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Euler-Maruyama Numerical solution of some stochastic functional differential equations

  • Ahmed, Hamdy M.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.11 no.1
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    • pp.13-30
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    • 2007
  • In this paper we study the numerical solutions of the stochastic functional differential equations of the following form $$du(x,\;t)\;=\;f(x,\;t,\;u_t)dt\;+\;g(x,\;t,\;u_t)dB(t),\;t\;>\;0$$ with initial data $u(x,\;0)\;=\;u_0(x)\;=\;{\xi}\;{\in}\;L^p_{F_0}\;([-{\tau},0];\;R^n)$. Here $x\;{\in}\;R^n$, ($R^n$ is the ${\nu}\;-\;dimenional$ Euclidean space), $f\;:\;C([-{\tau},\;0];\;R^n)\;{\times}\;R^{{\nu}+1}\;{\rightarrow}\;R^n,\;g\;:\;C([-{\tau},\;0];\;R^n)\;{\times}\;R^{{\nu}+1}\;{\rightarrow}\;R^{n{\times}m},\;u(x,\;t)\;{\in}\;R^n$ for each $t,\;u_t\;=\;u(x,\;t\;+\;{\theta})\;:\;-{\tau}\;{\leq}\;{\theta}\;{\leq}\;0\;{\in}\;C([-{\tau},\;0];\;R^n)$, and B(t) is an m-dimensional Brownian motion.

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Estimation for Mean Time Between Failures of a Repairable System. (수리가능한 시스템의 평균고장간격시간 추정에 관한 연구)

  • 이현우;김치용
    • The Korean Journal of Applied Statistics
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    • v.12 no.1
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    • pp.203-211
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    • 1999
  • 수리 가능한 시스템의 평균고장간격시간에 대한 많은 연구들이 진행되어 왔으며, 그 대부분은 n번째 고장발생시각 $T_n$을 관측한 후 그 다음 고장이 발생할 때까지의 평균시간, 즉 E($T_{n+1}$-$T_n$$\mid$$T_n$ = $t_n$)에 관한 연구들이었다. 본 연구에서는 수리가능한 시스템의 고장이 와이블과정을 따라 일어날 경우, n번째와 n+1번째 고장간의 평균고장간격시간 E($T_{n+1}$-$T_n$)에 대한 불편추정량을 구하고 일치성 및 근사적 정규성을 증명하였다.

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Characteristics of $TiN/TiSi_2$ Contact Barrier Layer by Rapid Thermal Anneal in $N_2$ Ambient (질소 분위기에서 순간역처리에 의해 형성시킨 $TiN/TiSi_2$ Contact Bsrrier Lauer의 특성)

  • 이철진;허윤종;성영권
    • The Transactions of the Korean Institute of Electrical Engineers
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    • v.41 no.6
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    • pp.633-639
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    • 1992
  • The physical and electrical properties of TiN/TiSiS12T contact barrier were studied. The TiN/TiSiS12T system was formed by rapid thermal anneal in NS12T ambient after the Ti film was deposited on silicon substrate. The Ti film reacts with NS12T gas to make a TiN layer at the surface and reacts with silicon to make a TiSiS12T layer at the interface respectively. It was found that the formation of TiN/TiSiS12T system depends on RTA temperature. In this experiment, competitive reaction for TiN/TiSiS12T system occured above $600^{\circ}C$. Ti-rich TiNS1xT layer and Ti-rich TiSiS1xT layer were formed at $600^{\circ}C$. stable structure TiN layer and TiSiS1xT layer which has CS149T phase and CS154T phase were formed at $700^{\circ}C$. Both stable TiN layer and CS154T phase TiSiS12T layer were formed at 80$0^{\circ}C$. The thickness of TiN/TiSiS12T system was increased as the thickness of deposited Ti film increased.

