• 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 Mathematical Society
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• v.47 no.2
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• pp.263-275
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• 2010

Let {$X_n;n\;\geq\;1$} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n\;=\;{\sum}^n_{k=1}X_k$, $M_n\;=\;max_{k{\leq}n}|S_k|$, $n\;{\geq}\;1$. Suppose $\sigma^2\;=\;EX^2_1+2{\sum}^\infty_{k=2}EX_1X_k$ (0 < $\sigma$ < $\infty$). We prove that for any b > -1/2, if $E|X|^{2+\delta}$(0<$\delta$$\leq$1), then $$lim\limits_{\varepsilon\searrow0}\varepsilon^{2b+1}\sum^{\infty}_{n=1}\frac{(loglogn)^{b-1/2}}{n^{3/2}logn}E\{M_n-\sigma\varepsilon\sqrt{2nloglogn}\}_+=\frac{2^{-1/2-b}{\sigma}E|N|^{2(b+1)}}{(b+1)(2b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2(b+1)}}$$ and for any b > -1/2, $$lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum^{\infty}_{n=1}\frac{(loglogn)^b}{n^{3/2}logn}E\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8loglogn}}-M_n\}_+=\frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2b+2'}}$$, where $\Gamma(\cdot)$ is the Gamma function and N stands for the standard normal random variable.

• Journal of the Chungcheong Mathematical Society
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• v.16 no.1
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• pp.81-96
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• 2003

Let ${\sigma}:k{\rightarrow}A{\otimes}B$ be a skew copairing on (A, B), where A and B are Hopf algebras of the same dimension n. Skew dual bases of A and B are introduced. If ${\sigma}$ is an invertible skew copairing then we can give a 2-cocycle bilinear form [${\sigma}$] on $A{\otimes}B$ and define a new Hopf algebra.

• Proceedings of the Korean Reliability Society Conference
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• 2005.06a
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• pp.141-145
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• 2005

Nowadays, most industries take $B_{6\sigma}$ quality level as a goal of the ultimate quality level of their products. On the other hand, $B_{10}$ to life indicates the time that $10\%$ of products are failed. There is no relation between the $B_{6\sigma}$ quality level and the $B_{10}$ to life. Therefore, some industries perform their quality control activities and reliability engineering activities separately. So I propose one measure which can express quality and reliability level simultaneously for the products to pursue quality and reliability activities together in the industry.

• Bulletin of the Korean Chemical Society
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• v.31 no.4
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• pp.959-964
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• 2010

In this work, equilibrium molecular dynamics (MD) simulations in a microcanonical ensemble are performed to evaluate the friction coefficient of a Brownian particle (BP) in a Lennard-Jones (LJ) solvent. The friction coefficients are determined from the time dependent friction coefficients and the momentum autocorrelation functions of the BP with its infinite mass at various ratios of LJ size parameters of the BP and solvent, ${\sigma}_B/{\sigma}_s$. The determination of the friction coefficients from the decay rates of the momentum autocorrelation functions and from the slopes of the time dependent friction coefficients is difficult due to the fast decay rates of the correlation functions in the momentum-conserved MD simulation and due to the scaling of the slope as 1/N (N: the number of the solvent particle), respectively. On the other hand, the friction coefficient can be determined correctly from the time dependent friction coefficient by measuring the extrapolation of its long time decay to t=0 and also from the decay rate of the momentum autocorrelation function, which is obtained by time integration of the time dependent friction coefficient. It is found that while the friction coefficient increases quadratically with the ratio of ${\sigma}_B/{\sigma}_s$ for all ${\sigma}_B$, for a given ${\sigma}_s$ the friction coefficient increases linearly with ${\sigma}_B$.

• ETRI Journal
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• v.33 no.6
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• pp.897-903
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• 2011

A hybrid ${\Delta}{\Sigma}$ modulator for audio applications is presented in this paper. The pulse generator for digital-to-analog converter alleviates the requirement of the external clock jitter and calibrates the coefficient variation due to a process shift and temperature changes. The input resistor network in the first integrator offers a gain control function in a dB-linear fashion. Also, careful chopper stabilization implementation using return-to-zero scheme in the first continuous-time integrator minimizes both the influence of flicker noise and inflow noise due to chopping. The chip is implemented in a 0.13 ${\mu}m$ CMOS technology (I/O devices) and occupies an active area of 0.37 $mm^2$. The ${\Delta}{\Sigma}$ modulator achieves a dynamic range (A-weighted) of 97.8 dB and a peak signal-to-noise-plus-distortion ratio of 90.0 dB over an audio bandwidth of 20 kHz with a 4.4 mW power consumption from 3.3 V. Also, the gain of the modulator is controlled from -9.5 dB to 8.5 dB, and the performance of the modulator is maintained up to 5 nsRMS external clock jitter.

• 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 ${\sigma}_{BW}(S{\oplus}T)={\sigma}_{BW}(S){\cup}{\sigma}_{BW}(T)$.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.349-355
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• 2016

For any integer $n{\geq}2$, each palindrome of n induces a circulant graph of order n. It is known that for each integer $n{\geq}2$, there is a one-to-one correspondence between the set of (resp. aperiodic) palindromes of n and the set of (resp. connected) circulant graphs of order n (cf. [2]). This bijection gives a one-to-one correspondence of the palindromes ${\sigma}$ with $gcd({\sigma})=1$ to the connected circulant graphs. It was also shown that the number of palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ is the same number of aperiodic palindromes of n. Let $a_n$ (resp. $b_n$) be the number of aperiodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). Let $c_n$ (resp. $d_n$) be the number of periodic palindromes ${\sigma}$ of n with $gcd({\sigma})=1$ (resp. $gcd({\sigma}){\neq}1$). In this paper, we calculate the numbers $a_n$, $b_n$, $c_n$, $d_n$ in two ways. In Theorem 2.3, we $n_d$ recurrence relations for $a_n$, $b_n$, $c_n$, $d_n$ in terms of $a_d$ for $d{\mid}n$ and $d{\neq}n$. Afterwards, we nd formulae for $a_n$, $b_n$, $c_n$, $d_n$ explicitly in Theorem 2.5.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.93-107
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• 1995

In their paper [2,3], Cameron and Storvick introduced some classes $S"+m$ and of functionals on classical Wiener spaces $C_0[a,b]$. For such functionals, they showed that the analytic Feynman integral exists and they gave some formulas for this integral. Moreover they obtained that the functionals of the form $$ (1.1) F(x) = exp {\int^b_a{\theta(s,x(x))dx} $$ are in S" where they assumbed that the potential $\delta : [a,b] \times R \to C$ satisfies (i) for each $s \in [a,b], \theta(s,\cdot)$ is the Fourier-Stieltjes transform of $\sigma_s \in M(R)$, (ii) for each Borel subset E of $[a,b] \times R, \sigma_s (E^{(s)})$ is a Borel measurable function of s on [a,b], and (iii) the total variation $\Vert \sigma_s \Vert$ of $\sigma_s$ is bounded as a function of s.tion of s.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1273-1283
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• 2019

In matrix analysis, the Wielandt-Mirsky conjecture states that $$dist({\sigma}(A),{\sigma}(B)){\leq}{\parallel}A-B{\parallel}$$ for any normal matrices $A,B{\in}{\mathbb{C}}^{n{\times}n}$ and any operator norm ${\parallel}{\cdot}{\parallel}$ on $C^{n{\times}n}$. Here dist(${\sigma}(A),{\sigma}(B)$) denotes the optimal matching distance between the spectra of the matrices A and B. It was proved by A. J. Holbrook (1992) that this conjecture is false in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt inequality). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.