We study the nonnegative part \overline {G_ {> 0}} of the De Concini-Procesi compactification of a semisimple algebraic group \overline {G_ {> 0}} , as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of \overline {G_ {> 0}} . This answers the question of Lusztig in Total positivity and canonical bases, Algebraic groups and Lie groups (ed. GI Lehrer), Cambridge Univ. Press, 1997, pp. 281-295. We will also prove that \overline {G_ {> 0}} has a cell decomposition which was conjectured by Lusztig.

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Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. The holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower quality. The curvature flow method is nonlinear and requires higher quality triangulations, but more flexible. The conformal geometric methods have been broadly applied in many engineering fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms are robust for surfaces scanned from real life, general for surfaces with different topologies. The efficiency and efficacy of the algorithms

Let X = (x_1,..,x_n) and Y = (y_1,...,y_m) be a pair of corresponding plaintext and ciphertext for a cryptosystem. We define an embedded surface of this cryptosystem as any polynomial equation: E(X,Y)=E(x_1,..,x_n,y_1,...,y_m)=0, which is satisfied by all such pairs. In this paper, we present a new attack on the multivariate public key cryptosystems from Diophantine equations developed by Gao and Heindl by using the embedded surfaces associated to this family of multivariate cryptosystems.