Questions tagged [computable-analysis]

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2
votes
1answer
118 views

LLPO as constructivity/computability for dense subsets

LLPO is the statement $\forall x \in \mathbb R. x \leq 0 \vee x \geq 0.$ The statement should be understood as a fragment of the Law of Excluded Middle, rather than a statement about the ordering of ...
16
votes
1answer
866 views

Is the one-point compactification of $\mathbb{N}$ computably countable?

The one-point compactification $\mathbb{N}_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of ...
6
votes
1answer
345 views

Are representations in computable analysis the equivalent to countably-generated condensed sets?

This is the first in a pair of questions. For the other see here. Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, ...
6
votes
1answer
278 views

Computing the complex roots of a monic polynomial

The map from monic complex polynomials to the unordered tuples of their roots (each appearing according to its multiplicity) is computable. This seems to have been known for a long time, and with ...
9
votes
0answers
222 views

Coding third-order objects via second-order ones

As is well-known, the language of second-order arithmetic only has variables for natural numbers and sets of natural numbers. Higher-order objects, like functions on $\mathbb{R}$, have to be ...
2
votes
0answers
43 views

Classification and computation of the entanglements of pairs of planar curves

Let $C$ is the set of continuous $f:[0;1]\to \mathbb R^2$ with $\|f\|=\max_t\|f(t)\|$. For $f,g\in C$ let $(f,g)\in E$ iff $\{f(0),f(1)\}\cap {\rm Range}(g)=\emptyset$ and $\{g(0),g(1)\}\cap {\rm ...
3
votes
1answer
644 views

Is the Hilbert space-filling curve bijective over computable numbers?

The Hilbert curve is a continuous space-filling curve that maps $I$ to $I^n$ where $I$ denotes the unit interval [1]. Like all other space-filling curves, it is not one-to-one. I am wondering if the ...
1
vote
1answer
142 views

Partial computability results on integrals over open intervals

It's well known in Type 2 Effectivity that integration over a compact interval is computable. So what about integration over an open interval? What rigorous computability results exist? My thoughts ...
20
votes
2answers
2k views

Is it possible to constructively prove that every quaternion has a square root?

Is it possible to constructively prove that every $q \in \mathbb H$ has some $r$ such that $r^2 = q$? The difficulty here is that $q$ might be a negative scalar, in which case there might be "too many"...
1
vote
0answers
85 views

Buridan's principle in computable analysis

In (Lamport, 2012), Lamport proposes the principle A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time. I think it could be ...
12
votes
1answer
289 views

Is there a computable homeomorphism between two different Cartesian powers of the computable real numbers?

It's well know that it is surprisingly difficult to prove that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic for $n\neq m$. Commonly proofs go through Brouwer's fixed point theorem, which is '...
2
votes
0answers
179 views

Banach spaces: A ball being a subset of the interior of the union of two balls

Let $X$ be a separable reflexive Banach space and let $A$, $B_1$, and $B_2$ be three closed balls in $X$. Is there a `handy' necessary and sufficient condition for checking whether $A \subseteq (B_1 \...
4
votes
1answer
175 views

Are these two definitions of arithmetical hierarchy of real numbers equivalent?

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:\...
4
votes
1answer
185 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-...
3
votes
1answer
359 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 ...
13
votes
2answers
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 ...
17
votes
1answer
782 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\...
12
votes
1answer
1k 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 ...
6
votes
2answers
374 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
405 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
952 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
311 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) = ...
5
votes
5answers
1k 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 ...
11
votes
3answers
1k 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 ...