• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1559-1568
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• 2015

The relative class number $H_d(f)$ of a real quadratic field $K=\mathbb{Q}(\sqrt{m})$ of discriminant d is the ratio of class numbers of $O_f$ and $O_K$, where $O_K$ denotes the ring of integers of K and $O_f$ is the order of conductor f given by $\mathbb{Z}+fO_K$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\mathbb{Q}(\sqrt{m})$ has been investigated using continued fraction in the special case when $(\sqrt{m})$ has a diagonal form. Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of $(\sqrt{m})$ is non-diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.

• 한국전산유체공학회:학술대회논문집
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• 2010.05a
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• pp.397-403
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• 2010

The Immersed boundary method(IBM) is one of CFD techniques which can simulate flow field around complex objectives using simple Cartesian grid system. In the previous studies the IBM has mostly been implemented to fractional step method based Navier-Stokes solvers. In these cases, pressure buildup near IB was found to occur when linear interpolation and stadard mass conservation is used and the interpolation scheme became complicated when higher order of interpolation is adopted. In this study, we implement the IBM to an incompressible Navier-Stokes solver which uses SIMPLE algorithm. Bi-linear and quadratic interpolation equations were formulated by using only geometric information of boundary to reconstruct velocities near IB. Flow around 2D circular cylinder at Re=40 and 100 was solved by using these formulations. It was found that the pressure buildup was not observed even when the bi-linear interpolation was adopted. The use of quadratic interpolation made the predicted aerodynamic forces in good agreement with those of previous studies.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.25-43
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• 2005

Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length $2^{n}$ generated by these idempotents are completely described. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n-1 idempotents are also obtained. The case n = 2,3 is dealt separately.

• Journal of the Korean Society of Industry Convergence
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• v.12 no.3
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• pp.149-155
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• 2009

A new finite element equation is derived by applying quadratic and cubic time integration scheme to the variational formulation in time-integral for the analysis of the transient elastodynamic problems to increase the numerical accuracy and stability. Emphasis is focused on methodology for cubic time integration scheme procedure which are never presented before. In this semidiscrete approximations of the field variables, the time axis is divided equally and quadratic and cubic time variation is assumed in those intervals, and space is approximated by the usual finite element discretization technique. It is found that unconditionally stable numerical results are obtained in case of the cubic time variation. Some numerical examples are given to show the versatility of the presented formulation.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.585-594
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• 2001

Let η be the complex upper half plane, let h($\tau$) be a cusp form, and let $\tau$ be an imaginary quadratic in η. If h($\tau$)$\in$$\Omega$( $g_{2}$($\tau$)$^{m}$ $g_{3}$ ($\tau$)$^{ι}$with $\Omega$the field of algebraic numbers and m. l positive integers, then we show that h($\tau$) is integral over the ring Q[h/$\tau$/n/)…h($\tau$+n-1/n)] (No Abstract.see full/text)

• Journal of the Korean Mathematical Society
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• v.58 no.6
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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.

• Honam Mathematical Journal
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• v.43 no.1
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• pp.88-99
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• 2021

The purpose of this paper is to assign a movable frame to an arbitrary point of a rational Bézier curve on the 2-sphere S2 in Euclidean 3-space R3 to provide a better understanding of the geometry of the curve. Especially, we obtain the formula of geodesic curvature for a quadratic rational Bézier curve that allows a curve to be characterized on the surface. Moreover, we give some important results and relations for the Darboux frame and geodesic curvature of a such curve. Then, in specific case, given characterizations for the quadratic rational Bézier curve are illustrated on a unit 2-sphere.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1035-1059
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• 2023

Let p be an odd prime. Let E be an elliptic curve defined over a quadratic imaginary field, where p splits completely. Suppose E has supersingular reduction at primes above p. Under appropriate hypotheses, we extend the results of [17] to ℤ²_p-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed 𝜇-invariants of one elliptic curve implies the vanishing of the signed 𝜇-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.11b
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• pp.306-311
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• 2002

Non-singular boundary element method (BEM) codes are developed in acoustics application. The BEM code is then used to calculate unknown boundary surface normal displacements and surface pressures from known exterior near Held pressures. And then the calculated surface normal displacements and surface pressures are again applied to the BEM in forward in order to calculate reconstructed field pressures. The initial exterior near field pressures are very well agreed with the later reconstructed field pressures. Only the same number of boundary surface nodes (1178) are used far the initial exterior pressures which are initially calculated by Finite Element Method (FEM) and BEM. Pseudo-inverse technique is used for the calculation of the unknown boundary surface normal displacements. The structural object is a tuning fork with 128.4 Hz resonant. The boundary element is a quadratic hexahedral element (eight nodes per element).

• Journal of the Korean Society for Advanced Composite Structures
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• v.4 no.1
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• pp.1-8
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• 2013

The finite element analysis (FEA) is a numerical technique to find solutions of field problems. A field problem is approximated by differential equations or integral expressions. In a finite element, the field quantity is allowed to have a simple spatial variation in terms of linear or polynomial functions. This paper represents a review and an accuracy-study of the finite element method comparing the FEA results with the exact solution. The exact solutions were calculated by solid mechanics and FEA using matrix stiffness method. For this study, simple bar and cantilever models were considered to evaluate four types of basic elements - constant strain triangle (CST), linear strain triangle (LST), bi-linear-rectangle(Q4),and quadratic-rectangle(Q8). The bar model was subjected to uniaxial loading whereas in case of the cantilever model moment loading was used. In the uniaxial loading case, all basic element results of the displacement and stress in x-direction agreed well with the exact solutions. In the moment loading case, the displacement in y-direction using LST and Q8 elements were acceptable compared to the exact solution, but CST and Q4 elements had to be improved by the mesh refinement.