Related papers: A uniform Dvoretzky-Kiefer-Wolfowitz inequality
Consider the Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse H\"older condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B…
We consider the equation $\e^{2}\Delta u=(u-a(x))(u^2-1)$ in $\Omega$, $\frac{\partial u}{\partial \nu} =0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in $\R^n$, $\nu$ the outer unit normal to $\pa\Omega$, and $a$ a…
In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})\Delta u = f(u) & \mbox{in $\Omega$,} u=0 & \mbox{on $\partial\Omega$,} \end{array}\right. \end{equation*} in a bounded domain,…
The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the values at zero of the concentration…
The problem of learning threshold functions is a fundamental one in machine learning. Classical learning theory implies sample complexity of $O(\xi^{-1} \log(1/\beta))$ (for generalization error $\xi$ with confidence $1-\beta$). The private…
Fox's H-function provide a unified and elegant framework to tackle several physical phenomena. We solve the space fractional diffusion equation on the real line equipped with a delta distribution initial condition and identify the…
We define the Heisenberg Kakeya maximal functions $M_{\delta}f$, $0<\delta<1$, by averaging over $\delta$-neighborhoods of horizontal unit line segments in the Heisenberg group $\mathbb{H}^1$ equipped with the Kor\'{a}nyi distance…
For $\alpha\ge 0$, let $\mathcal{W}(\alpha)$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0) = 0 $ and $ f'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + \alpha z f''(z)\} > 0$. This article…
We prove an entropy versus degree dichotomy for low-degree tests and the Sum-of-Squares (SoS) hierarchy on a calibrated window after a gadget layer. For a target distribution \(\mu\) and a product-like proxy \(u\), we study the low-degree…
We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $\epsilon>0$ there exists $\delta>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n}<kv-w> \ge \epsilon$ (where $<\cdot>$ denotes…
Let $H$ be a complex Hilbert space and let $\Omega\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Amp\`ere operator and no basis-free determinant of the Levi form. Hence, a determinant-type…
In this paper, we study the quasilinear inequality $ \Delta_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,\alpha>m-1 \quad and \quad f(t)> 0,\alpha f(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some…
We consider the behaviour at a boundary point of an open subset $U\subset\mathbb{C}$ of distributions that are holomorphic on $U$ and belong to what are called negative Lipschitz classes. The result explains the significance for holomorphic…
It is well known that the entropy $H(X)$ of a discrete random variable $X$ is always greater than or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tight…
We focus on the classification of positive solutions to $(-\Delta)^s u=\frac{x_n^{\alpha}}{u^\gamma}$ in the half space with $\gamma>0$, subject to the Dirichlet condition. We show that when $-2s<\alpha<(\gamma-1)s$, all positive solutions…
If $\mu$ is a distribution over the $d$-dimensional Boolean cube $\{0,1\}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $\mu$. Specifically, we consider the empirical mean estimator $\hat p_n$ and study the…
We introduce a new class of Frank-Wolfe algorithms for minimizing differentiable functionals over probability measures. This framework can be shown to encompass a diverse range of tasks in areas such as artificial intelligence,…
The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u<d\le e^{u+1}\}$. We prove that $\sum_{n\le x} \Delta(n) \ll x(\log\log x)^{11/4}$ for all $x\ge100$. This improves on…
Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such…
Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal…