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The notion of
preordering, which is a generalization of the notion of ordering, has been
introduced by Serre. On the other hand, the notion of round quadratic forms has
been introduced by Witt. Based on these ideas, it is here shown that 1) a field F is formally real n-pythagorean iff the nth
radical, RnF is a preordering (Theorem 2), and 2) a field F is n-pythagorean iff for any n-fold Pfister form ρ. There exists an odd integer l(>1) such that l×ρ is a round quadratic form (Theorem 8). By considering upper bounds for the number of squares
on Pfister’s interpretation, these results finally lead to the main result
(Theorem 10) such that the generalization of pythagorean fields coincides with
the generalization of Hilbert’s 17th Problem.
In this study, we estimated the growth area of aquatic
macrophytes that have expanded spontaneously in Lake
Shinji, located in eastern Shimane Prefecture, Japan, using Terra satellite
Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) data.
Visible and near infrared ASTER data from April, August, and September 2012
were used. The water depth at which ASTER can detect submersed aquatic
macrophytes using in situ spectral reflectance of aquatic macrophytes and a
bio-optical model was also examined. As a result, when the threshold value of a
normalized difference vegetation index (NDVI) was set to 0, only aquatic
macrophytes up to a depth of approximately 10 cm could be detected. The growth area of aquatic
macrophytes detected by NDVI from ASTER data was in relatively good agreement
with the growth area as observed by aerial photography.