Two mathematical models described in details are based upon: (1) full RANS equations written in the conservative form and (2) modification of the RANS equations by Boussinesq approximation re-casted in the non-conservative form.
First, dimensional analysis is carried out based on the governing differential equations with the Boussinesq approximation, and Re, Ri and Pr are found to be the governing dimensionless parameters in determining flow and temperature distribution.
The measured state duration statistic is used without any approximation for re-modelling, i.e., the state duration is a random realisation of the measured SDPDF.
The semi-Markov model offers some options to describe the SDPDF of each state: The measured state duration statistic is used without any approximation for re-modelling, i.e., the state duration is a random realisation of the measured SDPDF.
The thus obtained refractive index profiles turned out to represent efficient initial approximations for re-engineering purposes.
To check sensitivity to these approximations, we re-ran analyses excluding trials in which these were necessary and obtained broadly similar results (ESM Table 1).
In order to reach the desired band range, A1 called approximation band is re-separated, and procedures are continued consecutively until the intended frequency range is reached.
However, instead of using the exact mathematical log operation, profile C uses a piecewise linear approximation, defined by re-interpreting the bit-pattern of the half-logarithmic IEEE representation of floating-point numbers as integers, which is exactly invertible [8].
The linear approximation problem of re-projection of a signal represented in one basis onto another basis as addressed in this article, correlates with the problem considered in the compressive sensing (CS) literature [33].
It is important to note here that large-scale excursions along a specific low-frequency mode tend to violate the harmonic approximation, and a re-evaluation of normal modes at a different minimum may be needed in such cases.
Through the relaxation time approximation (RTA) with Debye model, the thermal conductivities of those RE pyrochlores were predicted, showing good agreement with experimental measurements.
