• 대한수학회지
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• 제45권5호
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• pp.1393-1404
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• 2008

Let K be a nonempty closed convex subset of a Banach space E. Suppose $\{T_{n}\}$ (n = 1,2,...) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ${\cap}_{n=1}^{\infty}$ F$\(T_n){\neq}{\phi}$. For $x_0{\in}K$, define $x_{n+1}={\lambda}_{n+1}x_{n}+(1-{\lambda}_{n+1})T_{n+1}x_{n},n{\geq}0$. If ${\lambda}_n{\subset}[0,1]$ satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n=0$, we proved that $\{x_n\}$ weakly converges to some $z{\in}F\;as\;n{\rightarrow}{\infty}$ in the framework of reflexive Banach space E which satisfies the Opial's condition or has $Fr{\acute{e}}chet$ differentiable norm or its dual $E^*$ has the Kadec-Klee property. We also obtain that $\{x_n\}$ strongly converges to some $z{\in}F$ in Banach space E if K is a compact subset of E or there exists one map $T{\in}\{T_{n};n=1,2,...\}$ satisfy some compact conditions such as T is semi compact or satisfy Condition A or $lim_{n{\rightarrow}{\infty}}d(x_{n},F(T))=0$ and so on.

• 대한수학회지
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• 제50권2호
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• pp.331-360
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• 2013

Let ${\sigma}_s(N)$ denote the sum of the sth powers of the positive divisors of a positive integer N and let $\tilde{\sigma}_s(N)={\sum}_{d|N}(-1)^{d-1}d^s$ with $d$, N, and s positive integers. Hahn [12] proved that $$16\sum_{k. In this paper, we give a generalization of Hahn's result. Furthermore, we find the formula ${\sum}_{k=1}^{N-1}\tilde{\sigma}_1(2^{n-m}k)\tilde{\sigma}_3(2^nN-2^nk)$ for $m(0{\leq}m{\leq}n)$.

• 호남수학학술지
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• 제30권2호
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• pp.233-246
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• 2008

In this paper, we obtain the general solution of the following cubic type functional equation and establish the stability of this equation (0.1) $kf({{\sum}\limits^{n-1}_{j=1}}x_j+kx_n)+kf({{\sum}\limits^{n-1}_{j=1}}x_j-kx_n)+2{{\sum}\limits^{n-1}_{j=1}}f(kx_j)+(k^3-1)(n-1)[f(x_1)+f(-x_1)]=2kf({\sum\limits^{n-1}_{j=1}}x_j)=K^3{\sum\limits^{n-1}_{j=1}[f(x_j+x_n)+f(x_j-x_n)]$ for any integers k and n with k ${\geq}$ 2 and n ${\geq}$ 3.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.375-382
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• 2007

We Study, firstly, the dynamics of the difference equation $x_{n+1}\;=\;{\alpha}\;+\;{\frac{x^p_n}{x^p_{n-1}}}$, with $p\;{\in}\;(0,\;1)\;and\;{\alpha}\;{\in}\;[0,\;{\infty})$. Then, we generalize our results to the (k + 1)th order difference equation $x_{n+1}\;=\;{\alpha}\;+\;{\frac{x^p_n}{nx^p_{n-k}}$, $k\;=\;2,\;3,\;{\cdots}$ with positive initial conditions.

• 대한수학회논문집
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• 제23권1호
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• pp.61-65
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• 2008

In this paper we prove that every positive solution of the third order rational difference equation $$x_{n+1}\;=\;\frac{{\beta}x_n\;+\;x_{n-2}}{A\;+\;Bx_n\;+\;x_{n-2}}$ converges to the positive equilibrium point $$\bar{x}\;=\;\frac{{\beta}\;+\;1\;-\;A}{B\;+\;1}$, where $0\;<\;{\beta}\;{\leq}\;B$, $1\;<\;A\;<\;{\beta}\;+\;1$

• 대한수학회보
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• 제53권4호
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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.

• 호남수학학술지
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• 제38권3호
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• pp.507-552
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• 2016

For a complex number q and a divisor function ${\sigma}_1(n)$ we define $$C(q):=q{\prod_{n=1}^{\infty}}(1-q^n)^{16}(1-q^{2n})^4,\\D(q):=q^2{\prod_{n=1}^{\infty}}(1-q^n)^8(1-q^{2n})^4(1-q^{4n})^8,\\L(q):=1-24{\sum_{n=1}^{\infty}}{\sigma}_1(n)q^n$$ moreover we obtain the number of representations of $n{\in}{\mathbb{N}}$ as sum of 24 squares, which are possible for us to deduce $L(q^4)C(q)$ and $L(q^4)D(q)$.

• Journal of Microbiology and Biotechnology
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• 제28권2호
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• pp.218-226
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• 2018

Nanometric Lactobacillus plantarum nF1 (nLp-nF1) is a biogenics consisting of dead L. plantarum cells pretreated with heat and a nanodispersion process. In this study, we investigated the immune-enhancing effects of nLp-nF1 in vivo and in vitro. To evaluate the immunostimulatory effects of nLp-nF1, mice immunosuppressed by cyclophosphamide (CPP) treatment were administered with nLp-nF1. As expected, CPP restricted the immune response of mice, whereas oral administration of nLp-nF1 significantly increased the total IgG in the serum, and cytokine production (interleukin-12 (IL-12) and tumor necrosis factor alpha (TNF-${\alpha}$)) in bone marrow cells. Furthermore, nLp-nF1 enhanced the production of splenic cytokines such as IL-12, TNF-${\alpha}$, and interferon gamma (IFN-${\gamma}$). In vitro, nLp-nF1 stimulated the immune response by enhancing the production of cytokines such as IL-12, TNF-${\alpha}$, and IFN-${\gamma}$. Moreover, nLp-nF1 given a food additive enhanced the immune responses when combined with various food materials in vitro. These results suggest that nLp-nF1 could be used to strengthen the immune system and recover normal immunity in people with a weak immune system, such as children, the elderly, and patients.

• 대한수학회지
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• 제45권4호
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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 $EX^2={\sigma}^2<{\infty}$.

• 대한수학회보
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• 제54권6호
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• pp.2001-2011
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• 2017

In 1997, H. Li [12] proposed a conjecture: if $M^n(n{\geqslant}3)$ is a complete spacelike hypersurface in de Sitter space $S^{n+1}_1(1)$ with constant normalized scalar curvature R satisfying $\frac{n-2}{n}{\leqslant}R{\leqslant}1$, then is $M^n$ totally umbilical? Recently, F. E. C. Camargo et al. ([5]) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ and also prove that $M^n$ is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface $M^n$ is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ satisfies $K(M^n)$ > 0, then $M^n$ is totally umbilical, i.e., Theorem 2.