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A scrapbook of inadmissible line complexes for the X-ray transform
Advances in Applied Mathematics ( IF 1.1 ) Pub Date : 2020-03-01 , DOI: 10.1016/j.aam.2020.102028
Eric Grinberg , Mehmet Orhon

We consider a finite field model of the X-ray transform that integrates functions along lines in dimension 3, within the context of finite fields. The admissibility problem asks for minimal sets of lines for which the restricted transform is invertible. Graph theoretic conditions are known which characterize admissible collections of lines, and these have been counted using a brute force computer program. Here we perform the count by hand and, at the same time, produce a detailed illustration of the possible structures of inadmissible complexes. The resulting scrapbook may be of interest in an artificial intelligence approach to enumerating and illustrating admissible complexes in arbitrary dimensions (arbitrarily large ambient spaces, with transforms integrating over subspaces of arbitrary dimensions.)

中文翻译:

用于 X 射线变换的不可接受线复合的剪贴簿

我们考虑 X 射线变换的有限场模型,该模型在有限场的背景下沿维度 3 中的线对函数进行积分。可接受性问题要求限制变换可逆的最小线集。图论条件是已知的,其表征可接受的线集合,并且这些已经使用蛮力计算机程序进行计数。在这里,我们手动进行计数,同时详细说明不可接受的复合物的可能结构。由此产生的剪贴簿可能对人工智能方法感兴趣,以枚举和说明任意维度(任意大的环境空间,在任意维度的子空间上集成变换)的可接受复合体。
更新日期:2020-03-01
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