• 대한수학회보
• /
• 제47권4호
• /
• pp.693-699
• /
• 2010

In this paper we show that the flat property of a left R-module does not imply (carry over) to the corresponding inverse polynomial module. Then we define an induced inverse polynomial module as an R[x]-module, i.e., given an R-linear map f : M $\rightarrow$ N of left R-modules, we define $N+x^{-1}M[x^{-1}]$ as a left R[x]-module. Given an exact sequence of left R-modules $$0\;{\rightarrow}\;N\;{\rightarrow}\;E^0\;{\rightarrow}\;E^1\;{\rightarrow}\;0$$, where $E^0$, $E^1$ injective, we show $E^1\;+\;x^{-1}E^0[[x^{-1}]]$ is not an injective left R[x]-module, while $E^0[[x^{-1}]]$ is an injective left R[x]-module. Make a left R-module N as a left R[x]-module by xN = 0. We show inj $dim_R$ N = n implies inj $dim_{R[x]}$ N = n + 1 by using the induced inverse polynomial modules and their properties.

• 한국전기전자재료학회논문지
• /
• 제22권8호
• /
• pp.677-681
• /
• 2009

$In_xGa_{1-x}N$ alloys with 20-nm-thickness were deposited onto Mg:GaN/AlN/SiC substrates by MOCVD at $800\;^{\circ}C$. TMGa, TMIn and $NH_3$ were used as the precursor of gallium, indium and nitrogen, respectively. The mole ratio of indium in $In_xGa_{1-x}N$ films varied between 0 and 0.2. The energy-band-gaps of the films were obtained from the photoluminescence and cathodoluminescence peaks. The mole ratios of $In_xGa_{1-x}N$ films were calculated by applying Vegard's law to XRD results. The energy-band-gap versus indium composition plot for $In_xGa_{1-x}N$ alloys were well fit with a bowing parameter of 2.27.

• 한국잠사곤충학회지
• /
• 제36권1호
• /
• pp.8-25
• /
• 1994

본 시험은 누에 일대잡종육종에 있어 양질다사량 우량품종을 육성하기 위한 유전적 정보를 얻고자 수행되었다. 공급재료는 특성이 각각 다른 일, 중, 구 3개지역 6개 품종을 정역으로 이면교잡시켜 F1, F2 각각 30조합으로서 실용형질에 대한 육종가를 얻기 위해 교배조합별 잡종강세 및 조합능력의 검정한 결과는 다음과 같았다. 1. 형질들의 잡종강세는 사질형질인 견사량, 견사장에서 각각 24.51% 및 23.43%로 높았고, 견질형질인 전사종 및 사층중은 15.56%~15.71%, 17.14%~19.01%로서 유의한 강세를 보였으나 유충경과일수 등(5령 및 전령)에서는 부의 잡종강세를 나타내어 그 이용이 유의하였다. 2. 교배조합간의 잡종강세는 5령경과일수의 경우 C70 X Romogua, N9 X Romogua 조합이 높은 부의 강세현상을 보여 사육일수가 단축되는 방향으로, 전견중은 N9 X Romogua, N9 X Romogua 조합이 높은 강세현상을 보여 사육일수가 단축되는 방향으로, 전견중은 N9 X Sansurian의 암, Romogua X Sansurian의 수, 견층중은 암수에 관계없이, N9 X Sansurian의 견사장과 견사량에서는 Sansurian X Romogua의 정역간 교잡에서 각각 높았다. 3. 잡종강세의 모본효과는 N9 X C5, N63 X C70이 전견중과 견층중에서, 견사장에서는 Sansurian이 N63, C5 및 C70과 교잡될 때, 또한 견사량에서는 N9 X C70, N63 X C70이 각각 큰 경향이었다. 4. F1의 조합능력에서 분사량은 GCA, SCA 및 RCA의 전형질에서 유의하여 F1의 조합능력은 상가적 효과와 비상가적 효과가 함께 작용하였으나 형질에 따른 정역간 차이도 컸었다. 5. GCA의 효과는 경과일수에서는 Sansurian이 N9, C5가 견질형질(견장, 견폭, 전견중, 전층중 및 견층비율)과 사질형질(견사장, 견자량 및 생사량비율)에서 세대에 관계없이 높은 정의 효과를 나타내었다. 6. F1의 SCA효과에서 5령경과일수는 Sansurian X C70, romogua X C70, Sansurian X C5, Romogua X C5 등 구주종계와 중국종계간의 교잡에서 전견중과 견층중은 N9 X C5, C70 X Sansurian 이 암수에 관계없이, 또한 Rogmogua X Sansurian, N9 X C5가 F1, F2 간에, 견사량은 Romogua X N63 조합에서, 생사량비율은 Sansurian X Romogua 조합에서 각각 높게 평가되었다.

