Film models mathematical relationships

Two of the relationships compiled in Table 8.1 (corresponding to two applications related to the Srinivasan-Gileadi model [47]) reflect the mathematical dependence shown in the graphics thus, the film formation process could be controlled by quasi-reversible adsorption in region A and by irreversible adsorption in region C. It is, therefore, advisable to examine the physicochemical processes involved, that is, to verify if the theoretical model is consistent with the observed phenomenon. In this way, complementary results are obtained. 

With the advent of the computer, numerous simulated unique patterns such as the rich Julia and Mandelbrot sets have been created this is computer graphical generation by pure mathematical models. However, these unique patterns were formed only in the computer by the mathematical means. The twin problems of how to realize these patterns in a real physical system and how to bridge the gap between the real physical system and the pure mathematical model have stimulated the interest of many natural scientists. Since the discovery of buckminsterfullerene its physical properties and interactions with atoms, molecules, polymers, and crystalline surfaces have also been the subject of intensive investigations. Moreover, fullerene-doped polymers show particular promise as new materials with novel electrical, optical, and/or optoelectrical properties. The present study focuses on the striking morphological properties of fullerene-TCNQ multilayered thin films formed under proper growth conditions and explores the relationship between the real physical system and the pure mathematical model. 

The most stringent need for wavenumber axis calibration is in determinations based on band position. For this reason, qualitative analyses are likely to be affected by drifts or inaccuracy in the wavenumber axis [14]. Likewise, quantitative determinations based on band position, such as strain in diamond films [6], will be affected similarly. Other quantitative analyses may also be affected by band-position error. It is common to use the raw spectral intensities (intensity at every wavenumber) in a multivariate analysis. Although this approach can be very powerful, any unexpected shift in wavenumber calibration can cause severe error in the model. In essence, the spectral pattern to which the model has been trained has been shifted. The mathematics of the model are expecting a particular relationship of intensity between adjacent variables (wavenumbers) and cannot usually account for shifts [31], To some extent, multivariate models can be desensitized to inaccuracy and imprecision by assuring that the calibration samples also exhibit some of the same shifting features, but model sensitivity may suffer as a result. Although not in common use, other deconvolution methods have been introduced which may be applicable to removing shift effects of inaccurate wavenumber calibrations [37]. 

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