• Journal of the Korean Statistical Society
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• v.24 no.1
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• pp.91-109
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• 1995

The rate of convergence to a random variable S for an almost certainly convergent series $S_n = \sum^n_{j=1} X_j$ of independent random variables is studied in this paper. More specifically, when $S_n$ converges to S almost certainly, the tail series $T_n = \sum^{\infty}_{j=n} X_j$ is a well-defined sequence of random variable with $T_n \to 0$ a.c. Various sets of conditions are provided so that for a given numerical sequence $0 < b_n = o(1)$, the tail series strong law of large numbers $b^{-1}_n T_n \to 0$ a.c. holds. Moreover, these results are specialized to the case of the weighted i.i.d. random varialbes. Finally, example are provided and an open problem is posed.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.55-60
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• 2014

In this paper, we will obtain Marcinkiewicz's type limit laws for fuzzy random sets as follows : Let {$X_n{\mid}n{\geq}1$} be a sequence of independent identically distributed fuzzy random sets and $E{\parallel}X_i{\parallel}^r_{{\rho_p}}$ < ${\infty}$ with $1{\leq}r{\leq}2$. Then the following are equivalent: $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ a.s. in the metric ${\rho}_p$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in probability in the metric ${\rho}_p$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in $L_1$ if and only if $S_n/n^{\frac{1}{r}}{\rightarrow}{\tilde{0}}$ in $L_r$ where $S_n={\Sigma}^n_{i=1}\;X_i$.

• Journal of the Institute of Convergence Signal Processing
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• v.1 no.2
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• pp.169-176
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• 2000

Deadbeat property is well established in digital control system design in time domain. But in continuous time system, deadbeat is impossible because of it's ripples between sampling points inspite of designs using the related digital control system design theory. But several researchers suggested delay elements. A delay element is made from the concept of finite Laplace Transform. From some specifications such as internal model stability, physical realizations as well as finite time settling, unknown coefficents and poles in error transfer functions with delay elements can be calulted so as to satisfy these specifications. For the application to the real system, robustness property can be added. In this paper, error transfer function is specified with 1 delay element and robustness condition is considered additionally. As the criterion of the robustness, a weighted sensitive function's $H_{infty}$ norm is used. For the minimum value of the criterion, error transfer function's poles are calculated optimally. In this sense, optimal design of the continuous time deadbeat controller is obtained.

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.815-834
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• 1995

Let $\Omega$ be a rectangular domain in $R^2$ with boundary $\partial\Omega$, and T be a positive real number such that $0 < T < \infty$.

• 한국전산유체공학회:학술대회논문집
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• 1999.05a
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• pp.186-191
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• 1999

An Euler solver is developed to predict accurate aerodynamic data such as lift coefficient, drag coefficient, and moment coefficient. The conservation law form of the compressible Euler equations are used in the generalized curvilinear coordinates system. The Euler solver uses a finite volume method and the second order Roe's flux difference splitting scheme with min-mod flux limiter to calculate the fluxes accurately. An implicit scheme which includes the boundary conditions is implemented to accelerate the convergence rate. The multi-block grid is integrated into the flow solver for complex geometry. The flowfields are analyzed around NACA 0012 airfoil in the cases of $M_{\infty}=0.75,\;\alpha=2.0\;and\;M_{\infty}=0.80,\;\alpha=1.25$. The numerical results are compared with other numerical results from the literature. The final goal of this research is to prepare a robust and an efficient Navier-Stokes solver eventually.

• East Asian mathematical journal
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• v.26 no.3
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• pp.351-363
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• 2010

We introduce a special class of real recurrent polynomials $f_m$$($m{\geq}1$) of degree m+1, with positive roots $s_m$, which are decreasing as m increases. The first root $s_1$, as well as the last one denoted by $s_{\infty}$ are expressed in closed form, and enclose all $s_m$ (m > 1). This technique is also used to find weaker than before [6] sufficient convergence conditions for some popular iterative processes converging to solutions of equations.

• Korean Journal of Mathematics
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• v.19 no.3
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• pp.279-289
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• 2011

Let E be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, C be a nonempty closed convex subset of E and f : $C{\rightarrow}C$ be a fixed bounded continuous strong pseudocontraction with the coefficient ${\alpha}{\in}(0,1)$. Let $\{{\lambda}_t\}_{0<t<1}$ be a net of positive real numbers such that ${\lim}_{t{\rightarrow}0}{\lambda}_t={\infty}$ and S = {$T(s)$ : $0{\leq}s$ < ${\infty}$} be a nonexpansive semigroup on C such that $F(S){\neq}{\emptyset}$, where F(S) denotes the set of fixed points of the semigroup. Then sequence {$x_t$} defined by $x_t=tf(x_t)+(1-t)\frac{1}{{\lambda}_t}{\int_{0}}^{{\lambda}_t}T(s)x{_t}ds$ converges strongly as $t{\rightarrow}0$ to $\bar{x}{\in}F(S)$, which solves the following variational inequality ${\langle}(f-I)\bar{x},\;p-\bar{x}{\rangle}{\leq}0$ for all $p{\in}F(S)$.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.753-762
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• 2010

Let K be a nonempty closed convex subset of a real Hilbert space H. Let T : K $\rightarrow$ K be a nonexpansive mapping with a nonempty fixed point set Fix(T). Let f : K $\rightarrow$ K be a contraction mapping. Let {$\alpha_n$} and {$\beta_n$} be sequences in (0, 1) such that $\lim_{x{\rightarrow}0}{\alpha}_n=0$, (0.1) $\sum_{n=0}^{\infty}\;{\alpha}_n=+{\infty}$, (0.2) 0 < a ${\leq}\;{\beta}_n\;{\leq}$ b < 1 for all $n\;{\geq}\;0$. (0.3) Then it is proved that the modified Krasnoselski-Mann iterative sequence {$x_n$} given by {$x_0\;{\in}\;K$, $y_n\;=\;{\alpha}_{n}f(x_n)+(1-\alpha_n)x_n$, $n\;{\geq}\;0$, $x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$, $n\;{\geq}\;0$, (0.4) converges strongly to a point p $\in$ Fix(T} which satisfies the variational inequality

$\leq$ 0, z $\in$ Fix(T). (0.5) This result improves and extends the corresponding results of Yao et al[Y.Yao, H. Zhou, Y. C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J Appl Math Com-put (2009)29:383-389.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.177-196
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• 2008

Using the results on the coefficients of Hermite-Fej$\acute{e}$r interpolations in [5], we investigate convergence of Hermite and Hermite-$Fej{\acute{e}}r$ interpolation of order m, m=1,2,... in $L_p(0<p<{\infty})$ and associated product quadrature rules for a class of fast decaying even $Erd{\H{o}}s$ weights on the real line.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.505-513
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• 2009

Let {$X_k;k{\in}N^d$} be centered and identically distributed random field which is asymptotically negative dependent in a certain case. In this note we prove that for $p{\alpha}$ > 1 and ${\alpha}$ > ${\frac{1}{2}}$ $E{\mid}X_1{\mid}^p(log^+{\mid}X_1{\mid}^{d-1})$ < ${\infty}$ if and only if ${\sum}_n{\mid}n{\mid}^{p{\alpha}-2}P$($max_{1{\leq}k{\leq}n{\mid}S_k{\mid}}$ > ${\epsilon}{\mid}n{\mid}$) < ${\infty}$ for all ${\epsilon}$ > 0, where log$^+$x = max{1,log x}.