• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.223-228
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• 2015

In this paper, we study the eigenvalue problem of Schr$\ddot{o}$dinger-type operator on bounded domains in strictly pseudoconvex CR manifolds and obtain some universal inequalities for lower order eigenvalues. Moreover, we will give some generalized Reilly-type inequalities of the first nonzero eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex CR manifold without boundary.

• International Journal of CAD/CAM
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• v.6 no.1
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• pp.89-97
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• 2006

This paper presents a novel approach for non-iterative surface smoothing with feature preservation on arbitrary meshes. Laplacian operator is performed in a global way over the mesh. The surface smoothing is formulated as a quadratic optimization problem, which is easily solved by a sparse linear system. The cost function to be optimized penalizes deviations from the global Laplacian operator while maintaining the overall shape of the original mesh. The features of the original mesh can be preserved by adding feature constraints and barycenter constraints in the system. Our approach is simple and fast, and does not cause surface shrinkage and distortion. Many experimental results are presented to show the applicability and flexibility of the approach.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1471-1484
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• 2020

It may very well be difficult to prove an eigenvalue inequality of Payne-Pólya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-Pólya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.

• Kyungpook Mathematical Journal
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• v.32 no.3_spc
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• pp.401-413
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• 1992

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.9 no.2
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• pp.139-149
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• 1991

In general, Digital Elevation Models(DEM) are constructed in a grid format for this form is advantageous to capture data automatically and is easy to manipulate. But, grid DEM requires vast volumes of data to represent terrain features finely and accurately as its data is sampled in a regular space. This paper aims at constructing compact DEM by selecting the significant points from grid DEM which affect well the terrain features. For the purpose, the significant points is detected by the high-pass filtering using Laplacian operator and gradient operator. The results of this study show that the Laplacian operator is more efficient than the gradient operator in selecting the significant points for compact DEM.

• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1131-1145
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• 2012

The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with p-Laplacian type operators $$\{({\alpha}(t,x^{\Delta}(t)))^{\Delta}-a(t){\phi}_p(x^{\sigma}(t))+f({\sigma}(t),x^{\sigma}(t))=0,\;{\Delta}-a.e.\;t{\in}I\\x^{\sigma}(0)=0,\\{\beta}_1x^{\sigma}(1)+{\beta}_2x^{\Delta}({\sigma}(1))=0,$$ where ${\beta}_1$, ${\beta}_2$ > 0, $I=[0,1]^{k^2}$, ${\alpha}({\cdot},x({\cdot}))$ is an operator of $p$-Laplacian type, $\mathbb{T}$ is a time scale. Some sufficient conditions for the existence of constant-sign solutions are obtained.

• Journal of the Korean Institute of Intelligent Systems
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• v.11 no.6
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• pp.454-458
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• 2001

Based on a fuzzy system representation of gray scale images, we derive and edge detection algorithm whose convolution kernel is different from the known kernels such as those of Robert's Prewitt's or Sobel's gradient. Our fuzzy system representation is an exact representation of the bicubic spline function which represents the gray scale image approximately. Hence the fuzzy system is a continuous function and it provides a natural way to define the gradient and the Laplacian operator. We show that the gradient at grid points can be evaluated by taking the convolution of the image with a 3$\times$3 kernel. We also that our gradient coupled with the approximate value of the continuous function generates an edge detection method which creates edge images clearer than those by other methods. A few examples of applying our methods are included.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.25 no.1B
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• pp.141-147
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• 2000

In the restoration of degraded noisy motion blurred image, we have trade-off problem between smoothing the noise and restoration of the edge region. While the noise is smoothed, die edge or details will be corrupted. On the other hand, restoring the edge will amplify the noise. To solve this problem we propose an adaptive algorithm which uses I- H regularization operator for flat region and Laplacian regularization operator for edge region. Through the experiments, we verify that the proposed method shows better results in the suppression of the noise amplification in flat region, introducing less ringing artifacts in edge region and better ISNR than those of the conventional ones.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.329-337
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• 2007

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\{^{\;(\phi_p(u'))'\;+\;q(t)f(t,u)=0,\;0\;<\;t\;<\;1,}_{\;u(0)\;-\;B(u'({\eta}))\;=\;0,\;u'(1)\;=\;0}$$ and $$\{^{\;(\phi_p(u'))'\;+\;q(t)f(t,u)=0,\;0\;<\;t\;<\;1,}_{\;u'(0)\;=\;0,\;u(1)+B(u'(\eta))\;=\;0.}$$. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0, 1.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.413-425
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• 2001

We use the branching rules on SU(3) to show that if two Aloff-Wallach spaces $M_{k,l}\;and\;M_{k',l'}$ are isospectral for the Laplacian acting on smooth functions, they are isometric. We also show that 1 is the non-zero smallest eigenvalue among all Aloff-Wallach spaces and compute the multiplicities.