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.

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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.

Motivated by an extra credit problem from our Linear Algebra class, we study the invertibility probability of binary matrices (the number of invertible binary matrices divided by the total number of binary matrices). Binary matrices are of interest in combinatorics, information theory, cryptology, and graph theory. It is known that the invertibility probability of n × n binary matrices goes to 1 as n→∞. We conjecture that this probability monotonically increases as the size of the binary matrix increases, and we investigate this by exploring how n×n binary matrices of rank n and rank (n−1) can be enlarged to (n + 1) × (n + 1) invertible binary matrices. Calculating this explicitly for the identity matrix, we obtain a probable bound that would show that, in a sense, our conjecture is asymptotically true. With the use of a computer, we also computed how many (n + 1) × (n + 1) invertible binary matrices can be enlarged from n×n matrices of rank n and rank (n−1) for small n. In addition, we study the invertibility probability of matrices with entries in Z_q.

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