• Journal of the Korean Mathematical Society
• /
• v.50 no.2
• /
• pp.331-360
• /
• 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 $m(0{\leq}m{\leq}n)$.

• Honam Mathematical Journal
• /
• v.39 no.2
• /
• pp.297-305
• /
• 2017

Let ${\alpha}$ > 0 be a real number and $(s_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$. We investigate Dirichlet series of the form ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-s}$. To do this, a generalization of Euler's totient function is required. For a real ${\alpha}$ > 0 and a positive integer n, an arithmetic function ${\varphi}{\alpha}(n)$ is defined to be the number of positive integers m for which gcd(m, n) = 1 and 0 < m/n < ${\alpha}$. Under a condition Re(s) > 1, this paper establishes an identity ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-S}=1+{\sum}^{\infty}_{n=1}{\varphi}_{\alpha}(n)({\zeta}(s)-{\zeta}(s,1+n^{-1}))n^{-s}$.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.4
• /
• pp.549-560
• /
• 1997

For positive integers $n_i \geq 2, i = 1, 2, 3, 4$, such that $\Sigma \frac{n_i}{1} < 2$, there exists a quadrilateral $P = P_1 P_2 P_3 P_4$ in the hyperbolic plane $H^2$ with the interior angle $\frac{n_i}{\pi}$ at $P_i$. Let $\Gamma \subset Isom(H^2)$ be the (discrete) group generated by reflections in each side of $P$. Then the quotient space $H^2/\gamma$ is a differentiable orbifold of type $(^* n_1 n_2 n_3 n_4)$. It will be shown that the deformation space of $Rp^2$-structures on this orbifold can be mapped continuously and bijectively onto the cell of dimension 4 - \left$\mid$ {i$\mid$n_i = 2} \right$\mid$$.

• Journal of the Korean Mathematical Society
• /
• v.40 no.3
• /
• pp.409-422
• /
• 2003

Let $F: \mathbb{C}^k\;{\rightarrow}\;\mathbb{C}^k$ be a dynamical system and let $\{x_n\}_{n{\geq}0}$ denote an orbit of F. We study the relation between $\{x_n\}$ and pseudoorbits $\{y_n}, y_0=x_0.\;Here\;y_{n+1}=F(y_n)+s_n.$ In general $y_n$ might diverge away from $x_n.$ Our main problem is whether there exists arbitrarily small $t_n$ so that if $\tilde{y}_{n+1}=F(\tilde{y}_n)+s_n+t_n,$ then $\tilde{y}_n$ remains close to $x_n.$ This leads naturally to the concept of sustainable orbits, and their existence seems to be closely related to the concept of hyperbolicity, although they are not in general equivalent.

• Proceedings of the Korea Information Processing Society Conference
• /
• 2000.10a
• /
• pp.633-636
• /
• 2000

본 논문에서는 하이퍼큐브보다 망비용이 개선된 HCN(n,n)과 2n-hypercube 사이의 임베딩을 분석한다. 2n-hypercube를 HCN(n.n)에 연장율 3에 임베딩 가능함을 보이고, HCN(n,n)을 2n-hypercube에 임베딩하는 비용이 O(n)임을 보인다.

• Journal of the Korean institute of surface engineering
• /
• v.35 no.6
• /
• pp.408-414
• /
• 2002

TiN and TiAlN coating layer were deposited on WC-Co steel substrates by an arc ion plating(AIP) technique. The crystallinity and morphology for the deposited coating layers were characterized by X-ray diffraction (XRD) and scanning electron microscopy (SEM). The impact behaviors of the deposited TiN and TiAlN coating layer were investigated with a ball-on-plate impact tester. Beyond $10^2$ impact cycle, TiAlN coating layer showed superior impact wear resistance compared to TiN coating layer. On the other hand, both TiN and TiAlN coating layers started to be partially failed between $10^2$ and $10^3$ impact cycle. Above $10^3$ impact cycle, TiN and TiAlN coating layers showed similar impact behavior because of the substrate effect.

• Proceedings of the Korean Institute of Surface Engineering Conference
• /
• 2013.05a
• /
• pp.119-120
• /
• 2013

공구, 다이몰드 등에 널리 쓰이는 TiC, TiN, CrN, TiCrN, TiAlN 코팅의 산화특성을 비교하기 위하여 $600^{\circ}C-900^{\circ}C$에서 대기중 산화시험을 실시하였다. 내산화성은 (TiC, TiN), TiAlN, TiCrN, CrN 코팅의 순서로 증가하였다. 코팅원소중 Ti는 $TiO_2$로, Cr은 $Cr_2O_3$로, Al은 $Al_2O_3$로 산화되었다.

• East Asian mathematical journal
• /
• v.31 no.1
• /
• pp.103-107
• /
• 2015

We use some known contiguous function relations for $_2F_1$ to show how simply the following three recurrence relations for Jacobi polynomials $P_n^{({\alpha},{\beta)}(x)$: (i) $({\alpha}+{\beta}+n)P_n^{({\alpha},{\beta})}(x)=({\beta}+n)P_n^{({\alpha},{\beta}-1)}(x)+({\alpha}+n)P_n^{({\alpha}-1,{\beta})}(x);$ (ii) $2P_n^{({\alpha},{\beta})}(x)=(1+x)P_n^{({\alpha},{\beta}+1)}(x)+(1-x)P_n^{({\alpha}+1,{\beta})}(x);$ (iii) $P_{n-1}^{({\alpha},{\beta})}(x)=P_n^{({\alpha},{\beta}-1)}(x)+P_n^{({\alpha}-1,{\beta})}(x)$ can be established.

• Journal of the Korean Institute of Gas
• /
• v.11 no.4
• /
• pp.73-78
• /
• 2007

The lower flash points for the n-pentanol+n-propionic acid and n-pentanol+n-butyric acid systems, in air at atmospheric pressure, were measured by using Pensky-Martens closed cup apparatus. The experimental data were compared with the values calculated by the Raoult's law and optimization method. The calculated values based on the optimization method were found to be better than those based on the Raoult's law.

• Journal of the Chungcheong Mathematical Society
• /
• v.21 no.2
• /
• pp.279-285
• /
• 2008

This paper presents some characterizations of the Weibull distribution by the independence of record values. We prove that $X{\sim}Weibull(1,{\alpha})$, ${\alpha}>0$ if and only if $\frac{X_{U(n+1)}}{X_{U(n+1)}-X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent. We show that $X{\sim}Weibull(1,{\alpha})$, ${\alpha}>0$ if and only if $\frac{X_{U(n+1)}}{X_{U(n+1)}-X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent. And we establish that $X{\sim}Weibull(1,{\alpha})$, ${\alpha}>0$ if and only if $\frac{X_{U(n+1)}+X_{U(n)}}{X_{U(n+1)}-X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent.