It appears that it also plays a role that near a change of the tendency (e.g. from falling to rising prices) there are typical "panic reactions" of the selling or buying agents with algebraically increasing bargain rapidities and volumes.See for example Preis, Mantegna, 2003. The "fat tails" are also observed in commodity markets.
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The familiar notion of vector addition for velocities in the Euclidean plane can be done in a triangular formation, or since vector addition is commutative, the vectors in both orderings geometrically form a parallelogram (see "parallelogram law"). This does not hold for relativistic velocity addition; instead a hyperbolic triangle arises whose edges are related to the rapidities of the boosts. Changing the order of the boost velocities, one does not find the resultant boost velocities to coincide.
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In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates. For one-dimensional motion, rapidities are additive whereas velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are proportional, but for higher velocities, rapidity takes a larger value, the rapidity of light being infinite.
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In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus they are conformal transformations with respect to the hyperbolic angle.
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An interesting channel to study pentaquarks in proton- nuclear collisions was suggested in This process has a large cross-section due to lack of electroweak intermediaries and gives access to pentaquark wave function. In the fixed-target experiments pentaquarks will be produced with small rapidities in laboratory frame and will be easily detected. Besides, if there are neutral pentaquarks, as suggested in several models based on flavour symmetry, these might be also produced in this mechanism. This process might be studied at future high-luminosity experiments like After@LHC and NICA.
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The energy and momentum of an object measured in two inertial frames in energy–momentum space – the yellow frame measures and while the blue frame measures and . The green arrow is the four-momentum of an object with length proportional to its rest mass . The green frame is the centre-of-momentum frame for the object with energy equal to the rest energy. The hyperbolae show the Lorentz transformation from one frame to another is a hyperbolic rotation, and and are the rapidities of the blue and green frames, respectively.
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This velocity-addition rule is easily derived from rapidities α and β, since sinh(α + β) = cosh α cosh β (tanh α + tanh β). For example, wAC refers to the proper speed of object A with respect to object C. Thus in calculating the relative proper speed, Lorentz factors multiply when coordinate speeds add. Hence each of two electrons (A and C) in a head-on collision at 45 GeV in the lab frame (B) would see the other coming toward them at vAC ~ c and wAC = 88,0002(1 + 1) ~ 1.55×1010 lightseconds per traveler second. Thus from the target's point of view, colliders can explore collisions with much higher projectile energy and momentum per unit mass.
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He is known for his four books on special relativity (1911, 1914, 1921, 1936) where he gave a spacetime derivation of the theory in an axiomatic-geometric way.A. J. Briginshaw (1979) "The axiomatic geometry of Space-Time: An assessment of the work of A. A. Robb", Centaurus 22: 315–323 Robb therefore was sometimes called the "Euclid of relativity". In the first of these works he used a hyperbolic angle ω to introduce the concept of rapidityRobb (1911) Optical Geometry of Motion and showed that the kinematic space of velocities is hyperbolic, so that "instead of a Euclidean triangle of velocities, we get a Lobachevski triangle of rapidities". However, contrary to the scientific mainstream, he believed that the works of Joseph Larmor and Hendrik Lorentz were more important for relativity than the works of Albert Einstein and Hermann Minkowski.
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