arXiv subjects

Abdulamin Ismailov

Publications and source records attributed to Abdulamin Ismailov.

3 recordsLinked to original sources

arXiv2026

With what probability does an inscribed triangle contain a given point?

Three points uniformly selected on the unit circle form a triangle containing a point $X$ at distance $r \in [0; 1]$ from its center with probability $P(r) = \frac{1}{4} - \frac{3}{2 π^2}\textrm{Li}_2(r^2)$, where $\textrm{Li}_2$ is the dilogarithm function (Jeremy Tan Jie Rui, 2018). In this paper we present an alternative proof of this fact. We also discuss a couple of other geometric probability problems where the dilogarithm function arises.

math.PR↗

arXiv2024

Lower bounds on the measure of the support of positive and negative parts of trigonometric polynomials

For a finite set of natural numbers $D$ consider a complex polynomial of the form $f(z) = \sum_{d \in D} c_d z^d$. Let $ρ_+(f)$ and $ρ_-(f)$ be the fractions of the unit circle that $f$ sends to the right($\operatorname{Re} f(z) > 0$) and left($\operatorname{Re} f(z) < 0$) half-planes, respectively. Note that $\operatorname{Re} f(z)$ is a real trigonometric polynomial, whose allowed set of frequencies is $D$. It turns out that $\min(ρ_+(f), ρ_-(f))$ is always bounded from below by a numerical characteristic $α(D)$ of our set $D$ which comes from a seemingly unrelated combinatorial problem. Furthermore, this result could be generalized to power series, almost periodic functions, functions of several variables and multivalued algebraic functions.

math.CA↗

arXiv2023

Dimension-free estimates on distances between subsets of volume $\varepsilon$ inside a unit-volume body

Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \varepsilon}$, for simplexes and hyperoctahedrons $-$ of order $-\ln \varepsilon$, for $\ell_p$ balls with $p \in [1;2]$ $-$ of order $(-\ln \varepsilon)^{\frac{1}{p}}$. These estimates are not dependent on the dimensionality $n$. The goal of the paper is to study this phenomenon. Isoperimetric inequalities will play a key role in our approach.

math.MG↗