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Principle of a group-contribution method. The Joback method is a group-contribution method.These kinds of methods use basic structural information of a chemical molecule, like a list of simple functional groups, add parameters to these functional groups, and calculate thermophysical and transport properties as a function of the sum of group parameters.
Consider the symmetric group acting on R 3 by permuting the coordinates. We add up the sum by group elements, as follows. Starting with the identity, we have = ().There is a three-element conjugacy class of , consisting of swaps of two coordinates.
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
The formula for an integration by parts is () ′ = [() ()] ′ (). Beside the boundary conditions , we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ...
To understand the normalization scale factor between the statistical measures, the following is the relevant statistical rule: For independent random variables X and Y, the variance (σ z 2) of a sum or difference (z = x − y) is the sum square of their variances (σ z 2 = σ x 2 + σ y 2).
If the sum is of the form = ()where ƒ is a smooth function, we could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum.
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In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically ():= (,) = ()where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R + into the unit circle, and χ is a group homomorphism of the unit group R × into the unit circle, extended to non-unit r, where it takes the ...
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