• Journal of the Korean Statistical Society
• /
• v.3 no.1
• /
• pp.13-16
• /
• 1974

In my former paper [3] I defined an almost periodicity of weakly sationary random processes (a.p.w.s.p.) and presented some basic results of it. In this paper I shall present some notes on the Fourier series of an a.p.w.s.p., resulting from [3]. All the conditions at the introduction of [3] are assumed to hold without repreating them here. The essential facts are as follows : The weakly stationary process $X(t,\omega), t\in(-\infty,\infty), \omega\in\Omega$, defined on a probability space $(\Omega,a,P)$, has a spectral representation $$X(t,\omega)=\int_{-\infty}^{infty}{e^{it\lambda\xi}(d\lambda,\omega)},$$ where $\xi(\lambda)$ is a random measure. Then, the continuous covariance $\rho(\mu) = E(X(t+u) X(t))$ has the form $$\rho(u)=\int_{-\infty}^{infty}{e^{iu\lambda}F(d\lambda)},$$ $E$\mid$\xi(\lambda+0)-\xi(\lambda-0)$\mid$^2 = F(\lambda+0) - F(\lambda-0) \lambda\in(-\infty,\infty)$, assumimg that $\rho(u)$ is a uniformly almost periodic function.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.631-638
• /
• 1996

The parabolic free boundary problem with Puschino dynamics is given by (see in [3]) $$ (1) { \upsilon_t = D\upsilon_{xx} - (c_1 + b)\upsilon + c_1 H(x - s(t)) for (x,t) \in \Omega^- \cup \Omega^+, { \upsilon_x(0,t) = 0 = \upsilon_x(1,t) for t > 0, { \upsilon(x,0) = \upsilon_0(x) for 0 \leq x \leq 1, { \tau\frac{dt}{ds} = C)\upsilon(s(t),t)) for t > 0, { s(0) = s_0, 0 < s_0 < 1, $$ where $\upsilon(x,t)$ and $\upsilon_x(x,t)$ are assumed continuous in $\Omega = (0,1) \times (0, \infty)$.

• Communications of the Korean Mathematical Society
• /
• v.13 no.1
• /
• pp.77-84
• /
• 1998

In this note we show that if $T_{\varphi}$ is a Toeplitz operator with quasicontinuous symbol $\varphi$, if $\omega$ is an open set containing the spectrum $\sigma(T_\varphi)$, and if $H(\omega)$ denotes the set of analytic fuctions defined on $\omege$, then the following statements are equivalent: (a) $T_\varphi$ is semi-quasitriangular. (b) Browder's theorem holds for $f(T_\varphi)$ for every $f \in H(\omega)$. (c) Weyl's theorem holds for $f(T_\varphi)$ for every $f \in H(\omega)$. (d) $\sigma(T_{f \circ \varphi}) = f(\sigma(T_varphi))$ for every $f \in H(\omega)$.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
• /
• 2003.09a
• /
• pp.6-6
• /
• 2003

In this work we consider the mathematical formulation and numerical resolution of the linear feedback control problem for Boussinesq equations. The controlled Boussinesq equations is given by $$\frac{{\partial}u}{{\partial}t}-{\nu}{\Delta}u+(u{\cdot}{\nabla}u+{\nabla}p={\beta}{\theta}g+f+F\;\;in\;(0,\;T){\times}\;{\Omega}$$, $${\nabla}{\cdot}u=0\;\;in\;(0,\;T){\times}{\Omega}$$, $$u|_{{\partial}{\Omega}=0,\;u(0,x)=\;u_0(x)$$ $$\frac{{\partial}{\theta}}{{\partial}t}-k{\Delta}{\theta}+(u{\cdot}){\theta}={\tau}+T,\;\;in(0,\;T){\times}{\Omega}$$ $${\theta}|_{{\partial}{\Omega}=0,\;\;{\theta}(0,X)={\theta}_0(X)$$, where $\Omega$ is a bounded open set in $R^{n}$, n=2 or 3 with a $C^{\infty}$ boundary ${\partial}{\Omega}$. The control is achieved by means of a linear feedback law relating the body forces to the velocity and temperature field, i.e., $$f=-{\gamma}_1(u-U),\;\;{\tau}=-{\gamma}_2({\theta}-{\Theta}}$$ where (U,$\Theta$) are target velocity and temperature. We show that the unsteady solutions to Boussinesq equations are stabilizable by internal controllers with exponential decaying property. In order to compute (approximations to) solution, semi discrete-in-time and full space-time discrete approximations are also studied. We prove that the difference between the solution of the discrete problem and the target solution decay to zero exponentially for sufficiently small time step.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.53-71
• /
• 2004

A certain subclass $T_{\Omega}(n,\;p,\;\lambda,\;\alpha)$ of starlike functions in the unit disk is introduced. The object of the present paper is to derive several interesting properties of functions belonging to the class $T_{\Omega}(n,\;p,\;\lambda,\;\alpha)$. Coefficient inequalities, distortion theorems and closure theorems of functions belonging to the class $T_{\Omega}(n,\;p,\;\lambda,\;\alpha)$ are determined. Also we obtain radii of convexity for the class $T_{\Omega}(n,\;p,\;\lambda,\;\alpha)$. Furthermore, integral operators and modified Hadamard products of several functions belonging to the class $T_{\Omega}(n,\;p,\;\lambda,\;\alpha)$ are studied here.

