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Using a Euclidean path integral representation for the canonical ensemble partition function we calculate the average value and fluctuations ofthe horizon area for thermal equilibrium black holes. The exciting prospect that black holes can have real thermodynamic properties was first seriously proposed by Bekenstein [l]. His proposals were later confirmed when Hawking 121 demonstrated that quantum effects lead to thermal radiation from black particular, Euclidean path integral representations for the partition functions have been developed. The pioneering contributions to these representations are those of Gibbons and Hawking [3] and York [4].The analysis by York [4] centres on the canonical ensemble. In the form appropriate for systems containing black holes, the canonical ensemble is defined by fixing the area A of some 2-sphere centred on the black hole and the inverse temperature p as measured on the same 2-sphere. From this definition a well defined canonical ensemble partition function Z(p, A) can be built. From Z ( p , A) can be extracted the canonical ensemble thermal equilibrium properties of black holes. The principal results of this paper are two formulae, one is for the average value of the area of the horizon (AH) and the other is for the fluctuations in (A,.,). The averaging is with respect to Z(p, A).Since Schwarzschild black holes have associated with them three types of horizons, me even1 nonzun, Lnc apparcnr nurrzun, an" LW LIIUC-IIXC-~iuiii suiia~ic {~C C L J J IUI

: The present study aimed to investigate the effects of drought stress on the accumulation of glucosinolates (GSLs) in the leaves of Kale cultivated in autumn and spring. HPLC analysis guided to identify seven GSLs including progoitrin, glucoraphanin, sinigrin, gluconapin, glucobrassicin, 4-methoxyglucobrassicin and neoglucobrasscin. Quantification of GSLs revealed that the contents of sigirin was the highest (45%) followed by the level of progoitrin (24%) in terms of total GSLs. The ranges of total GSL contents was 1.16 (84)-15.88 (89 DAS, µmol/g dry wt. (DW)) in treatment plot and 1.23 (84)-7.05 (74 DAS, µmol/g dry wt.) in control plot showed the enhancement in the contents of GSLs in treatment than in the control plot. The present results evidenced that the variation of total GSL contents were depending on the harvest period. In 105 DAS, comparatively no differences in the GSL contents on each sample in autumn season, whereas in spring season, although there was decrease in the GSLs tendency from 74 DAS to 84 DAS in both control and treatment plot, the GSL contents of treatment plot was dramatically increased in 89 DAS. In treatment plot, the GSL contents on 89 DAS (1.16) was 15 fold higher to 84 DAS (15.88 µmol/g DW). The variation in the contents of GSL in spring and autumn did not documented significant differences because of their differences in the growth time and cultivation conditions. In conclusion, the GSL contents in kale was likely to be affected by drought stress treatment. Scrutiny and further research for exact relation between drought stress and GSL contents in kale should be needed.

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