• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.437-448
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• 2007

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let $T_1,\;T_2\;and\;T_3\;:\;K{\rightarrow}E$ be nonexpansive mappings with nonempty common fixed points set. Let $\{\alpha_n\},\;\{\beta_n\},\;\{\gamma_n\},\;\{\alpha'_n\},\;\{\beta'_n\},\;\{\gamma'_n\},\;\{\alpha'_n\},\;\{\beta'_n\}\;and\;\{\gamma'_n\}$ be real sequences in [0, 1] such that ${\alpha}_n+{\beta}_n+{\gamma}_n={\alpha}'_n+{\beta'_n+\gamma}'_n={\alpha}'_n+{\beta}'_n+{\gamma}'_n=1$, starting from arbitrary $x_1{\in}K$, define the sequence $\{x_n\}$ by $$\{zn=P({\alpha}'_nT_1x_n+{\beta}'_nx_n+{\gamma}'_nw_n)\;yn=P({\alpha}'_nT_2z_n+{\beta}'_nx_n+{\gamma}'_nv_n)\;x_{n+1}=P({\alpha}_nT_3y_n+{\beta}_nx_n+{\gamma}_nu_n)$$ with the restrictions $\sum^\infty_{n=1}{\gamma}_n<\infty,\;\sum^\infty_{n=1}{\gamma}'_n<\infty,\; \sum^\infty_{n=1}{\gamma}'_n<\infty$. (i) If the dual $E^*$ of E has the Kadec-Klee property, then weak convergence of a $\{x_n\}$ to some $x^*{\in}F(T_1){\cap}{F}(T_2){\cap}(T_3)$ is proved; (ii) If $T_1,\;T_2\;and\;T_3$ satisfy condition(A'), then strong convergence of $\{x_n\}$ to some $x^*{\in}F(T_1){\cap}{F}(T_2){\cap}(T_3)$ is obtained.

• 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 Microbiology and Biotechnology
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• v.20 no.1
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• pp.63-70
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• 2010

Novel influenza A (H1N1) is a newly emerged flu virus that was first detected in April 2009. Unlike the avian influenza (H5N1), this virus has been known to be able to spread from human to human directly. Although it is uncertain how severe this novel H1N1 virus will be in terms of human illness, the illness may be more widespread because most people will not have immunity to it. In this study, we compared the codon usage bias between the novel H1N1 influenza A viruses and other viruses such as H1N1 and H5N1 subtypes to investigate the genomic patterns of novel influenza A (H1N1). Totally, 1,675 nucleotide sequences of the hemagglutinin (HA) and neuraminidase (NA) genes of influenza A virus, including H1N1 and H5N1 subtypes occurring from 2004 to 2009, were used. As a result, we found that the novel H1N1 influenza A viruses showed the most close correlations with the swine-origin H1N1 subtypes than other H1N1 viruses, in the result from not only the analysis of nucleotide compositions, but also the phylogenetic analysis. Although the genetic sequences of novel H1N1 subtypes were not exactly the same as the other H1N1 subtypes, the HA and NA genes of novel H1N1s showed very similar codon usage patterns with other H1N1 subtypes, especially with the swine-origin H1N1 influenza A viruses. Our findings strongly suggested that those novel H1N1 viruses seemed to be originated from the swine-host H1N1 viruses in terms of the codon usage patterns.

• East Asian mathematical journal
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• v.24 no.1
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• pp.35-43
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• 2008

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let ${\{T_i\}}^N_{i=1}$ be N nonexpansive self-mappings of K with $F\;=\;{\cap}^N_{i=1}F(T_i)\;{\neq}\;{\theta}$ (here $F(T_i)$ denotes the set of fixed points of $T_i$). Suppose that one of the mappings in ${\{T_i\}}^N_{i=1}$ is semi-compact. Let $\{{\alpha}_n\}\;{\subset}\;[{\delta},\;1-{\delta}]$ for some ${\delta}\;{\in}\;(0,\;1)$ and $\{{\beta}_n\}\;{\subset}\;[\tau,\;1]$ for some ${\tau}\;{\in}\;(0,\;1]$. For arbitrary $x_0\;{\in}\;K$, let the sequence {$x_n$} be defined iteratively by $\{{x_n\;=\;{\alpha}_nx_{n-1}\;+\;(1-{\alpha}_n)T_ny_n,\;\;\;\;\;\;\;\;\; \atop {y_n\;=\;{\beta}nx_{n-1}\;+\;(1-{\beta}_n)T_nx_n},\;{\forall}_n{\geq}1,}$, where $T_n\;=\;T_{n(modN)}$. Then {$x_n$} convergence strongly to a common fixed point of the mappings family ${\{T_i\}}^N_{i=1}$. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of ${\beta}_n\;{\equiv}\;1$ or N=1 are also new.