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A GENERAL VISCOSITY APPROXIMATION METHOD OF FIXED POINT SOLUTIONS OF VARIATIONAL INEQUALITIES FOR NONEXPANSIVE SEMIGROUPS IN HILBERT SPACES

  • Plubtieng, Somyot;Wangkeeree, Rattanaporn
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.4
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    • pp.717-728
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    • 2008
  • Let H be a real Hilbert space and S = {T(s) : $0\;{\leq}\;s\;<\;{\infty}$} be a nonexpansive semigroup on H such that $F(S)\;{\neq}\;{\emptyset}$ For a contraction f with coefficient 0 < $\alpha$ < 1, a strongly positive bounded linear operator A with coefficient $\bar{\gamma}$ > 0. Let 0 < $\gamma$ < $\frac{\bar{\gamma}}{\alpha}$. It is proved that the sequences {$x_t$} and {$x_n$} generated by the iterative method $$x_t\;=\;t{\gamma}f(x_t)\;+\;(I\;-\;tA){\frac{1}{{\lambda}_t}}\;{\int_0}^{{\lambda}_t}\;T(s){x_t}ds,$$ and $$x_{n+1}\;=\;{\alpha}_n{\gamma}f(x_n)\;+\;(I\;-\;{\alpha}_nA)\frac{1}{t_n}\;{\int_0}^{t_n}\;T(s){x_n}ds,$$ where {t}, {${\alpha}_n$} $\subset$ (0, 1) and {${\lambda}_t$}, {$t_n$} are positive real divergent sequences, converges strongly to a common fixed point $\tilde{x}\;{\in}\;F(S)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)\tilde{x},\;x\;-\;\tilde{x}{\rangle}\;{\leq}\;0$ for $x\;{\in}\;F(S)$.

CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE

  • Cho, Dong Hyun
    • Journal of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1105-1127
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    • 2013
  • Let C[0, $t$] denote the function space of real-valued continuous paths on [0, $t$]. Define $X_n\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\ldots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\ldots},x(t_n),x(t_{n+1}))$, respectively, where $0=t_0 <; t_1 <{\ldots} < t_n < t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form $fr((v_1,x),{\ldots},(v_r,x)){\int}_{L_2}_{[0,t]}\exp\{i(v,x)\}d{\sigma}(v)$ for $x{\in}C[0,t]$, where $\{v_1,{\ldots},v_r\}$ is an orthonormal subset of $L_2[0,t]$, $f_r{\in}L_p(\mathbb{R}^r)$, and ${\sigma}$ is the complex Borel measure of bounded variation on $L_2[0,t]$. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.

On the edge independence number of a random (N,N)-tree

  • J. H. Cho;Woo, Moo-Ha
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.119-126
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    • 1996
  • In this paper we study the asymptotic behavior of the edge independence number of a random (n,n)-tree. The tools we use include the matrix-tree theorem, the probabilistic method and Hall's theorem. We begin with some definitions. An (n,n)_tree T is a connected, acyclic, bipartite graph with n light and n dark vertices (see [Pa92]). A subset M of edges of a graph is called independent(or matching) if no two edges of M are adfacent. A subset S of vertices of a graph is called independent if no two vertices of S are adjacent. The edge independence number of a graph T is the number $\beta_1(T)$ of edges in any largest independent subset of edges of T. Let $\Gamma(n,n)$ denote the set of all (n,n)-tree with n light vertices labeled 1, $\ldots$, n and n dark vertices labeled 1, $\ldots$, n. We give $\Gamma(n,n)$ the uniform probability distribution. Our aim in this paper is to find bounds on $\beta_1$(T) for a random (n,n)-tree T is $\Gamma(n,n)$.

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A Study on the Properties of TiN/${TiSi}_{2}$ Bilayer by a Rapid Thermal Anneal in ${NH}_{3}$ Ambient (${NH}_{3}$ 분위기에서 급속열처리에 의한 TiN/${TiSi}_{2}$ 이중구조막의 특성에 대한 고찰)

  • 이철진;성영권
    • The Transactions of the Korean Institute of Electrical Engineers
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    • v.41 no.8
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    • pp.869-874
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    • 1992
  • The physical and electrical properties of TiN/TiSiS12T bilayer were studied. The TiN/TiSiS12T bilayer was formed by rapid thermal anneal in NHS13T ambient after the Ti film was deposited on silicon substrate. The Ti film reacts with NHS13T gas to make a TiN layer at the surface and reacts with silicon to make a TiSiS12T layer at the interface respectively. It was found that the formation of TiN/TiSiS12T bilayer depends on RTA temperature. In this experiment, competitive reaction for TiN/TiSiS12T bilayer occured above $600^{\circ}C$. Ti-rich TiNS1xT layer and Ti-rich TiSiS1xT layer and Ti-rich TiSiS1xT layer were formed at $600^{\circ}C$. stable structure TiN layer TiSiS12T layer which has CS149T phase and CS154T phase were formed at $700^{\circ}C$. Both stable TiN layer and CS154T phase TiSiS12T layer were formed at 80$0^{\circ}C$. The thickness of TiN/TiSiS12T bilayer was increased as the thickness of deposited Ti film increased.

