几类线性和非线性微分方程的数学可计算性问题起源于希尔伯特第十问题
Mathematical computability questions for some classes of linear and non-linear differential equations originated from Hilbert's tenth problem
摘要：
受量子力学的启发，我们将整数算术域中的希尔伯特第十问题转化为一组无限耦合的非线性微分方程或一类线性薛定谔方程的问题，并给出了相应的含时哈密顿量.然后，我们提出了这两类微分方程在可计算分析的某些计算模型中是否可计算的问题。这些都是重要的问题，因为：（i）与经典递归论不同，并非所有可计算分析的计算模型都是等价的;（ii）并非所有模型都必然地将实函数的可计算性降低到图灵机上的离散计算。无论我们的可计算性问题的肯定答案多么不可能，它们的存在都应该得到特别的关注和令人满意的解决，因为这样的肯定答案也可能在经典的递归论中为丘奇-图灵论题带来有趣的逻辑推论。
英文摘要：
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schrödinger equations with some appropriate time-dependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are non-trivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the Church-Turing thesis.
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