arXiv subjects

Linhang Huang

Publications and source records attributed to Linhang Huang.

5 recordsLinked to original sources

arXiv2026

Electrostatic skeletons and condition of strict descent

Given a precompact domain $\Omega \subseteq\mathbb{R}^2$, the electrostatic skeleton of $\Omega$ is defined as a positive measure inside $\Omega$, supported on a set with no simple loops, which generates $\partial \Omega$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons.

math.CV↗

arXiv2025

Low-Discrepancy Set Post-Processing via Gradient Descent

The construction of low-discrepancy sets, used for uniform sampling and numerical integration, has recently seen great improvements based on optimization and machine learning techniques. However, these methods are computationally expensive, often requiring days of computation or access to GPU clusters. We show that simple gradient descent-based techniques allow for comparable results when starting with a reasonably uniform point set. Not only is this method much more efficient and accessible, but it can be applied as post-processing to any low-discrepancy set generation method for a variety of standard discrepancy measures.

math.OC↗

arXiv2025

Many lemniscates with large diameter

We prove that for every $0 < c < 4$ and every $N \in \mathbb{N}$ there exists a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ such that the set $\{z \in \mathbb{C} : |p(z)| \leq 1\}$ has at least $N$ connected components with diameter at least $c$. This answers a question of Erd\H{o}s.

math.CV↗

arXiv2025

Collet-Eckmann type conditions and conformal welding of unicritical quadratic laminations

In this paper, we introduce a Collet-Eckmann type condition for the unicritical laminations on the unit circle. We prove that this condition implies the lamination admits a H\"older continuous conformal welding which produces a Julia set for some unicritical polynomial. In consequence, we present a new proof that almost all angles on the unit circle produce quadratic polynomials with H\"older Fatou components, without the use of Beurling's theorem.

math.DS↗

arXiv2025

On convex domains maximizing the gradient of the torsion function

We consider the solution of $-\Delta u = 1$ on convex domains $\Omega \subset \mathbb{R}^2$ subject to Dirichlet boundary conditions $u =0$ on $\partial \Omega$. Our main concern is the behavior of $\|\nabla u\|_{L^{\infty}}$, also known as the maximum shear stress in Elasticity Theory and first investigated by Saint Venant in 1856. We consider the two shape optimization problems $\| \nabla u\|_{L^{\infty}}/ |\Omega|^{1/2}$ and $\| \nabla u\|_{L^{\infty}}/ H^1( \partial \Omega)$. Numerically, the extremal domain for each functional looks a bit like the rounded letter `D'. We prove that (1) either the extremal domain does not have a $C^{2 + \varepsilon}$ boundary or (2) there exists an infinite set of points on $\partial \Omega$ where the curvature vanishes. Either scenario seems curious and is rarely encountered for such problems. The techniques are based on finding a representation of the functional using only conformal geometry and classic perturbation arguments.

math.AP↗