• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.669-688
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• 2021

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

• 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.

• 한국해양학회지
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• v.5 no.2
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• pp.41-51
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• 1970

Periodic characters of water temperature in the regions of the Mishima and the Okinoshima were derived through the analysis of the five days interval data during 1914 to 1970 mainly. In terms of ten days mean temperatures, annual variation function of the Mishima region, Korea Strait, is F($\theta_d$)=17.45-5.34 cos $\theta_d$-3.77 sin $\theta_d$+0.62 sin $2\theta_d$ -0.52 sin $3\theta_d$, where $\theta_d$=$\frac{\pi}{18}$(d-2), d is the order of ten days period 1 to 36. And in the region of Okinoshima, Tsushima Strait, we find F($\theta_d$)=18.88-5.39 cos $\theta_d$-3.60 sin $\theta_d$+0.52 sin $2\theta_d$. The annual mean temperature is 17.4$^{\circ}C$ in the Mishima region, 18.9$^{\circ}C$ in the Okinoshima region, and the amplitudes of annual variation functions are 7$^{\circ}C$ in both regions with minimum temperature in the middle ten days of February, maximum in the middle ten days of August. The long term variations of surface water temperature with 12 5 years period were observed in the annual mean temperature, monthly mean temperatures and the fixed day temperatures of every year. In addition to these, relatively short term variations were also found significant periods of 3 years, 4 years and 2 years, respectively.

• Structural Engineering and Mechanics
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• v.13 no.6
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• pp.671-694
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• 2002

This paper presents theoretical investigation on the cross correlation between torsional vibration ($u_{\theta}$) and translation vibration ($u_x$) of asymmetrical structure under white noise excitation. The formula reveals that the cross correlation coefficient (${\rho}$) is a function of uncoupled frequency ratio (${\Omega}={\omega}_{\theta}/{\omega}_x$), eccentricity, and damping ratio (${\xi}$). Simulations involving acceleration records from fifteen different earthquakes show correlation coefficients results similar to the theoretical correlation coefficients. The uncoupled frequency ratio is the dominating parameter to ${\rho}$; generally, ${\rho}$ is positive for ${\omega}_{\theta}/{\omega}_x$ > 1.0, negative for ${\omega}_{\theta}/{\omega}_x$ < 1.0, and close to zero for ${\omega}_{\theta}/{\omega}_x$ = 1.0. When the eccentricity or damping ratio increases, ${\rho}$ increases moderately for small ${\Omega}$ (< 1.0) only. The relation among $u_x$, $u_{\theta}$ and corner displacement are best presented by ${\rho}$; a simple way to hand-calculate the theoretical dynamic corner displacements from $u_x$, $u_{\theta}$ and ${\rho}$ is proposed as an alternative to dynamic analysis.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.17-23
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• 2009

Let F be a finite real abelian extension of a global function field k with G = Gal(F/k). Assume that F is an extension field of the Hilbert class field $K_e$ of k and is contained in a cyclotomic function field $K_n$. Let $\ell$ be any prime number not dividing $ph_k{\mid}G{\mid}$. In this paper, we show that if $\theta{\in}\mathbb{Z}[G]$ annihilates the Sylow $\ell$-subgroup of ${\mathcal{O}}^{\times}_F/{\mathcal{C}}_F$, then (q-1)$\theta$ annihilates the Sylow $\ell$-subgroup of ${\mathcal{Cl}}_F$.

• The Korean Journal of Applied Statistics
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• v.7 no.2
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• pp.113-130
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• 1994

The algorithm of Oh and Berger (1993) is extended to handle more general cases where the integrand $f(\theta)$ is not only multimodal but also skewed or has some undetected modes, each having curvature not much different from that of the nearest component. It runs Oh and Berger's algorithm in an iterative way, adding a component in each stage to the mixture importance function from previous stage for better approximation between $f(\theta)$ and the importance function.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.345-352
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• 2024

We introduce a new class of bi-partition function c_k(n), which counts the number of bi-color partitions of n in which the second color only appears at the parts that are multiples of k. We consider two partitions to be the same if they can be obtained by switching the color of parts that are congruent to zero modulo k. We show that the generating function for c_k(n) involves the partial theta function and obtain the following congruences: c₂(27n + 26) ≡ 0 (mod 3) and c₃(4n + 2) ≡ 0 (mod 2).

• Journal of Korean Ophthalmic Optics Society
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• v.10 no.4
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• pp.273-281
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• 2005

The weight of spectacles fined on is resolved into its components along the nose's slide plane and the normal to the nose plane where the nosepad is located. The equation and its numerical solution to determine the component force was derived as a function of splay angle ${\Psi}$, $sin{\Psi}$, $cos{\Psi}$, and $cot{\Psi}$, incorporated with ${\theta}$ and ${\Phi}$, the angles viewed from side and front of the face, respectively. Values of inclination angle ${\theta}$ and ${\Phi}$ could be obtained to fulfill the condition where the frictional force between the nose and pad is either greater than the normal pressure exerted by the spectacles on the nose. With the value of ${\theta}$ fixed the normal pressure increases as ${\Phi}$ increases. With ${\Phi}$ fixed, the effect of ${\theta}$ is the same.

• Journal of The Korean Astronomical Society
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• v.33 no.2
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• pp.89-95
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• 2000

As an efficient method to detect blending of general gravitational microlensing events, it is proposed to measure the shift of source star image centroid caused by microlensing. The conventional method to detect blending by this method is measuring the difference between the positions of the source star image point spread function measured on the images taken before and during the event (the PSF centroid shift, ${\delta}{\theta}$c,PSF). In this paper, we investigate the difference between the centroid positions measured on the reference and the subtracted images obtained by using the difference image analysis method (DIA centroid shift, ${\delta}{\theta}$c.DIA), and evaluate its relative usefulness in detecting blending over the conventional method based on ${\delta}{\theta}$c,PSF measurements. From this investigation, we find that the DIA centroid shift of an event is always larger than the PSF centroid shift. We also find that while ${\delta}{\theta}$c,PSF becomes smaller as the event amplification decreases, ${\delta}{\theta}$c.DIA remains constant regardless of the amplification. In addition, while ${\delta}{\theta}$c,DIA linearly increases with the increasing value of the blended light fraction, ${\delta}{\theta}$c,PSF peaks at a certain value of the blended light fraction and then eventually decreases as the fraction further increases. Therefore, measurements of ${\delta}{\theta}$c,DIA instead of ${\delta}{\theta}$c,PSF will be an even more efficient method to detect the blending effect of especially of highly blended events, for which the uncertainties in the determined time scales are high, as well as of low amplification events, for which the current method is highly inefficient.

• Journal of the Korean Mathematical Society
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• v.41 no.2
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• pp.265-294
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• 2004

In this paper, using a simple formula, we evaluate the conditional Fourier-Feynman transforms and the conditional convolution products of cylinder type functions, and show that the conditional Fourier-Feynman transform of the conditional convolution product is expressed as a product of the conditional Fourier-Feynman transforms. Also, we evaluate the conditional Fourier-Feynman transforms of the functions of the forms exp {$\int_{O}^{T}$ $\theta$(s,$\chi$(s))ds}, exp{$\int_{O}^{T}$ $\theta$(s,$\chi$(s))ds}$\Phi$($\chi$(T)), exp{$\int_{O}^{T}$ $\theta$(s,$\chi$(s))d${\zeta}$(s)}, exp{$\int_{O}^{T}$ $\theta$(s,$\chi$(s))d${\zeta}$(s)}$\Phi$($\chi$(T)) which are of interest in Feynman integration theories and quantum mechanics.