• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.9
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• pp.949-956
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• 2009

The C(t)-integral describes amplitude of stress and strain rate field near a tip of stationary crack under transient creep condition. Thus the C(t)-integral is a key parameter for the high-temperature crack assessment. Estimation formulae for C(t)-integral of the cracked component operating under mechanical load alone have been provided for decades. However, high temperature structures usually work under combined mechanical and thermal load. And no investigation has provided quantitative estimates for the C(t)-integral under combined mechanical and thermal load. In this study, 3-dimensional finite element analyses were conducted to calculate the C(t)-integral of elastic-creep material under combined mechanical and thermal load. As a result, redistribution time for the crack under combined mechanical and thermal load is re-defined through FE analyses to quantify the C(t)-integral. Estimates of C(t)-integral using this proposed redistribution time agree well with FE analyses results.

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.465-475
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• 1996

Fix t > 0. Denote by $C^t$ the space of $R$-valued continuous functions x on [0,t]. Let $C_0^t$ be the Wiener space - $C_0^t = {x \in C^t : x(0) = 0}$ - equipped with Wiener measure m. Let F be a function from $C^t to C$.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.763-775
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• 1996

For $Q = [0,S] \times [0,T]$ let C(Q) denote Yeh-Wiener space, i.e., the space of all real-valued continuous functions x(s,t) on Q such that x(0,t) = x(s,0) = 0 for every (s,t) in Q. Yeh [10] defined a Gaussian measure $m_y$ on C(Q) (later modified in [13]) such that as a stochastic process ${x(s,t), (s,t) \epsilon Q}$ has mean $E[x(s,t)] = \smallint_{C(Q)} x(s,t)m_y(dx) = 0$ and covariance $E[x(s,t)x(u,\upsilon)] = min{s,u} min{t,\upsilon}$. Let $C_\omega \equiv C[0,T]$ denote the standard Wiener space on [0,T] with Wiener measure $m_\omega$. Yeh [12] introduced the concept of the conditional Wiener integral of F given X, E(F$\mid$X), and for case X(x) = x(T) obtained some very useful results including a Kac-Feynman integral equation.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.35 no.6
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• pp.609-617
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• 2011

There is a trend towards the progressive use of higher operating temperatures and stresses to achieve improved efficiencies in power-generation equipment. It is important to perform the crack assessment under hightemperature and high-pressure conditions. The C(t)-integral is a key parameter in crack assessment for transient creep states. The estimation of the C(t)-integral is complex when considering the mechanical and thermal loads simultaneously. In this paper, we study estimation of C(t)-integral under combined mechanical and thermal load depending on loading conditions.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.10
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• pp.1065-1073
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• 2009

In this paper, the estimation method of C(t)-integral for combined mechanical and thermal loads is proposed for elastic-plastic-creep material via 3-dimensional FE analyses. Plasticity induced by initial loading makes relaxation rate different from those produced elastically. Moreover, the interactions between mechanical and thermal loads make the relaxation rate different from those produced under mechanical load alone. To quantify C(t)-integral for combined mechanical and thermal loads, the simplified formula are developed by modifying redistribution time in existing work done by Ainsworth et al..

• Kyungpook Mathematical Journal
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• v.52 no.4
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• pp.413-431
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• 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 $\mathbb{R}^r$.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1261-1273
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• 2016

In this paper, we analyze the necessary and sufficient condition introduced in [5]: that a functional F in $L^2(C_{a,b}[0,T])$ has an integral transform ${\mathcal{F}}_{{\gamma},{\beta}}F$, also belonging to $L^2(C_{a,b}[0,T])$. We then establish the inverse integral transforms of the functionals in $L^2(C_{a,b}[0,T])$ and then examine various properties with respect to the inverse integral transforms via the translation theorem. Several possible outcomes are presented as remarks. Our approach is a new method to solve some difficulties with respect to the inverse integral transform.

• Kyungpook Mathematical Journal
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• v.52 no.1
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• pp.33-38
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• 2012

The purpose of the present paper is to investigate mapping properties of an integral operator in which we show that the function g defined by $$g(z)=\{\frac{c+{\alpha}}{z^c}{\int}_{o}^{z}t^{c-1}(D^nf)^{\alpha}(t)dt\}^{1/{\alpha}}$$. belongs to the class $S(A,B)$ if $f{\in}S(n,A,B)$.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.791-815
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• 2014

We show amongst other that if $f,g:[a,b]{\rightarrow}\mathbb{C}$ are two functions of bounded variation and such that the Riemann-Stieltjes integral $\int_a^bf(t)dg(t)$ exists, then for any continuous functions $h:[a,b]{\rightarrow}\mathbb{C}$, the Riemann-Stieltjes integral $\int_{a}^{b}h(t)d(f(t)g(t))$ exists and $${\int}_a^bh(t)d(f(t)g(t))={\int}_a^bh(t)f(t)d(g(t))+{\int}_a^bh(t)g(t)d(f(t))$$. Using this identity we then provide sharp upper bounds for the quantity $$\|\int_a^bh(t)d(f(t)g(t))\|$$ and apply them for trapezoid and Ostrowski type inequalities. Some applications for continuous functions of selfadjoint operators on complex Hilbert spaces are given as well.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.609-631
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• 2021

Let C[0, T] denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval [0, T]. Define $Z_{\vec{e},n}$ : C[0, T] → ℝⁿ⁺¹ by $$Z_{\vec{e},n}(x)=\(x(0),\;{\int}_0^T\;e_1(t)dx(t),{\cdots},\;{\int}_0^T\;e_n(t)dx(t)\)$$, where e₁,…, e_n are of bounded variations on [0, T]. 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 $Z_{\vec{e},n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on C[0, T] which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval [0, T].