• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.697-707
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• 2022

It is well known that the continued fraction expansion of ${\sqrt{d}}$ has the form $[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}]$ and ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer l and a palindromic sequence of positive integers ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},$ we define the set $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})\;:=\;\{d{\in}{\mathbb{Z}}{\mid}d>0,\;{\sqrt{d}}=[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}],\;where\;{\alpha}_0={\lfloor}{\sqrt{d}}{\rfloor}\}.$ In this paper, we completely determine when $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})$ is not empty in the case that l is 4, 5, 6, or 7. We also give similar results for $(1+{\sqrt{d}})/2.$ For the case that l is 4, 5, or 6, we explicitly describe the fundamental units of the real quadratic field ${\mathbb{Q}}({\sqrt{d}}).$ Finally, we apply our results to the Mordell conjecture for the fundamental units of ${\mathbb{Q}}({\sqrt{d}}).$

• Communications of the Korean Mathematical Society
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• v.18 no.3
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• pp.501-513
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• 2003

The Matsumoto metric is an ($\alpha,\;\bata$)-metric which is an exact formulation of the model of Finsler space. Lately, this metric was expressed as an infinite series form for $$\mid$\beat$\mid$\;<\;$\mid$\alpha$\mid$$ by the first author. He introduced an approximate Matsumoto metric as the ($\alpha,\;\bata$)-metric of finite series form and investigated it in [11]. The purpose of the present paper is devoted to finding the condition for a Finsler space with an approximate Matsumoto metric to be projectively flat.

• Honam Mathematical Journal
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• v.40 no.1
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• pp.61-73
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• 2018

The main aim of this research paper is to evaluate the general integral of the form $${\int_{0}^{1}}x^{c-1}(1-x)^{c+{\ell}}[1+{\alpha}x+{\beta}(1-x)]^{-2c-{\ell}-1}\atop {\times}_3F_2\left\[ {a,\;b,\;2c+{\ell}+1} \\ {\frac{1}{2}(a+b+i+1),\;2c+j\;;\frac{(1+{\alpha})x}{1+{\alpha}x+{\beta}(1-x)} }\right]dx$$ in the most general form for any ${\ell}{\in}\mathbb{Z}$; and $i, j=0,{\pm}1,{\pm}2$. The results are established with the help of generalized Watson's summation theorem due to Lavoie, et al. Fifty interesting general integrals have also been obtained as special cases of our main findings.

• East Asian mathematical journal
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• v.4
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• pp.57-73
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• 1988

The subclasses S*($\alpha,\beta,\mu$) and C*($\alpha,\beta,\mu$) ($0\leqq\alpha<1,\;0<\beta\leqq1$ and $0\leqq\mu\leqq1$) of T the class of analytic and univalent functions of the form $$f(z)=z-\sum\limit^{\infty}_{n=2}\mid a_n\mid z^n$$ have been considered. Sharp results concerning coefficients, distortion of functions belonging to S*($\alpha,\beta,\mu$) and C*($\alpha,\beta,\mu$) are determined along with a representation formula for the functions in S*($\alpha,\beta,\mu$). Furthermore, it is shown that the classes S*($\alpha,\beta,\mu$) and C*($\alpha,\beta,\mu$) are closed under arithmetic mean and convex linear combinations. Also in this paper, we find extreme points and support points for these classes.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.363-375
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• 2004

In this paper, We show that the bandpass random signals of the form ∑$_{\alpha}$$\alpha$$_{\alpha}$ a Sin(2$\pi$f$_{\alpha}$t + b$_{\alpha}$) where a$_{\alpha}$ being a random number in [0,1], f$_{\alpha}$ a random integer in a given frequency band, and b$_{\alpha}$ a random number in [0, 2$\pi$], generate Gaussian white noise signals and hence they are adequate for simulating Continuous Markov processes. We apply the result to the fluctuation-dissipation formula for the Johnson noise and show that the probability distribution for the long term average of the power of the Johnson noise is a X$^2$ distribution and that the relative error of the long term average is (equation omitted) where N is the number of blocks used in the average.error of the long term average is (equation omitted) where N is the number of blocks used in the average.

• Applied Biological Chemistry
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• v.24 no.4
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• pp.252-259
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• 1981

The effects of tunicamycin (TM) on the growth and ${\alpha}-amylase$ productivity of B. amyloliquefaciens K were studied. The minimal growth inhibitory concentration was $0.25{\mu}\textrm{g}/m\ell$ and its ${\alpha}-amylase$ was stable up to $50^{\circ}C$. When the saking culture with $1{\mu}\textrm{g}/m\ell$ of Tunicamycin caused the change of cell shape from form rod to irregular circular form and the mycelium lysis. the grow th of His-, $TM^{\tau}$ mutant obtained by treatment of TM and ultraviolet ray was similar to that of the parent strain, but the productivity of ${\alpha}-amylase$, protease, and RNase was lower than that of the parent.

