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We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

No abstract uploaded!

No abstract uploaded!

Information Technology has enabled new levels of convenience to our daily life. While providing people with astonishingly good experiences, computer science has still a long way to meet the needs from the users for various scenarios. As a good example, path selection problem is an important and common problem which finds applications in location based services, internet routing and navigation systems. Choosing the most cost effective and time saving routines plays an essential role in both theory and engineering. Although a plethora of classical solutions have been made to the static shortest path problem, dynamic shortest path problem has not been thoroughly studied. To address this critical issue, this paper investigates the dynamic shortest path problem by optimizing existing work and conducting extensive experiments. The studies fall into three main variants of the problem: unweighted shortest paths, weighted shortest paths, and shortest paths on the map. We present a comprehensive comparison of our optimizations and existing solutions which demonstrates the effectiveness and efficiency of our algorithms.