Vertical thermal diffusion coefficients (Ks) were calculated in the surface layer by using tower data in and around Tokyo; then the Kz difference between urban and non-urban areas was examined on a seasonal and hourly basis. During the night-time (except for during summer), the urban Kz substantially exceeds the non-urban Kzi especially in December. The maximum Kz appearing in the midday is about 40 80 times the minimum Kz in the early morning in the non-urban area, and about 10-20 times in the urban center. The roughness length doesn't remarkably affect Kz as compared with the diurnal range of Kz, but it affects Kz more at night than during the day. Following this, simple numerical simulations of CO concentration were carried out from evening to midnight on highly polluted days. Through these simulations, urban and non-urban types of hourly concentration variation can be properly explained by the difference in the hourly trend of the Kz likely to be attributable to the fact that the surface inversion strength intensified more rapidly in the non-urban area than in the urban area during the early evening.

Formal verification using interactive theorem provers have been noticed as a method of verification of proofs that are too big for humans to check the validity of them. The purpose of this work is to verify the validity of Robertson-type uncertainty relation toward verifying unconditional security of quantum key distributions. We verify the validity of the relation by using proof assistant Coq and it is turned out that the theorem regarding the relation formally holds. The source code for Coq which represents the validity of the theorem is printed in Appendix.