• Journal of the Korean Mathematical Society
• /
• v.45 no.4
• /
• pp.993-1005
• /
• 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
• /
• v.47 no.3
• /
• pp.645-651
• /
• 2010

In this note we study positive solutions of the mth order rational difference equation $x_n=(a_0+\sum{{m\atop{i=1}}a_ix_{n-i}/(b_0+\sum{{m\atop{i=1}}b_ix_{n-i}$, where n = m,m+1,m+2, $\ldots$ and $x_0,\ldots,x_{m-1}$ > 0. We describe a sufficient condition on nonnegative real numbers $a_0,a_1,\ldots,a_m,b_0,b_1,\ldots,b_m$ under which every solution $x_n$ of the above equation tends to the limit $(A-b_0+\sqrt{(A-b_0)^2+4_{a_0}B}$/2B as $n{\rightarrow}{\infty}$, where $A=\sum{{m\atop{i=1}}\;a_i$ and $B=\sum{{m\atop{i=1}}\;b_i$.

• Kyungpook Mathematical Journal
• /
• v.53 no.1
• /
• pp.37-47
• /
• 2013

Let M and X be right R-modules. We introduce several modules relative to the class of B(M, X) and we investigate relation among these modules. In this note, we show if M is X-${\oplus}$-supplemented such that $M=M_1{\oplus}M_2$ implies $M_1$ and $M_2$ are relatively B-projective, then M is an X-H-supplemented module.

• Communications of the Korean Mathematical Society
• /
• v.9 no.3
• /
• pp.701-710
• /
• 1994

Suppose ${X_n}$ is a Markov process taking values in some arbitrary space $(S, \varphi)$ with n-stemp transition probability $$ P^{(n)}(x, B) = Prob(X_n \in B$\mid$X_0 = x), x \in X, B \in \varphi.$$ We shall call a Markov process with transition probabilities $P{(n)}(x, B)$ $\phi$-irreducible for some non-trivial $\sigma$-finite measure $\phi$ on $\varphi$ if whenever $\phi(B) > 0$, $$ \sum^{\infty}_{n=1}{2^{-n}P^{(n)}}(x, B) > 0, for every x \in S.$$ A non-trivial $\sigma$-finite measure $\pi$ on $\varphi$ is called invariant for ${X_n}$ if $$ \int{P(x, B)\pi(dx) = \pi(B)}, B \in \varphi $$.

• Journal of the Korean Magnetics Society
• /
• v.4 no.2
• /
• pp.126-129
• /
• 1994

In order to invetigate the effect of Pr additive on the magnetostriction of amorphous Fe-B alloys, amorphous $Fe_{86-x}B_{14}Pr_{x}(2{\leq}x{\leq}8\;at.%)$ alloys were prepared by a rapid solidification process. As the Pr content increased in the as-prepared amorphous $Fe_{84}B_{14}Pr_{2}$ alloy annealed at $300^{\circ}C$ for 2 hr increased to 70 ppm. Ac power loss and permeability$(f=50\;kHz,\;B_{m}=0.1\;T)$ of the annealed amorphous $Fe_{84}B_{14}Pr_{2}$ alloy were 15 W/kg and $5.5{\times}10^{3}$, respectively.

• Korean Journal of Materials Research
• /
• v.8 no.2
• /
• pp.141-146
• /
• 1998

Ball mill을 이용하여 Ar 분위기에서 기계적 합금법으로 $Fe_{ 5}$$Si_{x}$ $B_{5-x}$ 분말을 제조하고, 제조된 분말을 연속 진공 열처리 시킨 후 Si첨가에 따른 결정구조 및 자기적성질을 조사하였다. 250시간 볼밀처리한 Fe$_{5}$ $B_{5}$ 합금에서 전체적으로 비정질 구조가 형성되었으나 일부분에 결정질이 존재하고 있었으며, $800^{\circ}C$에서 2시간 열처리하면 FeB와 $Fe_{2}$B 상이 혼재된 구조를 얻었다. 250시간 볼밀처리한 $Fe_{5}$ $Si_{2}$$B_{3}$합금에서 전체적으로 비정질 구조를 얻을 수 있었고, 이 시료를 2시간, $800^{\circ}C$로 열처리 하였을 때 $Fe_{2}$B상은 사라지고, 대부분 FeB의 균질한 상을 나타내었다. $Fe_{5}$ $B_{5}$ /조성에서는 분말 입자크기가 약 $1\mu\textrm{m}$이었으나, Si이 첨가되면 분말 입자크기가 약 $10\mu\textrm{m}$로 커졌다. Si의 첨가에 의해서 비정질상의 형성을 촉진시켜 단일 FeB상의 합성시간을 단축시킬 수 있었다.

