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Fedor Nazarov

Publications and source records attributed to Fedor Nazarov.

At least 19 recordsLinked to original sources
arXiv2025

Convergence rate of $\ell^p$-energy minimization on graphs: sharp polynomial bounds and a phase transition at $p=3$

We consider the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Given $p>1$ and an initial opinion profile $f_0:V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v=v_t$ is selected, and the opinion there is updated to the value $f_{t}(v)$ that minimizes the sum $\sum_{w \sim v} |f_t(v)-f_{t-1}(w)|^p$ over neighbours $w$ of $v$. The case $p=2$ yields linear averaging dynamics, but for all $p \ne 2$ the dynamics are nonlinear. In the limiting case $p=\infty$ (known as Lipschitz learning), $f_t(v)$ is the average of the largest and smallest values of $f_{t-1}(w)$ among the neighbours $w$ of $v$. We show that the number of steps needed to reduce the oscillation of $f_t$ below $ε$ is at most $n^{β_p}$ (up to logarithmic factors in $n$ and $ε$), where $β_p:=max(\frac{2p}{p-1},3)$; we prove that the exponent $β_p$ is optimal. The phase transition at $p=3$ is a new phenomenon. We also derive matching upper and lower bounds for convergence time as a function of $n$ and the average degree; these are the most challenging to prove.

math.PR↗
arXiv2025

Small volume bodies of constant width

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

math.MG↗
arXiv2024

The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with at least $c\sqrt{n}$ distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood ($E_k=\{-1,1\}$) and Newman ($E_k=\{0,(-1)^k\}$) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence $E_k$ is periodic.

math.CA↗
arXiv2023

Fourier uniqueness and non-uniqueness pairs

Motivated by recent works by Radchenko and Viazovska and by Ramos and Sousa, we find sufficient conditions for a pair of discrete subsets of the real line to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are close to each other. The uniqueness result can be upgraded to an interpolation formula, which in turn produces an abundance of discrete measures with discrete Fourier transform.

math.CA↗
arXiv2023

On the maximal distance between the centers of mass of a planar convex body and its boundary

We prove that the length of the projection of the vector joining the centers of mass of a convex body on the plane and of its boundary to an arbitrary direction does not exceed $\frac{1}{6}$ of the body width in this direction. It follows that the distance between these centers of mass does not exceed $\frac16$ of the diameter of the body and $\frac{1}{12}$ of its boundary length. None of those constants can be improved.

math.MG↗
arXiv2022

A basic homogenization problem for the $p$-Laplacian in ${\mathbb R}^d$ perforated along a sphere: $L^\infty$ estimates

We consider a boundary value problem for the $p$-Laplacian, posed in the exterior of small cavities that all have the same $p$-capacity and are anchored to the unit sphere in $\mathbb{R}^d$, where $1 0$. We show that the problem possesses a critical window characterized by $τ:=\lim_{\varepsilon \downarrow 0}α/α_c \in (0,\infty)$, where $α_c=\varepsilon^{1/γ}$ and $γ= \frac{d-p}{p-1}.$ We prove that outside the unit sphere, as $\varepsilon\downarrow 0$, the solution converges to $A_*U$ for some constant $A_*$, where $U(x)=\min\{1,|x|^{-γ}\}$ is the radial $p$-harmonic function outside the unit ball. Here the constant $A_*$ equals 0 if $τ=0$, while $A_*=1$ if $τ=\infty$. In the critical window where $τ$ is positive and finite, $ A_*\in(0,1)$ is explicitly computed in terms of the parameters of the problem. We also evaluate the limiting $p$-capacity in all three cases mentioned above. Our key new tool is the construction of an explicit ansatz function $u_{A_*}^\varepsilon$ that approximates the solution $u^\varepsilon$ in $L^{\infty}(\mathbb{R}^d)$ and satisfies $\|\nabla u^\varepsilon-\nabla u_{A_*}^\varepsilon \|_{L^{p}(\mathbb{R}^d)} \to 0$ as $\varepsilon \downarrow 0$.

math.AP↗
arXiv2022

On a Bellman function associated with the Chang--Wilson--Wolff theorem: a case study

In this paper we estimate the tail of distribution (i.e., the measure of the set $\{f\ge x\}$) for those functions $f$ whose dyadic square function is bounded by a given constant. In particular we get a bit better estimate than the estimate following from the Chang--Wilson--Wolf theorem. In the paper we investigate the Bellman function corresponding to the problem. A curious structure of this function is found: it has jumps of the first derivative at a dense subset of interval $[0,1]$ (where it is calculated exactly), but it is of $C^\infty$-class for $x>\sqrt3$ (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists in the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to computer.

math.CA↗
arXiv2022

Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

We establish an instantaneous smoothing property for decaying solutions on the half-line $(0,+\infty)$ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of $H^1$ stable manifolds of such equations, showing that $L^2_{loc}$ solutions that remain sufficiently small in $L^\infty$ (i) decay exponentially, and (ii) are $C^\infty$ for $t>0$, hence lie eventually in the $H^1$ stable manifold constructed by Pogan and Zumbrun

math.AP↗
arXiv2021

A simple upper bound for Lebesgue constants associated with Leja points on the real line

