# Detection of Additives with the Help of Discrete Geometrical Invariants

^{1}

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## Abstract

**:**

## Featured Application

**The proposed methodology outlined in this work will be useful for detection of small additives in different complex fluids. It has a wide region of applicability in electrochemistry, chromatography etc., where different spectroscopic methods for detection of substance “traces” are used.**

## Abstract

^{−5}mol·L

^{−1}in a given solute (phosphate buffer solution (Na

_{2}HPO

_{4}+ KH

_{2}PO

_{4}) 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.

## 1. Introduction and Formulation of the Problem

- (1)
- Determination of some output parameters that are related to monotone concentration of the electroactive components in conditions of strongly fluctuating currents on the C(“active”) and Pt (“inertial”) electrodes, for construction of the desired calibration curves.
- (2)
- Comparison of the obtained characteristics for the C- and Pt-electrodes and derivation of the calibration/monotone curves with respect to the given d-tryptophan concentration.

- (a)
- Its generality (universality) implies the potential to compare any couple of random sets assigned by their coordinates (Y1
_{k}, Y2_{k}) in a 2D plane. - (b)
- All parameters are expressed in terms of the integer moments and their inter-correlations.
- (c)
- The number of parameters is finite, and they have clear geometrical interpretation.
- (d)
- The proposed method oriented for analysis of discrete sets is free from any unjustified supposition and treatment errors.
- (e)
- The proposed approach can compare random sequences having different sampling volumes in the frame of the same set of statistical parameters.

## 2. Materials and Methods

^{−3}M (SIGMA-ALDRICH, assay ≥98.0% (HPLC)) was prepared by dissolving an accurately weighed portion in the background electrolyte. As a background electrolyte, the standard phosphate buffer solution (pH 6.86, a mixture of Na

_{2}HPO

_{4}and KH

_{2}PO

_{4}) was used.

- (1)
- Electrochemical regeneration (precycling)—five successive cycles with a potential scan rate of 2.5 V/s in the range of potentials from 0–2.5 V.
- (2)
- Registration of the VAG curve in the analyzed solution at a potential scan rate of 0.5 V/s in the range of potentials from 0–1.5 V.

## 3. Description of the Treatment Procedure

- (a)
- Which curve is more sensitive to the presence of a Pt- or C-electrode?
- (b)
- How many monotone curves can be derived from the parameters that form the invariants of the second/fourth orders?

- (c)
- Which space is more sensitive to the presence of d-tryptophan in the solute: the data space or the measurement space?

_{m}(U) defines the initial data file corresponding to the measurement m, and the symbol <…> determines the arithmetic mean for each fixed measurement in the given space, calculated in accordance with the third line in Equation (1). The number of data points (j = 1, 2, …, N) for each measurement was equal to N = 1180. The symbol Integral(x, y) determines the conventional integral calculated by means of the recurrence trapezoid formulas follows:

_{0}= 0 and c

_{1}= 1 for C- and Pt-electrodes are given by Figure 2a,b for the normalized data (dJ/dU (derivative) and J(U)-usual VAG) in data space. Figure 3a,b demonstrate the behavior of the ranges (Rg(Y

_{m}) and Rg(JY

_{m}) in measurement space for the same concentrations c

_{0,1,9}(C-, Pt-electrodes), calculated for the derivatives (Figure 3a) and usual VAGs (Figure 3b). We remind the reader here that the range of a function f(x), defined in the region of a variable x from the interval [min(x), max(x)] and representing itself the maximal deviation is defined as Rg(f) = max(f) − min(f).

^{2}>/<DX

^{2}>), parameters A(c) and B(c), defined by Equations (A10) and (A11), and the angle α(c), derived from Equation (A9). It is instructive to realize this comparison for two types of electrodes (C and Pt) and for both spaces (data and measurement). The DGI approach allows using the same set of parameters in spite of the fact that the dimensions of two spaces (N and M) are different.

- We should take into account the behavior of parameter σ
_{B}_{,C}(c), which incorporates the influence of the higher moments. This dependence can be important for other applications. - The behavior of the value of the invariant I
_{4}(c) (derived from Equation (A17))) is also important and should be taken into account, as well.

