# Questions tagged [computable-analysis]

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Zheng and Weihrauch (http://www-sst.informatik.tu-cottbus.de/~wwwti/zheng/publications/1999/mfcs99.pdf) define a real number $x$ to be $\Sigma_n$ if and only if there is a computable function $f:\... 1answer 174 views ### Is the following product-like space a Polish space? Let$\mathcal{M}_1(\mathbb R)$denote the space of Borel probability measures on$\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-... 1answer 330 views ### floating point representation via the perspective of TTE/computable analysis Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ... 1answer 1k views ### Is this property equivalent to Lusin's property (N) for continuous functions? A function$F:[0,1]\rightarrow\mathbb{R}$satisfies Lusin's (N) property if for every measure zero set$A\subseteq [0,1]$,$F(A)$has measure zero. (This includes the assertion that$F(A)$is ... 1answer 751 views ### An NP-hard$n$fold integral We are given rational numbers$[c_1, c_2, \ldots, c_n]$and$v$from the interval$[0,1]$. Consider the$n$-fold integral$$J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} d\... 1answer 921 views ### Can nonstandard analysis be used to prove results in constructive or computable analysis? Nonstandard analysis is a useful tool which can be used to prove a number of results in analysis. Question Can it also be used to prove results in computable or constructive analysis? If so, what are ... 1answer 295 views ### tennenbaum phenomena for the reals? Let$\mathfrak{M} = \langle R, +,\times,> \rangle$be such that$R$is the set of real numbers and$\mathfrak{M} \models RA^1$(the first-order axioms for the reals). Do we have characterisations ... 1answer 401 views ### The complexity of the leading fractional bit of a power of a rational number On a mailing list (math-fun) that I subscribe to Dan Asimov asked what's the most efficient way to calculate the leading decimal digits (say 10 of them) of$(p/q)^n \bmod 1$where$p$and$q$are ... 2answers 901 views ### Simple example of a sequence without computable modulus of convergence Can anyone give a simple example of a sequence that converges, but there's no computable function that gives$N$as a function of$\epsilon$, i.e., the modulus of convergence is not computable? In ... 0answers 307 views ### Computable distribution on [0,1] with C-infinity distribution function Does anyone know of an easily-describable distribution on$[0,1]$with a density$p$(with respect to Lebesgue measure) that satisfies the following properties:$p$is$C^\inftyp(0) = a$,$p(1) = ...
Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows: (1) if $x = y$ then $f_{=}(x,y) = 1$; (2) $f_{x,y} = 0$ otherwise. I would like to have a proof for / a reference to a textbook ...
Call a computable function a total function $\mathbb{R} \to \mathbb{R}$, for which there exists a Turing machine outputting arbitrary close approximation to $f(x)$ given arbitrary close approximation ...