• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.257-268
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• 2008

Let G and K be compact subgroups of orthogonal groups and $0{\leq}r<x<{\infty}$. We prove that every topological fiber bundle over a definable $C^r$ manifold whose structure group is K admits a unique strongly definable $C^r$ fiber bundle structure up to definable $C^r$ fiber bundle isomorphism. We prove that every G vector bundle over an affine definable $C^rG$ manifold admits a unique strongly definable $C^rG$ vector bundle structure up to definable $C^rG$ vector bundle isomorphism.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1345-1356
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• 2017

For a given number field K, we show that the ranks of elliptic curves over K are uniformly finitely bounded if and only if the weak Mordell-Weil property holds in all (some) ultrapowers $^*K$ of K. We introduce the nonstandard weak Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*{\mathbb{Z}}$-module, where $^*{\mathbb{Z}}$ is an ultrapower of ${\mathbb{Z}}$, and we show that the nonstandard weak Mordell-Weil property is equivalent to the weak Mordell-Weil property in $^*K$. In a saturated nonstandard number field, there is a nonstandard ring of integers $^*{\mathbb{Z}}$, which is definable. We can consider definable abelian groups as $^*{\mathbb{Z}}$-modules so that the nonstandard weak Mordell-Weil property is well-defined, and we conclude that the nonstandard weak Mordell-Weil property and the weak Mordell-Weil property are equivalent. We have valuations induced from prime numbers in nonstandard rational number fields, and using these valuations, we identify two nonstandard rational numbers.

• Applied Microscopy
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• v.26 no.4
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• pp.389-399
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• 1996

In order to understand the morphological differences between two different organic loadings by its upstream, and to compare with other algal groups with references, the fine structure of blue-green algae, Microcystis aeruginosa Kitzing, taken from two branches, Tongbok and Bosung stream of Lake Chuam, Korea pennisula was examined. It showed extinct differences in most physicochemical factors between both branches, except water temperature and pH values. The concentrations of total phosphorus in Tongbok branch were twice as those of Bosung. M. aeruginosa cells were enumerated totally $1.2X10^4cells/ml$ and these individuals in branch of Tongbok were close to two times as much as Bosung. In light and electron microscopy, natural M. aeruginosa colonies formed irregular shape and non-directional array in amorphous matrix. They were consisted of many kinds of cells, youngs or olds in cell division, solitary, and various size of cells. Each cell ranged from 2.61 to $5.40{\mu}m$ in diameter, and averaged as $3.54{\pm}0.19{\mu}m$. In cytoplasm, they contained a number of inclusions in various size, shape and appearances. Among them, polyhedral bodies or carboxysomes, a structured granules, photosynthetic lamellae or thylakoids, and gas vacuoles were prominent and easy to recognize. Although it was failed to find the definable morphological variations in the ultrastructure of M. aeruginosa in terms of algal habitual environments, some useful characters were founded, outer layer of cell wall, polyhedral bodies and gas vacuoles, in blue-green algal classification and taxonomy.

• Korean Journal of Soil Science and Fertilizer
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• v.12 no.1
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• pp.1-14
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• 1979

The fertility which is the combined factor is one of the important capability determiants of paddy soils. In this study, we aimed at attaining a quantitative evaluation of soil fertility and further establishing an objective fertility classification on the basis of the fertility evaluation. The samples used in this series studies were collected from Korean paddy field. They include deltas, flood plains, coastal plains, valley plains, fans and low terraces. On the basis of correlation analysis, factor analysis was applied to a set of 15 variables. As a result of factor analysis, five mutually independent and clearly definable fertility component factors were extracted from the 15 variables for the whole 90 surface soil samples. The fertility status of each sample soil could be objectively designated by the score of the five factors. As a means of summarizing the information obtained, taxonomic distances between all pairs of the samples were computed from these five factor scores further to be subjected to numerical taxonomy. Seven fertility groups were formulated, each of which was characterized by one or more of the fertility components. As this fertility classification was based on the present state of soil properties, it would be useful in pointing to the proper direction of further fertility amelioration and improvement for each group to enhance potential productivity of Korean paddy fields.