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CENTRAL LIMIT TYPE THEOREM FOR WEIGHTED PARTICLE SYSTEMS

  • Cho, Nhan-Sook;Kwon, Young-Mee
    • Journal of the Korean Mathematical Society
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    • v.41 no.5
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    • pp.773-793
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    • 2004
  • We consider a system of particles with locations { $X_{i}$ $^{n}$ (t):t$\geq$0,i=1,…,n} in $R^{d}$ , time-varying weights { $A_{i}$ $^{n}$ (t) : t $\geq$0,i = 1,…,n} and weighted empirical measure processes $V^{n}$ (t)=1/n$\Sigma$$_{i=1}$$^{n}$ $A_{i}$ $^{n}$ (t)$\delta$ $X_{i}$ $^{n}$ (t), where $\delta$$_{x}$ is the Dirac measure. It is known that there exists the limit of { $V_{n}$ } in the week* topology on M( $R^{d}$ ) under suitable conditions. If { $X_{i}$ $^{n}$ , $A_{i}$ $^{n}$ , $V^{n}$ } satisfies some diffusion equations, applying Ito formula, we prove a central limit type theorem for the empirical process { $V^{n}$ }, i.e., we consider the convergence of the processes η$_{t}$ $^{n}$ ≡ n( $V^{n}$ -V). Besides, we study a characterization of its limit.t.

T-NEIGHBORHOODS IN VARIOUS CLASSES OF ANALYTIC FUNCTIONS

  • Shams, Saeid;Ebadian, Ali;Sayadiazar, Mahta;Sokol, Janusz
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.659-666
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    • 2014
  • Let $\mathcal{A}$ be the class of analytic functions f in the open unit disk $\mathbb{U}$={z : ${\mid}z{\mid}$ < 1} with the normalization conditions $f(0)=f^{\prime}(0)-1=0$. If $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and ${\delta}$ > 0 are given, then the $T_{\delta}$-neighborhood of the function f is defined as $$TN_{\delta}(f)\{g(z)=z+\sum_{n=2}^{\infty}b_nz^n{\in}\mathcal{A}:\sum_{n=2}^{\infty}T_n{\mid}a_n-b_n{\mid}{\leq}{\delta}\}$$, where $T=\{T_n\}_{n=2}^{\infty}$ is a sequence of positive numbers. In the present paper we investigate some problems concerning $T_{\delta}$-neighborhoods of function in various classes of analytic functions with $T=\{2^{-n}/n^2\}_{n=2}^{\infty}$. We also find bounds for $^{\delta}^*_T(A,B)$ defined by $$^{\delta}^*_T(A,B)=jnf\{{\delta}&gt;0:B{\subset}TN_{\delta}(f)\;for\;all\;f{\in}A\}$$ where A, B are given subsets of $\mathcal{A}$.

AN EVALUATION FORMULA FOR A GENERALIZED CONDITIONAL EXPECTATION WITH TRANSLATION THEOREMS OVER PATHS

  • Cho, Dong Hyun
    • Journal of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.451-470
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    • 2020
  • Let C[0, T] denote an analogue of Wiener space, the space of real-valued continuous functions on the interval [0, T]. For a partition 0 = t0 < t1 < ⋯ < tn < tn+1 = T of [0, T], define Xn : C[0, T] → ℝn+1 by Xn(x) = (x(t0), x(t1), …, x(tn)). In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on C[0, T] with the conditioning function Xn which has a drift and does not contain the present position of paths. As applications of the formula with Xn, we evaluate the Radon-Nikodym derivatives of the functions ∫0T[x(t)]mdλ(t)(m∈ℕ) and [∫0Tx(t)dλ(t)]2 on C[0, T], where λ is a complex-valued Borel measure on [0, T]. Finally we derive two translation theorems for the Radon-Nikodym derivatives of the functions on C[0, T].