_{A,B,C}(c) (defined by Equation (A16)), because the exhibited behavior repeats the behavior of parameters A(c) and B(c), considered above. We omit also the dependencies of curves from Equations (A7) and (A18) against the polar angle φ, because their forms are not important here for further analysis. However, we want to stress also the fact that the curve from Equation (A18) is more informative, because it can have both a real part Re(x(φ)), Re(y(φ)) and a complex conjugated part Im(x(φ)−<x>)), Im(y(φ)−<y>).

## 4. Results and Discussions

- (1)
- The C- and Pt-electrodes at the repeated functioning cycles exhibit the aging phenomenon that leads, in turn, to the temporal signal drift. This drift distorts the measured VAGs. This effect is expressed clearly in the analysis of the curves associated with the measurement space. For a quantitative description, as well as the control and elimination of this negative phenomenon, we use the DGI method that frees the system from the imposed model and errors related to the treatment procedure.
- (2)
- The integral curves in the measurement space because of the DGI application demonstrate the sensitivity of both electrodes to the amino-acid concentration variations (see the monotone curves obtained for the data space).
- (3)
- The aging effect confirms our previous conclusions; after achieving the activated state, the electrode behavior at the given potential sweep depends on the number of cycles to a lesser degree. This effect is expressed clearly for the more inertial Pt-electrodes. For C-electrodes, we presumably observe the varied behavior covering all 1000 measurements.
- (4)
- The ratio Rt(c) = <y(c)>/<x> (remember that the variable <x> = <y(0)> is always related to the buffer solution), while dispersion Dsp(c) = <DY
^{2}>/ <DX^{2}>, depicted in Figure 4a, characterizes the center of gravity and its standard deviation. They are more sensitive to the surface changes evoked by the aging phenomenon. Both parameters have monotone behavior for C- and Pt-electrodes in the presence/absence of the amino acid. For the Pt-electrode, this monotone dependence is weak; it confirms the high stability of the Pt-electrode during electro oxidation processes. We should stress here that Figure 4a,b are more informative and allow characterizing the d-tryptophan behavior, as well as comparing its electroactivity with the interaction of C- and Pt-electrodes.- (a)
- In data space, the dispersion curves Dsp for C- and Pt-electrodes differ by their mutual location with respect to each other; the value Dsp(c) (C-electrode) exceeds the curve Dsp(c) (Pt-electrode) for all values of c. The gravity ratios Rt(c) for both types of electrodes are strongly deviated from each other as well.
- (b)
- In measurement space, a significant difference is observed for the curve Dsp(c) (Pt). It is explained by the excess electroactivity of the tryptophan molecule in its interaction with the Pt-electrode. One can stress here that the clearly expressed peaks on the initial VAGs were not observed.

- (5)
- The behavior of parameters A(c) and B(c) (Figure 5a,b) is more sensitive to the surface changes; the behavior of parameter A(c) for both types of electrodes is characterized by a monotone behavior, which increases in the presence/absence of the amino acid.
- (6)
- The ellipse rotation (correlation) angle α for the C-electrode in the absence of amino acid does not correlate with the number of oxidation/reduction cycles; furthermore, it does not depend on the sensor duration and cannot be associated with the aging phenomenon. However, as shown in Figure 6, this parameter correlates strongly with d-tryptophan concentration. The calculated curve is monotone with respect to increasing the concentration of d-tryptophan. For the Pt-electrode after 300 cycles in the presence/absence of amino acid, this curve has an increasing trend. We relate this increasing tendency with additional processes of oxidation in the background solution. From our point of view, this parameter can be used for calibration purposes of some non-electroactive components of the background solute.
- (7)
- The parameters analyzed above reflect the specificity of electrode processes from different points of view. For the more active C-electrode, which has better oxidation characteristics, many output DGI parameters changed more intensively in comparison with the more inertial Pt-electrode. The Pt-electrode is more stable in oxidation processes. Being a more “stable” material with respect to the transfer electron process related to the oxidation of amino acids, it requires the calculation of an extended set of “fine” parameters that are sensitive to concentration variations of the electroactive substance. For the given case, we associate these variations with the behavior of parameter σ
_{B,C}. As it follows from the analysis of Figure 7a, these parameters have monotone and decreasing characteristics against the concentration in data space. In the measurement space shown in Figure 7b, these parameters characterize again the electroactivity of the C-electrode, which has variable characteristics, compared to the relatively stable behavior of the Pt-electrode. Again, the 300-cycle boundary is conserved. Only after this limit does the Pt-electrode tend to the aging regime. - (8)
- The proposed DGI approach is rather general and enables comparing random curves with different samples, using the same set of parameters expressed in terms of moments up to the fourth order. We showed also that these parameters have different sensitivity with respect to the input factor (in our case, this factor is associated with concentration of the amino acid). Comparing both spaces, one can say that the data space is more stable in comparison with the measurement space; the latter actually demonstrates the temporal evolution (each measurement together with “idle” time occupies 13 s) of the whole process of the interaction of the amino acid with the given electrode. The DGI approach allows obtaining a set of calibration curves and establishing a natural limit associated with parameter I
_{4}(c) (see Figure 8 for details). This parameter has a small value and loses its monotone behavior for the both spaces.

