• 대한수학회지
• /
• 제44권4호
• /
• pp.1025-1050
• /
• 2007

Let $X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))$ on the classical Wiener space, where ${{\alpha}_1,...,{\alpha}_k}$ is an orthonormal subset of $L_2$ [0, T] and ${\tau}:0 is a partition of [0, T]. In this paper, we establish a change of scale formula for conditional Wiener integrals $E[G_{\gamma}|X_k]$ of functions on classical Wiener space having the form $$G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))$$, for $F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))$, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and $\hat{M}(\mathbb{R}^{\gamma})$ is the space of Fourier transforms of measures of bounded variation over $\mathbb{R}^{\gamma}$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]$. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.

• Bulletin of the Korean Chemical Society
• /
• 제26권11호
• /
• pp.1723-1727
• /
• 2005

A generalized fractional diffusion equation (FDE) is presented, which describes the time-evolution of the spatial distribution of a particle performing continuous time random walk (CTRW) on a fractal lattice. For a case corresponding to the CTRW with waiting time distribution that behaves as $\psi(t) \sim (t) ^{-(\alpha+1)}$, the FDE is solved to give analytic expressions for the Green’s function and the mean squared displacement (MSD). In agreement with the previous work of Blumen et al. [Phys. Rev. Lett. 1984, 53, 1301], the time-dependence of MSD is found to be given as < $r^2(t)$ > ~ $t ^{2\alpha/dw}$, where $d_w$ is the walk dimension of the given fractal. A Monte-Carlo simulation is also performed to evaluate the range of applicability of the proposed FDE.

• Journal of applied mathematics & informatics
• /
• 제19권1_2호
• /
• pp.475-484
• /
• 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.

• 대한수학회보
• /
• 제53권5호
• /
• pp.1531-1548
• /
• 2016

Let C[0, t] denote the space of real-valued continuous functions on [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}\mathbb{R}^n$ by $Z_n(x)=(\int_{0}^{t_1}h(s)dx(s),{\ldots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $ t_n=t$ is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. Using a simple formula for a conditional expectation on C[0, t] with $Z_n$, we evaluate a generalized analytic conditional Wiener integral of the function $G_r(x)=F(x){\Psi}(\int_{0}^{t}v_1(s)dx(s),{\ldots},\int_{0}^{t}v_r(s)dx(s))$ for F in a Banach algebra and for ${\Psi}=f+{\phi}$ which need not be bounded or continuous, where $f{\in}L_p(\mathbb{R}^r)(1{\leq}p{\leq}{\infty})$, {$v_1,{\ldots},v_r$} is an orthonormal subset of $L_2[0,t]$ and ${\phi}$ is the Fourier transform of a measure of bounded variation over $\mathbb{R}^r$. Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of $G_r$ with the conditioning function $Z_n$.

• Kyungpook Mathematical Journal
• /
• 제52권4호
• /
• pp.413-431
• /
• 2012

Let $C^r[0,t]$ be the function space of the vector-valued continuous paths $x:[0,t]{\rightarrow}\mathbb{R}^r$ and define $X_t:C^r[0,t]{\rightarrow}\mathbb{R}^{(n+1)r}$ and $Y_t:C^r[0,t]{\rightarrow}\mathbb{R}^{nr}$ by $X_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}),\;x(t_n))$ and $Y_t(x)=(x(t_0),\;x(t_1),\;{\cdots},\;x(t_{n-1}))$, respectively, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n=t$. In the present paper, using two simple formulas for the conditional expectations over $C^r[0,t]$ with the conditioning functions $X_t$ and $Y_t$, we establish evaluation formulas for the analogue of the conditional analytic Fourier-Feynman transform for the function of the form $${\exp}\{{\int_o}^t{\theta}(s,\;x(s))\;d{\eta}(s)\}{\psi}(x(t)),\;x{\in}C^r[0,t]$$ where ${\eta}$ is a complex Borel measure on [0, t] and both ${\theta}(s,{\cdot})$ and ${\psi}$ are the Fourier-Stieltjes transforms of the complex Borel measures on $\mathbb{R}^r$.

• Transactions on Control, Automation and Systems Engineering
• /
• 제4권1호
• /
• pp.92-99
• /
• 2002

To obtain visual information of a target object, a camera should be placed within the visibility region. As the visibility region is dependent on the relative position of the target object and the surrounding object, the position change of the surrounding object during a task requires recalculation of the visibility region. For a fast computation of the visibility region so as to modify the camera position to be located within the visibility region, we propose a spherical projection method. After being projected onto the sphere the visibility region is represented in $\theta$-$\psi$ spaces of the spherical coordinates. The reduction of calculation space enables a fast modification of the camera location according to the motion of the surrounding objects so that the continuous observation of the target object during the task is possible.

