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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.
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 ...
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: = ¯
No one likes a bailout, particularly one caused by taking a simple and valuable business -- like, say, home loans -- and turning into a complex labyrinth of indecipherable derivative contracts.
Two complex Hadamard matrices are called equivalent, written , if there exist diagonal unitary matrices , and permutation matrices, such that =. Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.