## Author Contributions

## Funding

## Conflicts of Interest

## The Basic Abbreviations

DGI | Discrete geometrical invariant |

SVD | Single-valued decomposition |

VAG(s) | Voltammogram(s) |

## Appendix A

_{k}, y

_{k}) (k = 1, 2, 3, …, n). Let us consider the square of the distance connecting an arbitrary point M(x, y) with the kth point (x

_{k}, y

_{k}) belonging to two given sets.

^{2}≥ R

^{2}, where the equality sign corresponds to a circle with zero radius; it is convenient to consider the invariant circle with radius I

^{2}= 2R

^{2}. From another point of view, the requirement of Equation (A2) can be considered as the reduction of the given set of points to the continuous circle with four statistical parameters (<x

^{p}>, <y

^{p}>; p = 1, 2).

^{p}>, <y

^{p}>, p = 1, 2; <xy>) and three unknown parameters (A, B, C). We subordinate this combination to the following requirement:

^{2}= 2E

^{2}. In order to find three unknown parameters (A, B, C), it is convenient to use the obvious parameterization for the variables (x, y) relative to the angle φ.

_{k}= y

_{k}) that follows from the obvious requirement B = A

^{2}, (α = 0). Concluding this section, one can say that, with the help of the rotated counterclockwise ellipse in Equation (A7), we reduced 2n random points figuring in equation (A4) to eight statistical parameters Pr

_{8}

^{(2)}: (<x

^{p}>, <y

^{p}>, p = 1,2; <xy>, A, C, α). Equations (A7)–(A9) can be considered as the geometrical interpretation of the conventional Pearson correlation coefficient defined by Equation (A9).

_{4}and Inv from Equation (A13) are defined by Equation (A17)

^{q}(Δy)

^{p}> characterizing two compared sets are defined by Equation (A13). The polynomial K(X,Y) of the fourth order can be separated in the polar coordinate system. Using the notations in Equation (A16) and taking into account the fact that the constant A

_{4}figuring in K(X,Y) is an arbitrary proportion multiplier and, therefore, can be omitted, we present the desired curve in the form

_{2},

_{4}(φ) figuring in Equation (A18) are determined as follows:

_{8}

^{(4)}(<x>, <y>, σ

_{B}, σ

_{C}, S

_{A}

_{,B,C}, I

_{4}) determines the statistical proximity/difference of 2D random curves/sets located in the plane. What happens if two random curves are identical to each other (x

_{j}= y

_{j}) for all numbers of the discrete points j = 1, 2, …, N? In this case, as it is easy to see from Equation (A18), σ

_{B}= σ

_{C}= 1, I

_{4}= 0, S

_{A}

_{,B,C}= 4 <(Δx)

^{2}> and, therefore, from Equation (A18), it follows that r(φ) = 0. In this case, Equation (A18) is degenerated into a point with coordinates <x> = <y>, located on the line y = x.

_{m}

^{(2)}, Pr

_{m}

^{(4)}(m = 1, 2, …, 8) that reflect the different sensitivities with respect to a chemical additive presented in the measured VAG(s).

