近似与离散欧几里得向量丛
Approximate and discrete Euclidean vector bundles
AI总结：
本文提出欧几里得向量丛的ε-近似概念，证明小ε下可表示经典向量丛，并给出距离度量、重建定理及低维特征类的有效计算算法。
摘要：
我们针对ε≥0引入了欧几里得向量丛概念的ε-近似版本，当ε=0时，它恢复为经典的欧几里得向量丛概念。特别地，我们研究了系数在正交群中的满足近似上循环条件的切赫上链。我们证明了当ε>0足够小时，ε-近似向量丛可以用来表示经典向量丛。我们还引入了近似向量丛之间的距离，并利用这些距离证明了足够相似的近似向量丛表示相同的经典向量丛。这提供了一种使用有限数据在有限单纯复形上指定向量丛的方法，并且在实际应用中使用向量丛时也允许对噪声有一定的容忍度。作为例子，我们证明了从有限样本重建向量丛的重建定理。我们给出了直接从离散和近似表示有效计算向量丛低维特征类的算法，并通过计算示例说明了这些算法的使用。
英文摘要：
We introduce $\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of Euclidean vector bundle when $\varepsilon = 0$. In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon$-approximate vector bundles can be used to represent classical vector bundles when $\varepsilon > 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.
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