• Journal of the Korean Applied Science and Technology
• /
• v.18 no.3
• /
• pp.197-213
• /
• 2001

Conjugate trienoic acids (CTA) occurred in triacylglycerols (TGs) of the seed oils of Trichosanthes kirilowii, Momordica charantia and Aleurites fordii, and they were easily converted to their methyl esters in a mixture of sodium methoxide-methanol without any structural destruction. The main fatty acids in triacylglycerol (TG) fraction of the seed oils of Trichosanthes kirilowii are $C_{18:2{\omega}6}$ (32.2 mol %), $C_{18:3{\;}9c.11t,13c}$ (38.0 mol %) and $C_{18:1{\omega}9}$ (11.8 mol %), followed with $C_{16:0}$ (4.8 mol %) and $C_{18:0}$ (3.1 mol %). The TG fraction was resolved into 20 TG molecular species according to the partition number (PN) by reversed-phase (RP)-HPLC. The main TG species were $DT_{c2}$, $MDT_{c}$ and $D_{2}T_{c}$, of which amounts reached 63 mol % of total TG molecular species. The TG sample was fractionated into 11 fractions according to the number of double bond in the molecule by $Ag^{+}-HPLC$ and the species of $DT_{c2}$, $MDT_{c}$ and $D_{2}T_{c}$ were also eluted as main components. The TG species containing CTA showed unusual behaviours in the order of elution by HPLC ; first, TG moleular species of $DT_{c2}$ (D; dienoic acid, $T_{c}$; punicic acid, $T_{ci}$; ${\alpha}-eleostearic$ acid, M ; monoenoic acid, $S_{t}$; stearic acid) was eluted earlier than $Mt_{c2}$, although they have the same PN number of 40, and, secondly, the species of $DT_{ci2}$ with eight double bonds was eluted earlier than that of $D_2T_{ci}$ with seven double bonds. Intact TG of the seed oils of Momordica charantia contained mainly fatty acids such as $C_{18:3{\omega}9c,11t,13t}$ (57.7 mol %), $C_{18:1{\omega}9}$ (17.4 mol %), $C_{18:0}$ (12.3 mol %) and $C_{18:2{\omega}6}$ (10.6 mol %), and was classified into 13 fractions by RP-HPLC. The main TG species were as follows ; $MT_{ci2}$ [$(C_{18:1{\omega}9})(C_{18:3\;9c,11t,13t})_{2}$, 39.1 mol %] and $S_{t}T_{ci2}$ [$(C_{18:0})(C_{18:3\;9c,11t,13t})_2$, 33.9 mol %] comprising about 73 mol % of total TG species, accompanied by $DT_{ci2}$ [$(C_{18:2{\omega}6})(C_{18:3\;9c,11t,13t})_{2}$, 7.3 mol %], $D_{2}T_{ci}$ [$ (C_{18:2{\omega}6})_{2}(C_{18:3\;9c,11t,13t})$, 3.6 mol %] and $MDT_{ci}$ [$(C_{18:1{\omega}9})(C_{18:2{\omega}6})(C_{18:3\;9c,11t,13t})$, 3.5 mol %]. Simple TG species of $T_{ci3}$ [$(C_{18:3\;9c,11t,13t})_3]$ was present in a small amount of 1.4 mol %, but other simple TG species were not detected. The TG was also resolved into 11 fractions according to the number of double bond by $Ag^{+}-HPLC$, and the species were mainly occupied by $MT_{ci2}$ [$(C_{18:1{\omega}9})(C_{18:3\;9c,11t,13t})_{2}$, 39.4 mol %] and $S_tT-{ci2}$ [$(C_{18:0})(C_{18:3\;9c,11t,13t})_{2}$, 35.4 mol %] $DT_{ci2}$ species with eight double bonds was also developed faster than $D_2T_{ci}$ one with seven double bonds as indicated in the analysis of TG of the seed oils of T. kirilowii, and $MT_{ci2}$ species with cis, trans, trans-configurated double bond was eluted earlier than $MT_{c2}$ species with cis, trans, cis-configurated double bond. The main components of fatty acid in total TG fraction isolated from the seed oils of of Aleurites fordii were in the following order ; $C_{18:3\;9c,11t,13t}$ (81.2 mol %)> $C_{18:2{\omega}6}$ (8.5 mol %)> $C_{18:1{\omega}9}$ (5.4 mol %)$. With resolution of the TG by RP-HPLC, eight fractions such as $T_{ci3}$, $Dt_{ci2}$, $D_{2}T_{ci}$, $MT_{ci2}$, $PT_{ci2}$ (P; palmitic acid), $PMT_{ci}$, $PDT_{ci}$ and $S_{t}T_{ci2}$ ($S_{t}$; stearic acid) were isolated, respectively. TG species of $T_{ci3}$ [$(C_{18:3\;9c,11t,13t})_{3}$, 54.2 mol %], $DT_{ci2}$ [$(C_{18:2{\omega}6})(C_{18:3\;9c,11t,13t})_{2}$, 15.0 mol %] and $MT_{ci2}$ [$(C_{18:1{\omega}9})(C_{18:3 9c,11t,13t})_{2}$, 14.8 mol %] were present as main species.

