• Communications of the Korean Mathematical Society
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• v.30 no.1
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• pp.7-21
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• 2015

We give the characterization of H$\ddot{o}$lder differentiability points and non-differentiability points of the Riesz-N$\acute{a}$gy-Tak$\acute{a}$cs (RNT) singular function ${\Psi}_{a,p}$ satisfying ${\Psi}_{a,p}(a)=p$. It generalizes recent multifractal and metric number theoretical results associated with the RNT function. Besides, we classify the singular functions using the singularity order deduced from the H$\ddot{o}$lder derivative giving the information that a strictly increasing smooth function having a positive derivative Lebesgue almost everywhere has the singularity order 1 and the RNT function ${\Psi}_{a,p}$ has the singularity order $g(a,p)=\frac{a{\log}p+(1-a){\log}(1-p)}{a{\log}a+(1-a){\log}(1-a)}{\geq}1$.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.71-76
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• 1988

For each continuous convex function ${\phi}$ defined on an open convex subset $A_{\phi}$ of a Banach space X, if we define a positively homogeneous sublinear functional ${\sigma}_x$ on X by ${\sigma}_x(y)=\sup{\lbrace}f(y)\;:\;f{\in}{\partial}{\phi}(x){\rbrace}$, where ${\partial}{\phi}(x)$ is a subdifferential of ${\phi}$ at x, then we get the following characterization theorem of Gateaux differentiability (weak Asplund) sapce. THEOREM. For every ${\phi}$ above, $D_{\phi}={\lbrace}x{\in}A\;:\;\sup_{||u||=1}\;{\sigma}_x(u)+{\sigma}_x(-u)=0{\rbrace}$ contains dense (dense $G_{\delta}$) subset of $A_{\phi}$ if and only if X is a Gateaux differentiability (weak Asplund) space.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.3
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• pp.37-41
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• 2007

The semi-convexity for planar shapes has been recently introduced in [2]. As a generalization of the convextiy, semi-convexity is closed under the Minkowski sum. But the definition of semi-convexity requires that the shape boundary should satifisfy a differentiability condition $C^{1:1}$, which means that it should be possible to take the normal vector field along the domain's extended boundary. In view of the fact that the semi-convextiy is a most natural generalization of the convexity in many respects, this is a severe restriction for the semi-convexity, since the convexity requires no such a priori differentiability condition. In this paper, we generalize the semi-convexity to the closure of the class of semi-convex $\mathcal{M}$-domains for any Minkowski class $\mathcal{M}$, and show that this generalized semi-convexity is also closed under Minkowski sum.

• Journal of the Chungcheong Mathematical Society
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• v.12 no.1
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• pp.157-162
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• 1999

In this paper, we introduce approximate Perron-Stieltjes integral and approximate Roussel derivative and investigate the differentiability of indefinite approximate Perron-Stieltjes integral.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.327-337
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• 2015

This paper is concerned with the optimality conditions for optimal control problem of Belousov-Zhabotinskii reaction model. That is, we obtain the optimality conditions by showing the differentiability of the solution with respect to the control. We also show the uniqueness of the optimal control.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1037-1048
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• 2008

The purpose of this paper is to present a differential equation with delay from biological excitable medium. Existence, uniqueness and data dependence (monotony, continuity, differentiability with respect to parameter) results for the solution of the Cauchy problem of biological excitable medium are obtained using weakly Picard operator theory.

• Journal of the Korean Operations Research and Management Science Society
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• v.20 no.2
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• pp.125-137
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• 1995

This paper provides a locally convergent algorithm and a globally convergent algorithm for a variational inequality problem over convex polyhedral. The algorithm are based on the B (ouligand)-differentiability of the solution of a nonsmooth equation derived from the variational in-equality problem. Convergences of the algorithms are achieved by the results of Pang[3].

• East Asian mathematical journal
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• v.35 no.5
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• pp.547-557
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• 2019

This paper is concerned with the necessary conditions for optimal control problem governed by some ODE-PDE systems. That is, we obtain the necessary conditions for optimal control by showing the differentiability of the solution with respect to the control.

• Journal of the Korean School Mathematics Society
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• v.16 no.2
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• pp.435-457
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• 2013

The purpose of this paper is to discover effective teaching and learning methods for improving low level mathematic matriculants', who passed the early decision program, problem solving abilities by analyzing their error patterns in the special lecture for basic mathematics in P University. In this paper, we examine the matriculants' understanding and errors on the fundamental concepts of function, and continuity and differentiability of function based on the pre-examination. We also measure the their academic achievement in the special lecture for basic mathematics, and analyze the differences of error patterns between pre-test and post-test result on the concepts of continuity and differentiability of function.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.427-430
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• 2009

In this paper we introduce the intermediate differential of a Lipschitz map from a Banach space to another Banach space and prove that every locally Lipschitz function f defined on an open subset ${\Omega}$ of a superreflexive real Banach space X to a finite dimensional Banach space Y is uniformly intermediate differentiable at every point ${\Omega}/A$, where A is a ${\sigma}$-lower porous set.