• Bulletin of the Korean Chemical Society
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• v.35 no.3
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• pp.805-810
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• 2014

The shape invariant potentials are proved to be exactly solvable, i.e. the wave functions and energies of a particle moving under the influence of the shape invariant potentials can be algebraically determined without any approximations. It is well known that the SWKB quantization is exact for all shape invariant potentials though the SWKB quantization itself is approximate. This mystery has not been mathematically resolved yet and may not be solved in a concrete fashion even in the future. Therefore, in the present work, to understand (not prove) the mystery an attempt of deriving exactly solvable potentials directly from the SWKB quantization has been made. And it turns out that all the derived potentials are shape invariant. It implicitly explains why the SWKB quantization is exact for all known shape invariant potentials. Though any new potential has not been found in this study, this brute-force derivation of potentials helps one understand the characteristics of shape invariant potentials.

• East Asian mathematical journal
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• v.26 no.1
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• pp.39-47
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• 2010

Existence of common fixed points for generalized S-nonexpansive subcompatible mappings in convex metric spaces have been obtained. Invariant approximation results have also been derived by its application. These results extend and generalize various known results in the literature with the aid of more general class of noncommuting mappings.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.607-614
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• 2008

We present coincidence points and common fixed point results for (f, g)-contractive mapping and (f, g)-nonexpansive mappings. Our results generalize and complement various known results existing in the literature.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.551-564
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• 2013

In 2008, Al-Thaga and Shahzad [Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sinica, 24(5) (2008), 867-876] introduced the notion of occasionally weakly compatible mappings in metric spaces. In this paper, we prove some common fixed point theorems for families of occasionally weakly compatible mappings in Menger spaces using an implicit relation. We also give an illustrative example to support our main result.

• Bulletin of the Korean Space Science Society
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• 2003.10a
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• pp.40-40
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• 2003

When the magnetotail is disturbed by an impulsive input such as the substorm onset, compressional magnetohydrodynamic (MHD) waves play an important role in delivering perturbed energy and exciting various wave modes and currents. The plasmasheet, in which relatively hot plasmas exist, is surrounded by relatively cold plasmas at the plasma sheet boundary layer (PSBL) and the equatorial plasmasphere. Since the Alfven speed significantly varies near these regions, the compressional waves are expected to undergo mode conversion by inhomogeneity at the boundary between cold and hot plasma regions. We investigate how the initial compressional MHD wave energy is reflected, transmitted, and absorbed across that boundary by adopting the invariant imbedding method (IIM) which gives the exact reflection, transmission, and absorption coefficients without any theoretical approximations for given frequencies and wave numbers. The IIM method is very useful in quantifying the reflection and transmission of compressional waves in the sense that we can calculate how much fast mode wave energy is delievered into shear Alfven waves or field-aligned currents. Our results show that strongly localized absorption occurs at the boundary region. This feature suggests that localized field-aligned currents can be impulsively excited at such boundary regions by any compressional disturbances, which is highly associated with impulsive auroral brightening at the substorm onset. We compare our results with previous studies in cold inhomogeneous plasmas.