Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \omega with cusp singularities along a divisor D, we show the L^ 2 projective embedding of (X, D) defined by L^ k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Khler metrics with singularities.

We propose a finite-dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurstons gluing equation and Hakens normal surface equation. The action functional is the volume.

In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of \mathbb {R} P^\infty with the aid of a computer, which gives us its Adams \mathbb {R} P^\infty -page in the range of \mathbb {R} P^\infty . We also compute the transfer map on the Adams \mathbb {R} P^\infty -pages. These data are used in our computations of the stable homotopy groups of \mathbb {R} P^\infty in [6] and of the stable homotopy groups of spheres in [7].

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The aim of this study is to figure out the optimization of survival probability of the generals in the card game "Legends of the Three Kingdoms". First, based on combinatorics and probability theory, the PASCAL program was performed to analyze the probabilities of success when several popular generals perform their special skills in the game. The results are as follows: Second, seven typical generals, including Zhou Tai, Zhang Jiao, Zhen Ji, Lu Xun, Ma Chao, Xiahou Dun, and Guo Jia, were chosen, and their abilities including attacking and defense were arbitrarily quantified according to a model constructed in the present study. The scores are as follows: Card-drawing Changing Attacking Defense Blood-selling Zhen Ji 5.62 Zhou Tai 1.04 Ma Chao 1.68 Xiahou Dun 1.19 Guo Jia 3.36 Lu Xun 3.55 Zhang Jiao 2.55 The abilities of different generals were sorted, and by comparison with the statistical record online, the reliability of the model was proved. Thus, the irrationality of the game is identified in this study, and we recommend the game designer to modify the abilities of some generals and to obtain an optimized value according to the output of the model, by which the game would be more popular.