• East Asian mathematical journal
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• v.25 no.2
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• pp.179-196
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• 2009

Let E be a uniformly convex Banach space and K a nonempty closed convex subset which is also a nonexpansive retract of E. For i = 1, 2, 3, let $T_i:K{\rightarrow}E$ be an asymptotically nonexpansive mappings with sequence ${\{k_n^{(i)}\}\subset[1,{\infty})$ such that $\sum_{n-1}^{\infty}(k_n^{(i)}-1)$ < ${\infty},\;k_{n}^{(i)}{\rightarrow}1$, as $n{\rightarrow}\infty$ and F(T)=$\bigcap_{i=3}^3F(T_i){\neq}{\phi}$ (the set of all common xed points of $T_i$, i = 1, 2, 3). Let {$a_n$},{$b_n$} and {$c_n$} are three real sequences in [0, 1] such that $\in{\leq}\;a_n,\;b_n,\;c_n\;{\leq}\;1-\in$ for $n{\in}N$ and some ${\in}{\geq}0$. Starting with arbitrary $x_1{\in}K$, define sequence {$x_n$} by setting {$$x_{n+1}=P((1-a_n)x_n+a_nT_1(PT_1)^{n-1}y_n)$$ $$y_n=P((1-b_n)x_n+a_nT_2(PT_2)^{n-1}z_n)$$ $$z_n=P((1-c_n)x_n+c_nT_3(PT_3)^{n-1}x_n)$$. Assume that one of the following conditions holds: (1) E satises the Opial property, (2) E has Frechet dierentiable norm, (3) $E^*$ has Kedec -Klee property, where $E^*$ is dual of E. Then sequence {$x_n$} converges weakly to some p${\in}$F(T).

• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.397-409
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• 2018

In this paper, we investigate the generalized Hyers-Ulam stability of the functional equation $$f\({\sum\limits_{i=1}^{n}}x_i\)+{\sum\limits_{1{\leq}i<j{\leq}n}}f(x_i-x_j)-n{\sum\limits_{i=1}^{n}f(x_i)=0$$ for integer values of n such that $n{\geq}2$, where f is a mapping from a vector space V to a Banach space Y.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.17-23
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• 2023

For a locus with two alleles (I^A and I^B), the frequencies of the alleles are represented by $$p=f(I^A)={\frac{2N_{AA}+N_{AB}}{2N}},\;q=f(I^B)={\frac{2N_{BB}+N_{AB}}{2N}}$$ where N_AA, N_AB and N_BB are the numbers of I^AI^A, I^AI^B and I^BI^B respectively and N is the total number of populations. The frequencies of the genotypes expected are calculated by using p², 2pq and q². Choi defined the density and operator for the value of the frequency of one gene and found the adapted partial differential equation as a follow-up for the frequency of alleles and applied this adapted partial differential equation to several diploid model [1]. In this paper, we find adapted equations for the model for selection against recessive homozygotes and in case that the alley frequency changes after one generation of selection when there is no dominance. Also we consider the triploid model with three alleles I^A, I^B and i and determine whether six genotypes observed are in Hardy-Weinburg for equilibrium.

• The Journal of Natural Sciences
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• v.11 no.1
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• pp.15-17
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• 1999

Let {f(n)} be an increasing sequence such that f(n)>0 for each n and f(n)$\rightarrow$$\infty$. Let {X$_n$,n$\geq1$} be a sequence of pairwise independent random variables. In this paper we give sufficient conditions on {X$_n$,n$\geq1$} such that $sum_{i=1}^n$(X$_i$-EX$_i$)/f(n) converges to zero almost surely.

• Korean Journal of Materials Research
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• v.2 no.5
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• pp.330-336
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• 1992

Four distinctive hot pressed and heat treated S${i_3}{N_4}$ceramics, S${i_3}{N_4}$-8%${Y_2}{O_3}$, S${i_3}{N_4}$-6% ${Y_2}{O_3}$-2% $A{l_2}{O_3}$, S${i_3}{N_4}$-4% ${Y_2}{O_3}$-3% $A{l_2}{O_3}$, 그리고 S${i_3}{N_4}$-1% MgO-1% Si$O_2$(in wt%), were prepared and characterized by X-ray diffraction, scanning electron microscopy, image analysis and mechanical tests. The fracture toughness of S${i_3}{N_4}$-8% ${Y_2}{O_3}$specimens containing large elongated grains showed the highest value of about 9.8MPa$m^{1/2}$. Two out of four S${i_3}{N_4}$, ceramics(S${i_3}{N_4}$-6% ${Y_2}{O_3}$-2% $A{l_2}{O_3}$and S${i_3}{N_4}$-4% ${Y_2}{O_3}$-3% $A{l_2}{O_3}$) heat treated at 200 $0^{\circ}C$retained the fracture strength of over 900MPa and fracture toughness of over 8.0MPa$m^{1/2}$. Large ${\beta}$-S${i_3}{N_4}$grains having a diameter larger than 1${\mu}$m appeared to contribute to increase in fracture toughness.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.679-690
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• 2005

