• Journal for History of Mathematics
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• v.23 no.3
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• pp.57-73
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• 2010

Transcendental numbers are important in the history of mathematics because their study provided that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was insoluble. Liouville established in 1844 that transcendental numbers exist. In 1874, Cantor published his first proof of the existence of transcendentals in article [10]. Louville's theorem basically can be used to prove the existence of Transcendental number as well as produce a class of transcendental numbers. The number e was proved to be transcendental by Hermite in 1873, and $\pi$ by Lindemann in 1882. In 1934, Gelfond published a complete solution to the entire seventh problem of Hilbert. Within six weeks, Schneider found another independent solution. In 1966, A. Baker established the generalization of the Gelfond-Schneider theorem. He proved that any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental. This study aims to examine the concept and development of transcendental numbers and to present students with its open problems promoting a research on it any further.

• East Asian mathematical journal
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• v.23 no.1
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• pp.59-73
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• 2007

A new reverse of the generalised triangle inequality that complements the classical results of Diaz and Metcalf is obtained. Applications for inner product spaces and for complex numbers are provided.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.751-763
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• 2015

Let ${\alpha}$ be a positive integer, and let $p_1$, $p_2$ be two distinct prime numbers with $p_1$ < $p_2$. By using elementary methods, we give two equivalent conditions of all even near-perfect numbers in the form $2^{\alpha}p_1p_2$ and $2^{\alpha}p_1^2p_2$, and obtain a lot of new near-perfect numbers which involve some special kinds of prime number pairs. One kind is exactly the new Mersenne conjecture's prime number pair. Another kind has the form $p_1=2^{{\alpha}+1}-1$ and $p_2={\frac{p^2_1+p_1+1}{3}}$, where the former is a Mersenne prime and the latter's behavior is very much like a Fermat number.

• The Korean Journal of Applied Statistics
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• v.22 no.5
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• pp.1103-1113
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• 2009

We know that the first digits sequence of fibonacci numbers obey Benford's law. For the sequence in which the first two numbers are the arbitrary integers and the recurrence relation $a_{n+2}=a_{n+1}+a_n$ is satisfied, we can find that the first digits sequence of this sequence obey Benford's law. Also, we can find the stucture of the first digits sequence of this sequence with the exploratory data analysis tools.

• Korean Journal of Microbiology
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• v.37 no.3
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• pp.209-213
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• 2001

By FISH and spore staining methods, the changes of Bacillus spp. numbers were investigated at municipal waste water treatment system, which operated with B3 process. Total bacterial numbers, which is the sum ofDAPI bound cells and numbers of spore, were ranged TEX>$3.2{\times}10^8 ~ 4.4{\times}10^8 $ cells/ml. The Bacilli numbers whichis the sum of probe S-G-Bacill-0597-a-A-22 bound cells and numbers of spore were from $0.3{\times}10^7 ~ 8.4{\times}10^7 $ cells/ml. The ratios of Bacillus sp. to total bacteria were ranged from 8 to 60%. Among 4 aeration tanks, thefirst tank had shown the highest proportions of Bacillus sp. And third and fourth tanks, the numbers of sporewere increasing.

• Publications of The Korean Astronomical Society
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• v.18 no.1
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• pp.25-35
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• 2003

In this paper, we have re-examined the relative sunspot numbers from June 1987 to December 2002 observed at Korea Astronomy Observatory. For this we determined conversion factors (K) for each year data to derive the relative sunspot numbers. The estimated conversion factor ranges from 0.57 to 1.09 and has a trend to decrease with time, which seem to depend on the several effects such as observational system, observation methods, and experience of an observer. Our analysis shows that the newly-determined relative sunspot numbers are in much better agreements with the international sunspot numbers than the previously-determined ones in which the conversion factors were determined only four times. This result implies that we should determine the conversion factor for each year data. From these investigations, we also identified one and half solar cycles form our relative sunspot number.

• Animal cells and systems
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• v.15 no.1
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• pp.73-78
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• 2011

Estimating the habitat potential of inland forest patches for birds requires the modeling of species-area relationships, or relationships between habitat size and numbers of bird species in each patch. The accurate estimation of speciesarea relationships significantly reduces the effort required to recognize the number of species living in each patch. The objective of this study was to estimate the relationship between forest patch size and bird species diversity in Dangjin County, in northwest South Korea, based on the sizes of inland forest patches. KOMPSAT-2 images were obtained and ortho-rectified to construct a map of the target forest patches. The numbers of birds per patch were surveyed four times: August 2008, September 2008, February 2009 and May 2009. Regression models were derived to explain the relationships between the numbers of bird species and patch size. A model that was derived using data from all four observation periods had the highest coefficient of determination ($R^2$). According to these models, the numbers of bird species at first increased linearly with increasing patch size; however, the curve then plateaued. Our model including observations from four seasons will be useful for estimating the numbers of bird species in other inland forest patches in South Korea.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.71-84
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• 2007

J. Bernoulli first discovered the method which one can produce those formulae for the sum $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$ for any natural numbers k. After then, there has been increasing interest in Bernoulli and Euler numbers associated with Riemann zeta functions. Recently, Kim have been studied extended q-Bernoulli numbers and q-Euler numbers associated with p-adic q-integral on $\mathbb{Z}_p$, and sums of powers of consecutive q-integers, etc. In this paper, we investigate for the historical background and evolution process of the sums of powers of consecutive q-integers and discuss for Euler zeta functions subjects which are studying related to these areas in the recent.

• Korean Journal of Plant Taxonomy
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• v.48 no.4
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• pp.260-277
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• 2018

The evolution of chromosome numbers and the karyotype structure is a prominent feature of plant genomes contributing to or at least accompanying plant diversification and eventually leading to speciation. Polyploidy, the multiplication of whole chromosome sets, is widespread and ploidy-level variation is frequent at all taxonomic levels, including species and populations, in angiosperms. Analyses of chromosome numbers and ploidy levels of 252 taxa of Korean non-commelinid monocots indicated that diploids (ca. 44%) and tetraploids (ca. 14%) prevail, with fewer triploids (ca. 6%), pentaploids (ca. 2%), and hexaploids (ca. 4%) being found. The range of genome sizes of the analyzed taxa (0.3-44.5 pg/1C) falls well within that reported in the Plant DNA C-values database (0.061-152.33 pg/1C). Analyses of karyotype features in angiosperm often involve, in addition to chromosome numbers and genome sizes, mapping of selected repetitive DNAs in chromosomes. All of these data when interpreted in a phylogenetic context allow for the addressing of evolutionary questions concerning the large-scale evolution of the genomes as well as the evolution of individual repeat types, especially ribosomal DNAs (5S and 35S rDNAs), and other tandem and dispersed repeats that can be identified in any plant genome at a relatively low cost using next-generation sequencing technologies. The present work investigates chromosome numbers (n or 2n), base chromosome numbers (x), ploidy levels, rDNA loci numbers, and genome size data to gain insight into the incidence, evolution and significance of polyploidy in Korean monocots.