Related papers: On systems of complexity one in the primes

Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $\sigma$. From the…

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…

We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect…

Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$. We also show, as a corollary of the Green-Tao theorem, that there are arbitrarily long…

We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\}$ on a metric space $T$, with a parameter $\epsilon \downarrow 0$. We study the conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta \Big) =1…

Given a set X of finite strings, one interesting question to ask is whether there exists a member of X which is simple conditional to all other members of X. Conditional simplicity is measured by low conditional Kolmogorov complexity. We…

Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that the counting function of P is O(x^{1-c}) for some c>0, where c…

In this article, we prove the existence of global solutions to the inhomogeneous incompressible Navier--Stokes equations, whenever the initial velocity belongs to some subspace of $\mathrm{BMO}^{-1}$, and the initial density is sufficiently…

Let $A\subset \mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $\phi(x)=ax+b$ if \[|A \triangle \phi(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost…

We consider the exact sampling of the first passage of a stable subordinator across a non-increasing regular barrier. First, the sampling is reduced to one from a bivariate distribution parameterized by the index $\alpha$ of the…

We consider higher-dimensional versions of Kannan and Lipton's Orbit Problem---determining whether a target vector space V may be reached from a starting point x under repeated applications of a linear transformation A. Answering two…

We consider $n\times n$ real-valued matrices $A = (a_{ij})$ satisfying $a_{ii} \geq a_{i,i+1} \geq \dots \geq a_{in} \geq a_{i1} \geq \dots \geq a_{i,i-1}$ for $i = 1,\dots,n$. With such a matrix $A$ we associate a directed graph $G(A)$. We…

We perform a classification of third order integrable systems of evolution equations with respect to higher symmetries. Applying it, we consider polynomial systems that are 0-homogeneous under a suitable weighting of variables with main…

We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic solutions of such a problem with a normalization condition and set-theoretic constraints on their complex…

We present a method for finding large fixed-size primes of the form $X^2+c$. We study the density of primes on the sets $E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$, $c \in \mathbb{N}^*$. We describe an algorithm for generating…

We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only…

Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \mathbb{H} $ has a unique solution $ m $ with…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

This paper is devoted to the large time behavior of weak solutions to the three-dimensional Vlasov-Navier-Stokes system set on the half-space, with an external gravity force. This fluid-kinetic coupling arises in the modeling of…

This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly…