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An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor .
Thus 24/1000 (2.4%) of the total output will be defective and the likelihood that a randomly selected defective item was produced by machine C is 5/24 (~20.83%). This problem can also be solved using Bayes' theorem: Let X i denote the event that a randomly chosen item was made by the i th machine (for i = A,B,C).
Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. [78] His book, Elements, is widely considered the most successful and influential textbook of all time. [79]
Thurston's 24 questions [4] [5] 24-William Thurston: 1982 Smale's problems: 18: 14: Stephen Smale: 1998 Millennium Prize Problems: 7: 6 [6] Clay Mathematics Institute: 2000 Simon problems: 15 <12 [7] [8] Barry Simon: 2000 Unsolved Problems on Mathematics for the 21st Century [9] 22-Jair Minoro Abe, Shotaro Tanaka: 2001 DARPA's math challenges ...
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. For example, Br(k) is trivial if k is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem).
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé. [4]: 98 The theorem was first proved by Bienaymé in 1853 [5] and more generally proved by Chebyshev in 1867. [6] [7] His student Andrey Markov provided another proof in his 1884 Ph.D. thesis ...