In this paper, we study two types of the reducing subspaces for the weighted composition operator $W: frightarrow ucdot fcirc varphi$ on $L^2(Sigma)$. A necessary and sufficient condition is given for $W$ to possess the reducing subspaces of the form $L^2(Sigma_B)$ where $Bin Sigma_{sigma(u)}$. Moreover, we pose some necessary and some sufficient conditions under which the subspaces of the form...

Beineke and Robertson independently characterized line graphs in terms of nine forbidden induced subgraphs. In 1994, S8 olte s gave another characterization, which reduces the number of forbidden induced subgraphs to seven, with only five exceptional cases. A graph is said to be a dumbbell if it consists of two complete graphs sharing exactly one common edge. In this paper, we show that a graph...

Leaf area information is required in various horticultural and physiological studies and it will be more useful if done via non-destructive methods. The objective of this study was to establish equations to estimate leaf area (LA) using length (L), width (W), fresh weight (FW), dry weight (DW), length × length (L2), width × width (W2), length × width (L×W), length + width (L+W), fresh weight × ...