多邊形的剖分圖形數量之探討
從參考資料[1]可知,將凸n+2邊形利用n-1條不相交的對角線剖分成n個三角形的圖形數量即為卡特蘭數Cn。而我利用不相交的對角線把n+2邊形剖分成數個多邊形和三角形的組合,並從此類的剖分圖形與三角剖分圖形之關聯,進而由卡特蘭數的一般式推導出此類剖分圖形數量的一般式。在本研究中可得,若到把n+2邊形剖分成一個k+2邊形和多個三角形的圖形數量是(2n-k+1 n+1) ;把n+2邊形剖分成一個k+2邊形、一個m+2邊形和多個三角形的圖形數量,當m≠k,數量為n+2/2(2n-k-m+2 n+2) ,當m=k時,數量為n+2/2(2n-2k+2 n+2) ;把n+2邊形剖分成一個k1+2邊形、一個k2+2邊形、一個k3+2邊形、和n-k1-k2-k3 個三角形的剖分圖形,當k1,k2,k3兩兩相異時,數量為(n+2)(n+3)(2n-k1-k2-k3+3 n+3) ;把n+2邊形剖分成一個K1+2邊形、一個K2+2邊形、一個K3+2邊形、一個K4+2邊形和n-K1-k2-k3-k4個三角形的剖分圖形當k1,k2,k3,k4兩兩相異,數量為(n+2)(n+3)(n+4)(2n-k1-k2-k3-k4+4 n=4)。並猜測若k1,k2,...,ki兩兩相異時,把n+2邊形剖分成一個k1+2邊形、一個k2+2邊形、…、一個ki+2邊形、和n-Σkj 個三角形的剖分圖形數量為(n+i)!/(n+1)!(2n-Σkj+i n+i) 。
The Use of Brine Shrimp to Test for Water Pollutants
The use of brine shrimp nauplii to test for the overall toxicity of sediment samples is proposed. Brine shrimp nauplii were cultured with different concentrations of heavy metals, including chromium (III), copper (II), nickel, lead and zinc, and organic pollutants, including triclosan, oxybenzone, octinoxate and bisphenol A. The brine shrimp nauplii were observed under a dissection microscope to determine the death rate. Results showed that brine shrimp nauplii are more sensitive to copper, cadmium, bisphenol A and oxybenzone. The LC50 (24h) are 55.5, 24.9, 5.6 and 2.7 ppm respectively. Zinc is likely to have synergistic toxic effect with nickel or lead. The synergistic toxic effects of other heavy metals and organic pollutants should be confirmed with further investigations. Brine shrimp nauplii were treated with extracts from sediment samples collected from the oyster culture zone of the Deep Bay, namely Pak Nei, Sha Kiu Tsuen and Hang Hau Tsuen. The sediment samples were extracted with neutral sodium acetate to dissolve the exchangeable heavy metal ions and some organic pollutants. The death rate of brine shrimp nauplii treated with the sediment extract of Hang Hau Tsuen was similar to 1 ppm PBA. It was also about 10 to 20% higher than that of the other two sites (Pak Nei and Sha Kiu Tsuen). Since Hang Hau Tsuen is closer to the residential area and Lau Fau Shan Seafood Market than the other two sites, its sediment sample is likely to have a higher level of environmental pollutants. The results suggest that brine shrimp nauplii may be used as a biomarker to monitor the environmental changes in the overall level of pollutants in sediment samples.