Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
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数学、物理和精确实数计算领域
作者：
A. Edalat
本文综述了连续Domain在数学中为经典空间提供简单计算模型的应用，包括真实的直线、可数基局部紧空间、完备可分度量空间、可分Banach空间和概率分布空间。它显示了这些模型如何有一个逻辑和有效的介绍，以及它们如何被用来在数学和物理的几个领域给出一个计算框架。这些包括分形几何，其中新的结果的存在性和唯一性的吸引力和不变的分布已经获得，措施和整合理论，其中推广的黎曼理论的整合已经制定，和真实的算术，其中一个可行的设置，精确的计算机算术已经制定。我们给出了一些算法的理论计算的迭代函数系统与应用在统计物理和周期加倍路线的混乱，我们还展示了如何有效的算法已获得计算初等函数的精确真实的算术。
Abstract We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic.