• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.347-353
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• 2006

In this paper, we show that if the minimal basically disconnected cover ${\wedge}X_\imath\;of\;X_\imath$ is given by the space of fixed a $Z(X)^#$-ultrafilters on $X_\imath\;(\imath=1,2)\;and\;{\wedge}X_1\;{\times}\;{\wedge}X_2$ is a basically disconnected space, then ${\wedge}X_1\;{\times}\;{\wedge}X_2$ is the minimal basically disconnected cover of $X_1\;{\times}\;X_2$. Moreover, observing that the product space of a P-space and a countably locally weakly Lindelof basically disconnected space is basically disconnected, we show that if X is a weakly Lindelof almost P-space and Y is a countably locally weakly Lindelof space, then (${\wedge}X\;{\times}\;{\wedge}Y,\;{\wedge}_X\;{\times}\;{\wedge}_Y$) is the minimal basically disconnected cover of $X\;{\times}\;Y$.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.985-994
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• 1994

Boundary value problems are common in nature. Here we restrict our attention to the second order differential equations of the form $$ (1.1) \frac{d^2 y}{dx^2} = P(x)y(x) + Q(x), 0 \leq x \leq 1, $$ $$ y(0) = \alpha, $$ $$ y(1) = \beta, $$ where P(x) and Q(x) are continuous functions with $P(x) \geq 0, x \in [0, 1]$.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.9-20
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• 2008

Let ${X_i{\mid}i{\geq}1}$ be a strictly stationary sequence of associated random variables with mean zero and let ${\sigma}^2=EX_1^2+2\sum\limits_{j=2}^\infty{EX_1}{X_j}$ with 0 < ${\sigma}^2$ < ${\infty}$. Set $S_n={\sum\limits^n_{i=1}^\{X_i}$, the precise asymptotics for ${\varepsilon}^{{\frac{2(r-p)}{2-p}}-1}\sum\limits_{n{\geq}1}n^{{\frac{r}{p}}-{\frac{1}{p}}+{\frac{1}{2}}}P({\mid}S_n{\mid}{\geq}{\varepsilon}n^{{\frac{1}{p}}})$,${\varepsilon}^2\sum\limits_{n{\geq}3}{\frac{1}{nlogn}}p({\mid}Sn{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ and ${\varepsilon}^{2{\delta}+2}\sum\limits_{n{\geq}1}{\frac{(loglogn)^{\delta}}{nlogn}}p({\mid}S_n{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ as ${\varepsilon}{\searrow}0$ are established under the suitable conditions.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.703-712
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• 2011

In this paper, we investigate the stability of a cubic functional equation $$f(x+ny)+f(x-ny)+f(nx)=n^2f(x+y)+n^2f(x-y)+(n^3-2n^2+2)f(x)$$ in p-Banach spaces and in Banach modules, where $n{\geq}2$ is an integer.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.539-545
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• 1994

Let $q = p^r$, where p is a prime. A polynomial $f(x) \in GF(q)[x]$ is called a permutation polynomial (PP) over GF(q) if the numbers f(a) where $a \in GF(Q)$ are a permutation of the a's. In other words, the equation f(x) = a has a unique solution in GF(q) for each $a \in GF(q)$. More generally, $f(x_1, \cdots, x_n)$ is a PP in n variables if $f(x_1,\cdots,x_n) = \alpha$ has exactly $q^{n-1}$ solutions in $GF(q)^n$ for each $\alpha \in GF(q)$. Mullen ([3], [4], [5]) has studied the concepts of local permutation polynomials (LPP's) over finite fields. A polynomial $f(x_i, x_2, \cdots, x_n) \in GF(q)[x_i, \codts,x_n]$ is called a LPP if for each i = 1,\cdots, n, f(a_i,\cdots,x_n]$ is a PP in $x_i$ for all $a_j \in GF(q), j \neq 1$.Mullen ([3],[4]) found a set of necessary and three variables over GF(q) in order that f be a LPP. As examples, there are 12 LPP's over GF(3) in two indeterminates ; $f(x_1, x_2) = a_{10}x_1 + a_{10}x_2 + a_{00}$ where $a_{10} = 1$ or 2, $a_{01} = 1$ or x, $a_{00} = 0,1$, or 2. There are 24 LPP's over GF(3) of three indeterminates ; $F(x_1, x_2, x_3) = ax_1 + bx_2 +cx_3 +d$ where a,b and c = 1 or 2, d = 0,1, or 2.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.3
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• pp.245-261
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• 2006

In this paper, we prove the generalized Hyers-Ulam stability of ring homomorphisms over the p-adic field $\mathbb{Q}_p$ associated with the Cauchy functional equation f(x+y) = f(x)+f(y) and the Cauchy-Jensen functional equation $2f(\frac{x+y}{2}+z)=f(x)+f(y)+2f(z)$.

• Korean Journal of Food Science and Technology
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• v.35 no.4
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• pp.604-609
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• 2003

In order to investigate the transfer rate of calcium into rice and the effect of soaking conditions-temperature (20, 30, 40, 50 and $60^{\circ}C$, $X_1$), calcium concentration of soaking solution (0, 250, 500, 750 and 1000 mg/L, $X_2$) and soaking time (5, 10, 15, 20 and 25 min, $X_3$), moisture absorption rate, hardness and calcium content of rice after soaking were analyzed. When the soaking temperature and time were increased the moisture absorption rate also increased (p<0.01). Quadratic and interaction effects on the moisture absorption rate among variables were highly significant, within 5%. The predictive model for moisture absorption rate $(Y_1)$ was as follows: $Y=16.14+0.78X_1+1.24X_3-0.23X_1{^2}+0.14X_2{^2}-0.42X_3X_1-0.38X_3{^2}\;(R^2=0.9123)$. Hardness was decreased as soaking time and temperature increased. Calcium contents of soaked rice increased as temperature, the calcium concentration of soaking solution and soaking time were increased (p<0.01). Also the interactive effect between soaking time and calcium concentration of soaking solution was high. The predictive model for calcium content $(Y_3)$ was as follows: $Y_3=470.80+110.87X_1+123.47X_2+52.29X_3-7.42X_1{^2}-14.87X_2{^2}-9.49X_3X_2-20.57X_3{^2}\;(R^2=0.9840)$. From the results in this study, if adults and children take 400 g of calcium enriched rice with 600 mg/kg and $700{\sim}900\;mg/kg$, respectively, they can expect to take more calcium than the recommended intake without other calcium reinforcing agents.

• Journal of Integrative Natural Science
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• v.16 no.1
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• pp.39-45
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• 2023

P2X receptors are ATP-activated ion channels in the plasma membrane. P2X receptors have a role in a diverse range of disorders, making them a valuable therapeutic target. Hence, the present investigation employed homology modelling of the P2X receptor based on the crystal structure of 5SVJ, 6AH4, 5YVE and 5SVL. Twenty models, using both single- and multiple template-based methods, were developed, and the best model was chosen based on the validation result. We observed that a strategy based on multiple templates provided greater accuracy. Future studies involving binding site and docking analysis can make use of the produced structures.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1805-1821
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• 2016

In this paper, we are concerned with the nonlinear elliptic equations of the p(x)-Laplace type $$\{\begin{array}{lll}-div(a(x,{\nabla}u))+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u) && in\;{\Omega}\\(a(x,{\nabla}u)\frac{{\partial}u}{{\partial}n}={\lambda}{\theta}g(x,u) && on\;{\partial}{\Omega},\end{array}$$ which is subject to nonlinear Neumann boundary condition. Here the function a(x, v) is of type${\mid}v{\mid}^{p(x)-2}v$ with continuous function $p:{\bar{\Omega}}{\rightarrow}(1,{\infty})$ and the functions f, g satisfy a $Carath{\acute{e}}odory$ condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.161-171
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• 2012

We consider the fourth-order differential equation with one-dimensional $p$-Laplacian (${\phi}_p(x^{\prime\prime}(t)))^{\prime\prime}=f(t,x(t),x^{\prime}(t),x^{\prime\prime}(t)$) a.e. $t{\in}[0,1]$, subject to the boundary conditions $x^{\prime\prime}}(0)=0$, $({\phi}_p(x^{\prime\prime}(t)))^{\prime}{\mid}_{t=0}=0$, $x(0)={\sum}_{i=1}^n{\mu}_ix({\xi}_i)$, $x(t)=x(1-t)$, $t{\in}[0,1]$, where ${\phi}_p(s)={\mid}s{\mid}^{p-2}s$, $p$ > 1, 0 < ${\xi}_1$ < ${\xi}_2$ < ${\cdots}$ < ${\xi}_n$ < $\frac{1}{2}$, ${\mu}_i{\in}\mathbb{R}$, $i=1$, 2, ${\cdots}$, $n$, ${\sum}_{i=1}^n{\mu}_i=1$ and $f:[0,1]{\times}\mathbb{R}^3{\rightarrow}\mathbb{R}$ is a $L^1$-Carath$\acute{e}$odory function with $f(t,u,v,w)=f(1-t,u,-v,w)$ for $(t,u,v,w){\in}[0,1]{\times}\mathbb{R}^3$. We obtain the existence of at least one nonconstant symmetric solution by applying an extension of Mawhin's continuation theorem due to Ge. Furthermore, an example is given to illustrate the results.