• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.135-163
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• 2010

Motivated by [Science in China (Ser. A Mathematics) 36 (2006), no. 7, 721?732], this article deals with the following discrete type BVP $\LARGE\left\{{{\;{\Delta}[{\phi}({\Delta}x(n))]\;+\;f(n,\;x(n\;+\;1),{\Delta}x(n),{\Delta}x(n + 1))\;=\;0,\;n\;{\in}\;[0,N],}}\\{\;{x(0)-{\sum}^m_{i=1}{\alpha}_ix(n_i) = A,}}\\{\;{x(N+2)-\;{\sum}^m_{i=1}{\beta}_ix(n_i)\;=\;B.}}\right.$ The sufficient conditions to guarantee the existence of at least three positive solutions of the above multi-point boundary value problem are established by using a new fixed point theorem obtained in [5]. An example is presented to illustrate the main result. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multifixed-point theorems can be extended to treat nonhomogeneous BVPs. The emphasis is put on the nonlinear term f involved with the first order delta operator ${\Delta}$x(n).

• 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 $L^1(R^n) into L^{1, \infty}(R^n)$.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2006.05a
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• pp.1788-1792
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• 2006

The ordered weighted averaging (OWA) operator by Yager has received more and more attention since its appearance. One key point in the OWA operator is to determine its associated weights. Among numerous methods that have appeared in the literature, we notice the maximum entropy OWA (MEOWA) weights that are determined by taking into account two appealing measures characterizing the OWA weights. Instead of maximizing the entropy in the formulation for determining the MEOWA weights, the new method in the article tries to obtain the OWA weights which are evenly spread out around equal weights as much as possible while strictly satisfying the orness value provided in the program. This consideration leads to the least squared OWA (LSOWA) weighting method in which the program tries to obtain the weights that minimize the sum of deviations from the equal weights since entropy is maximized when the weights are equal. Above all, the LSOWA weights display symmetric allocations of weights on the basis of equal weights. The positive or negative allocations of weights from the median as a basis depend on the magnitude of orness specified. Further interval LSOWA weights are constructed when a decision-maker specifies his or her value of orness in uncertain numerical bounds.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1525-1532
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• 2011

The $m{\times}n$ Boolean matrix A is said to be of Boolean rank r if there exist $m{\times}r$ Boolean matrix B and $r{\times}n$ Boolean matrix C such that A = BC and r is the smallest positive integer that such a factorization exists. We consider the the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.623-632
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• 2010

A holomorphic function F defined on the unit disc belongs to $A^{p,{\alpha}}$ (0 < p < $\infty$, 1 < ${\alpha}$ < $\infty$) if $\int\limits_U|F(z)|^p \frac{1}{1-|z|}(1+log)\frac{1}{1-|z|})^{-\alpha}$ dxdy < $\infty$. For boundedness of the composition operator defined by $C_{fg}=g{\circ}f$ mapping Blochs into $A^{p,{\alpha}$ the following (1) is a sufficient condition while (2) is a necessary condition. (1) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha}M_p(r,\lambda{\circ}f)^p\;dr$ < $\infty$ (2) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha+p}(1-r)^pM_p(r,f^#)^p\;dr$ < $\infty$.

• Bulletin of the Korean Mathematical Society
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• v.36 no.1
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• pp.25-31
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• 1999

Let $\nu$ be a positive Borel measure on a space of homogeneous type (X, d, $\mu$), satisfying the doubling property. A condition on a weight $\omega$ for whixh a maximal operator $M\nu f$(x) defined by M$mu$f(x)=supr>0{{{{ { 1} over {ν(B(x,r)) } INT _{ B(x,r)} │f(y)│d mu (y)}}}}, is of weak type (p,p) with respect to (ν, $omega$), is that there exists a constant C such that C $omega$(y) for a.e. y$\in$B(x, r) if p=1, and {{{{( { 1} over { upsilon (B(x,r) } INT _{ B(x,r)}omega(y) ^ (-1/p-1) d mu (y))^(p-1)}}}} C, if 1$infty$.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1705-1723
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• 2008

Let n be a 2-step nilpotent Lie algebra which has an inner product <, > and has an orthogonal decomposition $n\;=z\;{\oplus}v$ for its center z and the orthogonal complement v of z. Then Each element z of z defines a skew symmetric linear map $J_z\;:\;v\;{\longrightarrow}\;v$ given by <$J_zx$, y> = for all x, $y\;{\in}\;v$. In this paper we characterize Jacobi fields and calculate all conjugate points of a simply connected 2-step nilpotent Lie group N with its Lie algebra n satisfying $J^2_z$ = A for all $z\;{\in}\;z$, where S is a positive definite symmetric operator on z and A is a negative definite symmetric operator on v.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.73-80
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• 1992

We discuss transverse harmonic fields on compact foliated Riemannian manifolds, and give a necessary and sufficient condition for a transverse field to be a transverse harmonic one and the non-existence of transverse harmonic fields. 1. On a foliated Riemannian manifold, geometric transverse fields, that is, transverse Killing, affine, projective, conformal fields were discussed by Kamber and Tondeur([3]), Molino ([5], [6]), Pak and Yorozu ([7]) and others. If the foliation is one by points, then transverse fields are usual fields on Riemannian manifolds. Thus it is natural to extend well known results concerning those fields on Riemannian manifolds to foliated cases. On the other hand, the following theorem is well known ([1], [10]): If the Ricci operator in a compact Riemannian manifold M is non-negative everywhere, then a harmonic vector field in M has a vanishing covariant derivative. If the Ricci operator in M is positive-definite, then a harmonic vector field other than zero does not exist in M.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.333-342
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• 2007

Suppose that ${\mu}$ is a finite positive Borel measure on the unit ball $B{\subset}C^n$. The boundary of B is the unit sphere $S=\{z:{\mid}z{\mid}=1\}$. Let ${\sigma}$ be the rotation-invariant measure on S such that ${\sigma}(S)=1$. In this paper, we will show that if $sup_{{\zeta}{\in}S}\;{\int}_{B}\;P(z,{\zeta})d{\mu}(z)$<${\infty}$ where $P(z,{\zeta})$ is the Poission-Szeg$\ddot{o}$ kernel for B, then ${\mu}$ is a Carleson measure. We will also show that if $sup_{{\zeta}{\in}S}\;{\int}_{B}\;P(z,{\zeta})d{\mu}(z)$<${\infty}$, then the operator T such that T(f) = P[f] is compact as a mapping from $L^p(\sigma)$ into $L^p(B,d{\mu})$.

• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.627-645
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• 2007

We investigate the eigenvalues of the semi-circulant preconditioned matrix for the finite difference scheme corresponding to the second-order elliptic operator with the variable coefficients given by $L_vu\;:=-{\Delta}u+a(x,\;y)u_x+b(x,\;y)u_y+d(x,\;y)u$, where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator $L_cu$ is constructed by using the leading term of $L_vu$ plus the constant reaction term such that $L_cu\;:=-{\Delta}u+d_cu$. Using the field of values arguments, we show that the eigenvalues of the preconditioned matrix are clustered at some number. Some numerical evidences are also provided.