Affine and projective universal geometry

仿射和射影语言中凸体的初等等价性

Elementary equivalence of convex bodies in affine and projective languages

研究紧致凸体在仿射和射影语言下的初等等价性，通过可定义紧致度量族等方法证明任意维度凸体在\(L_{\mathrm{aff}}\)中初等等价当且仅当仿射等价，还发展了\(L_B\)演算及得出相关结果并分类非紧致例子。

On affine motions and universal rigidity of tensegrity frameworks

Recently, Alfakih and Ye [Lin. Algebra Appl. 438:31--36, 2013] proved that if an $r$-dimensional bar framework $(G,p)$ on $n \geq r+2$ nodes in general position in $\R^r$ admits a positive semidefinite stress matrix with rank $n-r-1$, then $(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration $p$, each point and its neighbors in $G$ affinely span $\R^r$.