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Lorentzian Manifolds
Lorentzian may refer to * Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution * Lorentz lineshape (spectroscopy) * Lorentz transformation * Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ... See also * Lorentz (other) * Lorenz (other), spelled without the 't' {{Disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Distribution
The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f(x; x_0,\gamma) is the distribution of the -intercept of a ray issuing from (x_0,\gamma) with a uniformly distributed angle. It is also the distribution of the Ratio distribution, ratio of two independent Normal distribution, normally distributed random variables with mean zero. The Cauchy distribution is often used in statistics as the canonical example of a "pathological (mathematics), pathological" distribution since both its expected value and its variance are undefined (but see below). The Cauchy distribution does not have finite moment (mathematics), moments of order greater than or equal to one; only fractional absolute moments exist., Chapter 16. The Cauchy dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Line Shape
Spectral line shape or spectral line profile describes the form of an electromagnetic spectrum in the vicinity of a spectral line – a region of stronger or weaker intensity in the spectrum. Ideal line shapes include Lorentz distribution, Lorentzian, Normal distribution, Gaussian and Voigt function, Voigt functions, whose parameters are the line position, maximum height and half-width. Actual line shapes are determined principally by Doppler broadening, Doppler, Spectral line#Broadening and shift, collision and proximity broadening. For each system the half-width of the shape function varies with temperature, pressure (or concentration (chemistry), concentration) and phase. A knowledge of shape function is needed for spectroscopic curve fitting and deconvolution. Origins A spectral line can result from an electron transition in an atom, molecule or ion, which is associated with a specific amount of energy, ''E''. When this energy is measured by means of some spectroscopic te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v, representing a velocity confined to the -direction, is expressed as \begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end where and are the coordinates of an event in two frames with the spatial origins coinciding at , where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and \gamma = \frac is the Lorentz factor. When speed is much smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentzian Manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of Positive-definite bilinear form, positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as Causal structure, timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space that is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz (other)
Lorentz is a surname and a given name. Lorentz may also refer to: Things named for Hendrik Lorentz * Lorentz factor, Doppler effect *The Lorentz-Lorenz law, the law regarding the refractive index of a substance discovered independently by Hendrik Lorentz and Ludvig Lorenz * Lorentz force, the force exerted on a charged particle in an electromagnetic field * Lorentz transformation, the formula that provides the mathematical backbone for Einstein's theory of special relativity * The Lorentz group, the group containing all Lorentz transformations * Lorentz-Cauchy distribution, a distribution used in fitting peaks in a spectrum * Lorentz surface, a two-dimensional oriented smooth manifold with a conformal equivalence class of Lorentzian metrics. It is the analogue of a Riemann surface in indefinite signature * Lorentz curve (other), various meanings * Lorentz scalar, a scalar which is invariant under a Lorentz transformation * Lorentz Institute, Instituut-Lorentz in Dut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
