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A natural metric in 2-manifold surfaces is to use geodesic distance. If a 2-manifold surface is represented by a triangle mesh T , the geodesic metric on T can be computed exactly using computational geometry methods. Previous work for establishing the geodesic metric on T only supports using half-edge data structures; i.e., each edge e in T is split into two halves (he1, he2) and each half-edge corresponds to one of two faces incident to e. In this paper, we prove that the exact-geodesic structures on two halfedges of e can be merged into one structure associated with e. Four merits are achieved based on the properties which are studied in this paper: (1) Existing CAD systems that use edge-based data structures can directly add the geodesic distance function without changing the kernel to a half-edge data structure; (2) To find the geodesic path from inquiry points to the source, the MMP algorithm can be run in an onthe- fly fashion such that the inquiry points are covered by correct wedges; (3) The MMP algorithm is sped up by pruning unnecessary wedges during the wedge propagation process; (4) The storage of the MMP algorithm is reduced since fewer wedges need to be stored in an edge-based data structure. Experimental results show that when compared to the classic half-edge data structure, the edge-based implementation of the MMP algorithm reduces running time by 44% and storage by 29% on average.

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An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.

Mean-Field Games are games with a continuum of players that incorporate the time-dimension through a control-theoretic approach. Recently, simpler approaches relying on the Best Reply Strategy have been proposed. They assume that the agents navigate their strategies towards their goal by taking the direction of steepest descent of their cost function (i.e. the opposite of the utility function). In this paper, we explore the link between Mean-Field Games and the Best Reply Strategy approach. This is done by introducing a Model Predictive Control framework, which consists of setting the Mean-Field Game over a short time interval which recedes as time moves on. We show that the Model Predictive Control offers a compromise between a possibly unrealistic Mean-Field Game approach and the sub-optimal Best Reply Strategy.