斯特朗猜想：强可坍复形的正面结果及检验该猜想的代码
Strang's Conjecture: Positive Result on Strong Collapsible Complexes and a Code to Check the Conjecture
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中文总结 AI 辅助
本研究针对斯特朗猜想，建立二元C¹样条空间与其他函数空间的关联，得到等价复形并给出NGSolve代码用于检验，还证明强可坍单纯复形对应k=2时猜想成立。
中文摘要 AI 辅助
样条函数是具有给定光滑性的分段多项式函数，广泛应用于计算机辅助几何设计、代数几何及有限元方法实现等众多领域。在诸多应用中，了解区域上样条空间的维数至关重要，但这仍是一个未解决的问题。斯特朗猜想预测了无孔多边形区域上次数不超过k的二元C¹样条空间的维数，其中k=4和k≥5的情况已被证明成立，而k=2的情况，约翰·摩根与L·里奇韦·斯科特给出了反例。本研究首先将次数不超过k的二元C¹样条空间与其他各类函数空间建立关联，借助这些空间上的一些内在算子，得到了一个与斯特朗猜想等价的复形；该等价表述为检验猜想提供了计算途径，因此我们给出了一个NGSolve代码，用于针对给定网格检验斯特朗猜想。此外，我们证明：若与三角剖分关联的单纯复形是强可坍的（满足非共线条件），则斯特朗猜想对k=2成立。
英文摘要
Splines, which are piecewise polynomial functions with given smoothness, are used in numerous applications such as computer aided geometric design, algebraic geometry and for implementation of the finite element method. In many applications, it is important to know the dimension of the space of splines over a domain, but this continues to be an open problem. Strang's conjecture predicts the dimension of the space of bivariate $C^1$ splines of degree at most $k$ over a polygonal domain with no holes. For the cases, $k=4$ and $k \geq 5$ the conjecture has been proven true. For the case $k=2$, John Morgan and L. Ridgway Scott provided a counterexample. In this project, we begin by relating the space of bivariate $C^1$ splines of degree at most $k$ to various other function spaces. With the aid of some intrinsic operators on these spaces, we obtain a complex that yields an equivalence statement to Strang's conjecture. The equivalent statement admits a computational approach to the conjecture and thus we present an NGSolve code to check Strang's conjecture for a given mesh. In addition, we prove that if the simplicial complex associated to a triangulation is strongly collapsible (with a non-collinear condition), then Strang's conjecture holds for $k=2$.
