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SMALL DATA SCATTERING OF HARTREE TYPE FRACTIONAL SCHRÖDINGER EQUATIONS IN DIMENSION 2 AND 3

  • Cho, Yonggeun;Ozawa, Tohru
    • Journal of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.373-390
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    • 2018

  • In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.

  • https://doi.org/10.4134/JKMS.j170224 인용 PDF KSCI

ASUSD nuclear data sensitivity and uncertainty program package: Validation on fusion and fission benchmark experiments

  • Kos, Bor;Cufar, Aljaz;Kodeli, Ivan A.
    • Nuclear Engineering and Technology
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    • v.53 no.7
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    • pp.2151-2161
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    • 2021

  • Nuclear data (ND) sensitivity and uncertainty (S/U) quantification in shielding applications is performed using deterministic and probabilistic approaches. In this paper the validation of the newly developed deterministic program package ASUSD (ADVANTG + SUSD3D) is presented. ASUSD was developed with the aim of automating the process of ND S/U while retaining the computational efficiency of the deterministic approach to ND S/U analysis. The paper includes a detailed description of each of the programs contained within ASUSD, the computational workflow and validation results. ASUSD was validated on two shielding benchmark experiments from the Shielding Integral Benchmark Archive and Database (SINBAD) - the fission relevant ASPIS Iron 88 experiment and the fusion relevant Frascati Neutron Generator (FNG) Helium Cooled Pebble Bed (HCPB) Test Blanket Module (TBM) mock-up experiment. The validation process was performed in two stages. Firstly, the Denovo discrete ordinates transport solver was validated as a standalone solver. Secondly, the ASUSD program package as a whole was validated as a ND S/U analysis tool. Both stages of the validation process yielded excellent results, with a maximum difference of 17% in final uncertainties due to ND between ASUSD and the stochastic ND S/U approach. Based on these results, ASUSD has proven to be a user friendly and computationally efficient tool for deterministic ND S/U analysis of shielding geometries.

  • https://doi.org/10.1016/j.net.2021.01.034 인용 PDF KSCI