Related papers: Wadge degrees of $\Delta^0_2$ omega-powers

In this note we show that a set is provably $\Delta^0_2$ in the fragment $I\Sigma_n$ of arithmetic iff it is $I\Sigma_n$-provably in the class $D_\alpha$ of $\alpha$-r.e. sets in the Ershov hierarchy for an $\alpha <_{\epsilon_0}…

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $\Omega$ minimizes the functional \[ \mathcal…

The "similarity" degree of a unital operator algebra $A$ was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements:…

For a region $\Omega \subset\mathbb{C}$ denote by $\Psi(Q;\Omega)$ the number of complex algebraic numbers in $\Omega$ of degree $\leq n$ and naive height $\leq Q$. We show that $$…

For cyclic totally real number fields $K$ with odd prime degree $n$, odd class number, $2$ inert, and the property that every totally positive unit is a square, the density of rational primes $p$ that satisfy the spin relation…

Let $\Sigma = \mathbb B^n/\Gamma$ be a complex hyperbolic space with discrete subgroup $\Gamma$ of the automorphism group of the unit ball $\mathbb B^n$ and $\Omega $ be a quotient of $\mathbb B^n \times\mathbb B^n$ under the diagonal…

Recently data trees and data words have received considerable amount of attention in connection with XML reasoning and system verification. These are trees or words that, in addition to labels from a finite alphabet, carry data values from…

We study varieties of certain ordered $\Sigma$-algebras with restricted completeness and continuity properties. We give a general characterization of their free algebras in terms of submonads of the monad of $\Sigma$-coterms. Varieties of…

We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3,…

We study the state complexity of regular operations in the class of ideal languages. A language L over an alphabet Sigma is a right (left) ideal if it satisfies L = L Sigma* (L = Sigma* L). It is a two-sided ideal if L = Sigma* L Sigma *,…

We give an explicit construction of all complex continuous irreducible characters of the group ${\rm SL}_1(D)$, where $D$ is a division algebra of prime degree $\ell$ over a local field of odd residual characteristic different than $\ell$.…

Let $\mb w$ be a morphic word over a finite alphabet $\Sigma$, and let $\Delta$ be a nonempty subset of $\Sigma$. We study the behavior of maximal blocks consisting only of letters from $\Delta$ in $\mb w$, and prove the following: let…

Let R be a two-dimensional regular local ring with maximal ideal \mathfrak m, and let \wp be a simple complete \mathfrak m-primary ideal which is residually rational. Let R_0:= R\subsetneqq ...\subsetneqq R_r be the quadratic sequence…

We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=\delta n+(\delta+2).$$ We prove the similar result for $\varepsilon=\delta-2$ and…

The determinacy of lightface $\Delta^1_{2n+2}$ and boldface $\boldsymbol{\Pi}^1_{2n+1}$ sets implies the existence of an $(\omega, \omega_1)$-iterable $M_{2n+1}^{\#}$.

Commutative properties in formal languages pose problems at the frontier of computer science, computational linguistics and computational group theory. A prominent problem of this kind is the position of the language $O_n$, the language…

A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…

Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of \omega_1 can be made \Delta^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a \Delta^1_3…

Let omega(n) be the number of distinct prime factors dividing n and m > n natural numbers. We calculate a formula showing which prime numbers in which intervals divide a given binomial coefficient. From this formula we get an identity…

Classifying formal languages according to the expressiveness of grammars able to generate them is a fundamental problem in computational linguistics and, therefore, in the theory of computation. Furthermore, such kind of analysis can give…