• The Mathematical Education
• /
• v.22 no.1
• /
• pp.21-23
• /
• 1983

(1) Let f : XlongrightarrowX' be a homomorphism of BCK-algebras and let A,B be ideals of X and X' respectively such that f(A)⊂B. Then there is a unique homomorphism h : X/AlongrightarrowX'/B such that the diagram(equation omitted) commutes. (2) The class of all complexes of BCK-algebras becomes a category.

• Bulletin of the Korean Chemical Society
• /
• v.7 no.1
• /
• pp.29-35
• /
• 1986

An empirical formula for semiconductive metal oxides is proposed relating nonstoichiometric value x to a temperature or an oxygen partial pressure such that experimental data can be represented more accurately by the formula than by the well-known Arrhenius-type equation. The proposed empirical formula is log x = A + $B{\cdot}1000/T\;+\;C{\cdot}$exp$(-D{\cdot}1000/T)$ for a temperature dependence and $log\;{\times}\;=a\;+b{\cdot}log\;Po_2\;+\;c{\cdot}$exp$(-d{\cdot}log\;Po_2)$ for an oxygen partial pressure dependence. The A, B, C, D and a, b, c, d are parameters which are evaluated by means of a best-fitting method to experimental data. Subsequently, this empirical formula has been applied to the n-type metal oxides of $Zn_{1+x}O,\; Cd_{1+x}O,\;and\;PrO_{1.8003-x}$, and the p-type metal oxides of $CoO_{1+x},\; FeO_{1+x},\;and\;Cu_2O_{1+x}$. It gives a very good agreement with the experimental data through the best-fitted parameters within 6% of relative error. It is also possible to explain approximately qualitative characters of the parameters A, B, C, D and a, b, c, d from theoretical bases.

• The Pure and Applied Mathematics
• /
• v.16 no.4
• /
• pp.327-344
• /
• 2009

In this paper, we deal with the following system of nonlinear singular boundary value problems(BVPs) on time scale $\mathbb{T}$ $$\{{{{{{x^{\bigtriangleup\bigtriangleup}(t)+f(t,\;y(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}\atop{y^{\bigtriangleup\bigtriangleup}(t)+g(t,\;x(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}}\atop{\alpha_1x(a)-\beta_1x^{\bigtriangleup}(a)=\gamma_1x(\sigma(b))+\delta_1x^{\bigtriangleup}(\sigma(b))=0,}}\atop{\alpha_2y(a)-\beta_2y^{\bigtriangleup}(a)=\gamma_2y(\sigma(b))+\delta_2y^{\bigtriangleup}(\sigma(b))=0,}}$$ where $\alpha_i$, $\beta_i$, $\gamma_i\;{\geq}\;0$ and $\rho_i=\alpha_i\gamma_i(\sigma(b)-a)+\alpha_i\delta_i+\gamma_i\beta_i$ > 0(i = 1, 2), f(t, y) may be singular at t = a, y = 0, and g(t, x) may be singular at t = a. The arguments are based upon a fixed-point theorem for mappings that are decreasing with respect to a cone. We also obtain the analogous existence results for the related nonlinear systems $x^{\bigtriangledown\bigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangledown\bigtriangledown}(t)$ + g(t, x(t)) = 0, $x^{\bigtriangleup\bigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangleup\bigtriangledown}(t)$ + g(t, x(t)) = 0, and $x^{\bigtriangledown\bigtriangleup}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangledown\bigtriangleup}(t)$ + g(t, x(t)) = 0 satisfying similar boundary conditions.

• Kyungpook Mathematical Journal
• /
• v.49 no.1
• /
• pp.167-181
• /
• 2009

Based on Ricatti equation $XA^{-1}X=B$ for two (positive invertible) operators A and B which has the geometric mean $A{\sharp}B$ as its solution, we consider a cubic equation $X(A{\sharp}B)^{-1}X(A{\sharp}B)^{-1}X=C$ for A, B and C. The solution X = $(A{\sharp}B){\sharp}_{\frac{1}{3}}C$ is a candidate of the geometric mean of the three operators. However, this solution is not invariant under permutation unlike the geometric mean of two operators. To supply the lack of the property, we adopt a limiting process due to Ando-Li-Mathias. We define reasonable geometric means of k operators for all integers $k{\geq}2$ by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.