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

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that…

We consider the scalar second order ODE u + |u | $\alpha$ u + |u| $\beta$ u = 0, where $\alpha$, $\beta$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is…

We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the…

A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…

The regular languages with a neutral letter expressible in first-order logic with one alternation are characterized. Specifically, it is shown that if an arbitrary $\Sigma_2$ formula defines a regular language with a neutral letter, then…

Fix a natural $\alpha$. Let $n\ge \alpha$ be an integer. Consider the symmetric group $S_{\alpha+n}$ and its subgroup $S_n$. We consider the group algebra of $S_{\alpha+n}$ and its subalgebra $\mathbb{O}[\alpha;n]$ consisting of…

This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\times n$ binary matrix is at least $c\,2^n/n$ for $c=1/201$. That same quantity naturally lower bounds the number of distinct integers $D_n$ which…

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e.,…

Let $S$ be a complete star-omega semiring and $\Sigma$ be an alphabet. For a weighted $\omega$-restricted one-counter automaton $\mathcal{C}$ with set of states $\{1, \dots, n\}$, $n \geq 1$, we show that there exists a mixed algebraic…

The problem considered in this paper is whether an inequality of omega-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half)…

The class of omega-regular languages provides a robust specification language in verification. Every omega-regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens…

Two results on the completeness of maximal solutions to first and second order ordinary differential equations (or inclusions) over complete Riemannian manifolds, with possibly time-dependent metrics, are obtained. Applications to…

For two derived equivalent $k$-algebras $\bar\Lambda$ and $\bar\Gamma$, we introduce a correspondence between $\OO$-orders reducing to $\bar\Lambda$ and $\OO$-orders reducing to $\bar\Gamma$. We outline how this may be used to transfer…

In this short note, we provide OPEs for several affine W-algebras associated with Lie algebras of rank two and give some direct applications.

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…

Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element…

For an algebraic number $\alpha$ of degree $n$, let $\mathcal{M}_{\alpha}$ be the $\mathbb{Z}$-module generated by $1,\alpha ,\ldots ,\alpha^{n-1}$; then $\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\,…

This paper deals more generally with @-numbers defined as follows: Call `\textit{alpha number}' of order $(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2$, (denote its family by @$_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;…

By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…

It is usually argued that the Standard Model predicts slow D^0-\bar D^0 oscillations with \Delta M_D, \Delta \Gamma_D < 10^{-3}*\Gamma_D$ and that New Physics can reveal itself through \Delta M_D exceeding 10^{-3}*\Gamma_D. It is believed…