• Korean Journal of Mathematics
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• v.18 no.4
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• pp.395-407
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• 2010

In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations $$cf\(\sum_{i=1}^{n}x_i\)+\sum_{j=2}^{n}f\(\sum_{i=1}^{n}x_i-(n+c-1)x_j\)\\=(n+c-1)\(f(x_1)+c\sum_{i=2}^{n}f(x_i)+\sum_{i<j,j=3}^{n}\(\sum_{i=2}^{n-1}f(x_i-x_j\)\),\\Q\(\sum_{i=1}^{n}d_ix_i\)+\sum_{1{\leq}i<j{\leq}n}d_id_jQ(x_i-x_j)=\(\sum_{i=1}^{n}d_i\)\(\sum_{i=1}^{n}d_iQ(x_i)\)$$ in random normed spaces.

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.239-250
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• 2014

Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim$_RH_I^i(M){\leq}k$ for ${\forall}i$ < t, then $$\displaystyle\bigcup_{i=0}^{j}(Ass_RH_I^i(M))_{{\geq}k}=\displaystyle\bigcup_{i=0}^{j}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$$ for ${\forall}j{\leq}t$ and ${\forall}n$ >0. This shows that${\bigcup}_{n>0}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$ is a finite set for ${\forall}i{\leq}t$. Also, we prove that $\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1^{n_1},x_2^{n_2},{\ldots},x_i^{n_i})M)_{{\geq}k}=\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1,x_2,{\ldots},x_i)M)_{{\geq}k}$ if $x_1,x_2,{\ldots},x_r$ is M-sequences in dimension > k and $n_1,n_2,{\ldots},n_r$ are some positive integers. Here, for a subset T of Spec(R), set $T_{{\geq}i}=\{{p{\in}T{\mid}dimR/p{\geq}i}\}$.

• Proceedings of the Korean Society of Crop Science Conference
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• 2007.04a
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• pp.78.2-78.2
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• 2007

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.573-580
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• 2006

In this paper, we obtain sufficient conditions for the oscillation of every solution of the difference equation $$x_{n+1}-x_n+\sum_{i=1}^{m}p_ix_{n-k_i}+qx_{n-z}=0,\;n=0,1,2,{\cdots},$$ where $p_i{\in}\mathbb{R}$, $k_i{\in}\mathbb{Z}$ for $i=1,2,{\cdots},m$ and $z{\in}\{-1,0\}$. Furthermore, we obtain sufficient conditions for the oscillation of all solutions of the equation $${\Delta}^rx_n+\sum_{i=1}^{m}p_ix_{n-k_i}=0,\;n=0,1,2,{\cdots},$$ where $p_i{\in}\mathbb{R}$, $k_i{\in}\mathbb{Z}$ for $i=1,2,{\cdots},m$. The results are given terms of the $p_i$ and the $k_i$ for each $i=1,2,{\cdots},m$.

• 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 Medicinal Crop Science
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• v.12 no.5
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• pp.401-405
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• 2004

Karyotypes were established in the eight Korean native species of the genus Iris. Chromosome numbers were 2n=50 in I. koreana and 2n=42 in I. uniflora var. carinata and their karyotype formulas were K = 2n = 50 = 14m + 28sm + 8st and K = 2n = 42 = 16m + 26sm, respectively. I. dichotoma and I. pseudoacorus were diploids of 2n=34. However, they showed different karyotype formulas: K = 2n = 34 = 26m + 6sm + 2st in I. dichotoma and K = 2n = 34 = 8m + 24sm + 2st in I. pseudoacorus. I. setosa, and I. pallasii var. chinensis carried the same chromosome numbers of 2n=40, but they showed different patterns of karyotype formula: K = 2n = 40 = 22m + 14sm + 4st in I. setosa and K = 2n = 40 = 26m + 12sm + 2st in I. pallasii var. chinensis. I. sanguinea was a diploid of 2n=28 and the karyotype formula was K = 2n = 28 = 14m + 14sm. I. ensata var. spontanea was a diploid of 2n=24 and the karyotype formula was K = 2n = 24 = 10m + 14sm. Each species showed characteristic chromosome composition with a pair of satellite chromosome except I. koreana with three pairs of satellite chromosomes. The chromosomes of I. dichotoma and I. uniflora were comparatively short, while the chromosomes of I. ensata were remarkably bigger than those of other species. These cytological data will give a useful information for the identification and breeding program of the Iris plants.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1137-1146
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• 2011

