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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.
Many kinds of seaweed produce chloroform, and fungi are believed to produce chloroform in soil. [ 3 ] Most of the haloforms — specifically, chloroform ( CHCl 3 ), bromoform ( CHBr 3 ), and iodoform ( CHI 3 ) — are easy to prepare through the haloform reaction , although this method does not lend itself to bulk syntheses.
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.
The matrix element of the position acting on |l,m can only produce values of m which are bigger by one unit, so that if the coordinates are rotated so that the final state is |l′,l′ , the value of l′ can be at most one bigger than the biggest value of l that occurs in the initial state.
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 ...
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.
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.
The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet , gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change y = Q − 1 ( t ) x {\displaystyle \displaystyle y=Q^{-1}(t)x} with Q ( t + 2 T ) = Q ( t ) {\displaystyle \displaystyle Q(t+2T)=Q(t)} that ...