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Whenever a mathematical procedure is introduced, one of the most important questions is how to invert it.
A rigorous mathematical procedure is formulated for classifying the parametric behavior of the solution, including for viscous damping.
This mathematical procedure is based on orthogonal transformation from a set of correlated variables into a set of uncorrelated variables, the last ones called principal components.
Such a mathematical procedure is called inverse modeling which is described by an ill-conditioned system of linear equations based on the absorption factors method.
A simple mathematical procedure is further introduced for obtaining explicit overall rate expressions of reactants/products molecules involved in various elementary reactions.
To investigate the optical performance of such CPCs, an analytical mathematical procedure is developed to estimate daily and annual solar gain captured by such CPCs based on extraterrestrial radiation and monthly horizontal radiation.
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A number of verification checks performed to verify the mathematical procedure are described.
An original mathematical procedure was developed which involves two steps for each input method and image: robustly fitting a rectangle to the four points, and calculating a special kind of distance between estimated and real points.
A mathematical procedure was then employed to analyze the color images further, to discriminate between continuous radiation from the hot sand and discrete emission, mostly in the blue wavelength region, from bubbles exploding inside the bubbling bed or at its surface.
A mathematical procedure was performed by using (4 - 5) to find the best-fitting curves to data sets of polarized Raman intensity collected at different inplane orientation.
In order to compare directly the integral intensities of the Raman bands the following mathematical procedure was applied: Each spectrum was normalized in a way that the total area under the curve equals one; The spectra were deconvoluted with PeakFit v4.12 (SeaSolve Software Inc., SYSTAT Software Inc).
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