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- W3102580359 abstract "Abstract Let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒫</m:mi> </m:math> {mathcal{P}} be an n -dimensional convex polytope and let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒮</m:mi> </m:math> {mathcal{S}} be a hypersurface in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {mathbb{R}^{n}} . This paper investigates potentials to reconstruct <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒫</m:mi> </m:math> {mathcal{P}} , or at least to compute significant properties of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒫</m:mi> </m:math> {mathcal{P}} , if the modulus of the Fourier transform of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒫</m:mi> </m:math> {mathcal{P}} on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒮</m:mi> </m:math> {mathcal{S}} with wave length λ, i.e., <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo fence=true maxsize=210% minsize=210%>|</m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mi mathvariant=script>𝒫</m:mi> </m:msub> <m:mrow> <m:mpadded width=+1.7pt> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>λ</m:mi> </m:mfrac> <m:mo></m:mo> <m:mi>𝐬</m:mi> </m:mrow> <m:mo>⋅</m:mo> <m:mi>𝐱</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mpadded> <m:mo></m:mo> <m:mi>𝐝𝐱</m:mi> </m:mrow> </m:mrow> <m:mo fence=true maxsize=210% minsize=210%>|</m:mo> </m:mrow> <m:mo mathvariant=italic separator=true> </m:mo> <m:mrow> <m:mtext>for </m:mtext> <m:mo></m:mo> <m:mi>𝐬</m:mi> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant=script>𝒮</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> biggl{lvert}int_{mathcal{P}}e^{-ifrac{1}{lambda}mathbf{s}cdotmathbf{x% }},mathbf{dx}biggr{rvert}quadtext{for }mathbf{s}inmathcal{S}, is given, λ is sufficiently small and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒫</m:mi> </m:math> {mathcal{P}} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒮</m:mi> </m:math> {mathcal{S}} have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>𝒫</m:mi> </m:math> {mathcal{P}} with wave length λ when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>λ</m:mi> <m:mo>→</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {lambdarightarrow 0} . The theory of X-ray scattering of nanoparticles motivates this study, since the modulus of the Fourier transform of the reflected beam wave vectors is approximately measurable on a half sphere in experiments." @default.
- W3102580359 created "2020-11-23" @default.
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- W3102580359 date "2022-08-30" @default.
- W3102580359 modified "2023-09-27" @default.
- W3102580359 title "Reconstruction of polytopes from the modulus of the Fourier transform with small wave length" @default.
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- W3102580359 doi "https://doi.org/10.1515/jiip-2020-0144" @default.
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