We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.
With the development of cloud computing, service computing, IoT(Internet of Things) and mobile Internet, the diversity and sociality of services are increasingly apparent. To meet the customized user demands, service ecosystems begins to emerge with the formation of various IT services collaboration network. However, service ecosystem is a complex social-technology system with the characteristics of natural ecosystems, economic systems and complex networks. Hence, how to realize the multi-dimensional evaluation of service ecosystem is of great significance to promote its sound development. Based on this, this paper proposes a value entropy model to analyze the performance of service ecosystem, which is conducive to integrate evaluation indicators of different dimensions. In addition, a computational experiment system is constructed to verify the effectiveness of value entropy model. The result shows that our model can provide new means and ideas for the analysis of service ecosystem.
