• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1333-1354
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• 2019

In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

• Journal of Korean Society of Forest Science
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• v.90 no.4
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• pp.535-542
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• 2001

The objective of this study is to develop a spatial analysis modeling technique to evaluate the functional suitability of forest lands for land slide prevention. The functional suitability is classified into 3 categories of high, medium and low according to the potential of land slide on forest lands. The potential of land slide hazards is estimated using the measurements of 7 major site factors : slope, bed rock, soil depth, shape of slope, forest type and D.B.H. class of trees. The analytic hierarchical process is applied to determining the relative weight of site factors in estimating the potential of land slides. The spatial analysis modeling starts building base layers for the 7 major site factors by $25m{\times}25m$ grid analysis or TIN analysis, reclassifies them and produces new layers containing standardized attribute values, needed in estimating land slide potential. To these attributes, applied is the weight for the corresponding site factor to build the suitability classification map by map algebra analysis. Then, finally, cell-grouping operations convert the suitability classification map to the land unit function map. The whole procedures of the spatial analysis modeling are presented in this paper.