Given a point s and a set of h pairwise disjoint polygonal obstacles with a total of n vertices in the plane, after the free space is triangulated, we present an O(n + h log h) time and O(n) space algorithm for building a data structure (called shortest path map) of size O(n) such that for any query point t, the length of the L1 shortest obstacle-avoiding path from s to t can be reported in O(l...

Under natural conditions, we prove exponential ergodicity in the \( L_1\)-Wasserstein distance of two-type continuous-state branching processes Lévy random environments with immigration. Furthermore, express precisely parameters exponent. The coupling method and conditioned property play an important role approach. Using tool superprocesses, total variation is also proved.

‎We give a simple and independent axiomatization of reticulations on residuated lattices‎, ‎which were axiomatized by five conditions in [C‎. ‎Mureşan‎, ‎The reticulation of a residuated lattice‎, ‎Bull‎. ‎Math‎. ‎Soc‎. ‎Sci‎. ‎Math‎. ‎Roumanie‎ ‎51 (2008)‎, ‎no‎. ‎1‎, ‎47--65]‎. ‎Moreover‎, ‎we show that reticulations can be considered as lattice homomorphisms between residuated lattices and b...