• Structural Engineering and Mechanics
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• 제55권4호
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• pp.885-900
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• 2015

This paper introduces Romberg-Richardson's method as one of the numerical integration tools for computation of stress intensity factor in a pre-cracked specimen subjected to a complex stress field across the crack faces. Also, the computation of stress intensity factor for various stress fields using existing three methods: average stress over interval method, piecewise linear stress method, piecewise quadratic method are modified by using Richardson extrapolation method. The direct integration method is used as reference for constant and linear stress distribution across the crack faces while Gauss-Chebyshev method is used as reference for nonlinear distribution of stress across the crack faces in order to obtain the stress intensity factor. It is found that modified methods (average stress over intervals-Richardson method, piecewise linear stress-Richardson method, piecewise quadratic-Richardson method) yield more accurate results after a few numbers of iterations than those obtained using these methods in their original form. Romberg-Richardson's method is proven to be more efficient and accurate than Gauss-Chebyshev method for complex stress field.

• Journal of Mechanical Science and Technology
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• 제17권3호
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• pp.457-466
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• 2003

In this paper, the numerical finite element formulations were derived for the linearized Navier-Stokes' equations with assumptions of two-dimensional incompressible, homogeneous viscous fluid field, and small oscillation and the FAMD (Fluid Added Mass and Damping) code was developed for practical applications calculating the fluid added mass and damping. In formulations, a fluid domain is discretized with C$\^$0/-type quadratic quadrilateral elements containing eight nodes using a mixed interpolation method, i.e., the interpolation function for the velocity variable is approximated by a quadratic function based on all eight nodal points and the interpolation function for the pressure variable is approximated by a linear function based on the four nodal points at vertices. Using the developed code, the various characteristics of the fluid added mass and damping are investigated for the concentric cylindrical shell and the actual hexagon arrays of the liquid metal reactor cores.

• 대한수학회지
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• 제58권6호
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• pp.1529-1547
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• 2021

In this series, we investigate the calculation of mean values of derivatives of Dirichlet L-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For µ ≥ 1 an integer, we compute the mean value of the µ-th derivative of quadratic Dirichlet L-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.

• 대한수학회지
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• 제35권4호
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• pp.903-931
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• 1998

Since the modular curves X(N) = $\Gamma$(N)\(equation omitted)＊ (N =1,2,3) have genus 0, we have field isomorphisms K(X(l))(equation omitted)C(J), K(X(2))(equation omitted)(λ) and K(X(3))(equation omitted)( $j_3$) where J, λ are the classical modular functions of level 1 and 2, and $j_3$ can be represented as the quotient of reduced Eisenstein series. When N = 4, we see from the genus formula that the curve X(4) is of genus 0 too. Thus the field K(X(4)) is a rational function field over C. We find such a field generator $j_4$(z) = x(z)/y(z) (x(z) = $\theta$$_3$((equation omitted)), y(z) = $\theta$$_4$((equation omitted)) Jacobi theta functions). We also investigate the structures of the spaces $M_{k}$($\Gamma$(4)), $S_{k}$($\Gamma$(4)), M(equation omitted)((equation omitted)(4)) and S(equation omitted)((equation omitted)(4)) in terms of x(z) and y(z). As its application, we apply the above results to quadratic forms.rms.

• 대한수학회지
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• 제50권4호
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• pp.847-864
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• 2013

By a change of variables we obtain new $y$-coordinates of elliptic curves. Utilizing these $y$-coordinates as meromorphic modular functions, together with the elliptic modular function, we generate the fields of meromorphic modular functions. Furthermore, by means of the special values of the $y$-coordinates, we construct the ray class fields over imaginary quadratic fields as well as normal bases of these ray class fields.

• 한국해양공학회지
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• 제24권6호
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• pp.1-6
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• 2010

A far field approximate solution of the slowly varying force on a 3 dimensional offshore structure in gravity ocean waves is presented. The first order potential, or at least the far field form of the Kochin function, of each frequency wave is assumed to be known. The momentum flux of the fluid domain is formulated to find the time variant force acting on the floating body in bichromatic waves. The second order difference frequency force is identified and extracted from the time variant force. The final solution is expressed as the circular integration of the product of Kochin functions. The limiting form of the slowly varying force is identical to the mean drift force. It shows that the slowly varying force components caused by the body disturbance potential can be evaluated at the far field.

• 한국해양공학회지
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• 제33권3호
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• pp.229-235
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• 2019

In this study, a two-dimensional oscillating foil with forward speed in a propagating wave flow field was considered. The time-mean power to maintain the heaving and pitching motions of the foil was analyzed using the perturbation theory in an ideal fluid. The power, which was a non-linear quantity of the second-order, was expressed in terms of the quadratic transfer functions related to the mutual product of the heaving and pitching motions and incoming vertical flow. The effects of the pivot point and phase difference among the disturbances were studied. The negative power, which indicates energy extraction from the fluid, is shown as an example calculation.

• 대한수학회지
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• 제55권2호
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• pp.343-372
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• 2018

Let K be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$. Let E be an elliptic curve with complex multiplication by ${\mathcal{O}}_K$, and let $h_E$ be the Weber function on E. Let $N{\in}\{2,3,4,6\}$. We show that $h_E$ alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo $N{\mathcal{O}}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.

• 호남수학학술지
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• 제31권4호
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• pp.463-478
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• 2009

Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let ${\tau}{\in}k{\cap}$ℌ, q = exp(${\pi}i{\tau}$). And let n, t be positive integers with $1{\leq}t{\leq}n-1$. Then $q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $_{1{\psi}1}$ summation. These are also related to Rogers-Ramanujan continued fractions.

• 한국해양공학회지
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• 제22권2호
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• pp.7-12
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• 2008

A far field solution of the slowly varying force on an offshore structure by gravity ocean waves was shown as a function of the reflection and transmission of the body disturbed waves. The solution was obtained from the conservation of the momentum flux, which simply describes various wave forces, while making it unnecessary to compute complicated integration over a control surface. The solution was based on the assumption that the frequency difference of the bichromatic incident waves is small and its second order term is negligible. The final solution is expressed in term of the reflection and transmission waves, i.e. their amplitudes and phase angles. Consequently, it shows that not only the amplitudes but also the phase differences make critical contributions to the slowly varying force. In a limiting case, the slowly varying force solution gives the one of the mean drift force, which is only dependent on the reflection wave amplitude. An approximation is also suggested in a case where only the mean drift force information is available.