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We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometric properties of initial cuspidal edges., Comment: 9 pages, 2 figures

Seasonal changes in leaf photosynthetic rate and partitioning of 13C taken up by leaves were compared in relation to the development of early watercore using susceptible ‘Orin’ and resistant ‘Fuji’ apples (Malus domestica Borkh.). There was no difference between the cultivars in leaf photosynthetic rate from mid-July to mid-October. The extent of watercore in ‘Orin’ increased from mid-July and peaked in late-August, followed by a decrease through mid-October, while no watercore was detected during the experimental period in ‘Fuji’. Fruit fresh weight and soluble solids content (SSC) were higher in ‘Orin’ than ‘Fuji’ during the earlier stages. Leaf dry weight of ‘Orin’ was higher than that of ‘Fuji’ throughout the experimental period. Accumulation of dry matter in the fruit increased in both cultivars and ‘Orin’ kept significantly higher level than ‘Fuji’ from mid-July to mid-October. The amount of 13C distributed to ‘Orin’ fruit at 5 days after labeling was significantly higher than ‘Fuji’ in mid-July just before watercore started to occur. 13C taken up by ‘Orin’ fruit with prevalent watercore was also higher than ‘Fuji’ in late-August but the difference was not significant. In mid-October when early watercore was gradually reducing, 13C recovered by ‘Orin’ fruit decreased and was significantly lower than ‘Fuji’. The difference of partitioning pattern of 13C between the cultivars was associated with the susceptibility to early watercore as well as the changes in fruit growth and SSC. The seasonal changes in the amount of 13C distributed to ‘Orin’ fruit were also related to the fluctuations in watercore severity. The results suggest that active phloem transport might be the cause of early watercore in our previous work, which might be induced by the increased partitioning of assimilates to the fruit during summer in susceptible ‘Orin’ compared to resistant ‘Fuji’ and not by the higher potential of leaf photosynthesis.