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

• The Transactions of the Korea Information Processing Society
• /
• v.5 no.1
• /
• pp.40-48
• /
• 1998

Linear size expanders have been studied in many fields for the practical use, which make it possible to connect large numbers of device chips in both parallel communication systems and parallel computers. One major limitation on the efficiency of parallel computer designs has been the highly cost of parallel communication between processors and memories. Linear order concentrators can be used to construct theoretically optimal interconnection network schemes. Existing explicitly defined constructions are based on expanders, which have large constant factors, thereby rendering them impractical for reasonable sized networks. For these objectives, we use the more detailed matching points in permutation functions, to find out the bigger expansion constant from an equation, $\mid\Gamma_x\mid\geq[1+d(1-\midX\mid/n)]\midX\mid$. This paper presents an improvement of expansion constant on constructing concentrators using expanders, which realizes the reduction of the size in a superconcentrator by a constant factor. As a result, this paper shows an explicit construction of (n, 5, $1-\sqrt{3/2}$) expander. Thus, superconcentrators with 209n edges can be obtained by applying to the expanders of Gabber and Galil's construction.

• Communications of the Korean Mathematical Society
• /
• v.10 no.1
• /
• pp.145-153
• /
• 1995

A Jordan domain D in C is said to be a c-quasidisk if there exists a constant $c \geq 1$ such that each two points $z_1$ and $z_2$ in D can be joined by an arc $\tau$ in D such that $$ \ell(\tau) \leq c$\mid$z_1 - z_2$\mid$ $$ and $$ (1.1) min(\ell(\tau_1),\ell(\tau_2)) \leq c d(z, \partial D) $$ for all $z \in \tau$, where $\tau_1$ and $\tau_2$ are the components of $\tau\{z}$. Quasidisks have been extensively studied and can be characterized in many different ways [1],[2],[3].

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1651-1670
• /
• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

• Journal of the Korean Mathematical Society
• /
• v.57 no.5
• /
• pp.1119-1133
• /
• 2020

In this paper, we derive an upper bound for the distance from a point in the immediate basin of a root of a complex polynomial to the root itself. We establish that if z is a point in the immediate basin of a root α of a polynomial p of degree d ≥ 12, then ${\mid}z-{\alpha}{\mid}{\leq}{\frac{3}{\sqrt{d}}\(6{\sqrt{310}}/35\)^d{\mid}N_p(z)-z{\mid}$, where N_p is the Newton map induced by p. This bound leads to a new bound of the expected total number of iterations of Newton's method required to reach all roots of every polynomial p within a given precision, where p is normalized so that its roots are in the unit disk.

• Journal of Advanced Navigation Technology
• /
• v.21 no.4
• /
• pp.371-376
• /
• 2017

In this paper, we propose a magnetic resonant WPT(wireless power transfer) scenario using a large coil resonating at 6.78 MHz, and compare the characteristics through a three-dimensional electromagnetic field simulation and a magnetic resonant WPT equivalent circuit. The magnetic resonant WPT equivalent circuit proposed in this paper considers the parasitic capacitance generated between the coils in addition to the conventional equivalent circuit. Based on this analysis, we fabricated the magnetic resonant WPT coil and compared it with simulation prediction. As a result of comparison, the transfer characteristics and the resonance frequency shift can be predicted. Error proposed characteristics of equivalent circuit for the magnetic resonant WPT and the measured values are estimated to be ${\Delta}{\mid}S11{\mid}=1.31dB$ and ${\Delta}{\mid}S21{\mid}=1.21dB$, respectively.

• Animal Bioscience
• /
• v.34 no.5
• /
• pp.886-894
• /
• 2021

Objective: An experiment was conducted to study the effect of graded concentration of digestible lysine (dLys) on performance of layers fed diets containing sub-optimal level of protein. Methods: Five diets were formulated to contain graded concentrations of dLys (0.700%, 0.665%, 0.630%, 0.593%, and 0.563%), but similar levels of crude protein (15% CP), energy (10.25 MJ ME/kg) and other nutrients. A total of 3,520 hens (26 wk of age) with mean body weight of 1,215+12.65 g were randomly divided into 40 replicate groups of 88 birds in each and housed in an open sided colony cage house. Each diet was offered ad libitum to eight replicates from 27 to 74 wk of age. The performance was compiled at every 28 d and the data for each parameter were grouped into three phases, that is early laying phase (27 to 38 wk), mid laying phase (39 to 58 wk), and late laying phase (59 to 74 wk of age) for statistical analysis. Results: Egg production, egg mass and feed efficiency (feed required to produce an egg) were significantly improved by the dLys level during the early and mid laying phases but not during the late phase. Whereas feed intake was significantly reduced by dLys concentration during mid and late laying phases but not during early laying phase. The egg weight was not affected by dLys concentration in any of the three phases. Conclusion: Based on best fitted statistical models, dietary requirements of dLys worked out to be 0.685%, 0.640%, and 0.586% during early phase, mid phase, and late egg laying phase, respectively. The calculated requirement of dLys for the respective production phases are 727 mg/b/d during the early and mid laying phases and 684 mg/b/d during the late laying phase in diets containing 15% CP.

