• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1805-1821
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• 2016

In this paper, we are concerned with the nonlinear elliptic equations of the p(x)-Laplace type $$\{\begin{array}{lll}-div(a(x,{\nabla}u))+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u) && in\;{\Omega}\\(a(x,{\nabla}u)\frac{{\partial}u}{{\partial}n}={\lambda}{\theta}g(x,u) && on\;{\partial}{\Omega},\end{array}$$ which is subject to nonlinear Neumann boundary condition. Here the function a(x, v) is of type${\mid}v{\mid}^{p(x)-2}v$ with continuous function $p:{\bar{\Omega}}{\rightarrow}(1,{\infty})$ and the functions f, g satisfy a $Carath{\acute{e}}odory$ condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.

• Journal of the Korean Institute of Telematics and Electronics
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• v.12 no.2
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• pp.1-10
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• 1975

This paper presents an adaptive data compression algorithm for video data. The coling complexity due to the high correlation in the given data sequence is alleviated by coding the difference data, sequence rather than the data sequence itself. The adaptation to the nonstationary statistics of the data is confined within a code set, which consists of two constant length cades and six modified Shannon.Fano codes. lt is assumed that the probability distributions of tile difference data sequence and of the data entropy are Laplacian and Gaussion, respectively. The adaptive coding performance is compared for two code selection criteria: entropy and $P_r$[difference value=0]=$P_0$. It is shown that data compression ratio 2 : 1 is achievable with the adaptive coding. The gain by the adaptive coding over the fixed coding is shown to be about 10% in compression ratio and 15% in code efficiency. In addition, $P_0$ is found to he not only a convenient criterion for code selection, but also such efficient a parameter as to perform almost like entropy.

• Korean Journal of Radiology
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• v.22 no.5
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• pp.759-769
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• 2021

Objective: To evaluate the application of laplacian-regularized mean apparent propagator (MAPL)-MRI to brain glioma-induced corticospinal tract (CST) injury. Materials and Methods: This study included 20 patients with glioma adjacent to the CST pathway who had undergone structural and diffusion MRI. The entire CSTs of the affected and healthy sides were reconstructed, and the peritumoral CSTs were manually segmented. The morphological characteristics of the CST (track number, average length, volume, displacement of the affected CST) were examined and the diffusion parameter values, including fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD), radial diffusivity (RD), mean squared displacement (MSD), q-space inverse variance (QIV), return-to-origin probability (RTOP), return-to-axis probabilities (RTAP), and return-to-plane probabilities (RTPP) along the entire and peritumoral CSTs, were calculated. The entire and peritumoral CST characteristics of the affected and healthy sides as well as those relative CST characteristics of the patients with motor weakness and normal motor function were compared. Results: The track number, volume, MD, RD, MSD, QIV, RTAP, RTOP, and RTPP of the entire and peritumoral CSTs changed significantly for the affected side, whereas the AD and FA changed significantly only in the peritumoral CST (p < 0.05). In patients with motor weakness, the relative MSD of the entire CST, QIV of the entire and peritumoral CSTs, and the AD, MD, RD of the peritumoral CST were significantly higher, whereas the RTPP of the entire and peritumoral CSTs and the RTOP of the peritumoral CST were significantly lower than those in patients with normal motor function (p < 0.05 for all). In contrast, no significant changes were found in the CST morphological characteristics, FA, or RTAP (p > 0.05 for all). Conclusion: MAPL-MRI is an effective approach for evaluating microstructural changes after CST injury. Its sensitivity may improve when using the peritumoral CST features.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.917-926
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• 1994

Let ($M, G_M, F$) be a (p+q)-dimensional Riemannian manifold with a foliation F of codimension q and a bundle-like metric $g_M$ with respect to F ([9]). Aside from the Laplacian $\bigtriangleup_g$ associated to the metric g, there is another differnetial operator, the Jacobi operator $J_D$, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum isdiscrete as a consequence of the compactness of M. The study of the spectrum of $\bigtriangleup_g$ acting on functions or forms has attracted a lot of attention. In this point of view, the present authors [7] have studied the spectrum of the Laplacian and the curvature of a compact orientable cosymplectic manifold. On the other hand, S. Nishikawa, Ph. Tondeur and L. Vanhecke [6] studied the spectral geometry for Riemannian foliations. The purpose of the present paper is to study the relation between two spectra and the transversal geometry of cosymplectic foliations. We shall be in $C^\infty$-category. Manifolds are assumed to be connected.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.437-444
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• 2019

For c > -1, let ${\nu}_c$ denote a weighted radial measure on ${\mathbb{C}}$ normalized so that ${\nu}_c(D)=1$. For $c_1,c_2>-1$ and $f{\in}L^1(D^2,\;{\nu}_{c_1}{\times}{\nu}_{c_2})$, we define the weighted Berezin transform $B_{c_1,c_2}f$ on $D^2$ by $$(B_{c_1,c_2})f(z,w)={\displaystyle{\smashmargin2{\int\nolimits_D}{\int\nolimits_D}}}f({\varphi}_z(x),\;{\varphi}_w(y))\;d{\nu}_{c_1}(x)d{\upsilon}_{c_2}(y)$$. This paper is about the space $M^p_{c_1,c_2}$ of function $f{\in}L^p(D^2,\;{\nu}_{c_1}{\times}{\nu}_{c_2})$ ) satisfying $B_{c_1,c_2}f=f$ for $1{\leq}p<{\infty}$. We find the identity operator on $M^p_{c_1,c_2}$ by using invariant Laplacians and we characterize some special type of functions in $M^p_{c_1,c_2}$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.125-144
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• 2017

Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

• The Pure and Applied Mathematics
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• v.18 no.3
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• pp.245-253
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• 2011

Some new results on the intersection property of all nonzero solutions of a class of planar systems of Li$\acute{e}$nard type with vertical isoclines are obtained. The results of this paper generalize some previous results on this field.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.407-414
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• 2020

In this article, a second-order Sturm-Liouville problem with impulsive effects and involving the one-dimensional p-Laplacian is considered. The existence of at least three weak solutions via variational methods and critical point theory is obtained.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.401-416
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• 1994

We study the existence of an intermediate solution of nonlinear elliptic boundary value problems (BVP) of the form $$ (BVP) {\Delta u = f(x,u,\Delta u), in \Omega {Bu(x) = \phi(x), on \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $R^n, n \geq 1, and \partial\Omega \in C^{2,\alpha}, (0 < \alpha < 1), \Delta$ is the Laplacian operator, $\nabla u = (D_1u, D_2u, \cdots, D_nu)$ denotes the gradient of u and $$ Bu(x) = p(x)u(x) + q(x)\frac{d\nu}{du} (x), $$ where $\frac{d\nu}{du} denotes the outward normal derivative of u on $\partial\Omega$.

• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.339-358
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• 2000

Let be a bonded domain in N with smooth boundary . In this paper, we consider the existence of solutions of the following problem: (1.1)-div{} - + = , , , , , , where q > 1, p$\geq$1, $\delta$>0, , the Laplacian in N and is a positive function like as .