• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.427-436
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• 2005

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals $m=P_0\;{\supset}\;P_1\;{\supset}\;{\cdotS}\;{\supset}\;P_t=P$ and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by $n_v$. In this paper, if the m-adic order of P is 2, we show that $O(P_i)\;=\;1\;for\;1\;{\leq}\;i\; {\leq}\;{\lceil}\;{\frac{b+1}{2}}{\rceil}\;and\;O(P_i)\;=2\;for\;{\lceil}\;\frac{b+3}{2}\rceil\;{\leq}\;i\;\leq\;t,\;where\;b=n_v$. We also show that $n_w\;=\;n_v$ when w is the prime divisor associated to a simple v-ideal $Q\;{\supset}\;P$ of order 2 and that w(R) = v(R) as well.

• Journal of the Institute of Convergence Signal Processing
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• v.12 no.4
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• pp.341-347
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• 2011

This paper presents the performance evaluation. optimal design. and complex obstacle detection of an overlapped ultrasonic sensor ring by introducing a new concept of effective beam width. It is assumed that a set of ultrasonic sensors of the same type are arranged along a circle of nonzero radius at regular spacings with their beams overlapped. First, the global positional uncertainty of an overlapped ultrasonic sensor ring is expressed by the average value of local positional uncertainty over the entire obstacle detection range. The effective beam width of an overlapped ultrasonic sensor ring is assessed as the beam width of a single ultrasonic sensor having the same amount of global positional uncertainty, from which a normalized obstacle detection performance index is defined. Second. using the defined index, the design parameters of an overlapped ultrasonic sensor ring are optimized for minimal positional uncertainty in obstacle detection. For a given number of ultrasonic sensors, the optimal radius of an overlapped ultrasonic sensor ring is determined, and for a given radius of an overlapped ultrasonic sensor ring, the optimal number of ultrasonic sensors is determined. Third, the decision rules of positional uncertainty zone for multiple obstacle detection are provided based on the inequality relationships among obstacle distances by three adjacent ultrasonic sensors. Using the provided rules, the obstacle outline detection is performed in a rather complex environment consisting of several obstacles of different shapes.