Zobrazeno 1 - 10
of 96
pro vyhledávání: '"Sahasrabudhe, Julian"'
Autor:
Balister, Paul, Bollobás, Béla, Campos, Marcelo, Griffiths, Simon, Hurley, Eoin, Morris, Robert, Sahasrabudhe, Julian, Tiba, Marius
The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$, that $$R_
Externí odkaz:
http://arxiv.org/abs/2410.17197
Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) = (1+o(1)) \varep
Externí odkaz:
http://arxiv.org/abs/2405.20308
We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is the first
Externí odkaz:
http://arxiv.org/abs/2312.10026
Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij} \sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p = d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distrib
Externí odkaz:
http://arxiv.org/abs/2310.17635
Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p = p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n \cdot (pn)^{-1/2}$
Externí odkaz:
http://arxiv.org/abs/2310.17600
The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for
Externí odkaz:
http://arxiv.org/abs/2303.09521
Publikováno v:
Acta Mathematica Hungarica, 161 (2020), 540-549
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erd\H{o}s in 1950, and over the following decades he asked many questions about them. Most famously, h
Externí odkaz:
http://arxiv.org/abs/2211.01417
Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(\sigma_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C \varepsilon + e^{-cn
Externí odkaz:
http://arxiv.org/abs/2203.06141
Let $A$ be drawn uniformly at random from the set of all $n\times n$ symmetric matrices with entries in $\{-1,1\}$. We show that \[ \mathbb{P}( \det(A) = 0 ) \leq e^{-cn},\] where $c>0$ is an absolute constant, thereby resolving a well-known conjectu
Externí odkaz:
http://arxiv.org/abs/2105.11384
Autor:
Michelen, Marcus, Sahasrabudhe, Julian
Publikováno v:
Discrete Analysis 2022:13
This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\{0,\ldots,n\}$ with $\mathbb{P}(X = 0)\mathbb{P}
Externí odkaz:
http://arxiv.org/abs/2102.07699