• 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.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.91-95
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• 2012

In this paper, we show how to construct the first layer $k^{\alpha}_{1}$ of anti-cyclotomic ${\mathbb{{Z}}}_{3}$-extension of imaginary quadratic fields $k(=\;{\mathbb{{Q}}}(\sqrt{-d}))$ when the Sylow subgroup of class group of k is 3-elementary, and give an example. This example is different from the one we obtained before in the sense that when we write $k^{\alpha}_{1}\;=\;k({\eta}),{\eta}$ is obtained from non-units of ${\mathbb{{Q}}}({\sqrt{3d}})$.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.267-269
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• 2021

A finitely generated torsion module M for ℤ_p[[T, T₂, ⋯ , T_d]] is pseudo-null if M/TM is pseudo-null over ℤ_p[[T₂, ⋯ , T_d]]. This result is used as a tool to prove the generalized Greenberg's conjecture in certain cases. The converse may not be true. In this paper, we give examples of pseudo-null Iwasawa modules whose reduction are not pseudo-null.

• East Asian mathematical journal
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• v.38 no.5
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• pp.583-591
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• 2022

Let K be an imaginary quadratic field and 𝒪 be an order in K. Let H_𝒪 be the ring class field of 𝒪. Furthermore, for a positive integer N, let K_𝒪,N be the ray class field modulo N𝒪 of 𝒪. When the discriminant of 𝒪 is different from -3 and -4, we construct an extended form class group which is isomorphic to the Galois group Gal(K_𝒪,N/H_𝒪) and describe its Galois action on K_𝒪,N in a concrete way.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1994.10a
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• pp.205-209
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• 1994

The main purpose of this study is constructing new velocity fields on the base of shape function used in finite element method and showing the possibility of application it to metal forming processes. Utilizing the 8-node quadratic rectangular element, we expressed the velocity within the deformation region by interpolating the velocity of each nodal points. And the upper-bound formulation from this velocity fields was derived. In order to confirm the validity of this method we applied it to axisymmetic combined extrusion problem. the results of load show that this method is on better agreement with experiment than the conventional UBET, and also the flow pattern and profile of extruded part are reasonable.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.17 no.2
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• pp.181-186
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• 1992

The scattering property of TMz illuminated perfectly conducting and dielectric cylinders of arbitrary cross section are analyzed by the boundary element techniques. The boundary element equations are formulated via Maxwell’s equations, weighted residual or Green’s theorem, and the boundary conditions. The unknown surface fields on the boundaries are then calculated by the boundary element integral equations. Once the surface fields are found, the scattered fields in from a perfectly conducting circular and elliptic cylinders, a dielectric circular and elliptic cylinders are numerically analyzed. A general computer program has been developed using the quadratic elements(higher order boundary elements) and the Gaussian quadrature.

• Structural Engineering and Mechanics
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• v.66 no.5
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• pp.557-568
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• 2018

An investigation is made in the present work on the post-buckling and geometrically nonlinear behaviors of moderately thick perfect and imperfect rectangular plates made-up of functionally graded materials. Spectral collocation approach based on Legendre basis functions is developed to analyze the functionally graded plates while they are subjected to end-shortening strain. The material properties in this study are varied through the thickness according to the simple power law distribution. The fundamental equations for moderately thick rectangular plates are derived using first order shear deformation plate theory and taking into account both geometric nonlinearity and initial geometric imperfections. In the current study, the domain of interest is discretized with Legendre-Gauss-Lobatto nodes. The equilibrium equations will be obtained by discretizing the Von-Karman's equilibrium equations and also boundary conditions with finite Legendre basis functions that are substituted into the displacement fields. Due to effect of geometric nonlinearity, the final set of equilibrium equations is nonlinear and therefore the quadratic extrapolation technique is used to solve them. Since the number of equations in this approach will always be more than the number of unknown coefficients, the least squares technique will be used. Finally, the effects of boundary conditions, initial geometric imperfection and material properties are investigated and discussed to demonstrate the validity and capability of proposed method.

• Journal of computational fluids engineering
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• v.17 no.1
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• pp.44-53
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• 2012

Immersed boundary method(IBM) is a numerical scheme proposed to 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 this study, we implement the IBM to an incompressible Navier-Stokes solver which uses SIMPLE algorithm. The weight coefficients of the 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. For an analysis of moving boundary, we smulated an oscillating circular cylinder with Re=100 and KC(Keulegan-Carpenter) number of 5. The predicted flow fields were compared with experimental data and they also showed good agreements.

• The Journal of Korean Institute of Electromagnetic Engineering and Science
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• v.18 no.6 s.121
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• pp.577-583
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• 2007

The field uniformity inside the reverberation chamber has been improved using sets of quadratic residue diffuser (QRD). The electromagnetic field inside the reverberation chamber with the dimension of $100{\times}80{\times}80cm$ has been analyzed by the finite-difference time-domain(FDTD) method. The calculated fields in a $40{\times}30{\times}30cm$ test volume have been sampled to obtain a standard deviation and field uniformity. Results show that the standard deviation of the calculated field and uniformity have been improved by varying angles and orientation of the inclined surfaces of the QRDs installed inside the reverberation chamber.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.697-707
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• 2022

It is well known that the continued fraction expansion of ${\sqrt{d}}$ has the form $[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}]$ and ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer l and a palindromic sequence of positive integers ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},$ we define the set $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})\;:=\;\{d{\in}{\mathbb{Z}}{\mid}d>0,\;{\sqrt{d}}=[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}],\;where\;{\alpha}_0={\lfloor}{\sqrt{d}}{\rfloor}\}.$ In this paper, we completely determine when $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})$ is not empty in the case that l is 4, 5, 6, or 7. We also give similar results for $(1+{\sqrt{d}})/2.$ For the case that l is 4, 5, or 6, we explicitly describe the fundamental units of the real quadratic field ${\mathbb{Q}}({\sqrt{d}}).$ Finally, we apply our results to the Mordell conjecture for the fundamental units of ${\mathbb{Q}}({\sqrt{d}}).$