• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1027-1040
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• 2012

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum $S{\oplus}T$, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property ($gb$) and if ${\Pi}(T){\subset}{\sigma}_a(S)$, ${\PI}(S){\subset}{\sigma}_a(T)$, then $S{\oplus}T$ possesses property ($gb$) if and only if ${\sigma}_{SBF^-_+}(S{\oplus}T)={\sigma}_{SBF^-_+}(S){\cup}{\sigma}_{SBF^-_+}(T)$. Moreover, we prove that if T and S both satisfies generalized Browder's theorem, then $S{\oplus}T$ satis es generalized Browder's theorem if and only if ${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$.

• Honam Mathematical Journal
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• v.35 no.3
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• pp.445-506
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• 2013

Let ${\sigma}_s(N)$ denote the sum of the s-th power of the positive divisors of N and ${\sigma}_{s,r}(N;m)={\sum_{d{\mid}N\\d{\equiv}r\;mod\;m}}\;d^s$ with $N,m,r,s,d{\in}\mathbb{Z}$, $d,s$ > 0 and $r{\geq}0$. In a celebrated paper [33], Ramanuja proved $\sum_{k=1}^{N-1}{\sigma}_1(k){\sigma}_1(N-k)=\frac{5}{12}{\sigma}_3(N)+\frac{1}{12}{\sigma}_1(N)-\frac{6}{12}N{\sigma}_1(N)$ using elementary arguments. The coefficients' relation in this identity ($\frac{5}{12}+\frac{1}{12}-\frac{6}{12}=0$) motivated us to write this article. In this article, we found the convolution sums $\sum_{k<N/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(N-mk;2)$ for odd and even divisor functions with $i,j=0,1$, $m=1,2,4$, and $d{\mid}m$. If N is an odd positive integer, $i,j=0,1$, $m=1,2,4$, $s=0,1,2$, and $d{\mid}m{\mid}2^s$, then there exist $u,a,b,c{\in}\mathbb{Z}$ satisfying $\sum_{k& lt;2^sN/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(2^sN-mk;2)=\frac{1}{u}[a{\sigma}_3(N)+bN{\sigma}_1(N)+c{\sigma}_1(N)]$ with $a+b+c=0$ and ($u,a,b,c$) = 1(Theorem 1.1). We also give an elementary problem (O) and solve special cases of them in (O) (Corollary 3.27).

• Journal of the Korean Geotechnical Society
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• v.22 no.8
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• pp.63-72
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• 2006

In the ASTM and ISRM, an uniaxial compressive strength(${\sigma}_{c}$) has been estimated to be 23(ASTM) or $20{\sim}25$(ISRM) times of point load strength index using a diametral test regardless of the rock rating or geological conditions. This paper presents a relationship between $I_{s}$ and ${\sigma}_{c}$ of a weak sedimentary rocks on Ulsan of the Kyung-Sang Basin in Korea. In the results of 291 for ${\sigma}_{c}$ test and 2310 for $I_{s}$ test from 77 spots, the predicted errors of ${\sigma}_{c}$ determined by strength ratio of ${\sigma}_{c}/I_{s}$ have been relatively less than those determined by linear regression analysis. And in case of weak sedimentary rocks such as mudstones, shales and sandstones, ${\sigma}_{c}$ should be lower than those suggested by ISRM and ASTM.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.331-360
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• 2013

Let ${\sigma}_s(N)$ denote the sum of the sth powers of the positive divisors of a positive integer N and let $\tilde{\sigma}_s(N)={\sum}_{d|N}(-1)^{d-1}d^s$ with $d$, N, and s positive integers. Hahn [12] proved that $$16\sum_{k. In this paper, we give a generalization of Hahn's result. Furthermore, we find the formula ${\sum}_{k=1}^{N-1}\tilde{\sigma}_1(2^{n-m}k)\tilde{\sigma}_3(2^nN-2^nk)$ for $m(0{\leq}m{\leq}n)$.

• Journal of the Chungcheong Mathematical Society
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• v.17 no.1
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• pp.85-101
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• 2004

Let A be a Hopf algebra with a linear form ${\sigma}:k{\rightarrow}A{\otimes}A$, which is convolution invertible, such that ${\sigma}_{21}({\Delta}{\otimes}id){\tau}({\sigma}(1))={\sigma}_{32}(id{\otimes}{\Delta}){\tau}({\sigma}(1))$. We define Hopf algebras, ($A_{\sigma}$, m, u, ${\Delta}_{\sigma}$, ${\varepsilon}$, $S_{\sigma}$). If B and C are opposite skew copaired Hopf algebras and $A=B{\otimes}_kC$ then we find Hopf algebras, ($A_{[{\sigma}]}$, $m_B{\otimes}m_C$, $u_B{\otimes}u_C$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}B{\otimes}{\varepsilon}_C$, $S_{[{\sigma}]}$). Let H be a finite dimensional commutative Hopf algebra with dual basis $\{h_i\}$ and $\{h_i^*\}$, and let $A=H^{op}{\otimes}H^*$. We show that if we define ${\sigma}:k{\rightarrow}H^{op}{\otimes}H^*$ by ${\sigma}(1)={\sum}h_i{\otimes}h_i^*$ then ($A_{[{\sigma}]}$, $m_A$, $u_A$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}_A$, $S_{[{\sigma}]}$) is the dual space of Drinfeld double, $D(H)^*$, as Hopf algebra.

