Information about Integral Geometry
In mathematics, the term integral geometry is used in two ways, which, although related, imply different views of the content of the subject.
There is a sample space of lines, one on which the affine group of the plane acts. A probability measure is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, that solves the problem of formulating accurately what 'random line' means; and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some paradoxes.)
We can therefore say that integral geometry in the sense of Santalo, is the application of probability theory (as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of differential forms arising.
A very celebrated case is the problem of Buffon's needle: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various stochastic processes concerned with geometric and incidence questions.
One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem.
The more recent meaning of integral geometry is that of Israel Gelfand. It deals more specifically with integral transforms, modelled on the Radon transform. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as pullback onto the incidence graph and then push forward.
Cases
The more traditional usage is that of Santalo and Blaschke. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations.There is a sample space of lines, one on which the affine group of the plane acts. A probability measure is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, that solves the problem of formulating accurately what 'random line' means; and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some paradoxes.)
We can therefore say that integral geometry in the sense of Santalo, is the application of probability theory (as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of differential forms arising.
A very celebrated case is the problem of Buffon's needle: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various stochastic processes concerned with geometric and incidence questions.
One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem.
The more recent meaning of integral geometry is that of Israel Gelfand. It deals more specifically with integral transforms, modelled on the Radon transform. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as pullback onto the incidence graph and then push forward.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Luis Santaló (born October 09 1911 in Girona, Spain - died November 22, 2001 in Buenos Aires, Argentina) was an important Argentine Spanish mathematician.
He graduated from the University of Madrid and he studied in the University of Hamburg, where he received his Ph.D.
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He graduated from the University of Madrid and he studied in the University of Hamburg, where he received his Ph.D.
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Wilhelm Johann Eugen Blaschke (13 September1885 – 17 March1962) was an Austro-Hungarian differential and integral geometer. His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner.
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In mathematics, the Crofton formula, named after Morgan Crofton (1826—1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.
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Statement
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Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth
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In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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random is used to express lack of order, purpose, cause, or predictability in non-scientific parlance. A random process is a repeating process whose outcomes follow no describable deterministic pattern, but follow a probability distribution.
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In mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is the semidirect product of Kn and GL(n, K).
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In probability theory, the definition of the probability space is the foundation of probability theory. It was introduced by Kolmogorov in the 1930s. For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
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ParaDOX
(1997) Crimson
(1998)
"ParaDOX" is Nanase Aikawa's second album. The album reached #1 on Oricon charts.
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(1997) Crimson
(1998)
"ParaDOX" is Nanase Aikawa's second album. The album reached #1 on Oricon charts.
Track listing
- CAT on the Street
- Tenshi no You ni Odorasete
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Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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Andrey Kolmogorov
Born March 25 1903
Tambov, Imperial Russia
Died September 20 1987 (aged 84)
Moscow, USSR
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Born March 25 1903
Tambov, Imperial Russia
Died September 20 1987 (aged 84)
Moscow, USSR
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Erlangen Program (Erlanger Programm) — Klein was then at Erlangen — proposed a new kind of solution to the problems of geometry of the time.
At the time, geometry contained a very large number of theorems.
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At the time, geometry contained a very large number of theorems.
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Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).
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homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful.
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In mathematics, a Lie group (IPA pronunciation: [liː], sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an
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In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon: suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor.
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A stochastic process, or sometimes random process, is the opposite of a deterministic process (or deterministic system) in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as is the case, for example, for
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In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem states that the space of "measures" (see below) defined on finite unions of compact convex sets in Rn consists of one "measure" that is "homogeneous of degree
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Israïl Moiseevich Gelfand (Russian: Израиль Моисеевич Гельфанд
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In mathematics, the Radon transform in two dimensions, named after Johann Radon, is the integral transform consisting of the integral of a function over straight lines. The inverse of the Radon transform is used to reconstruct images from medical computed tomography scans.
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