Let {$X_i$, $i{\geq}1$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $S_n={\sum}_{i=1}^n\;X_i$, $M_n=max_{1{\leq}i{\leq}n}\;{\mid}S_i{\mid}$ and $V_n^2={\sum}_{i=1}^n\;X_i^2$. Then for d > -1, we showed that under some regularity conditions, $$\lim_{{\varepsilon}{\searrow}0}{\varepsilon}^2^{d+1}\sum_{n=1}^{\infty}\frac{(loglogn)^d}{nlogn}I\{M_n/V_n{\geq}\sqrt{2loglogn}({\varepsilon}+{\alpha}_n)\}=\frac{2}{\sqrt{\pi}(1+d)}{\Gamma}(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2d+2}}\;a.s.$$ holds in this paper, where If g denotes the indicator function.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.993-1005
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• 2008

Let {$X,\;X_n;n{\geq}1$} be a sequence of i.i.d. random variables. Set $S_n=X_1+X_2+{\cdots}+X_n,\;M_n=\max_{k{\leq}n}|S_k|,\;n{\geq}1$. Then we obtain that for any -1$\lim\limits_{{\varepsilon}{\searrow}0}\;{\varepsilon}^{2b+2}\sum\limits_{n=1}^\infty\;{\frac {(log\;n)^b}{n^{3/2}}\;E\{M_n-{\varepsilon}{\sigma}\sqrt{n\;log\;n\}+=\frac{2\sigma}{(b+1)(2b+3)}\;E|N|^{2b+3}\sum\limits_{k=0}^\infty\;{\frac{(-1)^k}{(2k+1)^{2b+3}$ if and only if EX=0 and $EX^2={\sigma}^2<{\infty}$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.959-970
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• 2016

In this paper, we generalize the stability for an n-dimensional cubic functional equation in Banach space to set-valued dynamics. Let $n{\geq}2$ be an integer. We define the n-dimensional cubic set-valued functional equation given by $$f(2{{\sum}_{i=1}^{n-1}}x_i+x_n){\oplus}f(2{{\sum}_{i=1}^{n-1}}x_i-x_n){\oplus}4{{\sum}_{i=1}^{n-1}}f(x_i)\\=16f({{\sum}_{i=1}^{n-1}}x_i){\oplus}2{{\sum}_{i=1}^{n-1}}(f(x_i+x_n){\oplus}f(x_i-x_n)).$$ We first prove that the solution of the n-dimensional cubic set-valued functional equation is actually the cubic set-valued mapping in [6]. We prove the Hyers-Ulam stability for the set-valued functional equation.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.331-346
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• 2009

Let ${\alpha}$ = (${\alpha}_1,\;{\alpha}_2$,...) and ${\beta}$ = (${\beta}_1,\;{\beta}_2$,...) be two sequences with ${\alpha}_1$ = ${\beta}_1$ and k and n be natural numbers. We denote by $A^{(k,{\pm})}_{{\alpha},{\beta}}(n)$ the matrix of order n with coefficients ${\alpha}_{i,j}$ by setting ${\alpha}_{1,i}$ = ${\alpha}_i,\;{\alpha}_{i,1}$ = ${\beta}_i$ for 1 ${\leq}$ i ${\leq}$ n and $${\alpha}_{i,j}=\{{\alpha}_{i-1,j-1}+{\alpha}_{i-1,j}\;if\;j{\equiv}$$2,3,4,..., k + 1 (mod 2k) $$\{{\alpha}_{i-1,j-1}-{\alpha}_{i-1,j}\;if\;j{\equiv}$$ k + 2,..., 2k + 1 (mod 2k) for 2 ${\leq}$ i, j ${\leq}$ n. The aim of this paper is to study the determinants of such matrices related to certain sequence ${\alpha}$ and ${\beta}$ and some natural numbers k.