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In chemistry, the haloform reaction (also referred to as the Lieben haloform reaction) is a chemical reaction in which a haloform (CHX 3, where X is a halogen) is produced by the exhaustive halogenation of an acetyl group (R−C(=O)CH 3, where R can be either a hydrogen atom, an alkyl or an aryl group), in the presence of a base.
Trihalomethanes with all the same halogen atoms are called haloforms. Many trihalomethanes find uses in industry as solvents or refrigerants . Some THMs are also environmental pollutants , and a few are considered carcinogenic .
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
So it is possible to shift any matrix element diagonally through the correspondence, (+) (+) where the right hand side is really only the (m − n) th Fourier component of dA / dJ at the orbit near m to this semiclassical order, not a full well-defined matrix. The semiclassical time derivative of a matrix element is obtained up to a ...
In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of ...
Halomethane compounds are derivatives of methane (CH 4) with one or more of the hydrogen atoms replaced with halogen atoms (F, Cl, Br, or I). Halomethanes are both naturally occurring, especially in marine environments, and human-made, most notably as refrigerants, solvents, propellants, and fumigants.
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators [1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives ...
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.