11 problems

Let $\mathscr{T}$ be a hierarchical T-mesh, and let $\mathscr{C}$ be its CVR (cross-vertex relationship) graph. For a spline space with the highest order of smoothness, write…

Schenck–Stiller's "2r+1" conjecture. For degree at least $2r+1$ and smoothness order $r$, the dimension of the spline space equals Schumaker's lower bound.

Let $\Delta$ be a triangulation of a simply connected polygonal domain in $\mathbb{R}^2$, and let $C^r_k(\Delta)$ be the vector space of splines of smoothness $r$ and degree at mos…

Let $\mathcal{T}$ range over triangulations of bounded polygonal domains in $\mathbb{R}^2$, let $\mathcal{T}$ have at most $N$ triangles, and let $P_{V(\mathcal{T})}$ denote the…

Dimensional absolute stability conjecture. $\mathscr{T}$ is dimensionally absolutely stable if and only if it is diagonalizable.

Let the Laplacian have periodic boundary conditions, and let the Galerkin method use the periodic $n$-dimensional spline space of degree $p$ with maximal smoothness $C^{p-1}$ on a…

Let $\overline{\Omega}\in\mathcal{A}$ be a randomly generated trilinear multi-patch volume with an inner edge of valency $\nu\in\{3,4,5\}$. Let $p$ be the spline degree, $r$ the re…

Interpolatory basis conjecture. There exist basis functions with the interpolatory property for any space of the form

Let $\Delta$ be a triangulation of a simply connected polygonal domain in $\mathbb{R}^2$, and let $C^1_3(\Delta)$ be the vector space of splines of smoothness $1$ and degree at mos…

Let $Q_m$ denote the spline generator and let $V(Q_m)$ be the shift-invariant spline space it generates. A sequence $\\{x_i:i\in\mathbb{Z}\\}$ is separated if its distinct points h…

The regularity conjecture. If $\mathcal{P}$ is central and complete, then