• Journal for History of Mathematics
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• v.20 no.4
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• pp.71-84
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• 2007

J. Bernoulli first discovered the method which one can produce those formulae for the sum $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$ for any natural numbers k. After then, there has been increasing interest in Bernoulli and Euler numbers associated with Riemann zeta functions. Recently, Kim have been studied extended q-Bernoulli numbers and q-Euler numbers associated with p-adic q-integral on $\mathbb{Z}_p$, and sums of powers of consecutive q-integers, etc. In this paper, we investigate for the historical background and evolution process of the sums of powers of consecutive q-integers and discuss for Euler zeta functions subjects which are studying related to these areas in the recent.

• Bulletin of the Society of Naval Architects of Korea
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• v.27 no.3
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• pp.38-46
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• 1990

A numerical solution method of the boundary integral equation for axisymmetric potential flows is presented. Those are represented by ring source and ring vorticity distribution. Strengths of ring source and ring vorticity are approximated by linear functions of a parameter $\zeta$ on a segment. The geometry of the body is represented by a cubic B-spline. Limiting integral expressions as the field point tends to the surface having ring source and ring vorticity distribution are derived upto the order of ${\zeta}ln{\zeta}$. In numerical calculations, the principal value integrals over the adjacent segments cancel each other exactly. Thus the singular part proportional to $\(\frac{1}{\zeta}\)$ can be subtracted off in the calculation of the induced velocity by singularities. And the terms proportional to $ln{\zeta}$ and ${\zeta}ln{\zeta}$ can be integrated analytically. Thus those are subtracted off in the numerical calculations and the numerical value obtained from the analytic integrations for $ln{\zeta}$ and ${\zeta}ln{\zeta}$ are added to the induced velocity. The four point Gaussian Quadrature formula was used to evaluate the higher order terms than ${\zeta}ln{\zeta}$ in the integration over the adjacent segments to the field points and the integral over the segments off the field points. The root mean square errors, $E_2$, are examined as a function of the number of nodes to determine convergence rates. The convergence rate of this method approaches 2.

• Korean Journal of Optics and Photonics
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• v.11 no.6
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• pp.385-389
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• 2000

A phase-retardation function which was derived from Wigner distribution function (WDF) is used to increase a focal depth of a radially symmetric optical system. The WDF for one-dimensional signal is represented as a two-dimensional function of phasespace ($\chi,\zeta$), and a normalized irradiance is described as a form of the Strehl ratio (SR). The increase in the focal depth is accomplished by delivering a shearing tilt a that represents a characteristic of free space propagation with simple manipulation in the WDF space. In this paper we propose a method for evaluating the focal depth quantitatively by representing the phaseretardation function in terms of the focal depth term. In order to verify the validity of the proposed method, we compared the numerically analyzed result with that of J. Sochki's study. study.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.28 no.7A
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• pp.503-510
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• 2003

In CDMA cellular system, because all users share the frequency resource the signals of other user becomes interference which influences the communication quality. The system capacity defined the number of connected users within a cell is determined by the amount of interference, therefore the exact estimation of interference is important to system performance evaluation. In this paper, we propose an approximated function which calculates other cell interference in terms of Riemann-Zeta function in CDMA cellular systems, and compare with simulation results in other to verify its usefulness. The upper and lower bounds of system capacity calculated with the proposed approximated function gives almost alike result with the simulation. The proposed interference bounds are useful to calculate system capacity and to evaluate some algorithm in a hierarchical cellular systems where various propagation environments are mixed.

• Journal of the Korea Society of Computer and Information
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• v.16 no.10
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• pp.101-109
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• 2011

