On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure

Dental Topography and Microwear Texture in Sapajus Apella

Dental microwear texture pattern has been associated with aspects of diet for a broad range of mammalian taxa. The basic idea is that soft, tough foods are sheared with a steeper angle of approach between opposing occlusal surfaces, whereas hard, brittle items are crushed with forces perpendicular to those surfaces; and this difference is manifested in anisotropic, striated microwear textures for tough foods, and complex, pitted ones for hard objects. Other factors may, however, influence microwear texture pattern and confound diet signals. For example, if tooth surface slope influences angle of approach between opposing teeth, then perhaps wear-related changes in tooth shape could affect microwear pattern. This study evaluates the effects of occlusal topography on microwear texture for a series of variably worn upper second molars of one primate species, Sapajus apella. Results indicate no significant covariation between any measured topographic attribute (average slope, angularity, relief) and microwear texture variable (complexity, anisotropy, textural fill volume). This suggests that, for this taxon at least, wear-related changes in tooth form do not affect microwear pattern in a consistent manner. This implies that variably worn teeth can be included in samples for comparisons aimed at distinguishing groups by diet

Demonstrating Additional Law of Relativistic Velocities based on Squeezed Light

Special relativity is foundation of many branches of modern physics, of which theoretical results are far beyond our daily experience and hard to realized in kinematic experiments. However, its outcomes could be demonstrated by making use of convenient substitute, i.e. squeezed light in present paper. Squeezed light is very important in the field of quantum optics and the corresponding transformation can be regarded as the coherent state of SU(1; 1). In this paper, the connection between the squeezed operator and Lorentz boost is built under certain conditions. Furthermore, the additional law of relativistic velocities and the angle of Wigner rotation are deduced as well

Methoden zur Analyse der vokalen Gestaltung populärer Musik

Although voice and singing play a crucial role in many genres of popular music, to date there are only few approaches to an in-depth exploration of vocal expression. The paper aims at presenting new ways for describing, analysing and visualizing several aspects of singing using computer-based tools. After outlining a theoretical framework for the study of voice and singing in popular music, some of those tools are introduced and exemplified by vocal recordings from various genres (blues, gospel music, country music, jazz). Firstly, pitch gliding (slurs, slides, bends, melismas) and vibrato are discussed referring to a computer-based visualization of pitch contour. Secondly, vocal timbre and phonation (e.g. vocal roughness) are explored and visualized using spectrograms

Listening and learning : the reciprocal relationship between worker and client

The relationship between worker and client has for the best part of 100 years been the mainstay of probation, and yet has recently been eroded by an increased emphasis on punishment, blame and managerialism. The views of offenders are in direct contradiction to these developments within the criminal justice system and this article argues that only by taking account of the views of those at the 'coal face' will criminologists, policy makers and practitioners be able to effect real change in crime rates. The article thus focuses on the views of a sample of previously persistent offenders in Scotland about offending, desistance and how the system can help them. It explores not only their need for friendship and support in youth but also the close association between relationships and the likelihood of offending. It also demonstrates the views of offenders themselves about the importance of the working relationship with supervising officers in helping them desist from crime. The article concludes that the most effective way of reducing offending is to re-engage with the message of the Probation Act of 100 years ago, namely, to 'advise, assist and befriend' offenders rather than to 'confront, challenge and change' offending behaviour

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs on the unit sphere $S^2$, in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group $SU(1,1) = Sp(2, R) = SL(2,R)$, the double cover of SO(2,1). The present work develops a theory of turns for $SL(2,C)$, the double and universal cover of SO(3,1) and $SO(3,C)$, rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late

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

Measurement error in the estimation of intake from herbage patches by non-fistulated heifers on apparently uniform swards

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