• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1451-1469
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• 2013

We present various new sharp assertions on multipliers in mixed norm, weighted Hardy and new Lizorkin-Triebel spaces of harmonic functions in higher dimension. Some results are new even in onedimensional case.

• Nuclear Engineering and Technology
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• v.52 no.6
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• pp.1137-1147
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• 2020

The discrete ordinates method (S_N) is one of the major shielding calculation method, which is suitable for solving deep-penetration transport problems. Our objective is to explore the available quadrature sets and to improve the accuracy in shielding problems involving strong anisotropy. The linear discontinuous finite-element (LDFE) quadrature sets based on the icosahedron (in short, ICLDFE quadrature sets) are developed by defining projected points on the surfaces of the icosahedron. Weights are then introduced in the integration of the discontinuous finite-element basis functions in the relevant angular regions. The multivariate secant method is used to optimize the discrete directions and their corresponding weights. The numerical integration of polynomials in the direction cosines and the Kobayashi benchmark are used to analyze and verify the properties of these new quadrature sets. Results show that the ICLDFE quadrature sets can exactly integrate the zero-order and first-order of the spherical harmonic functions over one-twentieth of the spherical surface. As for the Kobayashi benchmark problem, the maximum relative error between the fifth-order ICLDFE quadrature sets and references is only -0.55%. The ICLDFE quadrature sets provide better integration precision of the spherical harmonic functions in local discrete angle domains and higher accuracy for simple shielding problems.

• Journal of Electrical Engineering and Technology
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• v.13 no.6
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• pp.2158-2167
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• 2018

This paper analyzed the magnetic field produced by the cylindrical stator coils of permanent magnet spherical motor (PMSM). The elliptic equations about the vector magnetic potential were given. Given that the eddy current effects are neglected, the magnet field of the PMSM is regarded as irrotational field, which can be calculated by scalar magnetic potential. The current density of cylindrical stator coil was proposed based on the definition of current density. The expression of current density of stator coil was obtained according to the double Fourier series decomposition and spherical harmonic functions. Then the magnetic flux density for scalar magnetic potential was derived. Further, the influence of different parameters on radial flux density was also analyzed. Finally, the results by the analytical method in this paper were validated by finite element analysis (FEA).

• Economic and Environmental Geology
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• v.49 no.4
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• pp.269-280
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• 2016

I presented three vector crustal magnetic anomaly components and six gradients by using spherical Slepian functions over the cap area of $20^{\circ}$ of radius centered on the South Pole. The Swarm mission, launched by European Space Agency(ESA) in November of 2013, was planned to put three satellites into the low-Earth orbits, two in parallel in East-West direction and one in cross-over of the higher altitude. This orbit configuration will make the gradient measurements possible in North-South direction, vertical direction, as well as E-W direction. The gravity satellites, such as GRACE and GOCE, have already implemented their gradient measurements for recovering the accurate gravity of the Earth and its temporal variation due to mass changes on the subsurface. However, the magnetic gradients have little been applied since Swarm launched. A localized magnetic modeling method is useful in taking an account for a region where data availability was limited or of interest was special. In particular, computation to get the localized solutions is much more efficient and it has an advantage of presenting high frequency anomaly features with numbers of solutions fewer than the global ones. Besides, these localized basis functions that were done by a linear transformation of the spherical harmonic functions, are orthogonal so that they can be used for power spectrum analysis by transforming the global spherical harmonic coefficients. I anticipate in scientific and technical progress in the localized modeling with the gradient measurements from Swarm and here will do discussion on the results of the localized solution to represent the three vector and six gradient anomalies over the Antarctic area from the synthetic data derived from a global solution of the spherical harmonics for the crustal magnetic anomalies of Swarm measurements.

• Journal of the Korean earth science society
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• v.38 no.3
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• pp.173-181
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• 2017

The stability of the steady Rossby-Haurwitz wave (R-H wave) in the nondivergent barotropic model (NBM) on the sphere was investigated with the normal mode method. The linearized NBM equation with respect to the R-H wave was formulated into the eigenvalue-eigenvector problem consisting of the huge sparse matrix by expanding the variables with the spherical harmonic functions. It was shown that the definite threshold R-H wave amplitude for instability could be obtained by the normal mode method. It was revealed that some unstable modes were stationary, which tend to amplify without the time change of the spatial structure. The maximum growth rate of the most unstable mode turned out to be in almost linear proportion to the R-H wave amplitude. As a whole, the growth rate of the unstable mode was found to increase with the zonal- and total-wavenumber. The most unstable mode turned out to consist of more-than-one zonal wavenumber, and in some cases, the mode exhibited a discontinuity over the local domain of weak or vanishing flow. The normal mode method developed here could be readily extended to the basic state comprised of multiple zonalwavenumber components as far as the same total wavenumber is given.

• Journal of the Korean earth science society
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• v.35 no.5
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• pp.333-341
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• 2014

A new set of basis functions was constructed using the Rossby-Haurwitz waves, which are the eigenfunctions of nondivergent barotropic vorticity equations on the sphere. The basis functions were designed to be non-separable, that is, not factored into functions of either the longitude or the latitude. Due to this property, the nodal lines of the functions are aligned neither along with the meridian nor the parallel. The basis functions can be categorized into groups of which members have the same degree or the total wavenumber-like index on the sphere. The orthonormality of the basis functions were found to be close to the machine roundoffs, giving the error of $O(10^{-15})$ or $O(10^{-16})$ for double-precision computation (64 bit arithmetic). It was demonstrated through time-stepping procedure that the basis functions were also the eigenfunctions of the non-divergent barotropic vorticity equations. The projection of the basis functions was carried out onto the low-resolution geopotential field of Gaussian bell, and compared with the theory. The same projections were performed for the observed atmospheric-geopotential height field of 500 hPa surface to demonstrate decomposition into the fields that contain disturbance of certain range of horizontal scales. The usefulness of the new basis functions was thus addressed for application to the eigenmode analysis of the atmospheric motions on the global domain.

