• The Pure and Applied Mathematics
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• v.9 no.1
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• pp.31-37
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• 2002

In this paper, we find explicitly the eigenvalues and the eigenfunctions of Laplace operator on a triangle domain with a mixed boundary condition. We also show that a weaker form of the principle of spatial averaging holds for this domain under suitable boundary condition.

• Bulletin of the Korean Mathematical Society
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• v.39 no.3
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• pp.359-376
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• 2002

We give a new interpretation of Darboux transforms in the context of orthogonal polynomials and find conditions in or-der for any Darboux transform to yield a new set of orthogonal polynomials. We also discuss connections between Darboux trans-forms and factorization of linear differential operators which have orthogonal polynomial eigenfunctions.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.40 no.1
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• pp.34-42
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• 2003

In this paper, the image restoration process is interpreted as a backward diffusion process and the restored image is given as the solution of the backward diffusion equation (BDF). The ill-posedness of the backward diffusion if subdued by manipulating the exponentially increasing coefficients of the eigenfunctions. In manipulating the coefficients the spectral characteristics of an image is taken into account. The proposed scheme uses an exponentially decreasing function of the coefficients of the eigenfunctions beyond a certain threshold which is optimal with respect to the observation accuracy. The use of decreasing functions also improves the result compared with the constant bounded algorithm since it can include more low frequency components.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.721-745
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• 2020

In this paper we study fractional Sobolev spaces characterized by a norm based on eigenfunction expansions. The goal of this paper is twofold. The first one is to define fractional Sobolev spaces of order -1 ≤ s ≤ 2 equipped with a norm defined in terms of Neumann eigenfunction expansions. Due to the zero Neumann trace of Neumann eigenfunctions on a boundary, fractional Sobolev spaces of order 3/2 ≤ s ≤ 2 characterized by the norm are the spaces of functions with zero Neumann trace on a boundary. The spaces equipped with the norm are useful for studying cross-sectional traces of solutions to the Helmholtz equation in waveguides with a homogeneous Neumann boundary condition. The second one is to define fractional Sobolev spaces of order -1 ≤ s ≤ 1 for vector-valued functions in a simply-connected, bounded and smooth domain in ℝ². These spaces are defined by a norm based on series expansions in terms of eigenfunctions of the vector Laplacian with boundary conditions of zero tangential component or zero normal component. The spaces defined by the norm are important for analyzing cross-sectional traces of time-harmonic electromagnetic fields in perfectly conducting waveguides.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.21 no.12
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• pp.2038-2046
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• 1997

In this paper the instability of the system which has a disk and a mass-spring-damper system interacting through a medium having stiffness and damping is analyzed. To solve the equations of motion of this systme, it is assumed that the solution consists of the eigenfunctions which are the products of the Bessel functions and sine or cosine functions. The former represents the radial characteristics of the disk and the latter represents the circumferential characteristics. Using this assumed solution and the orthogonality of the eigenfunctions, the equations of motion can be transformed into a set of equations of motion with variables dependent only on the time. After this set is changed to the state equation, the eigenvalue problem can be made. Once the eigenvalues are calculated according to the angular velocity of the disk, the dynamic characteristics ofthis system is obtained. Because the thickness of the disk and the element characteristics of the mass-spring-damper system have important effects on the stability of the system, it will be understood how these factors affect the system and then a method to ameliorate the stability of the system with a disk will be presented.

• Journal of the Korean earth science society
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• v.35 no.5
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• pp.333-341
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• 2014

A new set of basis functions was constructed using the Rossby-Haurwitz waves, which are the eigenfunctions of nondivergent barotropic vorticity equations on the sphere. The basis functions were designed to be non-separable, that is, not factored into functions of either the longitude or the latitude. Due to this property, the nodal lines of the functions are aligned neither along with the meridian nor the parallel. The basis functions can be categorized into groups of which members have the same degree or the total wavenumber-like index on the sphere. The orthonormality of the basis functions were found to be close to the machine roundoffs, giving the error of $O(10^{-15})$ or $O(10^{-16})$ for double-precision computation (64 bit arithmetic). It was demonstrated through time-stepping procedure that the basis functions were also the eigenfunctions of the non-divergent barotropic vorticity equations. The projection of the basis functions was carried out onto the low-resolution geopotential field of Gaussian bell, and compared with the theory. The same projections were performed for the observed atmospheric-geopotential height field of 500 hPa surface to demonstrate decomposition into the fields that contain disturbance of certain range of horizontal scales. The usefulness of the new basis functions was thus addressed for application to the eigenmode analysis of the atmospheric motions on the global domain.

• Journal of Industrial Technology
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• v.27 no.B
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• pp.91-96
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• 2007

Electrical impedance tomography (EIT) is a technique for determining the electrical conductivity and permittivity distribution within the interior of a body from measurements made on its surface. One recent application area of the EIT is the detection of breast cancer by imaging the conductivity and permittivity distribution inside the breast. The present standard for breast cancer detection is X-ray mammography, and it is desirable that EIT and X-ray mammography use the same geometry. A forward model of a simplified mammography geometry for EIT imaging was proposed earlier. In this paper, we propose an iterative algorithm for computing the current pattern that will be applied to the electrodes. The current pattern applied to the electrodes influences the voltages measured on the electrodes. Since the measured voltage data is going to be used in the impedance imaging computation, it is desirable to apply currents that result in the largest possible voltage signal. We compute the eigenfunctions for a homogenous medium that will be applied as current patterns to the electrodes. The algorithm for the computation of the eigenfunctions is presented. The convergence of the algorithm is shown by computing the eigencurrent of the simplified mammography geometry.

• Journal of the Society of Naval Architects of Korea
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• v.30 no.1
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• pp.125-133
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• 1993

Most studies on the vibration analysis of a beam-column with ends elastically restrained and various intermediate constraints have been based on the Euler beam theory, which is inadequate for beam-columns of low slenderness ratios. In this paper, analytical methods for vibration and dynamic sensitivity of a Timoshenko beam-column with ends elastically restrained and various intermediate constraints are presented. Firstly, an exact solution method is shown. Since the exact method requires considerable computational effort, a Rayleigh-Ritz analysis is also investigated. In the latter two kinds of trial functions are examined for comparisions : eigenfunctions of the base system(the system without intermediate constraints) and polynomials having properties corresponding to the eigenfunctions of the base system. The results of some numerical Investigations show that the Rayleigh-Ritz analysis using the characteristic polynomials is competitive with the exact solutions in accuracy, and that it is much more efficient in computations than using the eigenfunctions of the base system, especially in the dynamic sensitivity analysis. In addition, the prediction of the changes of natural frequencies due to the changes of design variables based on the first order sensitivity is in good agreements with that by the ordinary reanalysis as long as the changes of design variables are moderate.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.129-147
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• 2011

In this paper we discuss the zeta-determinants of harmonic oscillators having general quadratic potentials defined on $\mathbb{R}^2$. By using change of variables we reduce the harmonic oscillators having general quadratic potentials to the standard harmonic oscillators and compute their spectra and eigenfunctions. We then discuss their zeta functions and zeta-determinants. In some special cases we compute the zeta-determinants of harmonic oscillators concretely by using the Riemann zeta function, Hurwitz zeta function and Gamma function.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.83-93
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• 2002

We investigate a history of reseahches of a nonlinear elliptic equation with jumping nonlinearity, under Dirichlet boundary condition. The investigation will be focussed on the researches by topological methods. We also add recent researches, relations between multiplicity of solutions and source terms of tile equation when the nonlinearity crosses two eigenvalues and the source term is generated by three eigenfunctions.