This work proposes a novel metric based algorithm for quadrilateral mesh generating. Each quad-mesh induces a Riemannian metric satisfying special conditions: the metric is a flat metric with cone singularities conformal to the original metric, the total curvature satisfies the Gauss–Bonnet condition, the holonomy group is a subgroup of the rotation group $\{e^{ik\pi/2}\}$, there is cross field obtained by parallel translation which is aligned with the boundaries, and its streamlines are finite geodesics. Inversely, such kind of metric induces a quad-mesh. Based on discrete Ricci flow and conformal structure deformation, one can obtain a metric satisfying all the conditions and obtain the desired quad-mesh. This method is rigorous, simple and automatic. Our experimental results demonstrate the efficiency and efficacy of the algorithm.

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.

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