The Riemann's zeta function $\zeta(s)$ has been known as answer for a number of primes $\pi$(x) less than given number x. In prime number theorem, there are another approximation function $\frac{x}{lnx}$,Li(x), and R(x). The error about $\pi$(x) is R(x) < Li(x) < $\frac{x}{lnx}$. The logarithmic integral function is Li(x) = $\int_{2}^{x}\frac{1}{lnt}dt$ ~ $\frac{x}{lnx}\sum\limits_{k=0}^{\infty}\frac{k!}{(lnx)^k}=\frac{x}{lnx}(1+\frac{1!}{(lnx)^1}+\frac{2!}{(lnx)^2}+\cdots)$. This paper shows that the $\pi$(x) can be represent with finite Li(x), and presents generalized prime counting function $\sqrt{{\alpha}x}{\pm}{\beta}$. Firstly, the $\pi$(x) can be represent to $Li_3(x)=\frac{x}{lnx}(\sum\limits_{t=0}^{{\alpha}}\frac{k!}{(lnx)^k}{\pm}{\beta})$ and $Li_4(x)=\lfloor\frac{x}{lnx}(1+{\alpha}\frac{k!}{(lnx)^k}{\pm}{\beta})}k\geq2$ such that $0{\leq}t{\leq}2k$. Then, $Li_3$(x) is adjusted by $\pi(x){\simeq}Li_3(x)$ with ${\alpha}$ and error compensation value ${\beta}$. As a results, this paper get the $Li_3(x)=Li_4(x)=\pi(x)$ for $x=10^k$. Then, this paper suggests a generalized function $\pi(x)=\sqrt{{\alpha}x}{\pm}{\beta}$. The $\pi(x)=\sqrt{{\alpha}x}{\pm}{\beta}$ function superior than Riemann's zeta function in representation of prime counting.

• Proceedings of the KIEE Conference
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• 2006.07b
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• pp.933-934
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• 2006

본 논문에서는 새로운 양방향 ZVS PWM Sepic/Zeta 컨버터를 제안한다. 제안한 컨버터는 PWM 제어가능한 컨버터로서 입력과 출력전압의 극성이 같은 비반전 컨버터의 특징을 지니며 DC 전압의 전달함수가 양방향 모두 M=D/(1-D)로 동일하다. 또한 각방향으로 전력전달시 다이오드와 병렬로 연결된 MOSFET가 다이오드의 'ON'시 동시에 'ON'되어 Sychronous Rectifier로 동작하므로 도통손을 저감하였으며, Auxiliary Resonant Commutated Pole를 사용하여 저감된 스위칭 손실을 갖는 특성을 지니고 있다. 또한 Transformer 버전이 존재하므로 입력과 출력사잉에 전기적 절연을 필요로 하는 실제응용에 유용하게 사용할 수 있다.

• Korean Journal of Fisheries and Aquatic Sciences
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• v.33 no.5
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• pp.448-454
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• 2000

This study was to examine the physicochemical characteristics of coagulation reaction between ignited oyster shell powder (IOSP) and red tide organisms (RTO), and its feasibility, in developing a technology for the removal of RTO bloom in coastal sea,IOSP was made from oyster shell and its physicochemical characteristics were examined for particle size distribution, surface characteristic by scanning electron microscope, zeta potential, and alkalinity and pH variations in sea water. Two kinds of RTO that were used in this study, Cylindrotheca closterium and Skeletonema costatum, were sampled in Masan bay and were cultured in laboratory. Coagulation experiments were conducted using various c(Incentrations of IOSP, RTO, and a jar tester. The supernatant and RTO culture solution were analyzed for pH, alkalinity, RTO cell number, IOSP showed positive zeta potentials of $11.1{\~}50.1\;mV\;at\;pH\;6.2{\~}12.7$, A positive zeta potential of IOSP slowly decreased with decreasing pNa 4,0 to 2,0. When pNa reached zero, the zeta potential approached zero, When a pMg value was decreased, the positive zeta potential of IOSP increased until pMg 3.0 and decreased below pMg 3.0. IOSP showed 4.8 mV of positive zeta potential while RTO showed -9.2 mV of negative zeta potential in sea water. A positive-negative EDL (electrical double-layer) interaction occurred between $Mg(OH)_2$ adsorption layer of IOSP and RTO in sea water so that EDL attractive force always worked between them. Hence, their coagulation reaction occurred at primary minimum on which an extreme attractive force acted because of charge neutralization by $Mg(OH)_2$ adsorption layer of IOSP. As a result, the coagulation reaction was rapidly processed and was irreversible according to DLVO (Deriaguin-Landau-Verwey-Overbeek) theory. Removal rates of RTO were exponentially increased with increasing both IOSP concentration and G-value. The removal rates were steeply increased until 50 mg/l of IOSP and reached $100{\%}\;at\;400\;mg/l$ of IOSP. Removal rates of RTO were $70.5,\;70.5,\;81.7,\;85.3{\%}$ for G-values of $1,\;6,\;29,\;139\;sec^(-1)$at IOSP 100 mg/l, respectively. This indicated that mixing (i.e., collision among particles) was very important for a coagulation reaction.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.30 no.3 s.246
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• pp.249-259
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• 2006

