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Cramer's rule is used in the Ricci calculus in various calculations involving the Christoffel symbols of the first and second kind. [14] In particular, Cramer's rule can be used to prove that the divergence operator on a Riemannian manifold is invariant with respect to change of coordinates. We give a direct proof, suppressing the role of the ...
The number of distinct terms (including those with a zero coefficient) in an n-th degree equation in two variables is (n + 1)(n + 2) / 2.This is because the n-th degree terms are ,, …,, numbering n + 1 in total; the (n − 1) degree terms are ,, …,, numbering n in total; and so on through the first degree terms and , numbering 2 in total, and the single zero degree term (the constant).
[2] [3] Cramer and Leonhard Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer. [4] It has become known as Cramer's paradox after featuring in his 1750 book Introduction à l'analyse des lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement. [5]
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, [1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be.
Cramér’s decomposition theorem, a statement about the sum of normal distributed random variable; Cramér's theorem (large deviations), a fundamental result in the theory of large deviations; Cramer's theorem (algebraic curves), a result regarding the necessary number of points to determine a curve
The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, [1] [2] [3] but has also been derived independently by Maurice Fréchet, [4] Georges Darmois, [5] and by Alexander Aitken and Harold Silverstone. [6] [7] It is also known as Fréchet-Cramér–Rao
The logarithmic moment generating function (which is the cumulant-generating function) of a random variable is defined as: = [ ()].Let ,, … be a sequence of iid real random variables with finite logarithmic moment generating function, i.e. () < for all .
It may be viewed as the association between two variables as a percentage of their maximum possible variation. φ c 2 is the mean square canonical correlation between the variables. [citation needed] In the case of a 2 × 2 contingency table Cramér's V is equal to the absolute value of Phi coefficient.