Related papers: On nonconvex constellations among primes I

Recently is has been proved that if $\sigma\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(\sigma_{ij})~(i\neq j,k\neq l)$ is a product of eight $E_n(R)$-conjugates of $\sigma$…

For a fixed exponent $0 < \theta \leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^\theta} \Lambda(n) \sim x^\theta$ as $x \to \infty$. From the recent zero density estimates of Guth and…

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f…

We analyse in this paper a conservative analogue of the celebrated model of dynamical percolation introduced by H\"aggstr\"om, Peres and Steif in [HPS97]. It is simply defined as follows: start with an initial percolation configuration…

As a refinement of the celebrated recent work of Yitang Zhang we show that any admissible k-tuple of integers contains at least two primes and almost primes in each component infinitely often if k is at least 181000. This implies that there…

We study first passage percolation on the plane for a family of invariant, ergodic measures on $\mathbb{Z}^2$. We prove that for all of these models the asymptotic shape is the $\ell$-$1$ ball and that there are exactly four infinite…

Observational investigations of the abundance of massive precursors of local galaxy clusters ("proto-clusters") allow us to test the growth of density perturbations, to constrain cosmological parameters that control it, to test the theory…

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincar\'e series converges at the critical exponent $\delta_\Gamma$. We…

Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of…

We study the evolution of the number density, as a function of the size, of passive early-type galaxies with a wide range of stellar masses 10^10<M*/Msun<10^11.5) from z~3 to z~1, exploiting the unique dataset available in the GOODS-South…

We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all natural number k a torsion group with growth function asymptotically…

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging omega-sequence or a non-trivial converging omega_1-sequence. We establish that this dichotomy holds in a variety of models;…

We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of…

We use the EAGLE hydrodynamical simulation to trace the quenching history of galaxies in its 10 most massive clusters. We use two criteria to identify moments when galaxies suffer significant changes in their star formation activity: {\it…

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…

We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f…

We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely…

Probabilistic models for the distribution of primes in the natural numbers are constructed in the article. The author found and proved the probabilistic estimates of the deviation $R(x)=|\pi(x)- Li(x)|$. The author has analyzed the…

We present the first results of the JWST Emission Line Survey (JELS). Utilising the first NIRCam narrow-band imaging at 4.7$\mu$m, over 63 arcmin$^{2}$ in the PRIMER/COSMOS field, we identified 609 emission line galaxy candidates. From…

We report on the study of a sample of 25 field early-type galaxies, in the redshift range z~0.1-0.5, selected on the basis of colours and morphology from the HST-MDS. Field early-type galaxies define a tight Fundamental Plane (FP) out to…