_{k}, y

_{k}) (k = 1, 2, …, n) of the compared random sets, their descriptive and geometric forms in the 2D plane, and their requirement of the constant Inv in each specific form. The consideration of other DGIs containing the moments of the fifth, sixth, etc. orders becomes problematic, because these invariants do not allow realizing the separation procedure in a polar coordinate system.

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**Figure 1.**The scheme of the measurement cycle. The precycling procedure included five successive cycling stages with a potential scan rate of 2.5 V/s covering the potential range from 0–2.5 V. The voltammogram (VAG) curve of the analyzed solution was registered at a potential scan rate of 0.5 V/s, and the potentials ranged from 0–1.5 V. The time interval between measurements was 10 s. The period of the single VAG registration occupied 3 s.

**Figure 2.**(

**a**) VAG derivatives (dJ/dU) for two types of electrodes (Pt and C). Minimal concentration (c = 1) corresponds to 6.54 × 10

^{−5}mol·L

^{−1}; see Table 2 for details. (

**b**) Normalized and integrated VAGs corresponding to the curves shown in the previous figure. These curves are more preferable for the detection of additives. The integration procedure was realized with the help of Equation (2), which eliminates the additional fluctuations and possible deviations evoked by the presence of an additive. Concentration c = 1 corresponds to the minimal concentration of 6.54 × 10

^{−5}mol·L

^{−1}; see Table 2 for details.

**Figure 3.**(

**a**) Distribution of the ranges for differential (dJ/dU) VAGs in the measurement space. One hundred measurements for c

_{0,1,9}were repeated in the same experimental (temperature, pressure, humidity, pH) conditions. The period of one VAG registration was 3 s. The duration between measurements was 10 s. One can notice that the chosen minimal concentration c

_{1}was very close to the curve, corresponding to the c

_{0}concentration for both types of electrodes. All measurements were reduced to the same interval [1,100]. (

**b**) Distribution of the ranges for the usual VAGs for the same values of concentrations c

_{0,1,9}. In comparison with the previous figure, these curves become more sensitive to the presence of d-tryptophan in the buffer solute and, hence, more preferable for further analysis.

**Figure 4.**(

**a**) Behavior of parameters Rt(c) = <y>/<x> and Dsp(c) = <DY

^{2}>/<DX

^{2}> for two types of electrodes in data space. All these curves changed monotonically with respect to the increase in d-tryptophan concentration. The true c-values are given in Table 2. (

**b**) Behavior of the same parameters Rt(c) = <y>/<x> and Dsp(c) = <DY

^{2}>/<DX

^{2}> is shown for two types of electrodes in the measurement space. Only two curves of Rt(c) for two types of electrodes kept their monotone behavior, while the two other curves for dispersion ratios lost (completely or partly) their monotone behavior.

**Figure 5.**(

**a**,

**b**) Behavior of parameters A(c) and B(c) for two types of electrodes combined for the data and measurement spaces. The scale of these parameters allows this. The monotone behavior of the A(c) and B(c) curves for the C-electrode in data space is clearly expressed, while, for the Pt-electrode, the calculated curves in data space kept their monotone behavior only in part. In the measurement space, four curves A(c) and B(c) for both types of electrodes completely lost their monotone behavior.

**Figure 6.**Behavior of the correlation angle α(c) for two types of electrodes in data and measurement spaces. We observe the monotone behavior for C- and Pt-electrodes in data space, while, for the measurement space, these curves for both types of electrode completely lost their monotone behavior.

**Figure 7.**(

**a**) Monotone behavior of the statistical parameters σ

_{B}(c) and σ

_{C}(c) in data space that can be chosen for calibration purposes. They independently confirm the behavior of the ellipse rotation (correlation) angle (see Figure 6) found for the discrete geometrical invariants (DGI) of the second order. (

**b**) Random-like (non-monotone) behavior of the statistical parameters σ

_{B}(c) and σ

_{C}(c) in the measurement space for the C-electrode, which cannot be chosen for calibration purposes. They reflect the strong fluctuation characteristic of the measurement space, confirmed also by the parameters characterizing the invariant of the second order. As for the Pt-electrodes, these curves demonstrate quasi-monotone behavior.