Let $K\subset \mathbb R$ be a regular compact set and let $g(z)=g_{\overline{\mathbb C}\setminus K}(z,\infty)$ be the Green function for $\overline{\mathbb C}\setminus K$ with pole at infinity. For $δ>0$, define $$ G(δ):=\max\{ g(z): z\in \mathbb C, \,\operatorname{dist}(z,K)\le 2δ\}. $$ Let $\{ x_n\}_{n=0}^\infty$ be a Leja sequence of points of $K$. Then the uniform norm $\|T_n\|=Λ_n, n=1,2,\ldots$ of the associated interpolation operator $T_n$, i.e., the $n$-th Lebesgue constant, is bounded from above by $$ \min_{δ>0}2n\left[\frac{\operatorname{diam}( K)}δe^{nG(δ)}\right]^{9/8}. $$ In particular, when $K$ is a uniformly perfect subset of $\mathbb R$, the Lebesgue constants grow at most polynomially in $n$. To the best of our knowledge, the result is new even when $K$ is a finite union of intervals.

math.CA↗
arXiv2021

On a factorization formula for the partition function of directed polymers

We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice $\mathbb{Z}^{d+1}$, subject to an i.i.d. random potential and in the regime of weak disorder. In particular, we show that the error term in the factorization formula is uniformly small for starting and end points $x, y$ in the sub-ballistic regime $\| x - y \| \leq t^σ$, where $σ< 1$ can be arbitrarily close to $1$. This extends a result of Sinai. We also derive asymptotics for spatial and temporal correlations of the field of limiting partition functions.

math.PR↗
arXiv2020

On Weissler's conjecture on the Hamming cube I

Let $1\leq p \leq q <\infty$, and let $w \in \mathbb{C}$. Weissler conjectured that the Hermite operator $e^{wΔ}$ is bounded as an operator from $L^{p}$ to $L^{q}$ on the Hamming cube $\{-1,1\}^{n}$ with the norm bound independent of $n$ if and only if \begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*} It was proved by Bonami (1970), Beckner (1975), and Weissler (1979) in all cases except $2<p\leq q <3$ and $3/2<p\leq q <2$, which stood open until now. The goal of this paper is to give a full proof of Weissler's conjecture in the case $p=q$. Several applications will be presented.

math.AP↗
arXiv2020

The local Tb theorem with rough test functions

We prove a local $Tb$ theorem under close to minimal (up to certain `buffering') integrability assumptions, conjectured by S. Hofmann (El Escorial, 2008): Every cube is assumed to support two non-degenerate functions $b^1_Q\in L^p$ and $b^2_Q\in L^q$ such that $1_{2Q}Tb^1_Q\in L^{q'}$ and $1_{2Q}T^*b^2_Q\in L^{p'}$, with appropriate uniformity and scaling of the norms. This is sufficient for the $L^2$-boundedness of the Calderon-Zygmund operator $T$, for any $p,q\in(1,\infty)$, a result previously unknown for simultaneously small values of $p$ and $q$. We obtain this as a corollary of a local $Tb$ theorem for the maximal truncations $T_{\#}$ and $(T^*)_{\#}$: for the $L^2$-boundedness of $T$, it suffices that $1_Q T_{\#}b^1_Q$ and $1_Q (T^*)_{\#}b^2_Q$ be uniformly in $L^0$. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov-Treil-Volberg.

math.CA↗
arXiv2020

Fluctuations in the number of nodal domains

We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step towards justification of the Bogomolny-Schmit heuristics.

math.PR↗
arXiv2020

Chemotaxis and Reactions in Biology

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions. One example is the immune system signalling, where infected tissues release chemokines attracting monocytes to fight invading bacteria. Another example is reproduction, where eggs release pheromones that attract sperm. A macro scale example is flower scent appealing to pollinators. In this paper we consider a system of PDE designed to model such processes. Our interest is to quantify the effect of chemotaxis on reaction rates compared to pure reaction-diffusion. We limit consideration to surface chemotaxis, which is well motivated from the point of view of many applications. Our results provide the first insight into situations where chemotaxis can be crucial for reaction success, and where its effect is likely to be limited. The proofs are based on new analytical tools; a significant part of the paper is dedicated to building up the linear machinery that can be useful in more general settings. In particular we establish precise estimates on the rates of convergence to ground state for a class of Fokker-Planck operators with potentials that grow at a logarithmic rate at infinity. These estimates are made possible by a new sharp weak weighted Poincaré inequality improving in particular a result of Bobkov and Ledoux.

math.AP↗
arXiv2019

Gelfand-type problem for turbulent jets

We consider the model of auto-ignition (thermal explosion) of a free round reactive turbulent jet. This model falls into the general class of Gelfand-type problems and constitutes a boundary value problem for a certain semi-linear elliptic equation that depends on two parameters: $α$ characterizing the flow rate and $λ$ (Frank-Kamentskii parameter) characterizing the strength of the reaction. Similarly to the classical Gelfand problem, this equation admits a solution when the Frank-Kametskii parameter $λ$ does not exceed some critical value $λ^*(α)$ and admits no solutions for larger values of $λ$. We obtain the sharp asymptotic behavior of the critical Frank-Kamenetskii parameter in the strong flow limit ($α\gg1$). We also provide a detailed description of the extremal solution (i.e., the solution corresponding to $λ^*$) in this regime.

math.AP↗
arXiv2018

Square functions and the Hamming cube: Duality

For $1<p\leq 2$, any $n\geq 1$ and any $f:\{-1,1\}^{n} \to \mathbb{R}$, we obtain $(\mathbb{E} |\nabla f|^{p})^{1/p} \geq C(p)(\mathbb{E}|f|^{p} - |\mathbb{E}f|^{p})^{1/p}$ where $C(p)$ is the smallest positive zero of the confluent hypergeometric function ${}_{1}F_{1}(\frac{p}{2(1-p)}, \frac{1}{2}, \frac{x^{2}}{2})$. Our approach is based on a certain duality between the classical square function estimates on the Euclidean space and the gradient estimates on the Hamming cube.

math.AP↗