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17
votes
1answer
500 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} ...
12
votes
1answer
516 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 ...
4
votes
1answer
205 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 ...
11
votes
1answer
339 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 ...
5
votes
2answers
518 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 ...
2
votes
0answers
273 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 $C^\infty$
$p(0) = a$, $p(1) = ...
2
votes
5answers
511 views
Uncomputability of the identity relation on computable real numbers
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 ...
10
votes
3answers
944 views
Differentiability of computable functions
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 ...
