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One of these coordinates is based on the radial flow behaviour.
The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself.
PCA of Procrustes coordinates is based on an eigenvalue decomposition of a covariance matrix, which transforms the Procrustes coordinates into scores along principal components (PCs) (Slice, 2005).
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Transformation of the magnetic field components from bird-based coordinates to geographical coordinates was based on the directional sensor data.
Rare CNVs were mapped to genes whenever at least one transcript overlapped the CNV; transcript coordinates were based on RefSeq hg18 (the same build as for the BeadChip arrays).
In two- and three-dimensional Euclidean space, the distance formulas for points in rectangular coordinates are based on the Pythagorean theorem.
It is worth noting that both the DEMs and gravity data coordinates are based on the Tokyo Datum.
Coordinates are based on the ITRF2008 (Altamimi et al., 2011) by constraining the daily coordinates of four IGS sites around Japan (aira, daej, khaj, and yssk).
When sources were not found in 2MASS, transformation from the image pixel coordinates to the sky coordinates was based on matching detected sources with the SAGE point source catalog (Meixner et al., 2006).
The coordinates are based on the International Terrestrial Reference Frame 2008 (Altamimi et al., 2011) and determined by constraining the daily coordinates of four IGS sites around Japan (AIRA, DAEJ, KHAI, and YSSK).
The deployment of the instruments at those exact geographical coordinates was based on ease of access to the sites, human activity level near these regions and sensitivity of the apparatuses in the water depths placed.
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