Dilation parameter, discretized

Usually, the dilation parameter 5 is discretized along the dyadic sequence,... 

The discretization of the dilation parameter need not be dyadic. Discretization schemes with integer or noninteger factors are possible and have been suggested. The reconstruction of Fit) is given by... 

In Eq. (12) each p(ry)a3 becomes p(iy) A3a3, in the potential that we have assumed to be radial we have a dilation of the distance i.e. v rij) is changed into v(Xrij) and an extra dependence in A appears in the logarithm term. Now we may shift the field into p(r, )A 1 that we may rename p(rj), as it is a dummy variable. Since the number of points in the discretization is fixed the new system has the same number of particles as before dilation. The scaling parameter A appears only in the pair potential and in the logarithm term. The derivative dln (A)/dA can be easily performed and after returning to the continuous expression we obtain,... 

The volume fraction of binder Vbinder(y.p. tp, rg) depends on four parameters. It can be shown that the volume fraction of binder is equal to the probability that the origin is covered by binder. Thus, if the profile of the discretization of the dilated fibers is a cube Qp with side length Irp, we have... 

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