Mathematical Sciences: Dynamical Systems and Complexity

这个项目的主要目的是增进对动力系统的理解，研究动力系统与拓扑和数值分析中的算法的关系，并研究这些算法的全局几何性质及其计算过程。主要研究人员提出了实数上的计算理论和计算模型。这一理论包括递归函数理论、通用机和NP完备性。例如，在分析多项式求根的迭代过程和线性规划的内点方法时，会出现退化不动点。线性规划问题的爆破技术涉及n球面的动力学问题。对这个问题的理解对于线性规划的离散型和投影重定标线性规划算法的最优解的渐近吸引率具有重要意义。

The main objectives of this project are to enhance the understanding of dynamical systems, to study the relation of these dynamical systems to topology and to algorithms in numerical analysis, and to study the global geometric properties of these algorithms and their computational process. The principal investigator proposes a theory of computation and a model for computation over the real numbers. This theory includes recursive function theory, universal machines, and NP completeness. For example, degenerate fixed points occur in the analysis of the iterative processes for polynomial root finding and interior point methods for linear programming. Blowing up techniques for the linear programming problem involve the dynamics of the n-sphere. An understanding of this problem has implications for the asymptotic rate of attraction to the optima for discrete versions of the linear and the projective rescaling algorithms of linear programming.