• Journal of Microbiology and Biotechnology
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• v.17 no.11
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• pp.1898-1903
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• 2007

We attempted to modulate the overall protein expression rate through the addition of a repressor against the araBAD promoter system of Escherichia coli, in which glucose was used as a repressor. Therefore, 0.5% L-arabinose was initially contained as an inducer in culture medium, and either 2% glucose or 2% glycerol was used as a carbon source, and it was found that the expression of recombinant interferon-${\alpha}$ could be observed at the beginning of the batch culture when glycerol was used as a carbon source. However, when glucose was used, the initiation of recombinant interferon-${\alpha}$ expression was delayed compared with that when glycerol was used. Furthermore, when the addition of 0.5% glucose was carried out once or twice after 0.5% L-arabinose induction during DO-stat fed-batch culture, the distributions of soluble and insoluble recombinant interferon-${\alpha}$ were modulated. When glucose was not added after the induction of L-arabinose, all of the expressed recombinant interferon-${\alpha}$ formed an inclusion body during the later half of culturing. However, when glucose was added after induction, the expressed recombinant interferon-${\alpha}$ did not all form an inclusion body, and about half of the total recombinant interferon-${\alpha}$ was expressed in a soluble form. It was deduced that the addition of glucose after the induction of L-arabinose might lower the cAMP level, and thus, CAP (catabolite activator protein) might not be activated. The transcription rate of recombinant interferon-${\alpha}$ in the araBAD promoter system might be delayed by the partial repression. This inhibition of the transcription rate probably resulted in more soluble interferon-${\alpha}$ expression caused by the reduction of the protein synthesis rate.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.229-234
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• 2005

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the form $x_{n+l} =\alpha+\frac{x_{n-1}^{p}}{x_{n}^{p}},\;\; n = 0, 1, ...$ is investigated, where all the coefficients are nonnegative real numbers.

• Korean Journal of Mathematics
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• v.8 no.1
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• pp.73-86
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• 2000

We study the Sobolev spaces to the generalized Sobolev spaces $H^s_{\mathcal{G}}$ of exponential type based on the Silva space $\mathcal{G}$ and investigate its properties such as imbedding theorem and structure theorem. In fact, the imbedding theorem says that for $s$ > 0 $u{\in}H^s_{\mathcal{G}}$ can be analytically continued to the set {$z{\in}\mathbb{C}^n{\mid}{\mid}Im\;z{\mid}$ < $s$}. Also, the structure theorem means that for $s$ > 0 $u{\in}H^{-s}_{\mathcal{G}}$ is of the form $$u={\sum_{\alpha}\frac{s^{{|\alpha|}}}{{\alpha}!}D^{\alpha}g{\alpha}$$ where $g{\alpha}$'s are square integrable functions for ${\alpha}{\in}\mathbb{N}^n_0$. Moreover, we introduce a classes of symbols of exponential type and its associated pseudo-differential operators of exponential type, which naturally act on the generalized Sobolev spaces of exponential type. Finally, a generalized Bessel potential is defined and its properties are investigated.

• Proceedings of the Korean Vacuum Society Conference
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• 2012.02a
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• pp.557-557
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• 2012

Silicatein-${\alpha}$, the enzyme extracted from silica spicules in glass sponges, has been studied extensively in the way of chemistry from 1999, in which the pioneering work by Morse, D. E. - the discovery of the enzymatic hydrolysis in Silicatein-${\alpha}$ - was published. Since its reaction conditions are physiologically favored, synthesis of various materials, such as gallium oxide, zirconium oxide, and silicon oxide, was achieved without any hazardous wastes. Although some groups synthesized oxide films and particles, they have not achieved yet controlled morphogenesis in the reaction conditions mentioned above. With the knowledge of catalytic triad involved in hydrolysis of silicone alkoxide and oligomerization of silicic acid, we designed the novel peptide amphiphiles to not only form self-assembled structure, but also display similar activities to silicatein-${\alpha}$. Designed templates were able to self-assemble into left-handed helices for the peptide amphiphiles with L-form amino acid, catalyzing polycondensation of silicic acids onto the surface of them. It led to the formation of silica helices with 30-50 nm diameters. These results were characterized by various techniques, including SEM, TEM, and STEM. Given the situation that nano-bio-technology, the bio-applicable technology in nanometer scale, has been attracting considerable attention; this result could be applied to the latest applications in biotechnology, such as biosensors, lab-on-a-chip, biocompatible nanodevices.