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The density-based clustering algorithm presented is different from the classical Density-Based Spatial Clustering of Applications with Noise (DBSCAN) (Ester et al., 1996), and has the following advantages: first, Greedy algorithm substitutes for R(*)-tree (Bechmann et al., 1990) in DBSCAN to index the clustering space so that the clustering time cost is decreased to great extent and I/O memory load is reduced as well; second, the merging condition to approach to arbitrary-shaped clusters is designed carefully so that a single threshold can distinguish correctly all clusters in a large spatial dataset though some density-skewed clusters live in it. Finally, authors investigate a robotic navigation and test two artificial datasets by the proposed algorithm to verify its effectiveness and efficiency.

Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n n . For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .