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Braid closure. In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links.The conditions are stated in terms of the group structures on braids.
The theorem was named after Carl Friedrich Gauss and Andrey Markov, although Gauss' work significantly predates Markov's. [3] But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above. [4] A further generalization to non-spherical errors was given by Alexander ...
In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of ...
The following representation, also referred to as the Riesz–Markov theorem, gives a concrete realisation of the topological dual space of C 0 (X), the set of continuous functions on X which vanish at infinity. Theorem Let X be a locally compact Hausdorff space.
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.
A single-move version of Markov's theorem, was published by in 1997. [7] Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid. The Markov theorem gives necessary and sufficient conditions under which the closures of two braids are equivalent links. [8]
This transformation effectively standardizes the scale of and de-correlates the errors. When OLS is used on data with homoscedastic errors, the Gauss–Markov theorem applies, so the GLS estimate is the best linear unbiased estimator for .
The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. [1] An example of a model for such a field is the Ising model.