• Korean Journal of Food Science and Technology
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• v.32 no.4
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• pp.974-978
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• 2000

The vane method was used to measure yield stresses of five commercial kochujang samples under a controlled shear-rate operating condition. Magnitudes of vane yield stress were higher than those of yield stress using Casson model with a concentric cylinder viscometer. Magnitudes of vane yield stresses showed great differences between the static $({\sigma}_s)$ and dynamic yield stresses $({\sigma}_d)$ of kochujang samples with undisturbed structure (UDS) and with broken down structure (BDS). A dimensionless yield number $(N{\sigma}_o)$ was determined from the ratio of ${\sigma}s$ to ${\sigma}d$ in order to describe the existence of temporary structure of kochujang.

• Journal of Korean Society for Quality Management
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• v.32 no.4
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• pp.140-155
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• 2004

Six Sigma Management has been introduced as a management strategy by leading companies and some small and medium-sized companies in Korea. But it is not easy for small and medium-sized companies to implement this new management system because of their business environment such as poor manpower or lack of time, etc. This paper proposes an effective implementation model of the Six Sigma Management for small and medium-sized companies using the management by process. First, We review the concept of Six Sigma Management and consider the problems of introducing the system for small and medium-sized companies. And then, Six Sigma Management model for small and medium-sized companies named SIX SIGMA-s is presented. Finally, we present a case study of applying SIX SIGMA-s for a medium- sized company.

• Atmosphere
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• v.25 no.1
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• pp.185-191
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• 2015

Ambient measurements of aerosol light absorption (${\sigma}_a$) and scattering coefficients (${\sigma}_s$) were done at Gosan climate observatory during summer 2008 using a 3-wavelength photoacoustic soot spectrometer (PASS). PASS was deployed photoacoustic method for light absorption and integrated nephelometry for light scattering measurements. The ${\sigma}_a$ and ${\sigma}_s$ from PASS were compared with those from co-located aethalometer and nephelometer measurements. The aethalometer measurements of ${\sigma}_a$ correlated reasonably well with photoacoustic measurements, but the slope of the linear fitting line indicated the PASS measurement values of ${\sigma}_a$ were larger by a factor of 1.53. The nephelometer measurement values of ${\sigma}_s$ correlated very well with PASS measurements of ${\sigma}_s$, with a slope of 1.12 and a small offset. Comparing to the aethalometer measurements, the photoacoustic measurements of ${\sigma}_a$ didn't exhibit a significant (i.e., the ratio between aethalometer and PASS increased) change with increasing relative humidity (RH). The ratio of ${\sigma}_s$ between nephelometer and PASS increased with increasing RH, especially when the RH increased beyond 80%. This apparent increase in ${\sigma}_s$ with RH may be due to the contribution of hygroscopic growth of aerosols.

• Journal of Korean Institute of Industrial Engineers
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• v.32 no.4
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• pp.268-278
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• 2006

Ten years have passed since Samsung SDI and LG Electronics introduced six sigma. Six sigma spreads on the wide industry areas such as service and public sectors as well as manufacturing industry. Six sigma speaks for Korean enterprise's management innovation now. Then, can we evaluate that Korean enterprise's six sigma has settled down successfully? This study compares Korean enterprise's six sigma with those of Japanese and American enterprises in the view of leadership, project, people, and methodology. We suggest five topics for the successful settlement of six sigma in Korean enterprises; 1) Role of executives and managers should be changed more actively, 2) Strategy-connected projects should be excavated, 3) Utilization and development plan should be made for black belt and master black belt, 4) Six sigma methodology should be advanced, especially DFSS methodology, and finally 5) Several innovation methodology should be integrated.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.195-200
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• 2010

Let C be a closed convex set in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$. Assume that ${\Sigma}$ is an n-dimensional compact minimal submanifold outside C such that ${\Sigma}$ is orthogonal to ${\partial}C$ along ${\partial}{\Sigma}{\cap}{\partial}C$ and ${\partial}{\Sigma}$ lies on a geodesic sphere centered at a fixed point $p{\in}{\partial}{\Sigma}{\cap}{\partial}C$ and that r is the distance in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ from p. We make use of a modified volume $M_p({\Sigma})$ of ${\Sigma}$ and obtain a sharp relative isoperimetric inequality $$\frac{1}{2}n^n{\omega}_nM_p({\Sigma})^{n-1}{\leq}Vol({\partial}{\Sigma}{\sim}{\partial}C)^n$$, where ${\omega}_n$ is the volume of a unit ball in ${\mathbb{R}}^n$ Equality holds if and only if ${\Sigma}$ is a totally geodesic half ball centered at p.