transformations between inertial frames made by Mansouri and Sexl is generalised to three dimensions for an arbitrary direction of the velocity. Assuming lenght contraction and time dilation to have their relativistic values, a set of transformations kinematically equivalent to special relativity is obtained. The ``clock hypothesis'' allows the derivation to be extended to accelerated systems. A theory of inertial transformations maintaining an absolute simultaneity is shown to be the only one logically consistent with accelerated movements. Algebraic properties of these transformations are discussed. Keywords: special relativity, synchronization, one-way velocity of light, ether, clock hypothesis.Comment: 16 pages (A5), Latex, one figure, to be published in Found. Phys. Lett. (1997