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, f($x_1,{\cdots},\;x_n$) a multilinear polynomial in n non-commuting variables over K, a $\in$ R. Supppose that, for any $x_1,{\cdots},\;x_n\;\in\;I,\;a[d(f(x_1,{\cdots},\;x_n)),\;f(x_1,{\cdots},\;x_n)]$ = 0. If $[f(x_1,{\cdots},\;x_n),\;x_{n+1}]x_{n+2}$ is not an identity for I and $$S_4(I,\;I,\;I,\;I)\;I\;\neq\;0$$, then aI = ad(I) = 0.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.293-300
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• 1995

Let $R = k[y_1,\cdots,y_n] \otimes E[x_1, \cdots, x_n]$ with characteristic $k = p > 2$ (odd prime), where $$\mid$y_i$\mid$ = 2, $\mid$x_i$\mid$ = 1$ and $y_i = \betax_i, \beta$ is the Bockstein homomorphism. Topologically, $R = H^*(B(Z/p)^n,k)$. For a symmetric group $\sum_n, R^{\sum_n} = k[\sigma_1,\cdots,\sigma_n] \otimes E[d\sigma_1, \cdots, d\sigma_n]$ where d is the derivation satisfying $d(y_i) = x_i$ and $d(x_iy_i) = x_iy_i + x_jy_i, 1 \leq i, j \leq n$. We give a direct proof of this theorem by using induction.

• The Journal of Natural Sciences
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• v.8 no.2
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• pp.5-7
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• 1996

Let {X,$X_n$,n$\geq$1} be i.i.d. random variables with mean zero and {$a_ni$, 1$\leq$i$\leq$n,n$\geq$1} a triangular array of constants. In this paper we give sufficient conditions on X and {$a_ni$} such that $sum_{i=1}^n$$a_{ni}$$X_i$ converges to zero almostly surely.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.205-215
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• 2009

Sufficient conditions for the existence of at least one solution of the boundary value problems for higher order nonlinear difference equations $\{{{{{\Delta^n}x(i-1)=f(i,x(i),{\Delta}x(i),{\cdots},\Delta^{n-2}x(i)),i{\in}[1,T+1],\atop%20{\Delta^m}x(0)=0,m{\in}[0,n-3],}\atop%20\Delta^{n-2}x(0)=\phi(\Delta^{n-1}(0)),}\atop%20\Delta^{n-1}x(T+1)=-\psi(\Delta^{n-2}x(T+1))}\$. are established.

• Journal of Physiology & Pathology in Korean Medicine
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• v.22 no.4
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• pp.709-717
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• 2008

$Ch\acute{e}n\;Xiuyu\acute{a}n$(陳修園) was a famous doctor and educator of the late Tang Dynasty. He was well known both for his books for beginners, and for his unique medical theories based on his profound research of <$Sh\bar{a}ngh\acute{a}nl\grave{u}n$(傷寒論)> and <$J\bar{i}nku\grave{i}y\grave{a}ol\ddot{u}e$>. He wrote <$Y\bar{i}xues\bar{a}nz\grave{i}j\bar{i}ng$(醫學三字經)> to establish the basic textbook for the beginners to set up right principles in pursuing their medical career. <$Y\bar{i}xues\bar{a}nz\grave{i}j\bar{i}ng$> was written in rhyme form, so that it can be easily memorized and used in future practices. There are quite many medical books in rhyme form, but this book is very unique as $Ch\acute{e}n\;Xiuyu\acute{a}n$ annotated his own notes, which is rare in this form of books. This feature makes <$Y\bar{i}xues\bar{a}nz\grave{i}j\bar{i}ng$> very outstanding, also with the fact that $Ch\acute{e}n\;Xiuyu\acute{a}n$ was the one with profound understanding and original theories based on medical bibles such as <$N\grave{e}ij\bar{i}ng$(內經)> and <$Sh\bar{a}ngh\acute{a}nl\grave{u}n$(傷寒論)>. We have translated this precious educational material into korean, hoping that this work could be of any help to students of korean medicine. And while doing this work, we have found followings: <$Y\bar{i}xues\bar{a}nz\grave{i}j\bar{i}ng$> covers the entire fields of medicine from theoretical discussions to practical clinical information. Nevertheless, as this is written in rhyme form, there are few phrases that are not easily understood for the sake of rhyme. Beginners probably may have difficulties in reading this book. To make this difficulty alleviated, and to develop our own educational material, we need to study further on the notes that $Ch\acute{e}n\;Xiuyu\acute{a}n$ annotated himself.