• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.331-351
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• 2007

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k$, $q=e^{{\pi}i\tau}$. We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.55-107
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• 2007

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k,\;q=e^{{\pi}i\tau}$. In this article, we obtain algebraic numbers from the 130 identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, $\frac{q^{1/8}}{1}+\frac{-q}{1+q}+\frac{-q^2}{1+q^2}+\cdots$ ([1]) is transcendental.

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.333-347
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• 2015

In this paper, by using the modular forms of weight nk ($2{\leq}n{\in}\mathbb{N}$ and $k{\in}\mathbb{Z}$), we construct a formula which generates modular forms of weight 2nk+4. This formula consist of some known results in [14] and [4]. Moreover, we obtain Fourier expansion of these modular forms. We also give some properties of an operator related to the derivative formula. Finally, by using the function $j_4$, we obtain the Fourier coefficients of modular forms with weight 4.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1379-1391
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• 2008

Let k be an imaginary quadratic field, ${\eta}$ the complex upper half plane, and let ${\tau}{\in}{\eta}{\cap}k,\;q=e^{{\pi}{i}{\tau}}$. For n, t ${\in}{\mathbb{Z}}^+$ with $1{\leq}t{\leq}n-1$, set n=${\delta}{\cdot}2^{\iota}$(${\delta}$=2, 3, 5, 7, 9, 13, 15) with ${\iota}{\geq}0$ integer. Then we show that $q{\frac}{n}{12}-{\frac}{t}{2}+{\frac}{t^2}{2n}{\prod}_{m=1}^{\infty}(1-q^{nm-t})(1-q^{{nm}-(n-t)})$ are algebraic numbers.

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.903-931
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• 1998

Since the modular curves X(N) = $\Gamma$(N)\(equation omitted)＊ (N =1,2,3) have genus 0, we have field isomorphisms K(X(l))(equation omitted)C(J), K(X(2))(equation omitted)(λ) and K(X(3))(equation omitted)( $j_3$) where J, λ are the classical modular functions of level 1 and 2, and $j_3$ can be represented as the quotient of reduced Eisenstein series. When N = 4, we see from the genus formula that the curve X(4) is of genus 0 too. Thus the field K(X(4)) is a rational function field over C. We find such a field generator $j_4$(z) = x(z)/y(z) (x(z) = $\theta$$_3$((equation omitted)), y(z) = $\theta$$_4$((equation omitted)) Jacobi theta functions). We also investigate the structures of the spaces $M_{k}$($\Gamma$(4)), $S_{k}$($\Gamma$(4)), M(equation omitted)((equation omitted)(4)) and S(equation omitted)((equation omitted)(4)) in terms of x(z) and y(z). As its application, we apply the above results to quadratic forms.rms.

• Laser Solutions
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• v.14 no.2
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• pp.24-29
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• 2011

The design of a telecentric f-${\theta}$ lens with a field of view (FOV) $30^{\circ}$ and an effective focal length of 1000mm is presented. The optical stop is placed at the front plane and the design is based on a geometric ray tracing technique, and the designed system consists of a series of convex and concave lenses. The designed f-${\theta}$ lens showed a considerable reduction in weight with a simplified structure and resulted in a good performance in the designated FOV. Detail analysis of rays is also presented. 653nm (red laser), 586nm (green laser), and 468nm (blue laser) were simulated as a light source and image illuminating source. The developed optical design requires 7 pieces of lenses made of SF1, N-FK56, N-LAK33, and BK7 glass materials. With optimal parametric design, the effective focal length was calculated to be 974.839mm which is very close to the initial design target. For the manufacturing purpose, the dimensions of lens curvature and thickness were truncated with error ranging 0.1% to 3.2%. As a result, the overall error was calculated to be 3.2% which can be still tolerable for display, laser material, and machining processing.

• International Journal of Reliability and Applications
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• v.10 no.1
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• pp.43-57
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• 2009

In this article, we study the reliability equivalence factor of a series system. The failure rates of the system components are functions of time t. we study two cases of non-constat failure rates (i) weibull distribution (ii) linear increasing failure rate distribution. There are two methods are used to improve the given system. Two types of reliability equivalence factors are discussed. Numerical examples are presented to interpret how one can utilize the obtained results.

