• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.65-72
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• 2012

Let T > 0 be given. Let $(C[0,T],m_{\varphi})$ be the analogue of Wiener measure space, associated with the Borel proba-bility measure ${\varphi}$ on ${\mathbb{R}}$, let $(L_{2}[0,T],\tilde{\omega})$ be the centered Gaussian measure space with the correlation operator $(-\frac{d^{2}}{dx^{2}})^{-1}$ and ${\el}_2,\;\tilde{m}$ be the abstract Wiener measure space. Let U be the space of all sequence $<c_{n}>$ in ${\el}_{2}$ such that the limit $lim_{{m}{\rightarrow}\infty}\;\frac{1}{m+1}\;\sum{^{m}}{_{n=0}}\;\sum_{k=0}^{n}\;c_{k}\;cos\;\frac{k{\pi}t}{T}$ converges uniformly on [0,T] and give a set function m such that for any Borel subset G of $\el_2$, $m(\mathcal{U}\cap\;P_{0}^{-1}\;o\;P_{0}(G))\;=\tilde{m}(P_{0}^{-1}\;o\;P_{0}(G))$. The goal of this note is to study the relationship among the measures $m_{\varphi},\;\tilde{\omega},\;\tilde{m}$ and $m$.

• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.677-683
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• 2003

In multi-allelic model $X\;=\;(x_1,\;x_2,\;\cdots\;,\;x_d),\;M_f(t)\;=\;f(p(t))\;-\;{\int_0}^t\;Lf(p(t))ds$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we examine the stochastic differential equation for model X and find the properties using stochastic differential equation.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.415-423
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• 2004

In allelic model X = ($x_1,\;x_2,...x_{d}$), $M_f(t)$= f(p(t)) - ${{\int}^{t}}_0$Lf(p(t))ds is a P-martingale for diffusion operator L under the certain conditions. In this note, we can show uniqueness of martingale problem associated with mean vector and obtain a complete description of ergodic property by using of the semigroup method.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.953-966
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• 2020

In this article we study the evolution and monotonicity of the first non-zero eigenvalue of weighted p-Laplacian operator which it acting on the space of functions on closed oriented Riemannian n-manifolds along the extended Ricci flow and normalized extended Ricci flow. We show that the first eigenvalue of weighted p-Laplacian operator diverges as t approaches to maximal existence time. Also, we obtain evolution formulas of the first eigenvalue of weighted p-Laplacian operator along the normalized extended Ricci flow and using it we find some monotone quantities along the normalized extended Ricci flow under the certain geometric conditions.

• Journal of the Korean Mathematical Society
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• v.57 no.1
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• pp.89-111
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• 2020

In the present paper, we give some characterization of the L₂ maximal estimate for the operator P^t_a,γf(Γ(x, t)) along curve with complex time, which is defined by $$P^t_{a,{\gamma}}f({\Gamma}(x,t))={\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}}}}\;e^{i{\Gamma}(x,t){\xi}}e^{it{\mid}{\xi}{\mid}^a}e^{-t^{\gamma}{\mid}{\xi}{\mid}^a}{\hat{f}}({\xi})d{\xi}$$, where t, γ > 0 and a ≥ 2, curve Γ is a function such that Γ : ℝ×[0, 1] → ℝ, and satisfies Hölder's condition of order σ and bilipschitz conditions. The authors extend the results of the Schrödinger type with complex time of Bailey [1] and Cho, Lee and Vargas [3] to Schrödinger operators along the curves.

• Honam Mathematical Journal
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• v.26 no.2
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• pp.137-145
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• 2004

we shall study some properties of a new class ($\sqrt[\kappa]{P}$) (defined below). Also we show that T may not be normaloid when $T{\in}(\sqrt[\kappa]{P})$, and that the class ($\sqrt{H}$) may not have the translation-invariant propety.

• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1131-1145
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• 2012

The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with p-Laplacian type operators $$\{({\alpha}(t,x^{\Delta}(t)))^{\Delta}-a(t){\phi}_p(x^{\sigma}(t))+f({\sigma}(t),x^{\sigma}(t))=0,\;{\Delta}-a.e.\;t{\in}I\\x^{\sigma}(0)=0,\\{\beta}_1x^{\sigma}(1)+{\beta}_2x^{\Delta}({\sigma}(1))=0,$$ where ${\beta}_1$, ${\beta}_2$ > 0, $I=[0,1]^{k^2}$, ${\alpha}({\cdot},x({\cdot}))$ is an operator of $p$-Laplacian type, $\mathbb{T}$ is a time scale. Some sufficient conditions for the existence of constant-sign solutions are obtained.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.53-71
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• 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 Chungcheong Mathematical Society
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• v.25 no.4
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• pp.655-668
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• 2012

An operator $T\;{\varepsilon}\;B(\mathcal{H})$ is said to be $k$-quasi-paranormal operator if $||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$ for every $x\;{\epsilon}\;\mathcal{H}$, $k$ is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of $k$-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically $k$-quasi-paranormal operator has Bishop's property ($\beta$), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for $f(T)$ for every $f\;{\epsilon}\;H({\sigma}(T))$; (ii) generalized a - Browder's theorem holds for $f(S)$ for every $S\;{\prec}\;T$ and $f\;{\epsilon}\;H({\sigma}(S))$; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.