• Journal of the Korean Mathematical Society
• /
• v.39 no.5
• /
• pp.801-819
• /
• 2002

In this note, we establish a translation theorem in an analogue of Wiener space (C[0,t],$\omega$$\phi$) and find formulas for the conditional $\omega$$\phi$-integral given by the condition X(x) ＝ (x(to), x(t$_1$),…, x(t$_{n}$)) which is the generalization of Chang and Chang's results in 1984. Moreover, we prove a translation theorem for the conditional $\omega$$\phi$-integral.l.

• Journal of Periodontal and Implant Science
• /
• v.44 no.4
• /
• pp.169-177
• /
• 2014

Purpose: We aimed to investigate the impact of nonsurgical periodontal treatment combined with one-year dietary supplementation with omega (${\omega}$)-3 on the serum levels of eicosapentaenoic acid (EPA), docosahexaenoic acid (DHA), docosapentaenoic acid (DPA), and arachidonic acid (AA). Methods: Fifteen patients with chronic generalized periodontitis were treated with scaling and root planing. The test group consisted of seven patients ($43.1{\pm}6.0$ years) supplemented with ${\omega}$-3, consisting of EPA plus DHA, three capsules, each of 300 mg of ${\omega}$-3 (180-mg EPA/120-mg DHA), for 12 months. The control group was composed of eight patients ($46.1{\pm}11.6$ years) that took a placebo capsule for 12 months. The periodontal examination and the serum levels of DPA, EPA, DHA, and AA were performed at baseline (T0), and 4 (T1), and 12 (T2) months after therapy. Results: In the test group, AA and DPA levels had been reduced significantly at T1 (P<0.05). AA and EPA levels had been increased significantly at T2 (P<0.05). The ${\Delta}EPA$ was significantly higher in the test compared to the placebo group at T2-T0 (P=0.02). The AA/EPA had decreased significantly at T1 and T2 relative to baseline (P<0.05). Conclusions: Nonsurgical periodontal treatment combined with ${\omega}$-3 supplementation significantly increased the EPA levels and decreased the AA/EPA ratio in serum after one year follow-up. However, no effect on the clinical outcome of periodontal therapy was observed.

• Korean Journal of Mathematics
• /
• v.24 no.3
• /
• pp.545-565
• /
• 2016

In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: $$T_{\lambda}(f;x)={\large\int_{\Omega}}K_{\lambda}(t,x,f(t))dt,\;x{\in}{\Psi},\;{\lambda}{\in}{\Lambda}$$, where ${\Psi}$ = and ${\Omega}$ = stand for arbitrary closed, semi-closed or open bounded intervals in ${\mathbb{R}}$ or these set notations denote $\mathbb{R}$, and ${\Lambda}$ is a set of non-negative numbers, to the function $f{\in}L_{p,{\omega}}({\Omega})$, where $L_{p,{\omega}}({\Omega})$ denotes the space of all measurable functions f for which $\|{\frac{f}{\omega}}\|^p$ (1 ${\leq}$ p < ${\infty}$) is integrable on ${\Omega}$, and ${\omega}:{\mathbb{R}}{\rightarrow}\mathbb{R}^+$ is a weight function satisfying some conditions.

• Bulletin of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.303-308
• /
• 1995

We consider a torus T, that is, a compact surface with genus 1 and $\Omega = D^2 \times S^1$ topologically with $\partial\Omega = T$, where $D^2$ is the open unit disk and $S^1$ is the unit circle. Let $\omega = (x,y)$ denote the generic point on T. For a smooth immersion $u : T \to R^3$, we define the Dirichlet functional by $$ E(u) = \frac{2}{1} \int_{T} $\mid$\nabla u$\mid$^2 d\omega $$ and the volume functional by $$ V(u) = \frac{3}{1} \int_{T} u \cdot u_x \Lambda u_y d\omege $$.