• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.371-385
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• 2004

In this paper, we investigate the uniqueness of positive solutions to weak competition models with self-cross diffusion rates under homogeneous Dirichlet boundary conditions. The methods employed are upper-lower solution technique and the variational characterization of eigenvalues.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.981-992
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• 2012

Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed to solving such equations. We introduce the NHPM for solving nonlinear integro-differential equations. Several examples for solving integro-differential equations are presented to illustrate the efficiency of the proposed NHPM.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.825-844
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• 2020

We obtain infinitely many solutions for a class of fractional Schrödinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.547-552
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• 2007

In this note, we extends some of the results of Liu [Fuzzy Sets and systems 157 (2006) 869-878]. This extension consists of a simple proof involving weighted functions and their preference index. We also give an elementary simple proof of the maximum entropy weighting function problem with a given preference index value without using any advanced theory like variational principles or without using Lagrangian multiplier methods.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.879-893
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• 2016

In this paper, we introduce three subgradient extragradient algorithms for solving pseudomonotone equilibrium problems. The paper originates from the subgradient extragradient algorithm for variational inequalities and the extragradient method for pseudomonotone equilibrium problems in which we have to solve two optimization programs onto feasible set. The main idea of the proposed algorithms is that at every iterative step, we have replaced the second optimization program by that one on a specific half-space which can be performed more easily. The weakly and strongly convergent theorems are established under widely used assumptions for bifunctions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.4
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• pp.296-310
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• 2023

In this paper, we derive the Γ-limit of functionals pertaining to some optimal material distribution problems that involve a variable exponent, as the exponent goes to infinity. In addition, we prove a relaxation result for supremal optimal design functionals with respect to the weak-∗ L^∞(Ω; [0, 1])× W^1,p0 (Ω;ℝ^m) weak topology.

• Bulletin of the Korean Chemical Society
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• v.34 no.1
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• pp.179-187
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• 2013

We describe newly developed software named KPACK for relativistic electronic structure computation of molecules containing heavy elements that enables the two-component ab initio calculations in Kramers restricted and unrestricted formalisms in the framework of the relativistic effective core potential (RECP). The spin-orbit coupling as relativistic effect enters into the calculation at the Hartree-Fock (HF) stage and hence, is treated in a variational manner to generate two-component molecular spinors as one-electron wavefunctions for use in the correlated methods. As correlated methods, KPACK currently provides the two-component second-order M${\o}$ller-Plesset perturbation theory (MP2), configuration interaction (CI) and complete-active-space self-consistent field (CASSCF) methods. Test calculations were performed for the ground states of group-14 elements, for which the spin-orbit coupling greatly influences the determination of term symbols. A categorization of three procedures is suggested for the two-component methods on the basis of spin-orbit coupling manifested in the HF level.

• Korean Journal of Optics and Photonics
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• v.13 no.1
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• pp.51-57
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• 2002

We have calculated the optimum refractive index and normalized thickness of a single layer antireflection coating on the facet of buried channel waveguides as a function of waveguide width for several waveguide depths using the angular spectrum method and field profiles obtained by the effective index method (EIM) and the variational method (VM), respectively, and discussed the results. In the area of large waveguide width, the optimum parameters of a single layer antireflection coating obtained by both methods are almost the same. However, as waveguide width decreases, the parameters obtained by the VM approach those of a single layer antireflection coating between cladding layer and air, while those obtained by the EIM do not approach those, and the difference between the two parameters is large. The tolerance maps of the quasi-TE and quasi-TM modes obtained by the VM for square waveguides are located in almost the same area regardless of refractive index contrast, while those obtained by the free space radiation mode (FSRM) method for refractive index contrast of 10% are located in the different area. Thus, we think that the tolerance maps obtained by the VM are more exact than those obtained by the FSRM method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.2
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• pp.143-156
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• 2014

This paper aims to give a quick view on denoising without comprehensive details. Denoising can be understood as removing unwanted parts in signals and images. Noise incorporates intrinsic random fluctuations in the data. Since noise is ubiquitous, denoising methods and models are diverse. Starting from what noise means, we briefly discuss a denoising model as maximum a posteriori estimation and relate it with a variational form or energy model. After that we present a few major branches in image and signal processing; filtering, shrinkage or thresholding, regularization and data adapted methods, although it may not be a general way of classifying denoising methods.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.747-775
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• 2020

In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, $$(P)\;\{(-{\Delta}_p)^su={\lambda}{\mid}u{\mid}^{q-2}u+{\frac{{\mid}u{\mid}^{p{^*_s}(t)-2}u}{{\mid}x{\mid}^t}}{\hspace{10}}in\;{\Omega},\\u=0{\hspace{217}}in\;{\mathbb{R}}^N{\backslash}{\Omega},$$ where Ω ⊂ ℝ^N is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < t < sp < N, 1 < q < p < p^∗_s where $p^*_s={\frac{N_p}{N-sp}}$, $p^*_s(t)={\frac{p(N-t)}{N-sp}}$, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-∆_p)^su with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by $\displaystyle(-{\Delta}_p)^su(x)=2{\lim_{{\epsilon}{\searrow}0}}\int{_{{\mathbb{R}}^N{\backslash}{B_{\epsilon}}}}\;\frac{{\mid}u(x)-u(y){\mid}^{p-2}(u(x)-u(y))}{{\mid}x-y{\mid}^{N+ps}}dy$, x ∈ ℝ^N. The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some α ∈ (0, 1), the weak solution to the problem (P) is in C^1,α(${\bar{\Omega}}$).