• Korean Journal of Mathematics
• /
• 제25권3호
• /
• pp.349-358
• /
• 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
• /
• 제30권3_4호
• /
• pp.455-462
• /
• 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
• /
• 제21권2호
• /
• pp.129-133
• /
• 1988

수유기간의 경과에 따른 모유중의 비중, 총고형분, 단백질 함량 변화를 살펴 보고자 수유부 27명을 대상으로 분만후 15일부터 150일까지 추적하여 채취한 총112개 모유시료로 부터 얻은 결과는 다음과 같다. 즉 모유 비중의 경시적인 변화는 15일경에 1.0298(S.D. 0.0044, n=25), 30일경에 1.0304(S.D. 0.0046, n=24), 60일경에 1.0270(S.D. 0.0035, n=20) 90일경에 1.0293(S.D. 0.0055, n=18), 120일경에 1.0254(S.D. 0.0038, n=16), 150일경에 1.0268(S.D. 0.0039, n=9)로서 나타났고 수유기간이 경과함에 따라 유의하게 감소하는 경향을 보였다. 총고형분은 모유 100$m\ell$중에 15일 경에 12.48g(S.D. 1.23, n=25), 30일경에 12.31g(S.D. 1.31, n=24) 60일경에 12.47g (S.D. 1.45, n=20), 90일경에 11.81g (S.D. 1.85, n=18), 120일경에 11.44g(S.D. 0.92, n=16) 150일경에 11.88 (S.D. 0.89, n=9)으로 나타났고 수유기간의 경과에 따른 유의한 차이는 보이지 않았다. 모유 100$m\ell$중 단백질함량은 1.07~1.46g의 범위였고, 수유기간별 변화는 15일경에 1.46g (S.D. 0.20, n=25), 30일 경에 1.32g (S.D. 0.19, n=24) 60일경에 1.15g (S.D. 0.10, n=20), 90일경에 1.09g (S.D. 0.17, n=18), 120일경에 1.11g (S.D. 0.17, n=16), 150일 경에 1.07g (S.D. 0.18, n=9)으로 수유기간이 경과 함에 따라 유의하게 감소하는 경향을 보였다. 또한 수유기간 15, 30, 50, 90, 120, 150일에 수집한 모유시료 전체의 총평균 단백질함량은 1.20g (S.D. 0.14, n=27, 총시료수 112)로 나타났다.

• 한국잠사곤충학회지
• /
• 제45권1호
• /
• pp.6-9
• /
• 2003

세리신잠의 성장발육과 실용형질 등 세리신잠의 특성을 알아보기 위하여 Nd-s잠과 N$d^{H}$잠을 대상으로 조사하여 얻은 실험결과는 다음과 같았다. 1. 최청기간은 Nd-s잠 10일 2시간, N$d^{H}$ 잠 10일 1시간, 백옥잠은 11일 1시간이었다. 실용부화율에 있어서 Nd-s잠과 Nd.$^{H}$ 잠은 83.9%와 83.3%였고, 백옥잠은 96%였다. 전령경과일수는 Nd-s잠 20일 1시간 N$d^{H}$잠 20일 5시간, 백옥잠 22일 12시간이었다. 2. 감잠비율은 Nd-s잠 12.0%, N$d^{H}$ 잠 18.7%, 백옥잠 5.3%이었으며, 화용비율은 Nd-s잠 83.0%, N$d^{H}$ 잠 76.3%, 백옥잠 92.3%였다. 3. 단견중은 Nd-s잠 1.39 g, N$d^{H}$ 잠 1.08g, 백옥잠 2.01 g, 전견중은 Nd-s 잠 13 cg, N$d^{H}$ 잠 3 cg, 백옥잠 48 cg이였으며, 견층비율은 Nd-s 잠 9.0%, N$d^{H}$ 잠 2.8%, 백옥잠 23.9%였다. 4. 고치의 크기는 Nd-s잠 장경 30.6mm, 단경 15.8mm, N$d^{H}$ 잠 장경 24.7mm, 단경 14.9mm, 백옥잠 장경 35.8mm, 단경 20.5mm이었다.5.8mm, 단경 20.5mm이었다.

• 대한수학회보
• /
• 제54권3호
• /
• pp.731-736
• /
• 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.

• 대한수학회지
• /
• 제46권1호
• /
• pp.41-49
• /
• 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.

• 대한수학회지
• /
• 제53권3호
• /
• pp.503-517
• /
• 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.

• 한국초전도ㆍ저온공학회논문지
• /
• 제18권2호
• /
• pp.5-7
• /
• 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.

• 대한수학회보
• /
• 제48권6호
• /
• pp.1137-1146
• /
• 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.

• 대한수학회보
• /
• 제44권1호
• /
• pp.131-150
• /
• 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.