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The first two steps of the Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.
In organic chemistry, the Schmidt reaction is an organic reaction in which an azide reacts with a carbonyl derivative, usually an aldehyde, ketone, or carboxylic acid, under acidic conditions to give an amine or amide, with expulsion of nitrogen.
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve r′(t), r′′(t), r′′′(t), and to apply the Gram-Schmidt process. The resulting ordered orthonormal basis is precisely the TNB frame. This procedure also generalizes to produce Frenet frames in higher dimensions.
In other words, the sequence is obtained from the sequence of monomials 1, x, x 2, … by the Gram–Schmidt process with respect to this inner product. Usually the sequence is required to be orthonormal , namely, P n , P n = 1 , {\displaystyle \langle P_{n},P_{n}\rangle =1,} however, other normalisations are sometimes used.
Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include On series expansions determined by the methods of least squares , and Investigations of the number of primes less than a given number .
In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. ...
Quantitative yields in Claisen–Schmidt reactions have been reported in the absence of solvent using sodium hydroxide as the base and plus benzaldehydes. [5] Because the enolizable nucleophilic carbonyl compound and the electrophilic carbonyl compound are two different chemicals, the Claisen–Schmidt reaction is an example of a crossed aldol ...
In geology, a rock composed of different minerals may be a compositional data point in a sample of rocks; a rock of which 10% is the first mineral, 30% is the second, and the remaining 60% is the third would correspond to the triple [0.1, 0.3, 0.6]. A data set would contain one such triple for each rock in a sample of rocks.