• East Asian mathematical journal
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• v.19 no.1
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• pp.27-39
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• 2003

Integral inequalities of Gruss type obtained via $P\acute{o}lya-Szeg\ddot{o}$ and Shisha-Mond results are given. Some applications for Taylor's generalized expansion are also provided.

• Journal of the Chungcheong Mathematical Society
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• v.37 no.1
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• pp.27-40
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• 2024

In this paper, we consider the higher-order HartreeFock equations. The higher-order linear Schrödinger equation was introduced in [5] as the formal finite Taylor expansion of the pseudorelativistic linear Schrödinger equation. In [13], the authors established global-in-time Strichartz estimates for the linear higher-order equations which hold uniformly in the speed of light c ≥ 1 and as their applications they proved the convergence of higher-order Hartree-Fock equations to the corresponding pseudo-relativistic equation on arbitrary time interval as c goes to infinity when the Taylor expansion order is odd. To achieve this, they not only showed the existence of solutions in L² space but also proved that the solutions stay bounded uniformly in c. We address the remaining question on the convergence of higherorder Hartree-Fock equations when the Taylor expansion order is even. The distinguished feature from the odd case is that the group velocity of phase function would be vanishing when the size of frequency is comparable to c. Owing to this property, the kinetic energy of solutions is not coercive and only weaker Strichartz estimates compared to the odd case were obtained in [13]. Thus, we only manage to establish the existence of local solutions in H^s space for s > $\frac{1}{3}$ on a finite time interval [-T, T], however, the time interval does not depend on c and the solutions are bounded uniformly in c. In addition, we provide the convergence result of higher-order Hartree-Fock equations to the pseudo-relativistic equation with the same convergence rate as the odd case, which holds on [-T, T].

• Journal of Korea Water Resources Association
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• v.51 no.spc
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• pp.1079-1089
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• 2018

The majority of existing studies for quantifying uncertainties in climate change impact assessments suggest only the uncertainties of each stage, and not the total uncertainty and its propagation in the whole procedure. Therefore, this study has proposed a new method, the Uncertainty Delta Method (UDM), which can quantify uncertainties using the variances of projections (as the UDM is derived from the first-order Taylor series expansion), to allow for a comprehensive quantification of uncertainty at each stage and also to provide the levels of uncertainty propagation, as follows: total uncertainty, the level of uncertainty increase at each stage, and the percentage of uncertainty at each stage. For quantifying uncertainties at each stage as well as the total uncertainty, all the stages - two emission scenarios (ES), three Global Climate Models (GCMs), two downscaling techniques, and two hydrological models - of the climate change assessment for water resources are conducted. The total uncertainty took 5.45, and the ESs had the largest uncertainty (4.45). Additionally, uncertainties are propagated stage by stage because of their gradual increase: 5.45 in total uncertainty consisted of 4.45 in emission scenarios, 0.45 in climate models, 0.27 in downscaling techniques, and 0.28 in hydrological models. These results indicate the projection of future water resources can be very different depending on which emission scenarios are selected. Moreover, using Fractional Uncertainty Method (FUM) by Hawkins and Sutton (2009), the major uncertainty contributor (emission scenario: FUM uncertainty 0.52) matched with the results of UDM. Therefore, the UDM proposed by this study can support comprehension and appropriate analysis of the uncertainty surrounding the climate change impact assessment, and make possible a better understanding of the water resources projection for future climate change.