• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1029-1036
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• 1997

Let ${A_t)}_{t>0}$ be a dilation group given by $A_t = exp(-P log t)$, where P is a real $n \times n$ matrix whose eigenvalues has strictly positive real part. Let $\nu$ be the trace of P and $P^*$ denote the adjoint of pp. Suppose that $K$ is a function defined on $R^n$ such that $$\mid$K(x)$\mid$ \leq k($\mid$x$\mid$_Q)$ for a bounded and decreasing function $k(t) on R_+$ satisfying $k \diamond $\mid$\cdot$\mid$_Q \in \cup_{\varepsilon >0}L^1((1 + $\mid$x$\mid$)^\varepsilon dx)$ where $Q = \int_{0}^{\infty} exp(-tP^*) exp(-tP)$ dt and the norm $$\mid$\cdot$\mid$_Q$ stands for $$\mid$x$\mid$_Q = \sqrt{}, x \in R^n$. For $f \in L^1(R^n)$, define $mf(x) = sup_{t>0}$\mid$K_t * f(x)$\mid$$ where $K_t(X) = t^{-\nu}K(A_{1/t}^* x)$. Then we show that $m$ is a bounded operator of $L^1(R^n) into L^{1, \infty}(R^n)$.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1327-1345
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• 2021

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type $$u(x)=\vec{\;l\;}+C_{\ast}{{\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}^n}}}\;{\frac{(1-{\mid}u(y){\mid}^2){\mid}u(y){\mid}^2u(y)-\frac{1}{2}(1-{\mid}u(y){\mid}^2)^2u(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}}dy.$$ Here u : ℝⁿ → ℝ^k is a bounded, uniformly continuous function with k ⩾ 1 and 0 < α < n, $\vec{\;l\;}{\in}\mathbb{R}^k$ is a constant vector, and C_* is a real constant. We prove that ${\mid}\vec{\;l\;}{\mid}{\in}\{0,\frac{\sqrt{3}}{3},1\}$ if u is the finite energy solution. Further, if u is also a differentiable solution, then we give a Liouville type theorem, that is either $u{\rightarrow}\vec{\;l\;}$ with ${\mid}\vec{\;l\;}{\mid}=\frac{\sqrt{3}}{3}$, when |x| → ∞, or $u{\equiv}\vec{\;l\;}$, where ${\mid}\vec{\;l\;}{\mid}{\in}\{0,1\}$.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.271-281
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• 2006

The commutative products of the distributions $x^r\;ln^p\;{\mid}x{\mid}\;and\;x^{-r-1}ln^q\;{\mid}x{\mid}$ and of sgn x $x^r\;ln^p\;{\mid}x{\mid}\;and\;sgn\;x\;x^{-r-1}ln^q\;{\mid}x{\mid}$ are evaluated for $r=0,\;\pm1,\;\pm2,...\;and\;p,\;q=0,1,2,....$

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1291-1308
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• 2016

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for intersection of two complex ellipsoids {$z{\in}\mathbb{C}^3:{\mid}z_1{\mid}^p+{\mid}z_2{\mid}^q$ < 1, ${\mid}z_1{\mid}^p+{\mid}z_3{\mid}^r$ < 1}. We consider cases p = 6, q = r = 2 and p = q = r = 2. We also investigate the Lu Qi-Keng problem for p = q = r = 2.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1139-1151
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• 2022

This paper studies the eigenvalues of the G(·)-Laplacian Dirichlet problem $$\{-div\;\(\frac{g(x,\;{\mid}{\nabla}u{\mid})}{{\mid}{\nabla}u{\mid}}{\nabla}u\)={\lambda}\;\(\frac{g(x,{\mid}u{\mid})}{{\mid}u{\mid}}u\)\;in\;{\Omega}, \\u\;=\;0\;on\;{\partial}{\Omega},$$ where Ω is a bounded domain in ℝ^N and g is the density of a generalized Φ-function G(·). Using the Lusternik-Schnirelmann principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues.

• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.797-805
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• 2022

Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ P_n, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.421-437
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• 2023

Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≥ 1. Then Malik [12] obtained the following inequality: $${_{max \atop {\mid}z{\mid}=1}{\mid}p{\prime}(z){\mid}{\leq}{\frac{n}{1+k}}{_{max \atop {\mid}z{\mid}=1}{\mid}p(z){\mid}.$$ In this paper, we shall first improve as well as generalize the above inequality. Further, we also improve the bounds of two known inequalities obtained by Govil et al. [8].

• The Korean Journal of Applied Statistics
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• v.5 no.2
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• pp.243-254
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• 1992

In the present study two sets of unbalanced two-way cross-classification data with and without empty cell(s) were used to evaluate empirically the various sums of squares in the analysis of variance table. Searle(1977) and Searle et.al.(1981) developed a method of computing R($\alpha$\mid$\mu, \beta$) and R($\beta$\mid$\mu, \alpha$) by the use of partitioned matrix of X'X for the model of no interaction, interchanging the columns of X in order of $\alpha, \mu, \beta$ and accordingly the elements in b. An alternative way of computing R($\alpha$\mid$\mu, \beta$), R($\beta$\mid$\mu, \alpha$) and R($\gamma$\mid$\mu, \alpha, \beta$) without interchanging the columns of X has been found by means of,$(X'X)^-$ derived, using $W_2 = Z_2Z_2-Z_2Z_1(Z_1Z_1)^-Z_1Z_2$. It is true that $R(\alpha$\mid$\mu,\beta,\gamma)\Sigma = SSA_W and R(\beta$\mid$\mu,\alpha,\gamma)\Sigma = SSB_W$ where $SSA_W$ and means analysis and $R(\gamma$\mid$\mu,\alpha,\beta) = R(\gamma$\mid$\mu,\alpha,\beta)\Sigma$ for the data without empty cell, but not for the data with empty cell(s). It is also noticed that for the datd with empty cells under W - restrictions $R(\alpha$\mid$\mu,\beta,\gamma)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W = R(\alpha$\mid$\mu) and R(\beta$\mid$\mu,\alpha,\gamma)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W = R(\beta$\mid$\mu) but R(\gamma$\mid$\mu,\alpha,\beta)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W \neq R(\gamma$\mid$\mu,\alpha,\beta)$. The hypotheses $H_o : K' b = 0$ commonly tested were examined in the relation with the corresponding sums of squares for $R(\alpha$\mid$\mu), R(\beta$\mid$\mu), R(\alpha$\mid$\mu,\beta), R(\beta$\mid$\mu,\alpha), R(\alpha$\mid$\mu,\beta,\gamma), R(\beta$\mid$\mu,\alpha,\gamma), and R(\gamma$\mid$\mu,\alpha,\beta)$ under the restrictions.

• Proceedings of the Optical Society of Korea Conference
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• 2001.08a
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• pp.34-35
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• 2001

• Proceedings of the Optical Society of Korea Conference
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• 2002.11a
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• pp.222-223
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• 2002