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Concept# Rotation (mathematics)

Summary

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.
Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transfo

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In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in thre

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One way to express this is
Q^\mathrm{T}

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Related lectures (16)

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

Context. Li is extensively known to be a good tracer of non-standard mixing processes occurring in stellar interiors. Aims. We present the results of a new large Li survey in red giant stars and combine it with surveys from the literature to probe the impact of rotation-induced mixing and thermohaline double-diffusive instability along stellar evolution. Methods. We determined the surface Li abundance for a sample of 829 giant stars with accurate Gaia parallaxes for a large sub-sample (810 stars) complemented with accurate HIPPARCOS parallaxes (19 stars). The spectra of our sample of northern and southern giant stars were obtained in three ground-based observatories (Observatoire de Haute-Provence, ESO-La Silla, and the Mc Donald Observatory). We determined the atmospheric parameters (T-eff, log(g) and [Fe/H]), and the Li abundance. We used Gaia parallaxes and photometry to determine the luminosity of our objects and we estimated the mass and evolution status of each sample star with a maximum-likelihood technique using stellar evolution models computed with the STAREVOL code. We compared the observed Li behaviour with predictions from stellar models, including rotation and thermohaline mixing. The same approach was used for stars from selected Li surveys from the literature. Results. Rotation-induced mixing accounts nicely for the Li behaviour in stars warmer than about 4200 K, independently of the mass domain. For stars with masses lower than 2M(circle dot) thermohaline mixing leads to further Li depletion below the T-eff of the RGB bump (about 4000 K), and on the early asymptotic giant branch, as observed. Depending on the definition we adopt, we find between 0.8 and 2.2% of Li-rich giants in our new sample. Conclusions.Gaia puts a new spin on the understanding of mixing processes in stars, and our study confirms the importance of rotation-induced processes and of thermohaline mixing. However asteroseismology is required to definitively pinpoint the actual evolution status of Li-rich giants.

The invention relates to a method and apparatus for automatic high-speed generation of digital angled halftone screens, specially suited for obtaining screens approximating the irrational angles which are generally required by high-quality colour reproduction. The method enables colour separations to be generated which minimize Moire effects, interferences and artifacts by applying discrete one-to-one rotations to digital halftone screens of the required period in order to reach the final screen angle. Dither tiles incorporating assemblies of the basic screen element are rotated by one-to-one discrete rotation and transformed into a new type of dither array, the scanning dither array. The scanning dither array is composed both of dither thresholds and of displacement vectors, providing the means to scan the dither array at image generation time. Several different discrete one-to-one rotation variants are proposed: a small angle rotation technique valid for a subset of rational rotation angles, a rigid band technique and an improved band technique valid for all rational rotation angles and a technique based on discrete shearing transformations. The high- quality of the so rotated dither tile is due to the fact that discrete one-to-one rotation preserves the exact number of elementary cells per screen element and their exact dither threshold values. Since discrete one-to-one rotation enables screen tiles generated by any existing or new method to be rotated, it provides a new range of solutions for obtaining high-quality digital angled halftone screens. In this range of solutions, high-quality solutions can be found for generating three digital angled halftone screens, each 30 deg. apart from each other, as known from traditional photographic colour screening techniques. Further solutions minimizing Moire effects may be obtained by halftone screens whose first order frequency component vectors sum up to zero. Since most of the proposed discrete one-to-one rotation variants can be accomplished by simple and incremental operations such as additions, subtractions, shifts, replications and table accesses, discrete one-to-one rotation is capable of generating angled halftone screens at high-speed. The invented process has turned out to be particularly effective when printing with color ink jet printers at resolutions between 150 and 800 dpi as well as with xerographic printers at resolutions between 300 and 1200 dpi.

1995