• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.737-748
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• 2012

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$-div(h(x){\nabla}u)=f(x,u)\;in\;{\Omega}$$ with Dirichlet boundary condition in a bounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, where $h(x){\in}L^1_{loc}({\Omega})$, $f(x,s)$ has asymptotically linear behavior. The solutions will be obtained in a subspace of the space $H^1_0({\Omega})$ and the proofs rely essentially on a variation of the mountain pass theorem in [12].

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.153-169
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• 2000

Let E be a real q-uniformly smooth Banach space which admits a weakly sequentially continuous duality map, and K a nonempty closed convex subset of E. Let T : K -> K be a mapping such that $F(T)\;=\;{x\;{\in}\;K\;:\;Tx\;=\;x}\;{\neq}\;0$ and (I - T) satisfies the accretive-type condition: $\;{\geq}\;{\lambda}$\mid$$\mid$x-Tx$\mid$$\mid$^2$, for all $x\;{\in}\;K,\;x^*\;{\in}\;F(T)$ and for some ${\lambda}\;>\;0$. The weak and strong convergence of the Ishikawa iteration method to a fixed point of T are investigated. An application of our results to the approximation of a solution of a certain linear operator equation is also given. Our results extend several important known results from the Mann iteration method to the Ishikawa iteration method. In particular, our results resolve in the affirmative an open problem posed by Naimpally and Singh (J. Math. Anal. Appl. 96 (1983), 437-446).

• Journal of The Korean Astronomical Society
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• v.34 no.4
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• pp.321-323
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• 2001

Compressible, magnetohydrodynamic (MHD) turbulence in two dimension is studied through high-resolution, numerical simulations with the isothermal equation of state. First, hydrodynamic turbulence with Mach number $(M)_{rms}\;\~$1 is generated by enforcing a random force. Next, initial, uniform magnetic field of various strengths with Alfvenic Mach number Ma $\gg$ 1 is added. Then, the simulations are followed until MHD turbulence is fully developed. Such turbulence is expected to exist in a variety of astrophysical environments including clusters of galaxies. Although no dissipation is included explicitly in our simulations, truncation errors produce dissipation which induces numerical resistivity. It mimics a hyper-resistivity in our second-order accurate code. After saturation, the resulting flows are categorized as SF (strong field), WF (weak field), and VWF (very weak field) classes respectively, depending on the average magnetic field strength described with Alfvenic Mach number, $(Ma)_{rms}{\ge}1$, $(Ma)_{rms}{\~}1$, and $(Ma)_{rms}{\gg}1$. The characteristics of each class are discussed.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.367-375
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• 1996

Suppose the kernel function $\kappa$ belongs to $S(R^2)$ and is symmetric such that $ < \otimes x, \kappa >\geq 0$ for all $x \in S'(R)$. Let A be the class of functions f such that the function f is measurable on $S'(R)$ with $\int_{S'(R)}$\mid$f((I + tK)^{\frac{1}{2}}x$\mid$^2d\mu(x) < M$ for some $M > 0$ and for all t > 0, where K is the integral operator with kernel function $\kappa$. We show that the \lambda$-potential $G_Kf$ of f is a weak solution of $(\lambda I - \frac{1}{2} \tilde{\Xi}_{0,2}(\kappa))_u = f$.

• Journal of the Chungcheong Mathematical Society
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• v.5 no.1
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• pp.159-165
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• 1992

In 1986, R. Huff [3] showed that a Dunford integrable function is Pettis integrable if and only if T : $X^*{\rightarrow}L_1(\mu)$ is weakly compact operator and {$T(K(F,\varepsilon))|F{\subset}X$, F : finite and ${\varepsilon}$ > 0} = {0}. In this paper, we introduce the notion of Bourgain property of real valued functions formulated by J. Bourgain [2]. We show that the class of pettis integrable functions is linear space and if lis bounded function with Bourgain property, then T : $X^{**}{\rightarrow}L_1(\mu)$ by $T(x^{**})=x^{**}f$ is $weak^*$ - to - weak linear operator. Also, if operator T : $L_1(\mu){\rightarrow}X^*$ with Bourgain property, then we show that f is Pettis representable.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.83-99
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• 2019

In this paper, we introduce a hybrid subgradient method for finding an element common to both the solution set of a class of pseudomonotone equilibrium problems, and the set of fixed points of a finite family of ${\kappa}$-strictly presudononspreading mappings in a real Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative method under some suitable conditions. These convergence theorems are investigated without the Lipschitz condition for bifunctions. Our results complement many known recent results in the literature.

• Communications for Statistical Applications and Methods
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• v.21 no.3
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• pp.245-251
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• 2014

Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called ${\psi}$-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The ${\psi}$-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

• Transactions on Control, Automation and Systems Engineering
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• v.2 no.2
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• pp.130-135
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• 2000

A two parameter model of a single fictitious weak acid with unknown dissociation constant has been successfully applied to design a neutralization system for many multi-component acid streams. But there are some processes for which above two parameter model is not satisfactory due to poor approxmation of the nonlinearity of pH process. Here, for etter control of wide class of multi-component acid streams, a three parameter model of a strong acid and a weak acid with unknown dissociation constant is proposed. The model approximates effectively three types of largest gain variation nonlinearities. Based on this model a nonlinear pH control system is designed. Parameters can eeasily estimated since their combinations appear linearly in the model equations and nonlinear adaptive control system may also be constructed just as with the two parameter model.

• Honam Mathematical Journal
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• v.42 no.3
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• pp.591-600
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• 2020

A module M is called ss-semilocal if every submodule U of M has a weak supplement V in M such that U∩V is semisimple. In this paper, we provide the basic properties of ss-semilocal modules. In particular, it is proved that, for a ring R, _RR is ss-semilocal if and only if every left R-module is ss-semilocal if and only if R is semilocal and Rad(R) ⊆ Soc(RR). We define projective ss-covers and prove the rings with the property that every (simple) module has a projective ss-cover are ss-semilocal.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.1039-1052
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• 2021

In this paper, we introduce and study the class of 𝜙-w-flat modules which are generalizations of both 𝜙-flat modules and w-flat modules. The 𝜙-w-weak global dimension 𝜙-w-w.gl.dim(R) of a commutative ring R is also introduced and studied. We show that, for a 𝜙-ring R, 𝜙-w-w.gl.dim(R) = 0 if and only if w-dim(R) = 0 if and only if R is a 𝜙-von Neumann ring. It is also proved that, for a strongly 𝜙-ring R, 𝜙-w-w.gl.dim(R) ≤ 1 if and only if each nonnil ideal of R is 𝜙-w-flat, if and only if R is a 𝜙-PvMR, if and only if R is a PvMR.