Related papers: On systems of complexity one in the primes

In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we…

Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O(\epsilon^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an…

We consider a process $\{X_t\}_{0\leq t\leq 1}$ in a fixed Wiener chaos $\mathcal{H}_n$. We establish some non-degenerate properties and related results for $\{X_t\}_{0\leq t\leq 1}$. As an application, we show that solution to SDE driven…

This work is a probabilistic study of the 'primes' of the Cram\'er model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\…

Hilberdink showed that there exists a constant $c_0>2$, such that there exists a continuous prim system satisfying $N(x)=c(x-1)+1$ if and only if $c\leq c_0$. Here we determine $c_0$ numerically to be $1.25479\cdot 10^{19}\pm2\cdot…

We consider a translation and dilation invariant system consisting of k diagonal equations of degrees 1,2,...,k with integer coefficients in s variables, where s is sufficiently large in terms of k. We show via the Hardy-Littlewood circle…

Systems of random linear equations may or may not have solutions with all components being non-negative. The question is, e.g., of relevance when the unknowns are concentrations or population sizes. In the present paper we show that if such…

This paper focuses on using the first curvature $\kappa(t)$ of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems [Y. Wang, H. Sun, Y. Song et al.,…

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal…

We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general…

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

We show that if a big set of integer points in [0,N]^d, d>1, occupies few residue classes mod p for many primes p, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of…

Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…

Str\"ombergsson and Venkatesh proved that a system of homogeneous linear congruence modulo prime has a positive probability to have a short non-trivial solution. We extend this result and show that the same holds for square-free moduli. In…

In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…