For the purpose of isogeometric analysis, one of the most common ways is to construct structured hexahedral meshes, which have regular tensor product structure, and fit them by volumetric T-Splines. This theoretic work proposes a novel surface quadrilateral meshing method, colorable quad-mesh, which leads to the structured hexahedral mesh of the enclosed volume for high genus surfaces. The work proves the equivalence relations among colorable quad-meshes, finite measured foliations and Strebel differentials on surfaces. This trinity theorem lays down the theoretic foundation for quadrilateral/hexahedral mesh generation, and leads to practical, automatic algorithms. The work proposes the following algorithm: the user inputs a set of disjoint, simple loops on a high genus surface, and specifies a height parameter for each loop; a unique Strebel differential is computed with the combinatorial type and the heights prescribed by the user’s input; the Strebel differential assigns a flat metric on the surface and decomposes the surface into cylinders; a colorable quad-mesh is generated by splitting each cylinder into two quadrilaterals, followed by subdivision; the surface cylindrical decomposition is extended inward to produce a solid cylindrical decomposition of the volume; the hexadhedral meshing is generated for each volumetric cylinder and then glued together to form a globally consistent hex-mesh. The method is rigorous, geometric, automatic and conformal to the geometry. This work focuses on the theoretic aspects of the framework, the algorithmic details and practical evaluations will be given in the future expositions.

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal Q = TX(TX)^{*}$, where $X$ is an $M_2\times N$ random matrix with $X_{ij}=N^{-1/2}q_{ij}$ such that $q_{ij}$ are $i.i.d.$ random variables with zero mean and unit variance, and $T$ is an $M_1 \times M_2$ deterministic matrix such that $T^* T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal Q$ when $M:=\min\{M_1,M_2\}$ and $N$ tend to infinity with $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$. We prove that the Tracy-Widom law holds for the largest eigenvalue of $\mathcal Q$ if and only if $\lim_{s \rightarrow \infty}s^4 \mathbb{P}(\vert q_{ij} \vert \geq s)=0$ under mild assumptions of $T$. The necessity and sufficiency of this condition for the edge universality was first proved for Wigner matrices by Lee and Yin [34].

It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. In this work, we present stronger and more general versions of such a trace theorem in a new nonlocal function space $\cS(\Omega)$ satisfying $H^1(\Omega)\subset \cS(\Omega)\subset L^2(\Omega)$. The new space $\cS(\Omega)$ is associated with a nonlocal norm characterized by a nonlocal interaction kernel that is defined heterogeneously with a special localization feature on the boundary. Through the heterogeneous localization, we are able to show that the $H^{1/2}$ norm of the trace on the boundary can be controlled by the nonlocal norm that are weaker than the classical $H^1$ norm. In fact, the trace theorems can be essentially shown without imposing any extra regularity of the function in the interior of the domain other than being square integrable. Implications of the new trace theorems to the coupling of local and nonlocal equations and possible further generalizations are also discussed.

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