• 한국분말재료학회지
• /
• 제4권2호
• /
• pp.106-112
• /
• 1997

$Cu-Al_20_3 $ composite powders were prepared by hydrogen reduction of $Cu^{2+}$ from ammoniacal copper sulfate solution on alumina core using autoclave. The copper reduction rate and the properties of copper layer were investigated using Scanning Electron Microscope(SEM), X-ray diffractometer, size and chemical analyzers. The reduction rate of $Cu^{2+}$ showed the maximum value when the molar ratio of [$NH_3$]/[$Cu^{2+}$] was 2. In order to prevent the agglomeration of Cu powder and ethane reduction rate, $Fe^{2+}$ and anthraquinone which act as catalysis were added in the solution. Catalysis was effectively chanced with the addition of two elemerts at a time. Optimum conditions obtained in this study were hydrogen reduction temperature of 205$^{\cire}C$, stirring speed of 500 rpm and hydrogen partial pressure of 300 psi. Obtained $Cu-Al_20_3 $ composite Powders were found to have the uniform and continuous copper coating layer of nodule shape with 3~5 $\mu$m thickness.

• Nonlinear Functional Analysis and Applications
• /
• 제27권3호
• /
• pp.533-551
• /
• 2022

In this paper, we introduce several types 𝛼-𝜓 interpolative proximal contractions and provide some sufficient conditions to prove the existence of best proximity points for these contractions in metric spaces. In the case of proximal contraction of the first kind, these metric spaces are not necessarily complete. Meanwhile, some new results can derive from our results. Finally, some examples are provided to show the validity of our results.

• 한국막학회:학술대회논문집
• /
• 한국막학회 1997년도 제5회 하계 Workshop (97 한,카 국제공동 Workshop, 고도 수처리를 위한 막분리 공정)
• /
• pp.143-158
• /
• 1997

Electrochemical deionization (EDI) offers continuous demineralization at higher water recovery rates (>90%), compared with mixed bed ion exchange, and without the use of chemical regenerants and the associated production of saline waste water. Although EDI technology has been used in some power generation applications, its wider application requires the satisfactory resolution of outstanding capital cost and performance issues. This paper reports on the field evaluation of a new cost-effective EDI technology in a power generation application. The E-Cell System$^{TM}$, which became commercially available in the fourth quarter of 1996, consists of a rugged, modular system, based on a new high-performance EDI stack. Starting in May 1996, a 100 gpm modular EDI pilot system, rated for operation at 100 psi, was evaluated at the TVA Brown's Ferry Nuclear Plant. The feed consisted of Reverse Osmosis (RO) permeate with a conductivity of 4-7 $\mu$S/cm. The pilot system reliably produced 17.8-18.0 M$\Omega$.cm water under design operating conditions, independent. Silica levels were reduced from ca. 50 ppb to 4 ppb, while TOC levels were reduced from ca. 120 ppb to 30 ppb.

• Membrane and Water Treatment
• /
• 제11권4호
• /
• pp.303-312
• /
• 2020

This study aims to investigate at a laboratory scale fluorides removal from an industrial wastewater having excessive F^- concentration through a hybrid process combining neutralization and membrane separation. For the membrane separation operation, both Reverse Osmosis (RO) and Nanofiltration (NF) were investigated and confronted. The optimized neutralization step with hydrated lime allowed reaching fluoride removal rates of 99.1± 0.4 %. To simulate continuous process, consecutive batch treatments with full recirculation of membrane process brines were conducted. Despite the relatively high super saturations with respect to CaF₂, no membrane cloaking was observed. The RO polishing treatment allowed decreasing the permeate fluoride concentration to 0.9± 0.3 mg/L with a fluoride rejection rate of 93± 2% at the optimal transmembrane pressure of around 100 psi. When NF membrane was used to treat neutralization filtrate, the permeate fluoride concentration dropped to 1.1± 0.4 mg/L with a fluoride rejection rate of 88± 5% at the optimal pressure of around 80 psi. Thus, with respect to RO, NF allowed roughly 20% decrease of the driving pressure at the expense of only 5% drop of rejection rate. Both NF and RO permeates at optimal operating transmembrane pressures respect environmental regulations for reject streams discharge into the environment.