Discrete Geometrical Invariants: How to Differentiate the Pattern Sequences from the Tested Ones?
Based on the new method (defined below as the discrete geometrical invariants—DGI(s)), one can show that it enables to find the statistical differences between random sequences that can be presented in the form of 2D curves. We generalized and considered the Weierstrass–Mandelbrot function and found the desired invariant of the fourth order that connects the WM-functions with different fractal dimensions. Besides, we consider an example based on real experimental data. A high correlation of the statistically significant parameters of the DGI obtained from the measured data (associated with reflection optical spectra of olive oil) with the sample temperature is shown. This new methodology opens wide practical applications in differentiation of the hidden interconnections between measured by the environment and external factors.
KeywordsWeerstrass–Mandelbrot function Discrete geometrical invariants Equipment calibration Nano-noise “reading”
2010 AMS Math. Subject Classification.Primary 40A05 40A25 Secondary 45G05.
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