• 대한수학회보
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• 제50권4호
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• pp.1109-1126
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• 2013

Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

• 한국식물학회:학술대회논문집
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• 한국식물학회 1987년도 식물생명공학 심포지움 논문집 Proceedings of Symposia on Plant Biotechnology
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• pp.149-155
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• 1987

Growth and development of a higher plant, or any living organism for that matter, could be defined as an orderly expression of the genome in time and space in close interaction with the environment. During differentiation and development of a tissue or organ a group of genes must be selectively turned on or turned off mainly by trans-acting regulators. In this general concept of regulation of regulation of gene expression, a DNA molecule is recognized at a specific nucleotide sequence by DNA-binding factors. Molecular biology of the regulatory factors such as hormones, and their receptors, target DNA sequences and DNA-binding proteins are well advanced. What is not clearly understood is the molecular basis of the interactions between DNA and binding factors, expecially of the usages of the dyad symmetry of the target DNA sequences and the dimeric nature of the DNA-binding proteins. A unique 3-dimensional structure of DNA has been proposed that may play an important role in the orderly expression of the gene. A foldback intercoil (FBI) DNA configuration which was originally found by electron microscopy among mtDNA molecules from pearl millet has some unique features. The FBI configuration of DNA is believed to be formed when a flexible double helix folds back and interwines in the widened major grooves resulting in a four stranded, intercoil DNA whose thickness is the same as that of double stranded DNA. More recently, the FBI structure of DNA has been also induced in vitro by a novel enzyme which was purified from pearl millet mitochondria. It has been proposed that the FBI DNA could be utillized in intramolecular recombination which leads to inversion or deletion, and in intermolecular recombination which can lead to either site-specific recombination, genetic recombination via single strand invasion, or cross strand recombination. The structure and function of DNA in 3-dimensional aspect is emphasized for better understanding orderly expression of genes during growth and development.