论文标题
球体上的傅立叶变换的模量决定了3维凸的多面体
The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes
论文作者
论文摘要
令$ \ Mathcal {p} $和$ \ Mathcal {p}'$为$ 3 $ -Dimensional convex polytopes in $ \ Mathbb {r}^3 $和$ s \ subseteq \ subseteq \ subseteq \ mathbb {r}^3 $是与Spheys的开放式的非空交易。由于结果有些更一般的结果，因此证明$ \ Mathcal {p} $和$ \ Mathcal {p}'$在$ | \ int _ {\ int _ {\ Mathcal {p}} e^e^e^{ - i \ Mathbf {s} e^{s} {s} {s} { \，\ Mathbf {dx} | = | \ int _ {\ Mathcal {p}'} e^{ - i \ mathbf {s} \ cdot \ cdot \ mathbf {x}}} \，\ Mathbf {dx} | $ for ALL $ \ MATHBF {S} \ in S $。这可以应用于晶体学领域，即有关以Ewald Spher上其X射线衍射模式的强度而唯一确定了以凸层模型的纳米颗粒。
Let $\mathcal{P}$ and $\mathcal{P}'$ be $3$-dimensional convex polytopes in $\mathbb{R}^3$ and $S \subseteq \mathbb{R}^3$ be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that $\mathcal{P}$ and $\mathcal{P}'$ coincide up to translation and/or reflection in a point if $|\int_{\mathcal{P}} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}| = |\int_{\mathcal{P}'} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}|$ for all $\mathbf{s} \in S$. This can be applied to the field of crystallography regarding the question whether a nanoparticle modelled as a convex polytope is uniquely determined by the intensities of its X-ray diffraction pattern on the Ewald sphere.