**Figure 8.**Behavior of the parameter I

_{4}(c) from Equation (A17). Having small values located in the interval 10

^{−6}to 5 × 10

^{−5}, it has a maximal sensitivity to the presence of an uncontrollable factor that can influence the interaction between the chosen electrode with d-tryptophan additive. The two curves below I

_{4}(c)-C (in data space) and I

_{4}(c)-Pt (in measurement space) do not have monotone characteristics. They are located in the interval 10

^{−9}to 10

^{−6}. Therefore, they appear as a couple of horizontal lines with values close to zero.

Method | Basic Limitations | Comments |
---|---|---|

Principal component analysis [10] | In single-valued decomposition (SVD), only 3–4 basic components are used. The influence of other components is not evaluated and explained properly. | This is a widely used method in chemometrics for the detection of possible pair correlations between two random sequences. |

Wavelet decomposition [11] | The criterion of the selection of the proper wavelet from the wide wavelet family is absent. Each chosen wavelet has its own treatment error, and, in many cases, it cannot be evaluated and applied for the analysis of nanonoises in the interval 10^{−9}–10^{−6} A. | There exist uncontrollable errors related to the application of continuous wavelets to discrete data. This method is important in the detection of small signals associated with the detection of “trace” substances. |

Timashev’s method (flicker-noise spectroscopy) [12]. | The basic supposition is related to a continuous representation of an initial sequence (with imposed definitions of intermittency, spikes, outliers, etc.) | The final expressions are derived from the apparatus of continuous mathematics and integral F-transform. These treatment errors are not properly evaluated in analysis of the chosen discrete data. |

Artificial neural network (ANN) [13] | There is a limited possibility for the physical interpretation of the model parameters. | Analytical possibilities of ANN are stipulated by a wide choice of different transfer functions, and by a wide variation of the number of “neurons” in the intermediate layer, which limits the possibilities of the optimal (well-educated) selection of the desired transfer function. |

Projection on latent structures, partial least squares (PLS) [14] | The PLS method does not allow deriving the reliable calibration model based on the measured voltammograms (VAGs) at multiple sensors because of the appearance of ellipse-like clusters in the score plots. | PLS regression is the generalization of the usual one-dimensional (1D) linear regression for the case of many independent variables. The fitting error cannot be properly evaluated. |

**Table 2.**The values of d-tryptophan added in the given buffer solute (the phosphate buffer solution (Na

_{2}HPO

_{4}+ KH

_{2}PO

_{4}) with pH 6.86).

Notation of the Solution Affected by d-Tryptophan Additive | Volume of Additive (mL) | d-Tryptophan Concentration (10^{−5} mol·L^{−1}) |
---|---|---|

“0”-buffer solution | 0 | 0 |

“1” | 0.7 | 6.54 |

“2” | 1.4 | 12.3 |

“3” | 2.1 | 17.4 |

“4” | 2.8 | 21.9 |

“5” | 3.5 | 25.9 |

“6” | 4.2 | 29.6 |

“7” | 4.9 | 32.9 |

“8” | 5.6 | 35.9 |

“9” | 6.3 | 38.7 |

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**MDPI and ACS Style**

Nigmatullin, R.; Vorobev, A.; Budnikov, H.; Sidelnikov, A.; Maksyutova, E.
Detection of Additives with the Help of Discrete Geometrical Invariants. *Appl. Sci.* **2019**, *9*, 926.
https://doi.org/10.3390/app9050926

**AMA Style**

Nigmatullin R, Vorobev A, Budnikov H, Sidelnikov A, Maksyutova E.
Detection of Additives with the Help of Discrete Geometrical Invariants. *Applied Sciences*. 2019; 9(5):926.
https://doi.org/10.3390/app9050926

**Chicago/Turabian Style**

Nigmatullin, Raoul, Artem Vorobev, Herman Budnikov, Artem Sidelnikov, and Elza Maksyutova.
2019. "Detection of Additives with the Help of Discrete Geometrical Invariants" *Applied Sciences* 9, no. 5: 926.
https://doi.org/10.3390/app9050926