• Journal of the Korean earth science society
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• v.39 no.5
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• pp.473-482
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• 2018

During the period of 2002 to 2017, the Gravity Recovery And Climate Experiment (GRACE) had observed time-varying gravity changes with unprecedented accuracy. The GRACE science data centers provide the monthly gravity solutions after removing the sub-monthly mass fluctuation using geophysical models. However, model misfit makes the solutions to be contaminated by aliasing errors, which exhibits peculiar north-south stripes. Two conventional filters are used to reduce the errors, but signals with similar spatial patterns to the errors are also removed during the filtering procedure. This would be particularly problematic for estimating the ice mass changes in Western Antarctic Ice Sheet (WAIS) and Antarctic Peninsula (AP) due to their similar spatial pattern to the elongated north-south direction. In this study, we introduce an alternative filter to remove aliasing errors using the Empirical Orthogonal Functions (EOF) analysis. EOF can decompose data into different modes, and thus is useful to separate signals from noise. Therefore, the aliasing errors are effectively suppressed through EOF method. In particular, the month-to-month mass changes in WAIS and AP, which have been significantly contaminated by aliasing errors, can be recovered using EOF method.

• Journal of Astronomy and Space Sciences
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• v.16 no.2
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• pp.159-166
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• 1999

As the spatial resolution of remote sensing satellites becomes higher, very accurate determination of the position of a LEO (Low Earth Orbit) satellite is demanding more than ever. Non-symmetric Earth gravity is the major perturbation force to LEO satellites. Since the orbit propagation is performed in the celestial frame while Earth gravity is defined in the terrestrial frame, it is required to convert the coordinates of the satellite from one to the other accurately. Unless the coordinate conversion between the two frames is performed accurately the orbit propagation calculates incorrect Earth gravitational force at a specific time instant, and hence, causes errors in orbit prediction. The coordinate conversion between the two frames involves precession, nutation, Earth rotation and polar motion. Among these factors, unpredictability and uncertainty of Earth rotation, called UTI-UTC, is the largest error source. In this paper, the effect of UTI-UTC on the accuracy of the LEO propagation is introduced, tested and analzed. Considering the maximum unpredictability of UTI-UTC, 0.9 seconds, the meaningful order of non-spherical Earth harmonic functions is derived.

• Investigative Magnetic Resonance Imaging
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• v.11 no.2
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• pp.95-102
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• 2007

Purpose: To acquire high-resolution spiral-scan images at higher magnetic field, high homogeneous magnetic field is needed. Field inhomogeneity mapping and in-vivo shimming are important for rapid imaging such as spiral-scan imaging. The rapid scanning sequences are very susceptible to inhomogeneity. In this paper, we proposed a higher-order shimming method to obtain homogeneous magnetic field. Materials and Methods: To reduce measurement time for field inhomogeneity mapping, simultaneous axial/ sagittal, and coronal acquisitions are done using multi-slice based Fast Spin echo sequence. Acquired field inhomogeneity map is analyzed using the spherical harmonic functions, and shim currents are obtained by the multiplication of the pseudo-inverse of the field pattern with the inhomogeneity map. Results: Since the field inhomogeneity is increasing in proportion to the magnetic field, higher order shimming to reduce the inhomogeneity becomes more important in high field imaging. The shimming technique in which axial, sagittal, and coronal section inhomogeneity maps are obtained in one scan is developed, and the shimming method based on the analysis of spherical harmonics of the imhomogenity map is applied. The proposed technique is applicable to a localized shimming as well. High resolution spiral-scan imaging was successfully obtained with the proposed higher order shimming. Conclusion: Proposed pulse sequence for rapid measurement of inhomogeneity map and higher order shimming based on the inhomogeneity map work very well at 3 Tesla MRI system. With the proposed higher order shimming and localized higher order shimming techniques, high resolution spiral-scan images are successfully obtained at 3 T MRI system.

• Journal of Astronomy and Space Sciences
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• v.32 no.3
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• pp.247-256
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• 2015

In this work, an efficient method with which to evaluate the high-degree-and-order gravitational harmonics of the non-sphericity of a central body is described and applied to state predictions of a lunar orbiter. Unlike the work of Song et al. (2010), which used a conventional computation method to process gravitational harmonic coefficients, the current work adapted a well-known recursion formula that directly uses fully normalized associated Legendre functions to compute the acceleration due to the non-sphericity of the moon. With the formulated algorithms, the states of a lunar orbiting satellite are predicted and its performance is validated in comparisons with solutions obtained from STK/Astrogator. The predicted differences in the orbital states between STK/Astrogator and the current work all remain at a position of less than 1 m with velocity accuracy levels of less than 1 mm/s, even with different orbital inclinations. The effectiveness of the current algorithm, in terms of both the computation time and the degree of accuracy degradation, is also shown in comparisons with results obtained from earlier work. It is expected that the proposed algorithm can be used as a foundation for the development of an operational flight dynamics subsystem for future lunar exploration missions by Korea. It can also be used to analyze missions which require very close operations to the moon.