Gauss theorem measurements

Here r is the distance from the atom centre. In this book the units are chosen so that Qi is measured in electron charges ( = valence units) and q is set equal to 1.0. According to Gauss theorem, if the electron density of the atom is spherically symmetric, fj- mono gives a fully correct description of the field generated by the atom in the region outside the atom itself, i.e. in the region where the electron density of the atom has fallen to zero. 

The probability density functions of the observations are generally unknown, but the Gauss-Markov theorem ensures that least-squares is always an acceptable estimator. However, the results of least squares are strongly influenced by discordant observations, so-called outliers. The robust-resistant techniques use weight-modification functions of O—Cy which progressively down-weight outliers. Tnese functions implicitly define probability functions p. They may alternatively be interpreted as an appreciation of the reliability of certain measurements. This approaches the frequently used option to simply omit discordant observations because they are judged to be unreliable. 

It is found that both Eqs. 21 and 23 give accurate results if the surface of the objects is very smooth and thus no defects are found in the triangulation process by the MCA. In other words, determination of x using the number of faces, edges, and vertices was found to be more sensitive to RI than the surface curvature measurements, e.g., PSM and SFM. In most cases, the Gauss-Bonnet theorem offers rehable results. 

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