Related papers: Pickands' constant $H_{\alpha}$ does not equal $1/…

For almost Mathieu operator $(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n$, the dry version of Ten Martini problem predicts that the spectrum $\Sigma_{\lambda,\alpha}$ of $ H_{\lambda,\alpha,\theta}$…

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to…

The triviality of the scalar sector of the standard one-doublet Higgs model implies that this model is only an effective low-energy theory valid below some cut-off scale Lambda. For a heavy higgs this scale must be relatively low (10 TeV or…

The Heisenberg uncertainty principle is one of the fundamental pillars of quantum mechanics and quantum field theory. It is normally introduced by postulating the commutation relations $[\hat{x}^i, \hat{p}^j] = i\hbar \delta^{ij}$. However,…

There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any $k,\ell\ge1$ and distinct integers $h_2,\ldots,h_k,a_1,\ldots,a_\ell$, we have $$\sum_{n\leq…

We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short…

The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations,…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right] $, write $A_{\alpha}\left( G\right) $ for the matrix \[ A_{\alpha}\left( G\right)…

A countable discrete group $\Gamma$ is said to be Frobenius stable if a function from the group that is "almost multiplicative" in the point Frobenius norm topology is "close" to a genuine unitary representation in the same topology. The…

In this paper a result of Latala about the tail behavior of Gaussian polynomials will be discussed. Latala proved an interesting result about this problem in paper [2]. But his proof applied an incorrect statement at a crucial point. Hence…

We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's…

Let $N$ be the number of copies of a small subgraph $H$ in an Erd\H{o}s-R\'enyi graph $G \sim \mathcal{G}(n, p_n)$ where $p_n \to 0$ is chosen so that $\mathbb{E} N = c$, a constant. Results of Bollob\'as show that for regular graphs $H$,…

Let $\alpha \colon \mathbb{N} \to S^1$ be the Steinhaus multiplicative function: a completely multiplicative function such that $(\alpha(p))_{p\text{ prime}}$ are i.i.d.~random variables uniformly distributed on the complex unit circle…

We prove that the tail probabilities of sums of independent uniform random variables, up to a multiplicative constant, are dominated by the Gaussian tail with matching variance and find the sharp constant for such stochastic domination.

Let $\Gamma$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $\Gamma$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on…

Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the "lower tail" of such a matrix, and prove that it is subgaussian under a simple fourth moment assumption on the…

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $\alpha$-stable L\'evy processes with $\alpha<1$. Our main result states that if the left tail of the L\'evy measure is regularly…

For the minimal $\lambda$-supersymmetry, it stays perturbative to the GUT scale for $\lambda \leq 0.7$. This upper bound is relaxed when one either takes the criteria that all couplings close to $\sim 4\pi$ for non-perturbation or allows…

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $\Gamma$ the generalized Tur\'an density $\pi_F(\Gamma)$ denotes the density of a maximum subgraph of $\Gamma$, which contains no copy of~$F$. Extending…

The operators $\Lambda_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $\Lambda_0$ is equal to…