In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.
Metric or metrical may refer to:
Radon is a chemical element; it has symbol Rn and atomic number 86. It is a radioactive noble gas and is colorless and odorless. Of the three naturally occurring radon isotopes, only radon-222 has a sufficiently long half-life for it to be released from the soil and rock, where it is generated. Radon isotopes are the immediate decay products of radium isotopes. Radon's most stable isotope, radon-222, has a half-life of only 3.8 days, making radon one of the rarest elements. Radon will be present on Earth for several billion more years, despite its half-life being a mere 3.8 days, because it is constantly being produced as a step in the decay chain of uranium-238, and that of thorium-232, each of which is an extremely abundant radioactive nuclide with a half-life of several billion years. The decay of radon produces many other short-lived nuclides, known as "radon daughters", ending at stable isotopes of lead. Radon-222 occurs in significant quantities as a step in the normal radioactive decay chain of uranium-238, also known as the uranium series, which slowly decays into a variety of radioactive nuclides and eventually decays into lead-206, which is stable. Radon-220 occurs in minute quantities as an intermediate step in the decay chain of thorium-232, also known as the thorium series, which eventually decays into lead-208, which is stable.
Value or values may refer to:
Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on the calculus of variations.
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations.
In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes. It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.
In mathematics, a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
The derivative of a function is the rate of change of the function's output relative to its input value.
In mathematics, a quasi-invariant measureμ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and μ is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of T on μ is locally expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of T.
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables of order 1, then X is of order and its law is approximately Gaussian.
In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
In measure theory, tangent measures are used to study the local behavior of Radon measures, in much the same way as tangent spaces are used to study the local behavior of differentiable manifolds. Tangent measures are a useful tool in geometric measure theory. For example, they are used in proving Marstrand's theorem and Preiss' theorem.
In mathematics, differentiation of measures may refer to:
In mathematics, the X-ray transform is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography because the X-ray transform of a function ƒ represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ƒ. Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ƒ from its known attenuation data.
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