• Journal of the Korean Mathematical Society
• /
• v.43 no.3
• /
• pp.579-591
• /
• 2006

In this paper we obtain a sufficient and necessary condition for an analytic function f on the unit ball B with Hadamard gaps, that is, for $f(z)\;=\;{\sum}^{\infty}_{k=1}\;P_{nk}(z)$ (the homogeneous polynomial expansion of f) satisfying $n_{k+1}/n_{k}{\ge}{\lambda}>1$ for all $k\;{\in}\;N$, to belong to the weighted Bergman space $$A^p_{\alpha}(B)\;=\;\{f{\mid}{\int}_{B}{\mid}f(z){\mid}^{p}(1-{\mid}z{\mid}^2)^{\alpha}dV(z) < {\infty},\;f{\in}H(B)\}$$. We find a growth estimate for the integral mean $$\({\int}_{{\partial}B}{\mid}f(r{\zeta}){\mid}^pd{\sigma}({\zeta})\)^{1/p}$$, and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space $H_{p,q,{\alpha}$(B) and weighted Bergman space on polydisc $A^p_{^{\to}_{\alpha}}(U^n)$ are also given.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.957-965
• /
• 2018

We study the class $C{\mathcal{V}} ({\Omega})$ of analytic functions f in the unit disk ${\mathbb{D}}=\{z{\in}{\mathbb{C}}$ : ${\mid}z{\mid}$ < 1} of the form $f(z)=z+{\sum}_{n=2}^{\infty}a_nz^n$ satisfying $$1+\frac{zf^{{\prime}{\prime}}(z)}{f^{\prime}(z)}{\in}{\Omega},\;z{\in}{\mathbb{D}}$$, where ${\Omega}$ is a convex and proper subdomain of $\mathbb{C}$ with $1{\in}{\Omega}$. Let ${\phi}_{\Omega}$ be the unique conformal mapping of $\mathbb{D}$ onto ${\Omega}$ with ${\phi}_{\Omega}(0)=1$ and ${\phi}^{\prime}_{\Omega}(0)$ > 0 and $$k_{\Omega}(z)={\displaystyle\smashmargin{2}{\int\nolimits_{0}}^z}{\exp}\({\displaystyle\smashmargin{2}{\int\nolimits_{0}}^t}{\zeta}^{-1}({\phi}_{\Omega}({\zeta})-1)d{\zeta}\)dt$$. Let $z_0,z_1{\in}{\mathbb{D}}$ with $z_0{\neq}z_1$. As the first result in this paper we show that the region of variability $\{{\log}\;f^{\prime}(z_1)-{\log}\;f^{\prime}(z_0)\;:\;f{\in}C{\mathcal{V}}({\Omega})\}$ coincides wth the set $\{{\log}\;k^{\prime}_{\Omega}(z_1z)-{\log}\;k^{\prime}_{\Omega}(z_0z)\;:\;{\mid}z{\mid}{\leq}1\}$. The second result deals with the case when ${\Omega}$ is the right half plane ${\mathbb{H}}=\{{\omega}{\in}{\mathbb{C}}$ : Re ${\omega}$ > 0}. In this case $CV({\Omega})$ is identical with the usual normalized class of convex univalent functions on $\mathbb{D}$. And we derive the sharp upper bound for ${\mid}{\log}\;f^{\prime}(z_1)-{\log}\;f^{\prime}(z_0){\mid}$, $f{\in}C{\mathcal{V}}(\mathbb{H})$. The third result concerns how far two functions in $C{\mathcal{V}}({\Omega})$ are from each other. Furthermore we determine all extremal functions explicitly.

• Journal of Advanced Navigation Technology
• /
• v.21 no.2
• /
• pp.193-199
• /
• 2017

In the paper, a dual band-notched monopole antenna is proposed for 2.4 GHz WLAN (2.4 ~ 2.484 GHz) and UWB (3.1 ~ 10.6 GHz) applications. The 3.5 GHz WiMAX band notched characteristic is achived by a pair of L-shaped slots instead of the previous U-shaped slot on the center of the radiating patch, whereas the 7.5 GHz band notched characteristic is achived by C-shaped strip resonator placed near to the microstrip feed line. The measured impedance bandwidth (${\mid}S_{11}{\mid}{\leq}-10dB$) is 8.62 GHz (2.38 ~ 11 GHz) which is sufficient to cover 2.4 GHz WLAN and UWB band, while measured band-notched bandwidths for 3.5 GHz WiMAX and 7.5 GHz bnad are 1.13 GHz (3.15 ~ 4.28 GHz) and 800 MHz (7.2 ~ 8 GHz) respectively. In particular, it has been observed that antenna has a good omnidirectional radiation patterns and higher gain of 2.51 ~ 6.81 dBi over the entire frequency band of interest.