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We study prefix-free presentations of computably enumerable reals. In 2, Calude et. al. proved that a real a is c.e. if and only if there is an infinite, computably enumerable prefix-free set V such that $$ \alpha = \sum _{\sigma \in V} 2^{ - \left| \sigma \right|} $$ . Following Downey and LaForte 5, we call V a prefixfree presentation of a. Each computably enumerable real has a computable presentation...
Let Csc and Cwc be classes of the semi-computable and weakly computable real numbers, respectively, which are discussed by Weihrauch and Zheng 12. In this paper we show that both Csc and Cwc are not closed under the total computable real functions of finite length on some closed interval, although such functions map always a semi-computable real numbers...
We study Turing computability of the nonlinear solution operator S of the Cauchy problem for the Schrödinger equation of the form $$ i\frac{{du}} {{dt}} = - \frac{{d^2 u}} {{dx^2 }} + mu + \left| u \right|^2 u $$ in ℝ.We prove that S is a computable operator from H1(ℝ) to C(ℝH1(ℝ))with respect to the canonical representations.
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