• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.281-292
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• 2005

In this paper a nonlinear alternative of Leray-Schauder type is proved in a Banach algebra involving three operators and it is further applied to a functional nonlinear integral equation of mixed type $$x(t)=k(t,x({\mu}(t)))+[f(t,x({\theta}(t)))]\(q(t)+{\int}_0{^{\sigma}^{(t)}}v(t,s)g(s,x({\eta}\(s)))ds\)$$ for proving the existence results in Banach algebras under generalized Lipschitz and $Carath{\acute{e}}odory$ conditions.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1969-1980
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• 2017

Let ${\Gamma}$ be a nonzero torsionless commutative cancellative monoid with quotient group ${\langle}{\Gamma}{\rangle}$, $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}R_{\alpha}$ be a graded integral domain graded by ${\Gamma}$ such that $R_{{\alpha}}{\neq}\{0\}$ for all ${\alpha}{\in}{\Gamma},H$ be the set of nonzero homogeneous elements of R, C(f) be the ideal of R generated by the homogeneous components of $f{\in}R$, and $N(H)=\{f{\in}R{\mid}C(f)_v=R\}$. In this paper, we introduce the notion of graded t-almost Dedekind domains. We then show that R is a t-almost Dedekind domain if and only if R is a graded t-almost Dedekind domain and RH is a t-almost Dedekind domains. We also show that if $R=D[{\Gamma}]$ is the monoid domain of ${\Gamma}$ over an integral domain D, then R is a graded t-almost Dedekind domain if and only if D and ${\Gamma}$ are t-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if ${\langle}{\Gamma}{\rangle}$ isatisfies the ascending chain condition on its cyclic subgroups, then $R=D[{\Gamma}]$ is a t-almost Dedekind domain if and only if R is a graded t-almost Dedekind domain.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.579-586
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• 2009

Let $\varphi$ be a complete probability measure on $\mathbb{R}$, let $m_{\varphi}$ be the analogue of Wiener measure over paths on [0, T] and let f(t) be continuously differentiable on [0, T]. In this note, we give the analogue of Wiener measure $m_{\varphi}$ of {x in C[0, T]$\mid$x(0) < f(0) and $x(s_0){\geq}f(s_{0})$ for some $s_{0}$ in [0, T]} by use of integral equation techniques. This result is a generalization of Park and Paranjape's 1974 result[1].

• Honam Mathematical Journal
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• v.32 no.3
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• pp.441-451
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• 2010

In this paper we evaluate the analogue of Wiener integral ${\int\limits}_{C[0,t]}x(t_1){\cdots}x(t_n)d\omega_\rho(x)$ where 0 = $t_0$ < $t_1$ $\cdots$ < $t_n$ $\leq$ t and the Paley-Wiener-Zygmund integral ${\int\limits}_{C[0,t]}$ exp $({\int\limits}_0^t h(s)\tilde{d}x(s))d\omega_\rho(x)$ is the analogue of Wiener measure space.

• Transactions of the Korean Society of Mechanical Engineers
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• v.11 no.3
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• pp.376-385
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• 1987

In this study, the plane stress fracture toughness and Tearing modulus are investigated for various crack ratios using the J integral. To evaluate the J integral and Tearing modulus, both experiments and estimation are used. The thickness of the low carbon steel specimens that is used in the experiments is 3mm. The type of specimen that is considered in the study is center-cracked-tension one. The measurements of crack length are performed by unloading compliance method. In the estimation of crack parameters such as the J integral and load line displacement, the Ramberg and Osgood stress strain law is assumed. Then simple formulas are given for estimating the crack parameters from contained yielding to fully plastic solutions. Obtained results are as follows; (1) When the crack ratio is in the range of 0.500 - 0.701, the plane stress fracture toughness is almost constant regardless of crack ratios. (2) The fracture toughness (J$\_$c/) and Tearing modulus (T) obtained are J$\_$c/=28.51kgf/mm, T=677.7 for base metal, J$\_$c/=31.85kgf/mm, T=742.0 for annealed metal. (3) Simpson's and McCabe's formulas which consider crack growth in estimating J integral are shown more conservative J and lower T than Rice's and Sumpter's. (4) Comparison of the prediction with the actual experimental measurements by Simpson's formula shows good agreement.

• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.339-350
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• 2005

We obtain the following two inequalities on a strongly pseudoconvex domain $\Omega\;in\;\mathbb{C}^n\;:\;for\;f\;{\in}\;O(\Omega)$ $$\int_{0}^{{\delta}0}t^{a{\mid}a{\mid}+b}M_p^a(t, D^{a}f)dt\lesssim\int_{0}^{{\delta}0}t^{b}M_p^a(t,\;f)dt\;\int_{O}^{{\delta}O}t_{b}M_p^a(t,\;f)dt\lesssim\sum_{j=0}^{m}\int_{O}^{{\delta}O}t^{am+b}M_{p}^{a}\(t,\;\aleph^{i}f\)dt$$. In [9], Shi proved these results for the unit ball in $\mathbb{C}^n$. These are generalizations of some classical results of Hardy and Littlewood.

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.999-1018
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• 1998

In this paper, we prove stability theorems for the operator-valued Feynman integral of certain functionals involving some Borel measures on (0, t) as a bounded linear operator from L$_1$(R) to $C_{0}$(R).

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1105-1112
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• 2016

Describing the group of units $U({\mathbb{Z}}G)$ of the integral group ring ${\mathbb{Z}}G$, for a finite group G, is a classical and open problem. In this note, we show that $$U_1({\mathbb{Z}}[T{\times}C_2]){\sim_=}[F_{97}{\rtimes}F_5]{\rtimes}[T{\times}C_2]$$, where $T={\langle}a,b:a^6=1,a^3=b^2,ba=a^5b{\rangle}$ and $F_{97}$, $F_5$ are free groups of ranks 97 and 5, respectively.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.179-200
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• 2013

Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $X_n:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ by $Xn(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t$ is a partition of $[0,t]$. In the present paper, using a simple formula for the conditional expectation given the conditioning function $X_n$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the form $$f((v_1,x),{\cdots},(v_r,x))\;for\;x{\in}C[0,t]$$, where {$v_1,{\cdots},v_r$} is an orthonormal subset of $L_2[0,t]$ and $f{\in}L_p(\mathbb{R}^r)$. We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.