7 problems

Arbitrary-dimensional finite element construction conjecture. The $C^r$ finite elements can be constructed with this local shape function space and these degrees of freedom.

Weak geometric T-junction conjecture. Then

Weak geometric analysis-suitability linear-independence conjecture. The generated splines are linearly independent.

Weak geometric analysis-suitability conjecture. Every ${\mathrm{W\mspace{-3mu}G\mspace{-2mu}A\mspace{-2mu}S}}$ mesh is ${\mathrm{W\mspace{-3mu}D\mspace{-2mu}C}}$.

Let $S^r_d(\Delta)$ denote the space of bivariate splines of smoothness $r$ and degree at most $d$ on a triangulation of a planar region $\Delta$. A lower-bound polynomial in $r$ a…

Let $\{\mathcal{M}^{\ell}\}_{\ell=0}^{\ell_{\max}}$ be the sequence of consecutively globally refined unstructured quad meshes described above, and let $\mathcal{B}^{\ell}$ be the…

Let $\Delta$ be a connected, pure $2$-dimensional simplicial complex embedded in $\mathbb{R}^2$, let $S_\bullet/J_\bullet$ be its spline complex, and let $r$ denote the smoothness…