• Journal for History of Mathematics
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• v.18 no.1
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• pp.85-92
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• 2005

We review the well known analogizes between variations and harmonic maps. Second section contains rapid summary of the harmonic maps. In the third section we establish some basic properties of the Laplacian on Sphere. In particular we compute its spectrum and eigenfunctions.

• Communications of the Korean Mathematical Society
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• v.16 no.3
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• pp.437-443
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• 2001

In this paper the inverse problems for the Dirac Operator are studied. A set of values of eigenfunctions in some internal point and spectrum are taken as a data. Uniqueness theorems are obtained. The approach that was used in the investigation of inverse problems for interior spectral data of the Sturm-Liouville operator is employed.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.153-167
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• 1996

In this paper we investigate the existence of solutions of a nonlinear wave equation $u_{tt} - u_{xx} = p(x, t, u)$$ in $H_0$, where $H_0$ is the Hilbert space spanned by eigenfunctions. If p satisfy condition $(p_1) - (p_3)$, this nonlinear gave equation has at least one solution.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.969-990
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• 2016

We derive a sampling theorem associated with first order self-adjoint eigenvalue problem with a finite rank perturbation. The class of the sampled integral transforms is of finite Fourier type where the kernel has an additional perturbation.

• East Asian mathematical journal
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• v.35 no.1
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• pp.41-50
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• 2019

This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

• Bulletin of the Society of Naval Architects of Korea
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• v.14 no.1
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• pp.1-10
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• 1977

The major objects of this report are to supplement data of natural frequencies of thin elastic rectangular plates to the available data, and to give an experimental verification for natural frequencies obtained by Rayleigh-Ritz method, the generation set of which are eigenfunctions of Euler beams. For the first object the following five models, for which data only for the fundamental mode or data only for square plates are available, are adopted; (1) two opposed edges are clamped and the other two opposed edges simply supported (C-C, S-S), (2) one edge is simply supported and the other three edges clamped (C-C, C-S), (3) one edge is free and the other three edges clamped (C-C, C-F), (4) two adjacent edges are clamped and the other two adjacent edges free (C-F, C-F). For the (C-C, S-S) model the frequency equation obtained with the mode shapes assumed as of a single trigonometric series is solved. And for the other four models Rayleigh-Ritz method taking eigenfunctions of Euler beams as the generating set is applied. The numerical examples are obtained up to the fourth, the fifth or the sixth order depending on the range of the aspect ratio (0.1-10.0). The number of terms in the generating set for Rayleigh-Ritz method is fifteen for all models. For the experiment three models made of 3.2mm thickness mild steel plate for general structure use were prepared in following size; $300mm{\times}600mm,\;600mm{\times}600mm\;and\;900mm{\times}600mm$. Their boundary conditions are made to fit (C-C, C-F) condition. From the experiment mechanical impedance curves based on the frequency response method were obtained together with phase relation diagrams. The experimental data are resulted in good conformity to calculated values.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.315-327
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• 2007

Mathematical aspects of the electromagnetic surface-wave propagation are examined for the dielectric core consisting of multiple sub-layers, which are embedded in the gap between the two bounding cladding metals. For this purpose, the linear problem with a partial differential wave equation is formulated into a nonlinear eigenvalue problem. The resulting eigenvalue is found to exist only for a certain combination of the material densities and the number of the multiple sub-layers. The implications of several limiting cases are discussed in terms of electromagnetic characteristics.

• Bulletin of the Korean Chemical Society
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• v.26 no.11
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• pp.1717-1722
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• 2005

The analytical transfer matrix method suggests a new quantization condition for calculating bound state eigenenergies exactly. In the quantization condition, the phase shifts of bound state wave functions scattered at classical turning points are explicitly introduced. We calculate the phase shifts of eigenfunctions of the Morse potential with various boundary conditions in order to understand the physical meaning of phase shifts. The Morse potential is known to adequately describe the interaction energy between two atoms and, therefore, it is frequently used to determine the vibrational energy levels of diatomic molecules. The variation of Morse potential eigenenergies influenced upon by changing boundary conditions is also investigated.

• Journal of the Korean Society for Aviation and Aeronautics
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• v.15 no.2
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• pp.9-17
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• 2007

A Green's function approach based on the laminate theory is adopted for solving the two-dimensional unsteady temperature field and the associated thermal stresses in an infinite plate made of functionally graded material (FGM). All material properties are assumed to depend only on the coordinate x (perpendicular to the surface). The unsteady heat conduction equation is formulated into an eigenvalue problem by making use of the eigenfunction expansion theory and the laminate theory. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the two-dimensional unsteady temperature. The associated thermoelastic field is analyzed by making use of the thermal stress function. Numerical analysis for a FGM plate is carried out and effects of material properties on unsteady thermoelastic behaviors are discussed.

• Communications for Statistical Applications and Methods
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• v.27 no.1
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• pp.149-161
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• 2020

In this paper we extend the functional principal component analysis for real-valued random functions to the case of Hilbert-space-valued functional random objects. For this, we introduce an autocovariance operator acting on the space of real-valued functions. We establish an eigendecomposition of the autocovariance operator and a Karuhnen-Loève expansion. We propose the estimators of the eigenfunctions and the functional principal component scores, and investigate the rates of convergence of the estimators to their targets. We detail the implementation of the methodology for the cases of compositional vectors and density functions, and illustrate the method by analyzing time-varying population composition data. We also discuss an extension of the methodology to multivariate cases and develop the corresponding theory.