• 대한수학회지
• /
• 제47권3호
• /
• pp.547-572
• /
• 2010

We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

• 대한수학회논문집
• /
• 제9권4호
• /
• pp.881-889
• /
• 1994

Let ($X, \Beta, \mu$) be a measure space with the $\sigma$-algebra $\Beta$ and the probability measure $\mu$. Throughouth this article set equalities and inclusions are understood as being so modulo measure zero sets. A transformation T defined on a probability space X is said to be measure preserving if $\mu(T^{-1}E) = \mu(E)$ for $E \in B$. It is said to be ergodic if $\mu(E) = 0$ or i whenever $T^{-1}E = E$ for $E \in B$. Consider the sequence ${x, Tx, T^2x,...}$ for $x \in X$. One may ask the following questions: What is the relative frequency of the points $T^nx$ which visit the set E\ulcorner Birkhoff Ergodic Theorem states that for an ergodic transformation T the time average $lim_{n \to \infty}(1/N)\sum^{N-1}_{n=0}{f(T^nx)}$ equals for almost every x the space average $(1/\mu(X)) \int_X f(x)d\mu(x)$. In the special case when f is the characteristic function $\chi E$ of a set E and T is ergodic we have the following formula for the frequency of visits of T-iterates to E : $$ lim_{N \to \infty} \frac{$\mid${n : T^n x \in E, 0 \leq n $\mid$}{N} = \mu(E) $$ for almost all $x \in X$ where $$\mid$\cdot$\mid$$ denotes cardinality of a set. For the details, see [8], [10].

• 대한수학회논문집
• /
• 제22권1호
• /
• pp.41-51
• /
• 2007

Let R be a ring with left identity e and suitably-restricted additive torsion, and Z(R) its center. Let H : $R{\times}R{\times}R{\rightarrow}R$ be a symmetric 3-additive mapping, and let h be the trace of H. In this paper we show that (i) if for each $x{\in}R$, $$n=<<\cdots,\;x>,\;\cdots,x>{\in}Z(R)$$ with $n\geq1$ fixed, then h is commuting on R. Moreover, h is of the form $$h(x)=\lambda_0x^3+\lambda_1(x)x^2+\lambda_2(x)x+\lambda_3(x)\;for\;all\;x{\in}R$$, where $\lambda_0\;{\in}\;Z(R)$, $\lambda_1\;:\;R{\rightarrow}R$ is an additive commuting mapping, $\lambda_2\;:\;R{\rightarrow}R$ is the commuting trace of a bi-additive mapping and the mapping $\lambda_3\;:\;R{\rightarrow}Z(R)$ is the trace of a symmetric 3-additive mapping; (ii) for each $x{\in}R$, either $n=0\;or\;<n,\;x^m>=0$ with $n\geq0,\;m\geq1$ fixed, then h = 0 on R, where denotes the product yx+xy and Z(R) is the center of R. We also present the conditions which implies commutativity in rings with identity as motivated by the above result.

