• Bulletin of the Korean Mathematical Society
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• 제54권3호
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• pp.731-736
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• 2017

It is well-known that there exists a constant-weight $[s{\theta}_{k-1},k,sq^{k-1}]_q$ code for any positive integer s, which is an s-fold simplex code, where ${\theta}_j=(q^{j+1}-1)/(q-1)$. This gives an upper bound $n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+n_q(k,d)$ for any positive integer d, where $n_q(k,d)$ is the minimum length n for which an $[n,k,d]_q$ code exists. We construct a two-weight $[s{\theta}_{k-1}+1,k,sq^{k-1}]_q$ code for $1{\leq}s{\leq}k-3$, which gives a better upper bound $n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+1+n_q(k-1,d)$ for $1{\leq}d{\leq}q^s$. As another application, we prove that $n_q(5,d)={\sum_{i=0}^{4}}{\lceil}d/q^i{\rceil}$ for $q^4+1{\leq}d{\leq}q^4+q$ for any prime power q.

• Bulletin of the Korean Mathematical Society
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• 제57권2호
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• pp.429-441
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• 2020

Let M(X, Y) denote the space of multipliers from X to Y, where X and Y are analytic function spaces. As we known, for Dirichlet-type spaces 𝓓_α^p, M(𝓓_p-1^p, 𝓓_q-1^q) = {0}, if p ≠ q, 0 < p, q < ∞. If 0 < p, q < ∞, p ≠ q, 0 < s < 1 such that p + s, q + s > 1, then M(𝓓_p-2+s^p, 𝓓_q-2+s^q) = {0}. However, X ∩ 𝓓_p-1^p ⊆ X ∩ 𝓓_q-1^q and X ∩ 𝓓_p-2+s^p ⊆ X ∩ 𝓓_q-2+s^p whenever X is a subspace of the Bloch space 𝓑 and 0 < p ≤ q < ∞. This says that the set of multipliers M(X ∩ 𝓓 _p-2+s^p, X∩𝓓_q-2+s^q) is nontrivial. In this paper, we study the multipliers M(X ∩ 𝓓_p-2+s^p, X ∩ 𝓓_q-2+s^q) for distinct classical subspaces X of the Bloch space 𝓑, where X = 𝓑, BMOA or 𝓗^∞.

• Geophysics and Geophysical Exploration
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• 제14권3호
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• pp.249-253
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• 2011

Based on intrinsic and scattering quality factor ($Q_i^{-1}$ and$Q_s^{-1}$) obtained from the seismic data of the southeastern Korean Peninsula, the expected coda quality factor (${Q_{Cexp}}^{-1}$) was theoretically calculated using multiple scattering model, and was compared with other quality factors such as $Q_i^{-1}$, $Q_s^{-1}$, and observed $Q_C^{-1}$ obtained by single scattering model. While the ${Q_{Cexp}}^{-1}$ values are typically comparable to the $Q_i^{-1}$ values, the $Q_C^{-1}$ values are different from the ${Q_{Cexp}}^{-1}$ values except for the higher frequency. Future works require to consider depth-dependent attenuation.

• Bulletin of the Korean Mathematical Society
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• 제47권6호
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• pp.1181-1188
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• 2010

q-Volkenborn integrals ([8]) and fermionic invariant q-integrals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25]) studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by $${\varsigma}E,q,\varepsilon(s)=[2]q \sum\limits_{n=0}^\infty\frac{(-1)^n\epsilon^nq^{sn}}{[n]_q}$$ where 0 < q < 1, $\mathfrak{R}$(s) > 1, $\varepsilon{\in}T_p$, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.

