• The Mathematical Education
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• v.26 no.1
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• pp.41-45
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• 1987

In this paper, we give several fixed point theorems in a complete metric space for two multi-valued mappings commuting with two single-valued mappings. In fact, our main theorems show the existence of solutions of functional equations f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ and $\chi$=f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ under certain conditions. We also answer an open question proposed by Rhoades-Singh-Kulsherestha. Throughout this paper, let (X, d) be a complete metric space. We shall follow the following notations : CL(X) = {A; A is a nonempty closed subset of X}, CB(X)={A; A is a nonempty closed and founded subset of X}, C(X)={A; A is a nonempty compact subset of X}, For each A, B$\in$CL(X) and $\varepsilon$>0, N($\varepsilon$, A) = {$\chi$$\in$X; d($\chi$, ${\alpha}$) < $\varepsilon$ for some ${\alpha}$$\in$A}, E$\sub$A, B/={$\varepsilon$ > 0; A⊂N($\varepsilon$ B) and B⊂N($\varepsilon$, A)}, and (equation omitted). Then H is called the generalized Hausdorff distance function fot CL(X) induced by a metric d and H defined CB(X) is said to be the Hausdorff metric induced by d. D($\chi$, A) will denote the ordinary distance between $\chi$$\in$X and a nonempty subset A of X. Let R$\^$+/ and II$\^$+/ denote the sets of nonnegative real numbers and positive integers, respectively, and G the family of functions ${\Phi}$ from (R$\^$+/)$\^$s/ into R$\^$+/ satisfying the following conditions: (1) ${\Phi}$ is nondecreasing and upper semicontinuous in each coordinate variable, and (2) for each t>0, $\psi$(t)=max{$\psi$(t, 0, 0, t, t), ${\Phi}$(t, t, t, 2t, 0), ${\Phi}$(0, t, 0, 0, t)} $\psi$: R$\^$+/ \longrightarrow R$\^$+/ is a nondecreasing upper semicontinuous function from the right. Before sating and proving our main theorems, we give the following lemmas:

• Economic and Environmental Geology
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• v.43 no.6
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• pp.615-623
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• 2010

Seismic refraction survey is a geophysical method that delineates subsurface velocity structure using direct wave and critically refracted wave. The generalized reciprocal method(GRM) is an inversion technique which uses travel-time data from several forward and reverse shots and which can provide the geometry of irregular inclined refractors and structures underlain by hidden layer such as low velocity zone and thin layer. In this study, a simple Excel-GRM routine was tested for fast mapping of the interface between weathering layer and bedrock during the survey, with employing a pair of forward and reverse shots. This routine was proved to control the maximum dip of approximately $30^{\circ}C$ and maximum velocity contrast of 0.6, based on the panel tests in terms of dipping angle and velocity contrast for the two-layer inclined models. In contrast with conventional operation of five to seven shots with sufficient offset distance and indoor data analysis thereafter, this routine was performed in the field shortly after data acquisition. Depth to the bedrock provided by Excel-GRM, during the field survey for Cheongju granite area, correlates well with the elevation of the surface of soft rock from the drill core and SPS logging data. This cost-effective routine developed for quickly delineating the bedrock surface in the field survey will be readily applicable to mapping of weathering zone in narrow zone with small variation of elevation of bedrock.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.39 no.6
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• pp.583-588
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• 2015

Asymptotic method has been often used to theoretically analyze the complete contact problem. The error of the asymptotic results increases as the distance from the contact edge increases. The singularity cannot be properly obtained from a finite element (FE) analysis owing to the finiteness of the element size. In the present work, the complete contact problem in bonded condition is analyzed using a combined experimental-numerical approach to assist and/or compare with the asymptotic results. Al and Cu alloys are used for the material combination of the punch and substrate. 120 and 135 degrees are used for the punch angle. The FE models are validated by comparison of displacement distributions obtained by the FE analysis and $moir{\acute{e}}$ experiment. Generalized stress intensity factors are evaluated using the validated FE models. Stress field in the vicinity of the sharp contact edges obtained from the FE and asymptotic analyses are compared. The discrepancies are also discussed.

• Journal of the Korean Institute of Telematics and Electronics C
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• v.36C no.11
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• pp.64-74
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• 1999

A new clock skew optimization for clock routing using link-edge insertion is proposed in this paper. It satisfies the given skew bound and prevent the total wire length from increasing. As the clock skew is the major constraint for high speed synchronous ICs, it must be minimized in order to obtain high performance. But clock skew minimization can increase total wire length, therefore clock routing is performed within the given skew bound which can not induce the malfunction. Clock routing under the specified skew bound can decrease total wire length Not only total wire length and delay time minimization algorithm using merging point relocation method but also clock skew reduction algorithm using link-edge insertion technique between two nodes whose delay difference is large is proposed. The proposed algorithm construct a new clock routing topology which is generalized graph model while previous methods uses only tree-structured routing topology. A new cost function is designed in order to select two nodes which constitute link-edge. Using this cost function, delay difference or clock skew is reduced by connecting two nodes whose delay difference is large and distance difference is short. Furthermore, routing topology construction and wire sizing algorithm is developed to reduce clock delay. The proposed algorithm is implemented in C programming language. From the experimental results, we can get the delay reduction under the given skew bound.