• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1425-1432
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• 2014

The Euler zeta function is defined by ${\zeta}_E(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^8}$. The purpose of this paper is to find formulas of the Euler zeta function's values. In this paper, for $s{\in}\mathbb{N}$ we find the recurrence formula of ${\zeta}_E(2s)$ using the Fourier series. Also we find the recurrence formula of $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2_{n-1})^{2s-1}}$, where $s{\geq}2({\in}\mathbb{N})$.

• Journal of the Korean Institute of Electrical and Electronic Material Engineers
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• v.13 no.1
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• pp.39-46
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• 2000

An analysis of the free vibration for the metal-piezoceramic laminated thin plates is described. The purpose of this study is to develop a equivalent method for the free vibration analysis of metal-piezoce-ramic laminated thin plate which are not sysmmetric about the adhered layer and the piezoelectric effect. In order to confirm the validity of the vibration analysis, double Fourier sine series is used as a modal displacement function of a metal-piezoceramic laminated thin plate and applied to the free vibration analysis of the plate under various boundary conditions.

• Communications for Statistical Applications and Methods
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• v.19 no.6
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• pp.885-898
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• 2012

In this paper, we consider Bayesian approaches to partially linear models, in which a regression function is represented by a semiparametric additive form of a parametric linear regression function and a nonparametric regression function. We make a comparative study on the performance of widely used Bayesian partially linear models in terms of empirical analysis. Specifically, we deal with three Bayesian methods to estimate the nonparametric regression function, one method using Fourier series representation, the other method based on Gaussian process regression approach, and the third method based on the smoothness of the function and differencing. We compare the numerical performance of three methods by the root mean squared error(RMSE). For empirical analysis, we consider synthetic data with simulation studies and real data application by fitting each of them with three Bayesian methods and comparing the RMSEs.

• Transactions of Materials Processing
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• v.4 no.4
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• pp.390-397
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• 1995

A simple kinematically admissible velocity field is proposed to determine the final-stage extrusion load and the average extruded length in the square-die forward extrusion of non-axisymmetric bars from circular billets. The proposed velocity field is applied to the square-die extrusion of trochoidal gear-shaped bars and rectangular-shaped bars. The profile function of a rectangle is approximated by using a Fourier series. Experiments have been carried out with hard solder billets at room temperature. The theoretical predictions of the extrusion load are in good agreements with the experimental results and there is generally reasonable agreements in average extruded length between theory and experiment.

• Applied Microscopy
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• v.32 no.2
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• pp.81-90
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• 2002

In this review, the author discussed how the Fresnel and Fraunhoffer Diffraction can be deduced from the Huygens-Fresnel principle and Kirchhoff Diffraction Theory. Fresnel diffraction became the basic theory of the CTEM image theory, and Fraunhoffer diffraction became the base for electron diffraction and HRTEM image theory by Fourier transformation. The author also discussed the diffraction based on Born series.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.12 no.2
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• pp.150-160
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• 2002

The free vibration characteristics of the triple cylindrical shells filled with fluid are investigated. The triple cylindrical shells are filled with compressible fluid. The boundary condition is clamped at both ends. Analytical method is developed to evaluate natural frequencies of triple cylindrical shells using Sanders' shell theory and courier series expansion by Stokes' transformation. Their results are compared with those of finite element method to verify the validation of the method developed. The modal characteristics of shells filled with fluid at region 1, 2 and 3 are evaluated.

• Structural Engineering and Mechanics
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• v.54 no.5
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• pp.937-953
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• 2015

This paper investigates the load model for single footfall trace of human walking. A large amount of single person walking load tests were conducted using the three-dimensional gait analysis system. Based on the experimental data, Fourier series functions were adopted to model single footfall trace in three directions, i.e. along walking direction, direction perpendicular to the walking path and vertical direction. Function parameters such as trace duration time, number of Fourier series orders, dynamic load factors (DLFs) and phase angles were determined from the experimental records. Stochastic models were then suggested by treating walking rates, duration time and DLFs as independent random variables, whose probability density functions were obtained from experimental data. Simulation procedures using the stochastic models are presented with examples. The simulated single footfall traces are similar to the experimental records.

• Structural Engineering and Mechanics
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• v.33 no.6
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• pp.697-708
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• 2009

Stresses are solved for two parallel cracks in an infinite orthotropic plate during passage of incoming shock stress waves normal to their surfaces. Fourier transformations were used to reduce the boundary conditions with respect to the cracks to two pairs of dual integral equations in the Laplace domain. To solve these equations, the differences in the crack surface displacements were expanded to a series of functions that are zero outside the cracks. The unknown coefficients in the series were solved using the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors were defined in the Laplace domain and were inverted numerically to physical space. Dynamic stress intensity factors were calculated numerically for selected crack configurations.

• Journal of the Korean Statistical Society
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• v.3 no.1
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• pp.13-16
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• 1974

In my former paper [3] I defined an almost periodicity of weakly sationary random processes (a.p.w.s.p.) and presented some basic results of it. In this paper I shall present some notes on the Fourier series of an a.p.w.s.p., resulting from [3]. All the conditions at the introduction of [3] are assumed to hold without repreating them here. The essential facts are as follows : The weakly stationary process $X(t,\omega), t\in(-\infty,\infty), \omega\in\Omega$, defined on a probability space $(\Omega,a,P)$, has a spectral representation $$X(t,\omega)=\int_{-\infty}^{infty}{e^{it\lambda\xi}(d\lambda,\omega)},$$ where $\xi(\lambda)$ is a random measure. Then, the continuous covariance $\rho(\mu) = E(X(t+u) X(t))$ has the form $$\rho(u)=\int_{-\infty}^{infty}{e^{iu\lambda}F(d\lambda)},$$ $E$\mid$\xi(\lambda+0)-\xi(\lambda-0)$\mid$^2 = F(\lambda+0) - F(\lambda-0) \lambda\in(-\infty,\infty)$, assumimg that $\rho(u)$ is a uniformly almost periodic function.

• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.781-795
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• 2020

In this paper, we obtain an analytical solution for an unsolved definite integral RC (m, n) from a 1915 paper of Srinivasa Ramanujan. We obtain our solution using the hypergeometric approach and an infinite series decomposition identity. Also, we give some generalizations of Ramanujan's integral RC (m, n) defined in terms of the ordinary hypergeometric function 2F3 with suitable convergence conditions. Moreover as applications of our result we obtain nine new infinite summation formulas associated with the hypergeometric functions 0F1, 1F2 and 2F3.