37 problems

Let $f$ be the $1$-periodic continuous function … A function is Hölder of exponent $1/2$ if there is a constant $C>0$ such that … for all $\theta_1,\theta_2$. Marmi-Moussa-Yoccoz c…

Computable-modulus conjecture. The function $f$ has a computable modulus of continuity.

Analytic-choice conjecture. The degree lies strictly above every $\lim^{(n)}$; hence $\mathrm{SCI}_{\mathrm{TTE}}=\infty$, and it may coincide with a known “analytic choice” princi…

Completeness conjecture. This index set is $\Sigma^1_1$-complete.

Negative-information noncomputability conjecture. Negative information does not suffice to prove computability, at least for the type of Whitney extension investigated in this pape…

Let $\mathsf{NON}$ be the problem discussed in the paper, and let $\mathsf{NON}^\omega$ and $\mathsf{NON}^\infty$ denote its countable and inverse-limit iterations, respectively. T…

Computable-measure non-effective-tightness conjecture. There exists a computable measure on an incomplete computable metric space that is not effectively tight.

Over ${\mathsf{PRA}^2}$, let $\mathsf{IntermediateValueTheorem}$ denote the intermediate value theorem and let $\Delta^0_1\text{-}\mathrm{CA}$ denote $Delta^0_1$ comprehension. Pri…

Decidability–clopenness conjecture. Every $\Lambda_{WP}$-decidable property of $\mathcal{G}_{WP}$ is clopen.

Main conjecture. Any $\Lambda_{WP}$-computable function defined on $\mathcal{G}_{WP}$, with image in any recursive metric space, is continuous.

Let $B_i$ be a computable Banach-space presentation, and let $\overline{B_i}$ denote its completion. The paper gives a first-order arithmetic formula $\psi$ such that … where…

Consider the formal language whose basic operations are the operations used in the proof of Corollary, together with multiplication as an additional operation. The proof is uniform…

Let $B_i$ be the computable Banach-space presentation whose completion is denoted by $\overline{B_i}$, and let $C[0,1]$ be the Banach space of continuous real-valued functions on…

Let $p$ be a right-c.e. real satisfying … An infinite-dimensional and separable $L^p$ space is a Lebesgue space with these properties, equipped with the relevant computable-present…

Let $\mathbf{f}$ be an $\alpha$-jump-computable function, and let a uniform procedure as in the uniform version of Lusin's theorem produce a continuous function $\mathbf{g}$ that a…

Computable-sequence conjecture. There exists a computable sequence of nonnegative real numbers having no computable metastable rate of asymptotic decreasingness.

Schnorr $K$-test conjecture. The appropriate test for Schnorr $K$-randomness is a lower semicomputable function $t$ such that the map

A decidable subshift is a subshift whose language is decidable, and its irreducibility rate measures its mixing strength. Consider decidable subshifts whose irreducibility rate is…

Assume that $\mathbb{X}$ and $\mathbb{Y}$ are computable metric spaces, that $\{\mu_a\}_{a\in\mathbb{N}^{\mathbb{N}}}$ is a uniformly computable family of probability measures on…

Let $E$ be a uniformly convex, smooth, computable Banach space of dimension two, and let $K\subseteq E$ be bounded, convex, and located, meaning that its distance function is compu…

Regular approximation conjecture. There exists a sequence $(\rho_n)_{n=1}^{\infty}$ such that $\rho_n\geq0$, $\rho_n\xrightarrow[n\to\infty]{}\rho$ in WOT, and

A real $x\in[0,1]^n$ is computably random if it avoids every effectively presented sequence of open sets forming a Martin-Löf test with the stronger computable-martingale randomnes…

Let $n$ and $m$ be the parameters for the corresponding Borel measurability levels, and let $\mathbf{X}$ and $\mathbf{Y}$ be Polish represented spaces. Write $\mathfrak{b}$ for the…

Let $n$ and $m$ be the parameters for the corresponding Borel measurability levels, and let $\boldsymbol{X}$ and $\boldsymbol{Y}$ be Polish represented spaces. Write $\mathfrak{b}$…

Brattka–Miller's conjecture. Dimension should be the crucial demarcation line for spaces with only Turing degrees.