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Although named for Edgar Buckingham, the π theorem was first proved by the French mathematician Joseph Bertrand [1] in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena.
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: Take a two-dimensional function f ( r ), project (e.g. using the Radon transform ) it onto a (one-dimensional) line, and do a Fourier transform of that projection.
Theorem 1 can now be restated in terms of Theorem 2, because the requirement that all the variables are positive is equivalent to pre-assigning each variable a 1, and asking for the number of solutions when each variable is non-negative. For example: + + + =
Stated by Claude Shannon in 1948, the theorem describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption. Shannon's theorem has wide-ranging applications in both communications and data storage. This theorem is of foundational importance to the modern field of information theory ...
Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles – a right kite.A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential.
It is uncertain who actually discovered the theorem; however, the oldest extant exposition appears in Spherics by Menelaus. In this book, the plane version of the theorem is used as a lemma to prove a spherical version of the theorem. [8] In Almagest, Ptolemy applies the theorem on a number of problems in spherical astronomy. [9]
Kőnig's theorem is equivalent to many other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem. [14]
In differential geometry, the slice theorem states: given a manifold on which a Lie group acts as diffeomorphisms, for any in , the map /, [] extends to an invariant neighborhood of / (viewed as a zero section) in / so that it defines an equivariant diffeomorphism from the neighborhood to its image, which contains the orbit of .