• Atmosphere
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• v.24 no.1
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• pp.49-68
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• 2014

This study analyzed atmospheric conditions for the convergent cloud band (Cu-Cb line) in developing stage and its neighbouring convections formed over the East Sea on 1 February 2012, by using synoptic, satellites data, and WRF numerical simulation output of high resolution. In both satellite images and the WRF numerical simulation outputs, the Cu-Cb line that stretched out toward northwest-southeast was shown in the East Sea, and cloud lines of the L mode were aligned in accordance with the prevailing surface wind direction. However, those of the T mode were aligned in the direction of NE-SW, which was nearly perpendicular direction to the surface winds. The directions of the wind shear vectors connecting top winds and bottom winds of the moist layers of the L mode and the T mode were identical with those of the cloud lines of L mode and T mode, respectively. From the WRF simulation convection circulations with a convergence in the lower layer of atmosphere and a divergence above 1.5 km ASL (Above Sea Level) were identified in the Cu-Cb line. A series of small sized vortexes (maximum vortex: $320{\times}10^{-5}s^{-1}$) of meso-${\gamma}$-scale formed by convergences was found along the Cu-Cb lines, suggesting that Cu-Cb lines, consisting of numerous convective clouds, were closely associated with a series of the small vortexes. There was an absolute unstable layer (${\partial}{\theta}/{\partial}z$ < 0) between sfc and ~0.3 km ASL, and a stable layer (${\partial}{\theta}/{\partial}z$ > 0) above ~2 km ASL over the Cu-Cb line and cloud zones. Not only convectively unstable layers (${\partial}{\theta}_e/{\partial}z$ < 0) but also neutral layers (${\partial}{\theta}_e/{\partial}z{\approx}=0$) in the lower atmosphere (sfc~1.5 km ASL) were scattered around over the cloud zones. Particularly, for the Cu-Cb line there were convectively unstable layers in the surface layer, and neutral layers (${\partial}{\theta}_e/{\partial}z{\approx}=0$) between 0.2 and ~1.5 km ASL over near the center of the Cu-Cb line, and the neutralization of unstable layers came from the release of convective instability.

• Carbon letters
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• v.8 no.2
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• pp.95-100
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• 2007

A series of activated carbons (ACs) were derived from sugarcane bagasse under two activation schemes: steam-pyrolysis at $600-800^{\circ}C$ and chemical activation with $H_3PO_4$ at $500^{\circ}C$. Some carbons were treated at 400, $600^{\circ}C$, or for 1-3 h, and/or in flowing air during pyrolysis of acid-impregnated mass. XRD profiles displayed two broad diffuse bands centered around $2{\theta}=23$ and $43^{\circ}$, currently associated with diffraction from the 002 and 100/101 set of planes in graphite, respectively. These correspond to the interlayer spacing, Lc, and microcrystallite lateral dimensions, La, of the turbostratic (fully disordered) graphene layers. Steam pyrolysis-activated carbons exhibit only the two mentioned broad bands with enhancement in number of layers, with temperature, and small decrease in microcrystallite diameter, La. XRD patterns of $H_3PO_4$-ACs display more developed and separated peaks in the early region with maxima at $2{\theta}=23$, 26 and $29^{\circ}$, possibly ascribed to fragmented microcrystallites (or partially organized structures). Diffraction within the $2{\theta}=43^{\circ}$ is still broad although depressed and diffuse, suggesting that the intragraphitic layers are less developed. Varying the conditions of chemical activation inflicts insignificant structural alterations. Circulating air during pyrolysis leads to enhancement of the basic graphitic structure with destruction and degradation in the lateral dimensions.

• Transactions of the Korean Society of Mechanical Engineers
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• v.11 no.3
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• pp.362-367
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• 1987

Cylinder pressure level is defined the value of sound pressure level derived from relations of pressure and frequency which are obtained Fourier series expansion of cylinder pressure. CPL is alone in general use in combustion analysis without regarding attenuation coefficient determined by engine structure because it is almost constant in various kinds of engines. Combustion pressure fluctuation and influence of CPL was investigated in this paper using Wiebe combustion function superposition th obtain the effect of premixed and diffusive combustion. The results are as follow. (1) Influence of combustion maximum pressure(P$\sub$max/) on CPL is correspond with total energy per cycle well. (2) P$\sub$max/, .DELTA.P$\sub$max/, (dp/d.theta.)$\sub$max/ and (d$\^$2/p/d.theta.$\^$2/)$\sub$max/ are very effective on CPL in the range of low frequency, mid-low frequency, mid-high frequency and high frequency respectively.