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In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand and later generalized to the mixed p-spin models by Panchenko. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi functional.

In the research of computer vision and machine perception, 3D objects are usually represented by 2-manifold triangular meshes M. In this paper, we present practical and efficient algorithms to construct iso-contours, bisectors, and Voronoi diagrams of point sites onM, based on an exact geodesic metric. Compared to euclidean metric spaces, the Voronoi diagrams onMexhibit many special properties that fail all of the existing euclidean Voronoi algorithms. To provide practical algorithms for constructing geodesicmetric- based Voronoi diagrams onM, this paper studies the analytic structure of iso-contours, bisectors, and Voronoi diagrams onM. After a necessary preprocessing of model M, practical algorithms are proposed for quickly obtaining full information about iso-contours, bisectors, and Voronoi diagrams on M. The complexity of the construction algorithms is also analyzed. Finally, three interesting applications—surface sampling and reconstruction, 3D skeleton extraction, and point pattern analysis—are presented that show the potential power of the proposed algorithms in pattern analysis.

Based on the double shockwave approximation procedure, a local Riemann solver for strongly nonlinear equations of state (EOS) such as the Jones-Wilkins-Lee (JWL) EOS is presented, which has suppressed successfully numerical oscillation caused by high-density ratio and high-pressure ratio across the interface between explosion products and air. The real ghost fluid method (RGFM) and the level set method have been used for converting multi-medium flows into pure flows and for implicitly tracking the interface, respectively. A fifth order finite difference weighted essentially non-oscillatory (WENO) scheme and a third order TVD Runge-Kutta method are utilized for the spatial discretization and the time advance, respectively. An enclosed-type MPI-based parallel methodology for the RGFM procedure on a uniform structured mesh is presented to realize the parallelization of three-dimensional {\color{red}(3D)} air explosion. The overall process of 3D air explosion of both TNT and aluminized explosives has been successfully simulated.

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