• 대한수학회지
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• 제45권6호
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• pp.1785-1801
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• 2008

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier- Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

• 충청수학회지
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• 제23권1호
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• pp.1-10
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• 2010

In this note, we investigate the infinitesimal generators of the transformation groups induced by the Fourier-Gauss and Fourier-Mehler transforms on white noise functionals.

• Journal of applied mathematics & informatics
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• 제31권5_6호
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• pp.825-834
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• 2013

In this paper, we study the Yeh convolution of white noise functionals. We first introduce the notion of Yeh convolution of test white noise functionals and prove a dual property of the Yeh convolution. By applying the dual object of the Yeh convolution, we study the Yeh convolution of generalized white noise functionals, which is a non-trivial extension. Finally, we study relations between the Yeh convolution and Fourier-Gauss, Fourier-Mehler transform.

• 대한수학회보
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• 제48권2호
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• pp.291-302
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• 2011

We first study the generalized Fourier-Gauss transforms of functionals defined on the complexification $\cal{B}_C$ of an abstract Wiener space ($\cal{H}$, $\cal{B}$, ${\nu}$). Secondly, we introduce a new class of convolution products of functionals defined on $\cal{B}_C$ and study several properties of the convolutions. Then we study various relations among the first variation the convolutions, and the generalized Fourier-Gauss transforms.

• 한국광학회지
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• 제8권1호
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• pp.1-6
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• 1997

역변환 문제(Inverse Problem)로 접근하여 상면에서의 최초 진폭임펄스 S$_{0}$(x)를 Gauss 함수 g$_{1}$(x), -1/2g$_{2-}$ (x+.DELTA.x$_{1}$), -1/2g$_{2+}$(x-.DELTA.x$_{1}$)를 중첩하여 설정하였다. 이를 Inverse Fourier Transform으로 동함수 A(.omega.)를 구하고, 유한구경 (-.omega.$_{0}$~+.omega.$_{0}$)에서 A(.omega)를 Fourier Transform하여 회절상의 진폭임펄스(Amplitude Impulse) S(x)를 구하였다. .lambda.=193nm, NA=0.5인 광학계에서 S(x)의 반치폭, 즉 1/2(FWHM)을 수치계산하여 49nm를 얻었다. 이는 Rayleigh 한계분해능 .epsilon.$_{R}$의 반, 1/2.epsilon.$_{R}$=96.5nm 보다 작으므로 초분해능 광학계임을 알 수 있다. OTF를 구하여 광학계의 성능을 평가한 결과 고주파영역에서 성능이 우수함을 알 수 있었다. 광학계는 Twyman-Green 간섭계를 포함하는 간섭결상계가 되고, Gauss 진폭변조판은 Polysiloxane Glass Resin을 사용하여 만들 수 있음을 제안하였다.제안하였다.

• Structural Engineering and Mechanics
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• 제83권1호
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• pp.67-77
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• 2022

In this study, frictionless continuous and discontinuous contact problems of a magneto-electro-elastic layer in the presence of the body force were discussed. The layer was indented by a rigid cylindrical insulating punch and supported by a rigid substrate without bond. Applying the Fourier integral transform technique, the general expressions of the problem were derived in the presence of body force. Thanks to the boundary conditions, the singular integral equations were obtained for both the continuous and the discontinuous contact cases. Gauss-Chebyshev integration formulas were used to transform the singular integral equations into a set of nonlinear equations. Contact width under the punch, initial separation distance, critical load, separation regions and contact stress under the punch and between the layer, and substrate were given as a result.

• Structural Engineering and Mechanics
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• 제48권2호
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• pp.241-255
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• 2013

The receding contact problem for two elastic layers whose elastic constants and heights are different supported by two elastic quarter planes is considered. The lower layer is supported by two elastic quarter planes and the upper elastic layer is subjected to symmetrical distributed load whose heights are 2a on its top surface. It is assumed that the contact between all surfaces is frictionless and the effect of gravity force is neglected. The problem is formulated and solved by using Theory of Elasticity and Integral Transform Technique. The problem is reduced to a system of singular integral equations in which contact pressures are the unknown functions by using integral transform technique and boundary conditions of the problem. Stresses and displacements are expressed depending on the contact pressures using Fourier and Mellin formula technique. The singular integral equation is solved numerically by using Gauss-Jacobi integration formulation. Numerical results are obtained for various dimensionless quantities for the contact pressures and the contact areas are presented in graphics and tables.

• 지구물리와물리탐사
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• 제21권1호
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• pp.1-7
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• 2018

전기탐사 2차원 모델링에서는 다수의 파수영역 전위를 계산하고 이를 푸리에 역변환하여 공간영역 전위를 계산한다. 푸리에 역변환은 여러 개의 서로 다른 파수에서의 파수영역 전위를 사용하여 수치적으로 얻어진다. 적분의 정확도를 향상시키기 위하여 파수의 크기에 따라 적분 구간을 지수 근사와 대수 근사 구간으로 분할하는 방법이 널리 사용되고 있다. 푸리에 역변환에는 크게 구간 적분법과 가우스 적분법이 사용되고 있다. 그러나 이들 방법은 송수신 간격을 고려하지 못하므로 송수신 간격에 따른 오차를 피할 수 없다. 특히 송수신 간격이 매우 작거나 클 경우 오차가 급격하게 증가하는 문제점을 가지고 있다. 이 연구에서는 송수신 간격을 고려하여 가우스 좌표값 및 가중값을 적용하는 새로운 수치 적분법을 개발하였다. 반무한 공간에 대한 수치 실험 결과, 개발된 수치 적분법은 송수신 간격에 관계없이 0.4% 이하의 정밀도를 나타내었다.

• 한국생산제조학회지
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• 제7권1호
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• pp.41-50
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• 1998

The detemination of sound pressure radiated from peoriodic plate structures is fundamental in the estimation of noise levels in aircraft fuselages and ship hull structures. As a robust approach to this problem, here a very general and comprehensive analytical model for predicting the sound radiated by a vibrating plate stiffened by periodically spaced orthogonal symmetric beams subjected to a sinusoidally time varying point load is developed. The plate is assumed to be infinite in extent, and the beams are considered to exert both line force and moment reactions on it. Structural damping is included in both plate and beam materials. A space harmonic series representation of the spatial variables is used in conjunction with the Fourier transform to find the sound pressure in terms of harmonic coefficients. From this theoretical model. the sound pressure levels on axis in a semi-infinite fluid (water) bounded by the plate with the variation in the locations of an external time harmonic point force on the plate can be calculated efficiently using three numerical tools such as the Gauss-Jordan method, the LU decomposition method and the IMSL numerical package.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.185-220
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• 2016

We study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator defined on the half plane. Next, we establish a principle of maximum for the singular partial differential operator $${\Delta}_{\alpha}={\frac{{\partial}^2}{{\partial}r^2}+{\frac{2{\alpha}+1}{r}{\frac{\partial}{{\partial}r}}+{\frac{{\partial}^2}{{\partial}x^2}}+{\frac{{\partial}^2}{{\partial}t^2}}};\;(r,x,t){\in}]0,+{\infty}[{\times}{\mathbb{R}}{\times}]0,+{\infty}[$$. Later, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every $p{\in}]1,+{\infty}[$, there exists a positive constant $C_p$ such that for every $f{\in}L^p(d{\nu}_{\alpha})$, $${\frac{1}{C_p}}{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}{\leqslant}{\parallel}g(f){\parallel}_{p,{\nu}_{\alpha}}{\leqslant}C_p{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}$$.