Lí Lian

This paper describes a theoretical framework of ecological phase transitions for modeling tree-grass dynamics and analyzing the shifts or phase transitions from one vegetation structure to another in the southern Texas landscape. This framework implements the integration of percolation theory, fractal geometry and phase transition theory as a method for modeling the spatial patterns of tree-grass dynamics, and nonlinear Markov non-equilibrium thermodynamic stability theory as a method for characterizing temporal tree-grass dynamics and phase transition. An historical sequence of aerial photographs at a Prosopis - thornscrub savanna parkland site in southern Texas was used to determine the parameters of the models. The preliminary analytical result accords well with current understanding and field survey of vegetation dynamics in the southern Texas landscape. The potential of such approaches and other relevant theories such as self-organized criticality and synergetics to vegetation dynamics is also discussed.

By PFM, we mean a finitely generated module over a principal ideal domain; a linear sentence is a sentence that contains no disjunctive and negative symbols. In this paper, we present an algorithm which decides the truth for linear sentences on a given PFM, and we discuss its time complexity. In particular, when the principal ideal domain is the ring of integers or a univariate polynomial ring over the field of rationals, the algorithm is polynomial-time. Finally, we consider some applications to Abelian groups