• 대한수학회보
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• 제47권6호
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• pp.1235-1250
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• 2010

In this paper we establish the existence of at least three weak solutions for Neumann doubly eigenvalue elliptic systems driven by a ($p_1,\ldots,p_n$)-Laplacian. Our main tool is a recent three critical points theorem of B. Ricceri.

• East Asian mathematical journal
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• 제31권5호
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• pp.727-735
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• 2015

We introduce the fundamental theorem of upper and lower solutions for a class of singular ($p_1,\;p_2$)-Laplacian systems and give the proof by using the Schauder fixed point theorem. It will play an important role to study the existence of solutions.

• East Asian mathematical journal
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• 제38권3호
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• pp.331-338
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• 2022

In this paper, we prove that the first nonzero eigenvalues λ₁ of the Laplacian and the p-Laplacian are decreasing along the inverse mean curvature flow with forced term in Euclidean space.

• Korean Journal of Mathematics
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• 제29권1호
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• pp.13-24
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• 2021

For a finite commutative ring R with identity 1 ≠ 0, the zero divisor graph ��(R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs ��(ℤn) for various values of n. Also, we find signless Laplacian spectrum of ��(ℤn) for n = pz, z ≥ 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph ��(ℤn) are signless Laplacian integral.

• 대한수학회지
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• 제58권6호
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• pp.1449-1460
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• 2021

In this paper, we consider the following nonlocal fractional Laplacian equation with a singular nonlinearity (-∆)^su(x) = λu^β (x) + a₀u^-γ (x), x ∈ ℝⁿ, where 0 < s < 1, γ > 0, $1<{\beta}{\leq}\frac{n+2s}{n-2s}$, λ > 0 are constants and a₀ ≥ 0. We use a direct method of moving planes which introduced by Chen-Li-Li to prove that positive solutions u(x) must be radially symmetric and monotone increasing about some point in ℝⁿ.

• Journal of applied mathematics & informatics
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• 제41권5호
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• pp.1115-1128
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• 2023

In this paper, a novel hybrid method for multi-focus image fusion is proposed. The method combines the advantages of wavelet transform-based methods and focus-measure-based methods to achieve an improved fusion result. The input images are first decomposed into different frequency sub-bands using the discrete wavelet transform (DWT). The focus measure of each sub-band is then calculated using the Laplacian of Gaussian (LoG) operator, and the sub-band with the highest focus measure is selected as the focused sub-band. The focused sub-band is sharpened using an unsharp masking filter to preserve the details in the focused part of the image.Finally, the sharpened focused sub-bands from all input images are fused using the maximum intensity fusion method to preserve the important information from all focus images. The proposed method has been evaluated using standard multi focus image fusion datasets and has shown promising results compared to existing methods.

• East Asian mathematical journal
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• 제34권5호
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• pp.643-649
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• 2018

Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigenvectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we proved that a connected 2-type hypersurface M in $E^{n+1}$ whose postion vector x satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle}$, ${\rangle}$ is the usual inner product in $E^{n+1}$, is of null 2-type and has constant mean curvature and scalar curvature.

• 대한수학회지
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• 제57권6호
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• pp.1451-1470
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• 2020

We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)^s_p is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L^∞(Ω) of any possible weak solution by applying the bootstrap argument.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제15권5호
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• pp.1761-1777
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• 2021

An image with infrared features and visible details is obtained by processing infrared and visible images. In this paper, a fusion method based on Laplacian pyramid and generative adversarial network is proposed to obtain high quality fusion images, termed as Laplacian-GAN. Firstly, the base and detail layers are obtained by decomposing the source images. Secondly, we utilize the Laplacian pyramid-based method to fuse these base layers to obtain more information of the base layer. Thirdly, the detail part is fused by a generative adversarial network. In addition, generative adversarial network avoids the manual design complicated fusion rules. Finally, the fused base layer and fused detail layer are reconstructed to obtain the fused image. Experimental results demonstrate that the proposed method can obtain state-of-the-art fusion performance in both visual quality and objective assessment. In terms of visual observation, the fusion image obtained by Laplacian-GAN algorithm in this paper is clearer in detail. At the same time, in the six metrics of MI, AG, EI, MS_SSIM, Q_abf and SCD, the algorithm presented in this paper has improved by 0.62%, 7.10%, 14.53%, 12.18%, 34.33% and 12.23%, respectively, compared with the best of the other three algorithms.

• 대한수학회보
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• 제58권5호
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• pp.1109-1127
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• 2021

We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃₁ and 𝜃₂. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.