Stress and displacement fields of a propagating Mode III crack in an orthotropic functionally gradient material (OFGM), which has (1) linear variation of shear modulus with a constant density, and (2) an exponential variation of shear modulus and density, are derived. The equations of motion in OFGM are developed and solution to the displacement and stress fields fer a propagating crack at constant speed though an asymptotic analysis. The stress terms associated with $\gamma^{-1/2}\;and\;\gamma^{0}$ are not affected by the FGM constant $\zeta$ which is nonhomogeneous parameter, only on the higher order terms, the influences of nonhomogeneity on the stress are explicitly brought out. When the FGM constant $\zeta\;is\;zero\;or\;\gamma{\rightarrow}0$, the fields for OFGM are almost same as the those for homogeneous orthotropic material. Using the stress components, the effects of nonhomogeneity on stress components are discussed.

• Korean Journal of Fisheries and Aquatic Sciences
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• v.33 no.5
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• pp.455-462
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• 2000

The objective of this study was to examine the physicochemical characteristics of coagulation reaction between loess and red tide organisms (RTO) and its feasibility, in developing a technology for the removal of RTO bloom in coastal sea. The physicochemical characteristics of loess were examined for a particle size distribution, surface characteristics by scanning electron microscope, zeta potential, and alkalinity and pH variations in sea water. Two kinds of RTO that were used in this study, Cylindrothen closterium and Skeietonema costatum, were sampled in Masan bay and were cultured in laboratory. Coagulation experiments were conducted using various concentrations of loess, RTO, and a jar tester. The supernatant and RTO culture solution were analyzed for pH, alkalinity, RTO cell number. A negative zeta potential of loess increased with increasing pH at $10^(-3)M$ NaCl solution and had -71.3 mV at pH 9.36. Loess had a positive zeta potential of +1,8 mV at pH 1.98, which resulted in a characteristic of material having an amphoteric surface charge. In NaCl and $CaCl_2$, solutions, loess had a decreasing negative zeta potential with increasing $Na^+\;and\;Ca^(+2)$ ion concentration and then didn't result in a charge reversal due to not occurring specific adsorption for $Na^+$ ion while resulted in a charge reversal due to occurring specific adsorption for $Ca^(+2)$ ion. In sea water, loess and RTO showed the similar zeta potential values of -112,1 and -9.2 mV, respectively and sea sand powder showed the highest zeta potential value of -25.7 mV in the clays. EDLs (electrical double-layers) of loess and RTO were extremely compressed due to high concentration of salts included in sea water, As a result, there didn't almost exist EDL repulsive force between loess and RTO approaching each other and then LVDW (London-yan der Waals) attractive force was always larger than EDL repulsive force to easily form a floe. Removal rates of RTO exponentially increased with increasing a loess concentration. The removal rates steeply increased until $800 mg/l$ of loess, and reached $100{\%}$ at 6,400 mg/l of loess. Removal rates of RTO exponentially increased with increasing a G-value. This indicated that mixing (i.e., collision among particles) was very important for a coagulation reaction. Loess showed the highest RTO removal rates in the clays.

• Journal of the Korean Ceramic Society
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• v.39 no.11
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• pp.1042-1047
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• 2002

The sedimentation characteristics of ceramic fiber were analyzed when viscosity of the slurry for ceramic paper formation was varied and zeta potential change and degree of dispersion with pH were studied as well. The proper viscosity of the slurry for dispersion of fibers was between 28 and 31 cps. Zeta potential of the slurry was sensitively changed with pH adjustment and showed maximum value of -35~-36 mV at ph 7.5~9.5, which indicated better dispersion of ceramic fiber as zeta potential of the slurry was increased. The sedimentation rate of ceramic fiber in a slurry was reported minimum at the maximum zeta potential. Water content of the casted paper should be lower than 83% after vacuum dehydration for retention of binder and lower than 62% after press rolling for wet paper handling. The obtained ceramic paper had tensile strength and basis weight, $102 kgf/cm^2$ and $98 g/m^2$, respectively.