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An example of a spontaneous (without addition of an external energy source) decomposition is that of hydrogen peroxide which slowly decomposes into water and oxygen (see video at right): 2 H 2 O 2 → 2 H 2 O + O 2. This reaction is one of the exceptions to the endothermic nature of decomposition reactions.
That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage subproblem for a given first-stage solution.
For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix, so that all the entries of its main diagonal are set to one. Then the system of equations has the following solution:
Comment: One can always normalize the eigenvectors to have length one (see the definition of the eigenvalue equation) Comment: Every normal matrix A (that is, matrix for which A A ∗ = A ∗ A {\displaystyle AA^{*}=A^{*}A} , where A ∗ {\displaystyle A^{*}} is a conjugate transpose ) can be eigendecomposed.
Classically, these reactions result in the precipitation of one product. In older literature, the term double decomposition is common. The term double decomposition is more specifically used when at least one of the substances does not dissolve in the solvent, as the ligand or ion exchange takes place in the solid state of the reactant. For ...
The following simplified example shows the economy one gets from the Cholesky decomposition: suppose the goal is to generate two correlated normal variables and with given correlation coefficient . To accomplish that, it is necessary to first generate two uncorrelated Gaussian random variables z 1 {\textstyle z_{1}} and z 2 {\textstyle z_{2 ...
As an example, Bayesian network methods attempt to decompose a joint distribution along its causal fault lines, thus "cutting nature at its seams". The essential motivation behind these methods is again that within most systems (natural or artificial), relatively few components/events interact with one another directly on equal footing. [ 1 ]
Another example of thermal decomposition is 2Pb(NO 3) 2 → 2PbO + O 2 + 4NO 2. Some oxides , especially of weakly electropositive metals decompose when heated to high enough temperature. A classical example is the decomposition of mercuric oxide to give oxygen and mercury metal .