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We analyze this map by applying the method of multiple scales upto first order.
Then, approximate solutions are determined by applying the method of multiple time scales.
This difficulty is eventually overcome by applying the method of matched asymptotic expansions.
An exact analytical solution of the problem is obtained by applying the "method of images".
The numerical robustness is further illustrated by applying the method to highly deformed mesh elements.
The second stage is to construct a PDE model by applying the method of gray relational analysis with programming languages.
The response of the system at the frequency of excitation is found analytically by applying the method of averaging.
Fibre damage accumulation caused by composite tensile straining is also measured by applying the method to previously stressed wire samples.
Theoretical validation and related error analysis are conducted by applying the method to a simple lumped parameter system.
This method has also been used for a bricks-and-mortar application by applying the method to a company's promotion and pricing, in various retail channels, with similar results.
The influence of matrix on extraction recovery was also evaluated by applying the method to different finger-paints, and recoveries were similar or even higher in some cases.
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