• Journal of the Korean Geophysical Society
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• 제3권3호
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• pp.193-200
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• 2000

The attenuation characteristics($Q^{-1}$) are important factors representing the physical properties of the Earth interiors, and are essential for the quantitative prediction of strong ground-motion. Based on 156 earthquakes including 76 single-station record on the seismic station located Deok-jung Ri, southeastern Korea, we made the simultaneous measurement of P and S wave attenuation($Q_P^{-1}\;and\;Q_S^{-1}$) by means of extended coda-normalization method. Estimated $Q_P^{-1}\;and\;Q_S^{-1}$ decreased from $1{\times}10^{-2}\;and\;9{\times}10^{-3}$ at 1.5 Hz to $6{\times}10^{-4}\;and\;5{\times}10^{-4}$ at 24 Hz, respectively. This can be expressed by $Q_P^{-1}=0.01\;f^{-1.07}\;and\;Q_S^{-1}=0.01\;f^{-1.03}$ which indicate strong frequency dependence.

• Journal of the Korean earth science society
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• 제22권6호
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• pp.500-511
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• 2001

For the southeastern Korea aound the Yangsan fault we measured Q$_P^{-1}$ and Q$_S^{-1}$ simultaneously by using the extended coda-normalization method for seismograms registered at 9 stations deployed by KIGAM. We analyzed 707 seismograms of local earthquakes that occurred between December 1994 and February 2000. From seismograms, bandpass filtered traces were made by applying Butterworth filter with frequency-bands of 1${\sim}$2, 2${\sim}$4, 4${\sim}$8, 8${\sim}$16 and 16${\sim}$32 Hz. Estimated Q$_P^{-1}$ and Q$_S^{-1}$ values decrease from (7${\pm}$2)${\times}$10$^{-3}$ and (5${\pm}$4)${\times}$10$^{-4}$ at 1.5 Hz to (5${\pm}$4)${\times}$10$^{-3}$ and (5${\pm}$2)${\times}$10$^{-4}$ at 24 Hz, respectively. By fitting a power-law frequency dependent to estimated values over the whole stations, we obtained 0.009 (${\pm}$0.003)f$^{-1.05({\pm}0.14)$ for Q$_P^{-1}$ and 0.004 (${\pm}$0.001)f$^{-0.75({\pm}0.14)$) for Q$_S^{-1}$, where f is frequency in Hz.

• Journal of the Korean earth science society
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• 제27권4호
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• pp.475-485
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• 2006

The physical properties of the central and southwestern crust of South Korea were estimated by comparing values of ${Q_P}^{-1}\;and\;{Q_S}^{-1}$ in the Kimcheon and Mokpo areas. In order to get ${Q_P}^{-1}\;and\;{Q_S}^{-1}$ values, seismic data were collected from two stations of the KIGAM network (KMC and MUN) and four stations of the KMA network (CPN, KUC, MOP, and WAN). An extended coda-normalization method was applied to these data. Estimates of ${Q_P}^{-1}\;and\;{Q_S}^{-1}$ show variations depending on frequency. As frequencies vary from 3 Hz to 24 Hz, the estimates decrease from $(1.4{\pm}3.9){\times}10^{-3}\;to\;(2.3{\pm}3.5){\times}10^{-4}\;for\;{Q_P}^{-1}\;and\;(1.8{\pm}1.3){\times}10^{-3}\;to\;(1.9{\pm}1.5){\times}10^{-4}\;for\;{Q_S}^{-1}$ in central South Korea, and $(5.9{\pm}4.8){\times}10^{-3}\;to\;(2.2{\pm}3.8){\times}10^{-4}\;for\;{Q_P}^{-1}\;and\;(0.5{\pm}2.8){\times}10^{-3}\;to\;(1.8{\pm}1.6){\times}10^{-4}\;for\;{Q_S}^{-1}$ in southwestern South Korea. According that a frequency-dependent power law is applied to the data, the best fits of ${Q_P}^{-1}\;and\;{Q_S}^{-1}\;are\;0.003f^{-0.49}\;and\;0.005f^{-1.03}$ in central South Korea, and $0.026f^{-1.47}$ and $0.001f^{-0.49}$ in southwestern South Korea, respectively. These values almost correspond to those of seismically stable regions although ${Q_P}^{-1}$ values of southwestern South Korea are a little high due to lack of data used.

