• Korean Journal of Mathematics
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• v.25 no.3
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• pp.349-358
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• 2017

The cyclic group $C_n={\langle}(12{\cdots}n){\rangle}$ acts on the set $(^{[n]}_k)$ of all k-subsets of [n]. In this action of $C_n$ the number of orbits of size d, for d | n, is $$O^{n,k}_d={\frac{1}{d}}{\sum\limits_{{\frac{n}{d}}{\mid}s{\mid}n}}{\mu}({\frac{ds}{n}})(^{n/s}_{k/s})$$. Stanton and White [6] generalized the above identity to construct the orbit polynomials $$O^{n,k}_d(q)={\frac{1}{[d]_{q^{n/d}}}}{\sum\limits_{{\frac{n}{d}}{\mid}s{\mid}n}}{\mu}({\frac{ds}{n}})[^{n/s}_{k/s}]_{q^s}$$ and conjectured that $O^{n,k}_d(q)$ have non-negative coefficients. In this paper we give a constructive proof for the positivity of coefficients of the orbit polynomial $O^{n,2}_d(q)$.

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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• pp.455-462
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• 2012

The cyclic group $Cn={\langle}(12{\cdots}n){\rangle}$ acts on the set ($^{[n]}_k$) of all $k$-subsets of [$n$]. In this action of $C_n$ the number of orbits of size $d$, for $d|n$, is $$O^{n,k}_d=\frac{1}{d}\sum_{\frac{n}{d}|s|n}{\mu}(\frac{ds}{n})(^{n/s}_{k/s})$$. Stanton and White[7] generalized the above identity to construct the orbit polynomials $$O^{n,k}_d(q)=\frac{1}{[d]_{q^{n/d}}}\sum_{\frac{n}{d}|s|n}{\mu}(\frac{ds}{n})[^{n/s}_{k/s}]{_q}^s$$ and conjectured that $O^{n,k}_d(q)$ have non-negative coefficients. In this paper we give a combinatorial proof for the positivity of coefficients of the orbit polynomial $O^{n,3}_d(q)$.

• Journal of Nutrition and Health
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• v.21 no.2
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• pp.129-133
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• 1988

The lonitudinal determination of specific gravity, total solid and protein contents of human milk were carried out in 27 Korean women. Human milk samples from the subjects were collected at 15, 30, 60, 90, 120, 150th days of lactation. 1) The average values of specific gravity of the milk was 1, 0281 (S.D. 0.0018, n=12), with a range of 1.0200-1.0383. 2) The average values of total soild of the milk was 12.07g/100ml(S.D. 0.38, n=112), with a range of 9.36-15.88g/100ml. 3) The average values of protein content of the milk was 1.20g/100ml (S.D. 0.14, n=112), with a range of 1.09-1.46g/100ml. A slight decrease of specific gravity and protein content in human milk was found during the course of lactation, but significant decrease was not found in total solid content.

• Journal of Sericultural and Entomological Science
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• v.45 no.1
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• pp.6-9
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• 2003

This experiment was done to know developmental characteristics of Sericin jam. Incubation periods were 10 day 2 hr, and 10 day 1 hr. 11 day 1 hr for Nd-s jam, N $d^{H}$ jam, and Baegok jam, respectively. Hatching rates were 83.9, 83.3 and 96.0% for Nd-s jam, N $d^{H}$ jam, and Baegok jam. Larval periods were, 20 days 1 hr for Nd-s jam, 20 days 5 hrs for N $d^{H}$ jam, and 22 days 12 hrs for Baegok jam. Death rate of larvae was highest in N $d^{H}$ iam, followed by Nd-s jam and Baegok jam. Pupation rate was highest in Baegok Jam followed by Nd-s jam and that of N $d^{H}$ jam was the lowest among the three. Cocoon weight was 1.39, 1.08, and 2.01 g for Nd-s jam, N $d^{H}$ jam, and Baegok jam, respectively. Shell weight were 13, 3, and 48 cg for Nd-s jam, N $d^{H}$ jam, and Baegok jam. Cocoon shell ratios were 9.0% for Nd-s jam, 2.8% for N $d^{H}$ jam and 23.9% for Baegok jam. Cocoon sizes were 30.6${\times}$15.8 mm for Nd-s jam, 24.7${\times}$14.9 mm for N $d^{H}$ jam and 35.8 ${\times}$ 20.5 mm(1${\times}$w) for Baegok jam.w) for Baegok jam.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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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.

• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.41-49
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• 2009

Let $\zeta$ be a fixed complex number. In this paper, we study the quantity $S(\zeta,\;n):=mas_{f{\in}{\Lambda}_n}\;|f(\zeta)|$, where ${\Lambda}_n$ is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given ${\zeta}\;{\in}\;{\mathbb{C}}$ and $n\;{\in}\;{\mathbb{N}}$, the quantity S($\zeta$, n) can be calculated. Then we compute the limit $lim_{n{\rightarrow}{\infty}}\;S(\zeta,\;n)/n$ for every ${\zeta}\;{\in}\;{\mathbb{C}}$ of modulus 1. It is equal to 1/$\pi$ if $\zeta$ is not a root of unity. If $\zeta\;=\;\exp(2{\pi}ik/d)$, where $d\;{\in}\;{\mathbb{N}}$ and k $\in$ [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin($\pi$/d)) and 1/(2d sin($\pi$/2d)) for d = 1, d even and d > 1 odd, respectively.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.503-517
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• 2016

For positive integers d and n, let $[n]^d$ be the set of all vectors ($a_1,a_2,{\cdots},a_d$), where ai is an integer with $0{\leq}a_i{\leq}n-1$. A subset S of $[n]^d$ is called a Sidon set if all sums of two (not necessarily distinct) vectors in S are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $[n]^d$. First, let $\mathcal{Z}_{n,d}$ be the number of all Sidon sets in $[n]^d$. We show that ${\log}(\mathcal{Z}_{n,d})={\Theta}(n^{d/2})$, where the constants of ${\Theta}$ depend only on d. Next, we estimate the maximum size of Sidon sets contained in a random set $[n]^d_p$, where $[n]^d_p$ denotes a random set obtained from $[n]^d$ by choosing each element independently with probability p.

• Progress in Superconductivity and Cryogenics
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• v.18 no.2
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• pp.5-7
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• 2016

We present an experimental investigation of the superconducting transition temperatures, $T_c$, of superconductor/ferromagnet bilayers with varying the thickness of ferromagnetic layer. FeN was used for the ferromagnetic (F) layer, and NbN and Nb were used for the superconducting (S) layer. The results were obtained using three different-thickness series of the S layer of the S/F bilayers: NbN/FeN with NbN thickness, $d_{NbN}{\approx}9.3nm$ and $d_{NbN}{\approx}10nm$, and Nb/FeN with Nb thickness $d_{Nb}{\approx}15nm$. $T_c$ drops sharply with increasing thickness of the ferromagnetic layer, $d_{FeN}$, before maximal suppression of superconductivity at $d_{FeN}{\approx}6.3nm$ for $d_{NbN}{\approx}10nm$ and at $d_{FeN}{\approx}2.5nm$ for $d_{Nb}{\approx}15nm$, respectively. After shallow minimum of $T_c$, a weak $T_c$ oscillation was observed in NbN/FeN bilayers, but it was hardly observable in Nb/FeN bilayers.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1137-1146
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• 2011

Let {$X_i$, $i{\geq}1$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $S_n={\sum}_{i=1}^n\;X_i$, $M_n=max_{1{\leq}i{\leq}n}\;{\mid}S_i{\mid}$ and $V_n^2={\sum}_{i=1}^n\;X_i^2$. Then for d > -1, we showed that under some regularity conditions, $$\lim_{{\varepsilon}{\searrow}0}{\varepsilon}^2^{d+1}\sum_{n=1}^{\infty}\frac{(loglogn)^d}{nlogn}I\{M_n/V_n{\geq}\sqrt{2loglogn}({\varepsilon}+{\alpha}_n)\}=\frac{2}{\sqrt{\pi}(1+d)}{\Gamma}(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2d+2}}\;a.s.$$ holds in this paper, where If g denotes the indicator function.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.131-150
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• 2007

In this paper, we will define direct producted $W^*-porobability$ spaces over their diagonal subalgebras and observe the amalgamated free-ness on them. Also, we will consider the amalgamated free stochastic calculus on such free probabilistic structure. Let ($A_{j},\;{\varphi}_{j}$) be a tracial $W^*-porobability$ spaces, for j = 1,..., N. Then we can define the corresponding direct producted $W^*-porobability$ space (A, E) over its N-th diagonal subalgebra $D_{N}\;{\equiv}\;\mathbb{C}^{{\bigoplus}N}$, where $A={\bigoplus}^{N}_{j=1}\;A_{j}\;and\;E={\bigoplus}^{N}_{j=1}\;{\varphi}_{j}$. In Chapter 1, we show that $D_{N}-valued$ cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the $D_{N}-freeness$ is characterized by the direct sum of scalar-valued freeness. As application, the $D_{N}-semicircularityrity$ and the $D_{N}-valued$ infinitely divisibility are characterized by the direct sum of semicircularity and the direct sum of infinitely divisibility, respectively. In Chapter 2, we will define the $D_{N}-valued$ stochastic integral of $D_{N}-valued$ simple adapted biprocesses with respect to a fixed $D_{N}-valued$ infinitely divisible element which is a $D_{N}-free$ stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the $D_{N}-valued$ case.