Related papers: On nonconvex constellations among primes I

We prove that for every nonnegative integer $m$ there exists an $\varepsilon>0$ such that if $\lambda\in (0,\varepsilon]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$ for which the interval…

In 2017, Geordie Williamson proved the existence of counterexamples to James's conjecture on the decomposition matrices of symmetric groups and their Hecke algebras. The smallest counterexample detectable by Williamson's method occurs in…

We investigate a spectroscopic sample of 48 early-type (E+S0) galaxies in the rich cluster Abell 2390 (z=0.23) and 48 E+S0 galaxies from a previously published survey of Abell 2218 (z=0.18). The A2390 spectra are based on Multi Object…

The world of primes has many gaps between evidence and theorems. Here, we review Legendre's conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming…

In this paper we show that a certain subset of the Carmichael numbers contains good Euler pseudoprimes, composite numbers that for many bases survive the Solovay-Strassen primality test. We present a classification of Carmichael numbers,…

A minimal counterexample to the Erd\H{o}s-Gy\'arf\'as conjecture is a graph of minimum possible order and size with minimum degree at least 3 that contains no cycle whose length is a power of 2. Markstr\"om observed that any such graph must…

It is natural to ask, given a permutation with no three-term ascending subsequence, at what index the first ascent occurs. We shall show, using both a recursion and a bijection, that the number of 123-avoiding permutations at which the…

Projected axis ratio measurements of 880 early-type galaxies at redshifts 1<z<2.5 selected from CANDELS are used to reconstruct and model their intrinsic shapes. The sample is selected on the basis of multiple rest-frame colors to reflect…

In this paper we present a statistical study of the environments of 49 galaxies in which there is gas- or stellar- counterrotation. The number of possible companions in the field (to apparent magnitude 22), their size and concentration were…

We present new results on the Fundamental Plane of galaxies in two rich clusters, Cl1358+62 at $z=0.33$ and MS2053--04 at $z=0.58$, based on Keck and HST observations. Our new data triple the sample of galaxies with measured Fundamental…

Motivated by the search for a counterexample to the Poincar\'e conjecture in three and four dimensions, the Andrews-Curtis conjecture was proposed in 1965. It is now generally suspected that the Andrews-Curtis conjecture is false, but small…

The hot ($>10^6$ K) phase of the circumgalactic medium (CGM) contains a large fraction of baryons in galaxies. It also retains signatures of the processes that shaped the galaxies, such as feedback from active galactic nuclei (AGNs) and…

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such…

In the first part of the paper, we study the inversion statistic of random permutations under the family $(\mathbb{P}_\theta^{(n)})_{\theta \ge 0}$ of Ewens sampling distributions on $S_n$. We obtain a rather simple exact formula for the…

Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is…

Blind spectroscopy of massive lensing galaxy clusters with MUSE has revealed large numbers of gravitationally-lensed Lyman-$ \alpha $ emitters exhibiting asymmetric profiles at $ 2.9 \leq z \leq 6.7 $, suggesting abundant outflows from…

We completely classify the asymptotic behavior of the number of alternating sign matrices classically avoiding a single permutation pattern, in the sense of [Johansson and Linusson 2007]. In particular, we give a uniform proof of an…

A fundamental feature of galaxies is their structure, yet we are just now understanding the evolution of structural properties in quantitative ways. As such, we explore the quantitative non-parametric structural evolution of 16,778 galaxies…

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ and $\text{dist}(…

We present a comprehensive study of the star formation histories of massive-quenched galaxies at $z=3$ in 3 semi-analytic models (SHARK, GAEA, GALFORM) and 3 cosmological hydrodynamical simulations (EAGLE, IllustrisTNG, Simba). We study the…