• 충청수학회지
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• 제22권2호
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• pp.155-159
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• 2009

Using the Binet's formula, we show that the quotient related ratio $l_{1(x)}\;\neq\;0$ for the eventually periodic continued fraction x. Using this ratio, we also show that the derivative of the Minkowski question mark function at the simple periodic continued fraction is infinite or 0. In particular, $l_1({[\bar{1}]})$ = 2 log $\gamma$ where $\gamma$ is the golden mean $(1+\sqrt{5})/2$ and the derivative of the Minkowski question mark function at the simple periodic continued fraction $[\bar{1}]$ is infinite.

• 대한수학회보
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• 제50권5호
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• pp.1623-1630
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• 2013

The aim of this paper is to prove that every quadratic formal power series ${\omega}$ can be expressed as a periodic non-simple continued fraction having period length one.

• 충청수학회지
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• 제18권1호
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• pp.95-100
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• 2005

In this paper, we consider the finite form of the continued fraction found in the unorganized material, Entry 9 [2], and give the sum of n terms. We give the $n^{th}$ convergent series in two ways-one by simple expansion and the other by using partition theory.

• 대한수학회보
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• 제53권3호
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• pp.699-709
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• 2016

Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.

• 한국수학사학회지
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• 제13권2호
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• pp.49-64
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• 2000

Every real number can be expressed as a simple continued fraction. In particular, a number is rational if and only if its simple continued fraction has a finite number of terms. Owing to this property, continued fractions have been a powerful tool which determines a real number to be rational or not. Continued fractions provide not only a series of best estimate for a real number, but also a useful method for finding near commensurabilities between events with different periods. In this paper, we investigate the history and some properties of continued fractions, and then consider their applications in several examples. Also we explain why the Fibonacci numbers and the Golden section appear in nature in terms of continued fractions, with some examples such as the arrangements of petals round a flower, leaves round branches and seeds on seed head.

• 호남수학학술지
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• 제36권1호
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• pp.103-112
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• 2014

There are several theories to say that 'Mathematics is beautiful', but the typical one of them is a theory about the golden ratio. Often the golden ratio apt to be considered only as the geometric shapes or the simple number of ratio used in buildings and arts. However in this paper, we studied to consider the mathematical theories which are contained in their inside. In particular, we investigate the various expressions of the continued fraction which are represented by the golden ratio.

• 한국수학교육학회지시리즈A:수학교육
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• 제47권4호
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• pp.447-465
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• 2008

The purpose of this study is to suggest effective teaching methods on the concept of infinity for students to obtain the right concept in the middle school curriculum. Many people have thought that infinity is something vouge and unapproachable. But, nowadays it is rather something with a precise definition that lies at the core of modern mathematics. To understand mathematics and science very well, it is necessary to comprehend the concept of infinity. But students tend to figure out the properties of infinite objects and limit concepts only through their experience closely related to finite process, and so they are apt to have their spontaneous intuition and misconception about it. Since most of them have cognitive obstacles in studying the infinite concepts and misconception, mathematics teachers need to help them overcome the obstacles and establish the right secondary intuition for the concepts through good examples and appropriate explanation. In this study, we consider the developing process of the concept of infinity in human history and give some comments and suggestions in teaching methods relative to that concept with new suitable examples.

• 식물조직배양학회지
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• 제27권6호
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• pp.423-428
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• 2000

In 1997 when cloned sheep Dolly and soon after Polly were born, it had become head-line news because in the former the nucleus that gave rise to the lamb came from cells of six-year-old adult sheep and in the latter case a foreign gene was inserted into the donor nucleus to make the cloned sheep produce human protein, factor IX, in e milk. In the last few years, once the realm of science fiction, cloned mammals especially in livestock have become almost commonplace. What the press accounts often fail to convey, however, is that behind every success lie hundreds of failures. Many of the nuclear-transferred egg cells fail to undergo normal cell divisions. Even when an embryo does successfully implant in the womb, pregnancy often ends in miscarriage. A significant fraction of the animals that are born die shortly after birth and some of those that survived have serious developmental abnormalities. Efficiency remains at less than one % out of some hundred attempts to clone an animal. These facts show that something is fundamentally wrong and enormous hurdles must be overcome before cloning becomes practical. Cloning researchers now tent to put aside their effort to create live animals in order to probe the fundamental questions on cell biology including stem cells, the questions of whether the hereditary material in the nucleus of each cell remains intact throughout development, and how transferred nucleus is reprogrammed exactly like the zygotic nucleus. Stem cells are defined as those cells which can divide to produce a daughter cell like themselves (self-renewal) as well as a daughter cell that will give rise to specific differentiated cells (cell-differentiation). Multicellular organisms are formed from a single totipotent stem cell commonly called fertilized egg or zygote. As this cell and its progeny undergo cell divisions the potency of the stem cells in each tissue and organ become gradually restricted in the order of totipotent, pluripotent, and multipotent. The differentiation potential of multipotent stem cells in each tissue has been thought to be limited to cell lineages present in the organ from which they were derived. Recent studies, however, revealed that multipotent stem cells derived from adult tissues have much wider differentiation potential than was previously thought. These cells can differentiate into developmentally unrelated cell types, such as nerve stem cell into blood cells or muscle stem cell into brain cells. Neural stem cells isolated from the adult forebrain were recently shown to be capable of repopulating the hematopoietic system and produce blood cells in irradiated condition. In plants although the term$\boxDr$ stem cell$\boxUl$is not used, some cells in the second layer of tunica at the apical meristem of shoot, some nucellar cells surrounding the embryo sac, and initial cells of adventive buds are considered to be equivalent to the totipotent stem cells of mammals. The telomere ends of linear eukaryotic chromosomes cannot be replicated because the RNA primer at the end of a completed lagging strand cannot be replaced with DNA, causing 5' end gap. A chromosome would be shortened by the length of RNA primer with every cycle of DNA replication and cell division. Essential genes located near the ends of chromosomes would inevitably be deleted by end-shortening, thereby killing the descendants of the original cells. Telomeric DNA has an unusual sequence consisting of up to 1,000 or more tandem repeat of a simple sequence. For example, chromosome of mammal including human has the repeating telomeric sequence of TTAGGG and that of higher plant is TTTAGGG. This non-genic tandem repeat prevents the death of cell despite the continued shortening of chromosome length. In contrast with the somatic cells germ line cells have the mechanism to fill-up the 5' end gap of telomere, thus maintaining the original length of chromosome. Cem line cells exhibit active enzyme telomerase which functions to maintain the stable length of telomere. Some of the cloned animals are reported prematurely getting old. It has to be ascertained whether the multipotent stem cells in the tissues of adult mammals have the original telomeres or shortened telomeres.