• Korean Journal of Crystallography
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• v.5 no.2
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• pp.78-84
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• 1994

The crystal structure of N-11-(benzotriazol-1-yl)butyl]-P-nitroaniline ( C16H17N502) has been determinedfromsingle crystal x-ray diffractionstudy:C16H17N502 monoclinic, P21/n, a=17542(2)A, b=10.755(3)A, c=8.891(1)A, β=104.58(1)˚, V=1623.4(5)A3, 7=293(2)K, Z=4, Cuka(A = 1.5418A) , The molecular structure was solved was by direct meshed refined by full-matrix least squares to a final R =0.0411 for 2248 unique observed [F≥4o(p) ] reflections and 255 Parameters. The crystal structure is stabilized by intermolecular N (11) -Hl 1 (Nl 1) ‥‥N (3) hydrogen bond with N(11) ‥‥ N(3) =3.136(2)A and N(11)-Hll(Nll)‥‥N(3) =164.1(15) ˚.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.321-328
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• 2004

In this paper, we will prove the following: Let D be a nonempty of a normed linear space X and T : D -> X be a nonexpansive mapping. Let ${x_n}$ be a sequence in D and ${t_n}$, ${s_n}$ be real sequences such that (i) $0\;{\leq}\;t_n\;{\leq}\;t\;<\;1\;and\;{\sum_{n=1}}^{\infty}\;t_n\;=\;{\infty},\;(ii)\;(a)\;0\;{\leq}\;s_n\;{\leq}\;1,\;s_n\;->\;0\;as\;n\;->\;{\infty}\;and\;{\sum_{n=1}}^{\infty}\;t_ns_n\;<\;{\infty}\;or\;(b)\;s_n\;=\;s\;for\;all\;n\;{\geq}\;1\;and\;s\;{\in}\;[0,1),\;(iii)\;x_{n+1}\;=\;(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n)\;for\;all\;n\;{\geq}\;1.$ Then, if the sequence {x_n} is bounded, then $lim_{n->\infty}\;$\mid$$\mid$x_n-Tx_n$\mid$$\mid$\;=\;0$. This result improves and complements a result of Deng [2]. Furthermore, we will show that certain conditions on D, X and T guarantee the weak and strong convergence of the Ishikawa iterative sequence to a fixed point of T.

• Journal of the Korean Mathematical Society
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• v.41 no.3
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• pp.479-487
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• 2004

Let G be a finite group and N be a normal subgroup of G. We denote by ncc(N) the number of conjugacy classes of N in G and N is called n-decomposable, if ncc(N) = n. Set $K_{G}\;=\;\{ncc(N)$\mid$N{\lhd}G\}$. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X. In this paper we characterise the {1, 3, 4}-decomposable finite non-perfect groups. We prove that such a group is isomorphic to Small Group (36, 9), the $9^{th}$ group of order 36 in the small group library of GAP, a metabelian group of order $2^n{2{\frac{n-1}{2}}\;-\;1)$, in which n is odd positive integer and $2{\frac{n-1}{2}}\;-\;1$ is a Mersenne prime or a metabelian group of order $2^n(2{\frac{n}{3}}\;-\;1)$, where 3$\mid$n and $2\frac{n}{3}\;-\;1$ is a Mersenne prime. Moreover, we calculate the set $K_{G}$, for some finite group G.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.549-582
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• 2016

All rings are commutative with $1{\neq}0$ and n is a positive integer. Let ${\phi}:{\Im}(R){\rightarrow}{\Im}(R){\cup}\{{\emptyset}\}$ be a function where ${\Im}(R)$ denotes the set of all ideals of R. We say that a proper ideal I of R is ${\phi}$-n-absorbing primary if whenever $a_1,a_2,{\cdots},a_{n+1}{\in}R$ and $a_1,a_2,{\cdots},a_{n+1}{\in}I{\backslash}{\phi}(I)$, either $a_1,a_2,{\cdots},a_n{\in}I$ or the product of $a_{n+1}$ with (n-1) of $a_1,{\cdots},a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of ${\phi}$-n-absorbing primary ideals.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.341-349
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• 2003

Let {$x_{nk}\;$\mid$1\;\leq\;k\;\leq\;n,\;n\;\geq\;1$} be an array of random varianbles and $\{a_n$\mid$n\;\geq\;1\}\;and\;\{b_n$\mid$n\;\geq\;1} be a sequence of constants with $a_n\;>\;0,\;b_n\;>\;0,\;n\;\geq\;1. In this paper, for array of row negatively associated(NA) random variables, we establish a general weak law of large numbers (WLLA) of the form (${\sum_{\kappa=1}}^n\;a_{\kappa}X_{n\kappa}\;-\;\nu_{n\kappa})\;/b_n$ converges in probability to zero, as $n\;\rightarrow\;\infty$, where {$\nu_{n\kappa}$\mid$1\;\leq\;\kappa\;\leq\;n,\;n\;\geq\;1$} is a suitable array of constants.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.89-96
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• 2009

In this article, we show a generalization of Clement's theorem on the pair of primes. For any integers n and k, integers n and n + 2k are a pair of primes if and only if 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) whenever (n, (2k)!) = (n + 2k, (2k)!) = 1. Especially, n or n + 2k is a composite number, a pair (n, n + 2k), for which 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) is called a pair of pseudoprimes for any positive integer k. We have pairs of pseudorimes (n, n + 2k) with $n{\leq}5{\times}10^4$ for each positive integer $k(4{\leq}k{\leq}10)$.