The two-point correlation function is a powerful statistical tool for describing galaxy clustering.
Now, if you decompose the separation of each galaxy pair into a radial component and a transversal (across-the-sky) component, you can plot the two-point correlation function for each component separately.
However, the results depend much on the threshold value [s.sub.0] suggesting a large contribution from the meson-meson continuum, and so in Section 5 we use only the connected parts of the two-point correlation function to perform the QCD sum rule analyses.
The imaginal part of the two-point correlation function is
To solve this problem, we shall use only the connected parts of the two-point correlation function to perform the QCD sum rule analysis in the next section [72-74].
We use [S.sup.q.sub.ab](x) to denote the quark propagator (q = u for up quark, and q = d for down quark), and the contracted two-point correlation function is
These inequalities are given in terms of certain combinations of two-point correlation functions
. It is worth pointing out that although Bell posited the existence of some hidden variables, there is a proof of Bell's inequalities without any assumption about existence and properties of hidden variables (see [63, 64]).
For comparison, results are presented in terms of the normalized two-point correlation functions where all functions are scaled to a range between 1 and 0, regardless of the value of S(0) in each image.
A graph of the normalized two-point correlation functions N(r) for four of the Figure 11 images is shown in Figure 13.