Related papers: A Computably Enumerable $tt$-Degree Without Comput…

It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial $\Pi^0_1$ subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise…

In this article, we study an analogue of $tt$-reducibility for points in computable metric spaces. We characterize the notion of the metric $tt$-degree in the context of first-level Borel isomorphism. Then, we study this concept from the…

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of…

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…

We prove that there exist Salem numbers with trace $-3$ and every even degree $\geq 34$. Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it…

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\E$, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the…

Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive curvature. Assume that M has no locally Euclidean factors and no factors locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any continuous map…

We are studying the degrees in which a computable structure is relatively computably categoricity, i.e., computably categorcial among all non-computable copies of the structure. Unlike the degrees of computable categoricity we can bound the…

We construct a nonrecursive set \(A\le_T\emptyset'\) and a uniformly computable family of sets \(C_0,C_1,\dots\), all bounded finite-one equivalent to \(A\), such that the corresponding \(1\)-degrees form a copy of the dense linear order…

A result of the second-named author states that there are only finitely many CM-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke's Equidistribution Theorem and is hence non-effective. In…

We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the…

We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment…

The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to study some extensions of $q$-groups by nilpotent groups and the power of these groups in a certain computational model. In its simplest formulation, it…

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree…

We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with…

A computable structure $\mathcal{A}$ has degree of categoricity $\mathbf{d}$ if $\mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $\mathcal{A}$. Fokina, Kalimullin, and Miller…

Let $\mathfrak g$ be a reductive Lie algebra, and $m$ a positive integer. There is a natural density of irreducible representations of $\mathfrak g$, whose degrees are not divisible by $m$. For $\mathfrak g=\mathfrak{gl}_n$, this density…

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras.…

A set $A$ is coarsely computable with density $r \in [0,1]$ if there is an algorithm for deciding membership in $A$ which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least $r$. To any Turing…

Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…