Related papers: Ulam numbers have zero density

Let (M, g) be an asymptotically flat static vacuum initial data set with non-empty compact boundary. We prove that (M, g) is isometric to a spacelike slice of a Schwarzschild spacetime under the mere assumption that the boundary of (M, g)…

Equilibrium particle densities near a hard wall are studied for a quantum fluid made of point charges which interact via Coulomb potential without any regularization. In the framework of the grand-canonical ensemble, we use an equivalence…

In this note, we study the set $\mathcal{D}$ of values of the quadruplet $(\underline{\mathrm{d}}(A),\overline{\mathrm{d}}(A),\underline{\mathrm{d}}(2A),\overline{\mathrm{d}}(2A))$ where $A\subset\mathbb{N}$ and…

Let X_{d,n} be an n-element subset of {0,1}^d chosen uniformly at random, and denote by P_{d,n} := conv X_{d,n} its convex hull. Let D_{d,n} be the density of the graph of P_{d,n} (i.e., the number of one-dimensional faces of P_{d,n}…

Consider a random matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ ($m \geq n$) containing independent complex Gaussian entries with zero mean and unit variance, and let $0<\lambda_1\leq \lambda_{2}\leq ...\leq \lambda_n<\infty$ denote the…

We study de Rham prismatic crystals on $(\mathcal{O}_K)_{\bbold{\Delta}}$. We show that a de Rham crystal is controlled by a sequence of matrices $\{A_{m,1}\}_{m \geq 0}$ with $A_{0,1}$ "nilpotent". Using this, we prove that the natural…

For each integer $m\ge0$ we study the $m$-ideal energy \[ E_m[\gamma]:=\frac12\int_\gamma k_{s^m}^2\,ds \] on closed immersed planar curves, where $k$ is signed curvature and $s$ is arclength; $k^2_{s^m} := (k_{s^m})^2$. The $m$-ideal…

In this article, we prove some subsets of the set of natural numbers $\mathbb{N}$ and any non-zero ideals of an order of imaginary quadratic fields are fractionally dense in $\mathbb{R}_{>0}$ and $\mathbb{C}$ respectively.

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so…

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p…

The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq…

We obtain a necessary and sufficient condition in order that a semi-plane of the form $\Re(s)>r$, $r\in \mathbb{R}$, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalization of a well-known…

Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof…

We compute the one-level density of zeros of order $\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting…

In this paper we give a new proof of the fact that for all pairs of positive integers (d, m) with d/m < 117/37, the linear system of plane curves of degree d with ten general base points of multiplicity m is empty.

We prove that in a large collection of naturally defined sets of permutations of fixed length, the numbers of permutations at Ulam distance k from the identity form a log-concave sequence in k.

Motzkin posed the problem of finding the maximal density $\mu(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In…

We report a new detection of neutral deuterium in the sub Damped Lyman Alpha system with low metallicity [O/H]\,=\,$-2.042 \pm 0.005$ at $z_{\rm abs}=2.437$ towards QSO~J\,1444$+$2919. The hydrogen column density in this system is…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets $P_{M/K}(\sigma)$, with $M/K$ Galois and $\sigma \in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary small)…