• 자원환경지질
• /
• 제4권4호
• /
• pp.225-234
• /
• 1971

With the advance of civilization and steadily increasing population rivalry and competition for the use of the sewage, culverts, farm irrigation and control of various types of flood discharge have developed and will be come more and more keen in the future. The author has tried to calculated a formula that could adjust these conflicts and bring about proper solutions for many problems arising in connection with these conditions. The purpose of this study is to find out effective sewage, culvert, drainage, farm irrigation, flood discharge and other engineering needs in the Taegu area. If demands expand further a new formula will have to be calculated. For the above the author estimated methods of control for the probable expected rainfall using a formula based on data collected over a long period of time. The formula is determined on the basis of the maximum daily rainfall data from 1921 to 1971 in the Taegu area. 1. Iwai methods shows a highly significant correlation among the variations of Hazen, Thomas, Gumbel methods and logarithmic normal distribution. 2. This study obtained the following major formula: ${\log}(x-2.6)=0.241{\xi}+1.92049{\cdots}{\cdots}$(I.M) by using the relation $F(x)=\frac{1}{\sqrt{\pi}}{\int}_{-{\infty}}^{\xi}e^{-{\xi}^2}d{\xi}$. ${\xi}=a{\log}_{10}\(\frac{x+b}{x_0+b}\)$ ($-b<x<{\infty}$) ${\log}(x_0+b)=2.0448$ $\frac{1}{a}=\sqrt{\frac{2N}{N-1}}S_x=0.1954$. $b=\frac{1}{m}\sum\limits_{i=1}^{m}b_s=-2.6$ $S_x=\sqrt{\frac{1}{N}\sum\limits^N_{i=1}\{{\log}(x_i+b)\}^2-\{{\log}(x_0+b)\}^2}=0.169$ This formule may be advantageously applicable to the estimation of flood discharge, sewage, culverts and drainage in the Taegu area. Notation for general terms has been denoted by the following. Other notations for general terms was used as needed. $W_{(x)}$ : probability of occurranec, $W_{(x)}=\int_{x}^{\infty}f_{(n)}dx$ $S_{(x)}$ : probability of noneoccurrance. $S_{(x)}=\int_{-\infty}^{x}f_(x)dx=1-W_{(x)}$ T : Return period $T=\frac{1}{nW_{(x)}}$ or $T=\frac{1}{nS_{(x)}}$ $W_n$ : Hazen plot $W_n=\frac{2n-1}{2N}$ $F_n=1-W_x=1-\(\frac{2n-1}{2N}\)$ n : Number of observation (annual maximum series) P : Probability $P=\frac{N!}{{t!}(N-t)}F{_i}^{N-t}(1-F_i)^t$ $F_n$ : Thomas plot $F_n=\(1-\frac{n}{N+1}\)$ N : Total number of sample size $X_l$ : $X_s$ : maximum, minumum value of total number of sample size.

• 대한수학회논문집
• /
• 제9권2호
• /
• pp.419-421
• /
• 1994

Recently Hong and Oh [5] provided a fairly general weak law for arrays in the following form: Let {(X/sub ni/, l ≤ i ≤ k/sub n/), n ≥ l}, k/sub n/ → ∞ as n → ∞, be an array of random variables on (Ω, F, P) and set F/sub nj/ = σ{X/sub ni/, 1 ≤ i ≤ j}, 1 ≤ j ≤ k/sub n/, n ≥ 1, and F/sub n0/ = {ø, Ω}, n ≥ 1. Suppose that (equation omitted) aP { X/sub ni/ /sup p/ ＞ a} → 0 as a → ∞ uniformly in n for some 0 < p < 2. Then S/sub n//(equation omitted) → 0 in probability as n → ∞ where S/sub n/ = (equation omitted)(X/sub ni/ - E(X/sib ni/I( X/sub ni/ /sub p/ ≤ k/sub n/) F/sub n,i-l/)). In this note, we will prove the following result under the same domination condition of Hong and Oh [5].(omitted)

• Journal of applied mathematics & informatics
• /
• 제24권1_2호
• /
• pp.437-448
• /
• 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.

• 대한수학회논문집
• /
• 제19권2호
• /
• pp.321-328
• /
• 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.