• Korean Journal of Mathematics
• /
• v.30 no.2
• /
• pp.239-247
• /
• 2022

Let C₀[0, T] denote the Wiener space, the space of continuous functions x(t) on [0, T] such that x(0) = 0. Define a random vector $Z_{\vec{e},k}:C_0[0,\;T] {\rightarrow}{\mathbb{R}}^k$ by $$Z_{\vec{e},k}(x)=({\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^T}\;e_1(t)dx(t),\;{\ldots},\;{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^T}\;ek(t)dx(t))$$ where e_j ∈ L₂[0, T] with e_j ≠ 0 a.e., j = 1, …, k. In this paper we study the conditional Fourier-Feynman transform and a conditional convolution product for a cylinder type functionals defined on C0[0, T] with a general vector valued conditioning functions $Z_{\vec{e},k}$ above which need not depend upon the values of x at only finitely many points in (0, T] rather than a conditioning function X(x) = (x(t₁), …, x(t_n)) where 0 < t₁ < … < t_n = T. In particular we show that the conditional Fourier-Feynman transform of the conditional convolution product is the product of conditional Fourier-Feynman transforms.

• Korean Journal of Mathematics
• /
• v.22 no.1
• /
• pp.57-69
• /
• 2014

We obtain a formula for the conditional Wiener integral of the first variation of functionals and establish several integration by parts formulas of conditional Wiener integrals of functionals on a function space. We then apply these results to obtain various integration by parts formulas involving conditional integral transforms and conditional convolution products on the function space.

• Honam Mathematical Journal
• /
• v.35 no.1
• /
• pp.51-71
• /
• 2013

In the present paper, we evaluate the analytic conditional Fourier-Feynman transforms and convolution products of unbounded function which is the product of the cylinder function and the function in a Banach algebra which is defined on an analogue o Wiener space and useful in the Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the function with their relationships and finally prove that th analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the product of the conditional Fourier-Feynman transforms of each function.

• Korean Journal of Mathematics
• /
• v.20 no.1
• /
• pp.1-18
• /
• 2012

We study various relationships that exist among generalized conditional integral transform, generalized conditional convolution and generalized first variation for a class of functionals defined on K[0, T], the space of complex-valued continuous functions on [0, T] which vanish at zero.

• Korean Journal of Mathematics
• /
• v.24 no.3
• /
• pp.573-586
• /
• 2016

We study the conditional integral transforms and conditional convolutions of functionals defined on K[0, T]. We consider a general vector-valued conditioning functions $X_k(x)=({\gamma}_1(x),{\ldots},{\gamma}_k(x))$ where ${\gamma}_j(x)$ are Gaussian random variables on the Wiener space which need not depend upon the values of x at only finitely many points in (0, T]. We then obtain several relationships and formulas for the conditioning functions that exist among conditional integral transform, conditional convolution and first variation of functionals in $E_{\sigma}$.

• Journal of the Korean Mathematical Society
• /
• v.50 no.5
• /
• pp.1105-1127
• /
• 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.

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

• Proceedings of the Korean Institute of Intelligent Systems Conference
• /
• 2003.05a
• /
• pp.18-21
• /
• 2003

We propose some properties of fuzzy conditional expectation of hybrid number the addition of fuzzy number and random variable using Cartesian product distance for ${\alpha}$-level sets.

• Nuclear Engineering and Technology
• /
• v.56 no.1
• /
• pp.141-146
• /
• 2024

Station blackout (SBO) risk is one of the most significant contributors to nuclear power plant risk. In this paper, the sequence probability formulas derived by the convolution approach are compared with those derived by the conventional event tree/fault tree (ET/FT) approach for the SBO situation in which emergency diesel generators fail to start. The comparison identifies what makes the ET/FT approach more conservative and raises the issue regarding the mission time of a turbine-driven auxiliary feedwater pump (TDP), which suggests a possible modeling improvement in the ET/FT approach. Monte Carlo simulations with up-to-date component reliability data validate the convolution approach. The sequence probability of an alternative alternating current diesel generator (AAC DG) failing to start and the TDP failing to operate owing to battery depletion contributes most to the SBO risk. The probability overestimation of the scenario in which the AAC DG fails to run and the TDP fails to operate owing to battery depletion contributes most to the SBO risk overestimation determined by the ET/FT approach. The modification of the TDP mission time renders the sequence probabilities determined by the ET/FT approach more consistent with those determined by the convolution approach.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• v.15 no.9
• /
• pp.3204-3220
• /
• 2021

We perform the task of video object segmentation by incorporating a conditional random field (CRF) and convolutional neural networks (CNNs). Most methods employ a CRF to refine a coarse output from fully convolutional networks. Others treat the inference process of the CRF as a recurrent neural network and then combine CNNs and the CRF into an end-to-end model for video object segmentation. In contrast to these methods, we propose a novel higher-order CRF model to solve the problem of video object segmentation. Specifically, we use CNNs to establish a higher-order dependence among pixels, and this dependence can provide critical global information for a segmentation model to enhance the global consistency of segmentation. In general, the optimization of the higher-order energy is extremely difficult. To make the problem tractable, we decompose the higher-order energy into two parts by utilizing auxiliary variables and then solve it by using an iterative process. We conduct quantitative and qualitative analyses on multiple datasets, and the proposed method achieves competitive results.