• 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 < ⋯ < t_n < t_n+1 = T of [0, T], define X_n : C[0, T] → ℝⁿ⁺¹ by X_n(x) = (x(t₀), x(t₁), …, x(t_n)). 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 X_n which has a drift and does not contain the present position of paths. As applications of the formula with X_n, we evaluate the Radon-Nikodym derivatives of the functions ∫₀^T[x(t)]^mdλ(t)(m∈ℕ) and [∫₀^Tx(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].

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.17-33
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• 2007

Suppose F is an arbitrary field. Let ${\mid}F{\mid}$ be the number of the elements of F. Let $T_{n}(F)$ be the space of all $n{\times}n$ upper-triangular matrices over F. A map ${\Psi}\;:\;T_{n}(F)\;{\rightarrow}\;T_{n}(F)$ is said to preserve idempotence if $A-{\lambda}B$ is idempotent if and only if ${\Psi}(A)-{\lambda}{\Psi}(B)$ is idempotent for any $A,\;B\;{\in}\;T_{n}(F)$ and ${\lambda}\;{\in}\;F$. It is shown that: when the characteristic of F is not 2, ${\mid}F{\mid}\;>\;3$ and $n\;{\geq}\;3,\;{\Psi}\;:\;T_{n}(F)\;{\rightarrow}\;T_{n}(F)$ is a map preserving idempotence if and only if there exists an invertible matrix $P\;{\in}\;T_{n}(F)$ such that either ${\Phi}(A)\;=\;PAP^{-1}$ for every $A\;{\in}\;T_{n}(F)\;or\;{\Psi}(A)=PJA^{t}JP^{-1}$ for every $P\;{\in}\;T_{n}(F)$, where $J\;=\;{\sum}^{n}_{i-1}\;E_{i,n+1-i}\;and\;E_{ij}$ is the matrix with 1 in the (i,j)th entry and 0 elsewhere.

• 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 $\mathcal{A}$.

• Analytical Science and Technology
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• v.14 no.3
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• pp.238-243
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• 2001

The study was carried out to analysis of the pollutant $COD_{Mn}$, T-N, T-P for water quality proximated to sediment in lake of K river basin. water extracted from sediment showed higher $COD_{Mn}$, T-N, T-P datas than water proximated to sediment. Also, water proximated to sediment and water 5-10cm proximated to sediment showed the following data : $COD_{Mn}$, 1.2~1.9mg/L, T-N, 1.3~6.2mg/L, TP, 0.05~0.26mg/L, respectively. From this results, we have known the fact that the pollution degree of sediment have an effect on the water quality in lake and stream.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.323-342
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• 2013

Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $Xn:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}:C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\cdots},x(t_n),x(t_{n+1}))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t_{n+1}=t$. 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 $${\int}_{L_2[0,t]}{{\exp}\{i(v,x)\}d{\sigma}(v)}{{\int}_{\mathbb{R}^r}}\;{\exp}\{i{\sum_{j=1}^{r}z_j(v_j,x)\}dp(z_1,{\cdots},z_r)$$ for $x{\in}C[0,t]$, where $\{v_1,{\cdots},v_r\}$ is an orthonormal subset of $L_2[0,t]$ and ${\sigma}$ and ${\rho}$ are the complex Borel measures of bounded variations on $L_2[0,t]$ and $\mathbb{R}^r$, respectively. We then investigate the inverse transforms of the function with their relationships and finally prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transforms of each function.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.591-603
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• 2006

For a set $X{\subset}{\mathbb{Z}}^n$ let $(X,\;T^n_X)$ be the subspace of the Khalimsky n-dimensional space $({\mathbb{Z}}^n,\;T^n)$, $n{\in}N$. Considering a k-adjacency of $(X,\;T^n_X)$, we use the notation $(X,\;k,\;T^n_X)$. In this paper for a map $$f:(X,\;k,\;T^n_X){\rightarrow}(Y,\;2\;T_Y)$$, we find the condition that weak (k, 2)-continuity of the map f implies strong (k, 2)-continuity of f.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.475-484
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• 2005

Let W(t) be a Wiener path, let $\xi\;:\;[0,\;{\infty})\;\to\;\mathbb{R}$ be a continuous and increasing function satisfying $\xi$(0) > 0, let $$T_{/xi}=inf\{t{\geq}0\;:\;W(t){\geq}\xi(t)\}$$ be the first-passage time of W over $\xi$, and let F denote the distribution function of $T_{\xi}$. Then the first passage problem has a unique continuous solution as following $$F(t)=u(t)+{\sum_{n=1}^\infty}\int_0^t\;H_n(t,s)u(s)ds$$, where $$u(t)=2\Psi(\xi(t)/\sqrt{t})\;and\;H_1(t,s)=d\Phi\;(\{\xi(t)-\xi(s)\}/\sqrt{t-s})/ds\;for\;0\;{\leq}\;s.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.677-685
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• 2023

For n ≥ 2 and a real Banach space E, 𝓛(ⁿE : E) denotes the space of all continuous n-linear mappings from E to itself. Let Π (E) = {[x^*, (x₁, . . . , x_n)] : x^*(x_j) = ||x^*|| = ||x_j|| = 1 for j = 1, . . . , n }. An element [x^*, (x₁, . . . , x_n)] ∈ Π(E) is called a numerical radius point of T ∈ 𝓛(ⁿE : E) if |x^*(T(x₁, . . . , x_n))| = v(T), where the numerical radius v(T) = sup_{[y^*,y1,...,yn]∈Π(E)}|y^*(T(y₁, . . . , y_n))|. For T ∈ 𝓛(ⁿE : E), we define Nradius(T) = {[x^*, (x₁, . . . , x_n)] ∈ Π(E) : [x^*, (x₁, . . . , x_n)] is a numerical radius point of T}. T is called a numerical radius peak n-linear mapping if there is a unique [x^*, (x₁, . . . , x_n)] ∈ Π(E) such that Nradius(T) = {±[x^*, (x₁, . . . , x_n)]}. In this paper we present explicit formulae for the numerical radius of T for every T ∈ 𝓛(ⁿE : E) for E = c₀ or l_∞. Using these formulae we show that there are no numerical radius peak mappings of 𝓛(ⁿc₀ : c₀).

• Journal of the Korean Institute of Navigation
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• v.16 no.1
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• pp.94-97
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• 1992

In this paper, the individual number of the future has depended not only upon the present individual number but upon the present individual age, considering the stochastic process model of individual number when the life span of each individual number and the individual age as a set, this becomes a Markovian. Therefore, in this paper the individual is treated as invariable, without depending upon the whole record of each individual since its birth. As a result, suppose {N(t), t>0} be a counting process and also suppose $Z_n$ denote the life span between the (n-1)st and the nth event of this process, (n{$geq}1$) : that is, when the first individual is established at n=1(time, 0), the Z$Z_n$ at time nth individual breaks, down. Random walk $Z_n$ is $Z_n=X_1+X_2+{\cdots}{\cdots}+X_A, Z_0=0$ So, fixed time t, the stochastic model is made up as follows ; A) Recurrence (Regeneration)number between(0.t) $N_t=max{n ; Z_n{\leq}t}$ B) Forwardrecurrence time(Excess life) $T^-I_t=Z_{Nt+1}-t$ C) Backward recurrence time(Current life) $T^-_t=t-Z_{Nt}$