N ( Skwarnicki 1986 ) is a normalization factor and \ alpha, n, \ bar x and \ sigma are parameters which are fitted with the data . erf is the error function.
22.
The normalization factor on the bottom is called the " partition function " for the field, and it coincides with the statistical mechanical partition function at zero temperature when rotated into imaginary time.
23.
Leaving aside normalization factors, this formula asserts that S-matrix elements are the residues of the poles that arise in the Fourier transform of the correlation functions as four-momenta are put on-shell.
24.
On the other hand, the interacting field can also connect many-particle states to the vacuum, thanks to interaction, so the expectation values on the two sides of the last equation are different, and need a normalization factor in between.
25.
The grand potential serves two roles : to provide a normalization factor for the probability distribution ( the probabilities, over the complete set of microstates, must add up to one ); and, many important ensemble averages can be directly calculated from the function.
26.
Specifically, the eigenstate is written in the form \ sqrt { Z } e ^ T | \ phi \ rangle, where \ sqrt { Z } is a normalization factor and | \ phi \ rangle is a state with the minimal number of constituents.
27.
The free energy serves two roles : first, it provides a normalization factor for the probability distribution ( the probabilities, over the complete set of microstates, must add up to one ); second, many important ensemble averages can be directly calculated from the function.
28.
The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space : up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anticommutes with all reflections of the real line.
29.
The values of the normalization factor M _ t, as well as of the lower and upper bound of summation L _ t and U _ t, differ between the central part of the time series and the vicinity of its endpoints ( Ghil et al ., 2002 ).
30.
Where " J " is the nucleation rate, " k " 1 and " k " 2 are rate constants based on the concentrations of H 2 O and NH 3 and " g s " is the normalization factor based on the amount of silica precursor.
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