• Journal of the Korean earth science society
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• 제22권2호
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• pp.112-119
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• 2001

The Yangsan fault in the southeastern Korea has been receiving increasing attention in its seismic activity. In this fault region, by using the extended coda-normalization method for 707 seismograms of local earthquakes, were obtained 0.009f$^{-1.05}$ and 0.004f$^{-0.70}$ for fitting values of Q$_p^{-1}$ and Q$_s^{-1}$, respectively. These results indicate that Q$_p^{-1}$ and Q$_s^{-1}$ in the southeastern Korea is the lowest level in the world although the exponent values agree well with those in the other areas. The low Q-1 is not related to the movement of the Yangsan fault but to the tectonically inactive status like a shield area.

• Journal of the Korean Mathematical Society
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• 제54권5호
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• pp.1483-1504
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• 2017

We consider weak solutions of the instationary Navier-Stokes system in a smooth bounded domain ${\Omega}{\subset}{\mathbb{R}}^3$ with initial value $u_0{\in}L^2_{\sigma}({\Omega})$. It is known that a weak solution is a local strong solution in the sense of Serrin if $u_0$ satisfies the optimal initial value condition $u_0{\in}B^{-1+3/q}_{q,s_q}$ with Serrin exponents $s_q$ > 2, q > 3 such that ${\frac{2}{s_q}}+{\frac{3}{q}}=1$. This result has recently been generalized by the authors to weighted Serrin conditions such that u is contained in the weighted Serrin class ${{\int}_0^T}({\tau}^{\alpha}{\parallel}u({\tau}){\parallel}_q)^s$ $d{\tau}$ < ${\infty}$ with ${\frac{2}{s}}+{\frac{3}{q}}=1-2{\alpha}$, 0 < ${\alpha}$ < ${\frac{1}{2}}$. This regularity is guaranteed if and only if $u_0$ is contained in the Besov space $B^{-1+3/q}_{q,s}$. In this article we consider the limit case of initial values in the Besov space $B^{-1+3/q}_{q,{\infty}}$ and in its subspace ${{\circ}\atop{B}}^{-1+3/q}_{q,{\infty}}$ based on the continuous interpolation functor. Special emphasis is put on questions of uniqueness within the class of weak solutions.

• Journal of the Korean earth science society
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• 제28권6호
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• pp.720-732
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• 2007

For the Seoul metropolitan area and the eastern Kyeongsang Basin, we simultaneously calculated $Q_P^{-1}$ and $Q_S^{-1}$ by applying the extended coda-normalization method for 98 seismograms of local Earthquakes. As frequency increases from 1.5 Hz to 24 Hz, the result decreased from $(4.0{\pm}9.2){\times}10^{-3}$ to $(4.1{\pm}4.2){\times}10^{-4}$ for $Q_P^{-1}$ and $(5.5{\pm}5.6){\times}10^{-3}$ to $(3.4{\pm}1.3){\times}10^{-4}$ for $Q_S^{-1}$ in Seoul Metropolitan Area. The result of eastern Kyeongsang basin also decreased from $(5.4{\pm}8.8){\times}10^{-3}$ to $(3.7{\pm}3.4){\times}10^{-4}$ for $Q_P^{-1}$ and $(5.7{\pm}4.2){\times}10^{-3}$ to $(3.5{\pm}1.6){\times}10^{-4}$ for $Q_S^{-1}$. If we fit a frequency-dependent power law to the data, the best fits of $Q_P^{-1}$ and $Q_S^{-1}$ are $0.005f^{-0.89}$ and $0.004f^{-0.88}$ in Seoul metropolitan Area, respectively. The value of $Q_P^{-1}$ and $Q_S^{-1}$ in the eastern Kyeongsang basin are $0.007f^{-1.02}$ and $0.006f^{-0.99}$, respectively. The $Q_S^{-1}$ value of the eastern Kyeongsang basin is almost similar to Seoul metropolitan area. But the $Q_P^{-1}$ value of the eastern Kyeongsang basin is a little higher than that of Seoul metropolitan area. This may be that the crustal characteristics of the eastern Kyeongsang basin is seismologically more heterogeneous. However, these $Q_P^{-1}$ values in Korea belong to the range of seismically stable regions all over the world.