Detection of Additives with the Help of Discrete Geometrical Invariants

Abstract

In this paper, we propose a general mathematical method for the detection of electrochemical additives in a given solute with the help of discrete geometrical invariants (DGI). This idea is based on the generalization of Pythagor’s theorem that can be proved for two random sets located in the two-dimensional (2D) plane. This statement follows from the previous ideas proposed by Babenko, who essentially modernized the well-known theorem and propagated it on a wide class of “right” discrete sets with different symmetry. However, attentive analysis of these results shows that there is a possibility for their further generalization. For practical purposes, it is important to have discrete and deterministic curve(s) with the limited number of parameters that enables comparing two random sets of any nature if their quantitative description expressed in terms of the “best-fit” model is absent. Under the best-fit model, we imply the microscopic model that enables describing the measured data in terms of the minimal set of the fitting parameters. We propose at least two invariants: (a) the curve of the second order that coincides with the classical ellipse oriented at an arbitrary direction relative to the X- and Y-axes, and (b) the curve of the fourth order that has eight quantitative parameters and includes the cross-combination of the integer moments. In this paper, the DGIs of both types were used. These curves are made useful for the solution of a key problem in electrochemistry, i.e., the detection of small concentrations of d-tryptophan (6.54 ÷ 38.7) 10−5 mol·L−1 in a given solute (phosphate buffer solution (Na2HPO4 + KH2PO4) with pH = 6.86) that was activated by electrodes of two types—Pt (platinum) and C (carbon). The DGI method is free from treatment errors and model suppositions; therefore, it can be applied for the detection of small additives in a given solute and a further description can be attained with the help of a monotone/calibration curve expressed by means of parameters associated with the DGI.