• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.189-212
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• 2006

Let $F^{\alpha}G$ be a twisted group algebra with basis ${{\mu}g|g\;{\in}\;G}$ and $P\;=\;{C_g|g\;{\in}\;G}$ be a partition of G. A projective class algebra associated with P is a subalgebra of $F^{\alpha}G$ generated by all class sums $\sum\limits{_{x{\in}C_g}}\;{\mu}_x$. A main object of the paper is to find interrelationships of projective class algebras in $F^{\alpha}G$ and in $F^{\alpha}H$ for H < G. And the a-spherical function will play an important role for the purpose. We find functional properties of a-spherical functions and investigate roles of $\alpha-spherical$ functions as characters of projective class algebras.

• Honam Mathematical Journal
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• v.33 no.1
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• pp.1-9
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• 2011

In this paper we study oscillatory integrals with analytic homogeneous phase functions for smooth radial functions. We give their sharp asymptotic behavior in terms of spherical Newton distance.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1339-1351
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• 2015

In this paper, using pseudo-spherical Frenet frame of pseudo-spherical curves in hyperbolic space, we define the notion of the structure functions on the non-developable ruled surfaces with timelike ruling. Then we obtain the properties of the structure functions and a complete classification of the non-developable ruled surfaces with timelike ruling in Minkowski 3-space by the theories of the structure functions.

• Journal for History of Mathematics
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• v.36 no.2
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• pp.21-34
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• 2023

Spherical trigonometry refers to the geometry related to spherical triangles. It has been an important tool for studying astronomy since ancient times. In trigonometry, concepts such as trigonometric functions naturally emerge from the relationship between arcs and chords of a circle. In this paper, we briefly examine the origin of spherical trigonometry. To introduce the basics of spherical trigonometry, we present fundamental and important theorems such as Menelaus's theorem, the law of sines and the law of cosines on a sphere, along with their proofs. Furthermore, we discuss the educational value and potential applications of spherical trigonometry.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.601-603
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• 2013

We show that a compact embedded hypersurface with constant ratio of mean curvature functions in a convex cone $C{\subset}\mathbb{R}^{n+1}$ is part of a hypersphere if it has a point where all the principal curvatures are positive and if it is perpendicular to ${\partial}C$.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.62 no.9
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• pp.1260-1263
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• 2013

Plane wave scattering from a spherical resonator is calculated by solving the combined field integral equation (CFIE) with Rao-Wilton-Glisson (RWG) basis functions and the moment method. The calculations show that magnetic and electric dipoles are found at resonant modes. These characteristics are confirmed by radiation patterns in the far field region. In addition, an analysis of a magnetodielectric sphere is discussed.

• Journal of the Korean earth science society
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• v.38 no.2
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• pp.105-114
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• 2017

Two dimensional finite element method with quadrilateral basis functions was applied to the spherical high order filter on the spherical surface limited area domain. The basis function consists of four shape functions which are defined on separate four grid boxes sharing the same gridpoint. With the basis functions, the first order derivative was expressed as an algebraic equation associated with nine point stencil. As the theory depicts, the convergence rate of the error for the spherical Laplacian operator was found to be fourth order, while it was the second order for the spherical Laplacian operator. The accuracy of the new high order filter was shown to be almost the same as those of Fourier finite element high order filter. The two-dimension finite element high order filter was incorporated in the weather research and forecasting (WRF) model as a hyper viscosity. The effect of the high order filter was compared with the built-in viscosity scheme of the WRF model. It was revealed that the high order filter performed better than the built in viscosity scheme did in providing a sharper cutoff of small scale disturbances without affecting the large scale field. Simulation of the tropical cyclone track and intensity with the high order filter showed a forecast performance comparable to the built in viscosity scheme. However, the predicted amount and spatial distribution of the rainfall for the simulation with the high order filter was closer to the observed values than the case of built in viscosity scheme.

• Journal of the Semiconductor & Display Technology
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• v.10 no.4
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• pp.47-51
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• 2011

We investigated the substrate surface polishing by the spherical rigid abrasive under the compression using classical molecular dynamics modeling. We performed three-dimensional molecular dynamic simulations using the Morse potential functions for the various slide-to-roll ratios, from 0 to 1, and then, the compressive forces acting on the spherical rigid abrasive were calculated as functions of the time and the slide-to-roll ratio. The friction coefficients obtained from the classical molecular dynamics simulations were compared to those obtained from the experiments; and found that the molecular dynamic simulation results with the slide-to-roll ratio of 0 value were in good agreement with the experimental results.

• Steel and Composite Structures
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• v.27 no.2
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• pp.201-216
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• 2018

In this paper, three-dimensional (3D) elasticity theory in conjunction with nonlocal strain gradient theory (NSGT) is developed for mechanical analysis of anisotropic nanoparticles. The present model incorporates two scale coefficients to examine the mechanical characteristics much accurately. All the elastic constants are considered and assumed to be the functions of (r, ${\theta}$, ${\varphi}$), so all kind of anisotropic structures can be modeled. Moreover, all types of functionally graded spherical structures can be investigated. To justify our model, our results for the radial vibration of spherical nanoparticles are compared with experimental results available in the literature and great agreement is achieved. Next, several examples of the radial vibration and wave propagation in spherical nanoparticles including nonlocal strain gradient parameters are presented for more than 10 different anisotropic nanoparticles. From the best knowledge of authors, it is the first time that 3D elasticity theory and NSGT are used together with no approximation to derive the governing equations in the spherical coordinate. Moreover, up to now, the NSGT has not been used for spherical anisotropic nanoparticles. It is also the first time that all the 36 elastic constants as functions of (r, ${\theta}$, ${\varphi}$) are considered for anisotropic and functionally graded nanostructures including size effects. According to the lack of any common approximations in the displacement field or in elastic constant, present theory can be assumed as a benchmark for future works.

• Nuclear Engineering and Technology
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• v.52 no.6
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• pp.1137-1147
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• 2020

The discrete ordinates method (S_N) is one of the major shielding calculation method, which is suitable for solving deep-penetration transport problems. Our objective is to explore the available quadrature sets and to improve the accuracy in shielding problems involving strong anisotropy. The linear discontinuous finite-element (LDFE) quadrature sets based on the icosahedron (in short, ICLDFE quadrature sets) are developed by defining projected points on the surfaces of the icosahedron. Weights are then introduced in the integration of the discontinuous finite-element basis functions in the relevant angular regions. The multivariate secant method is used to optimize the discrete directions and their corresponding weights. The numerical integration of polynomials in the direction cosines and the Kobayashi benchmark are used to analyze and verify the properties of these new quadrature sets. Results show that the ICLDFE quadrature sets can exactly integrate the zero-order and first-order of the spherical harmonic functions over one-twentieth of the spherical surface. As for the Kobayashi benchmark problem, the maximum relative error between the fifth-order ICLDFE quadrature sets and references is only -0.55%. The ICLDFE quadrature sets provide better integration precision of the spherical harmonic functions in local discrete angle domains and higher accuracy for simple shielding problems.