• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.861-868
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• 2006

We present a Choquet-Deny-type theorem for downward filtering convex sets of continuous functions and show that the Identity Korovkin cone of a downward filtering convex cone S is exactly the uniform closure of S.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.345-353
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• 2021

In this paper, first, we present new fixed point theorems for Mönch type multimaps on abstract convex uniform spaces and, also, a fixed point theorem for Mönch type multimaps in Hausdorff KKM L𝚪-spaces. Second, we show that Mönch type multimaps in the better admissible class defined on an L𝚪-space have fixed point properties whenever their ranges are Klee approximable. Finally, we obtain fixed point theorems on 𝔎ℭ-maps whose ranges are 𝚽-sets.

• Bulletin of the Korean Mathematical Society
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• v.5 no.1
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• pp.61-62
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• 1968

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.275-285
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• 1997

Let E be a uniformly convex Banach space with a uniformly G$\hat{a}teaux differentiable norm, C a nonempty closed convex subset of $E, T : C \to E$ a nonexpansive mapping, and Q a sunny nonexpansive retraction of E onto C. For $u \in C$ and $t \in (0,1)$, let $x_t$ be a unique fixed point of a contraction $R_t : C \to C$, defined by $R_tx = Q(tTx + (1-t)u), x \in C$. It is proved that if ${x_t}$ is bounded, then the strong $lim_{t\to1}x_t$ exists and belongs to the fixed point set of T. Furthermore, the strong convergence of ${x_t}$ in a reflexive and strictly convex Banach space with a uniformly G$\hat{a}$teaux differentiable norm is also given in case that the fixed point set of T is nonempty.

• Journal of the Korean Institute of Telematics and Electronics C
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• v.34C no.7
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• pp.51-60
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• 1997

Prototype based methods are commonly used in cluster analysis and the results may be highly dependent on the prototype used. In this paper, we propose a fuzzy clustering method that involves adaptively expanding convex polytopes. Thus, the dependency on the use of prototypes can be eliminated. The proposed method makes it possible to effectively represent an arbitrarily distributed data set without a priori knowledge of the number of clusters in the data set. Specifically, nonlinear membership functions are utilized to determine whether a new cluster is created or which vertex of the cluster should be expanded. For this, the membership function of a new vertex is assigned according to not only a distance measure between an incoming pattern vector and a current vertex, but also the amount how much the current vertex has been modified. Therefore, cluster expansion can be only allowed for one cluster per incoming pattern. Several experimental results are given to show the validity of our mehtod.

• Journal of Institute of Control, Robotics and Systems
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• v.19 no.6
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• pp.489-494
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• 2013

This paper tackles the problem of designing a dynamic output-feedback control for linear discrete-time norm-bounded uncertain systems with input saturation. By employing a LPV (Linear Parameter Varying) instead of LTI (Linear Time-Invariant) control, the useful information on interpolation parameters appearing in the procedure of representing saturation nonlinearity as a convex polytope is additionally applied in the control design procedure. By solving the addressed problem that can be recast into a convex optimization problem characterized by LMIs (Linear Matrix Inequalities) with one prescribed scalar, the vertices of convex set containing an LPV output-feedback control gain and the associated maximal invariant set of initial states are simultaneously obtained.

• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.153-162
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• 2001

In this Paper, the notion of $\varepsilon$-Birkhoff orthogonality introduced by Dragomir [An. Univ. Timisoara Ser. Stiint. Mat. 29(1991), no. 1, 51-58] in normed linear spaces has been extended to metric linear spaces and a decomposition theorem has been proved. Some results of Kainen, Kurkova and Vogt [J. Approx. Theory 105 (2000), no. 2, 252-262] proved on e-near best approximation in normed linear spaces have also been extended to metric linear spaces. It is shown that if (X, d) is a convex metric linear space which is pseudo strictly convex and M a boundedly compact closed subset of X such that for each $\varepsilon$>0 there exists a continuous $\varepsilon$-near best approximation $\phi$ : X → M of X by M then M is a chebyshev set .

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.175-184
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• 2013

In this paper, the reverse Bonnesen style inequalities for convex domain in the Euclidean plane $\mathbb{R}^2$ are investigated. The Minkowski mixed convex set of two convex sets K and L is studied and some new geometric inequalities are obtained. From these inequalities obtained, some isoperimetric deficit upper limits, that is, the reverse Bonnesen style inequalities for convex domain K are obtained. These isoperimetric deficit upper limits obtained are more fundamental than the known results of Bottema ([5]) and Pleijel ([22]).

• Transactions on Control, Automation and Systems Engineering
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• v.4 no.4
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• pp.324-329
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• 2002

In this paper, the guaranteed cost filtering design method for linear time delay systems with convex bounded uncertainties in discrete-time case is presented. The uncertain parameters are assumed to be unknown but belonging to known convex compact set of polytotype less conservative than norm bounded parameter uncertainty. The main purpose is to design a stable filter which minimizes the guaranteed cost. The sufficient condition for the existence of filter, the guaranteed cost filter design method, and the upper bound of the guaranteed cost are proposed. Since the proposed sufficient conditions are LMI(linear matrix inequality) forms in terms of all finding variables, all solutions can be obtained simultaneously by means of powerful convex programming tools with global convergence assured. Finally, a numerical example is given to check the validity of the proposed method.

• Journal of the Korean Institute of Intelligent Systems
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• v.18 no.1
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• pp.145-149
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• 2008

The similarity measure is constructed for non-convex fuzzy membership function using well known Hamming distance measure. Comparison with convex fuzzy membership function is carried out, furthermore characteristic analysis for non-convex function are also illustrated. Proposed similarity measure is proved and the usefulness is verified through example. In example, usefulness of proposed similarity is pointed out.