Related papers: Multiple points on the boundaries of Brownian loop…

In this paper, we prove the following version of the famous Bernstein's theorem: Let $X\subset \mathbb R^{n+k}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$.…

The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop clusters associated with a rotation invariant…

We study properties of distributions of random variables with independent identically distributed symbols of generalized L\"{u}roth series (GLS) expansions (the family of GLS-expansions contains L\"{u}roth expansion and $Q_{\infty}$- and…

The well-known clustering algorithm of Miller, Peng, and Xu (SPAA 2013) is useful for many applications, including low-diameter decomposition and low-energy distributed algorithms. One nice property of their clustering, shown in previous…

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does not contain…

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with…

We propose a new method to accurately approximate the Pompeiu-Hausdorff distance from a triangle soup A to another triangle soup B up to a given tolerance. Based on lower and upper bound computations, we discard triangles from A that do not…

Let $X$ be a compact Gromov-Hausdorff limit space of a collapsing sequence of compact $n$-manifolds, $M_i$, of Ricci curvature $\text{Ric}_{M_i}\ge -(n-1)$ and all points in $M_i$ are $(\delta,\rho)$-local rewinding Reifenberg points, or…

Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums…

In a statistical cluster or loop model such as percolation, or more generally the Potts models or O(n) models, a pinch point is a single bulk point where several distinct clusters or loops touch. In a polygon P harboring such a model in its…

Robert Kaufman's proof that the set of rapid points of Brownian motion has a Fourier dimension equal to its Hausdorff dimension was first published in 1974. A study of the proof of the original paper revealed several gaps in the arguments…

Given two sets of points $A$ and $B$ in a normed plane, we prove that there are two linearly separable sets $A'$ and $B'$ such that $\mathrm{diam}(A')\leq \mathrm{diam}(A)$, $\mathrm{diam}(B')\leq \mathrm{diam}(B)$, and $A'\cup B'=A\cup B.$…

In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge $c \in (0, 1]$ in both chordal and radial cases, by relating these measures to the Brownian loop…

We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,\lambda\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson…

It is shown that in a $\lambda \phi^4$ theory of one real scalar field with spontaneous breaking of symmetry a calculation of the amplitudes of production by a virtual field $\phi$ of $n$ on-mass-shell bosons all being exactly at rest is…

An isotropic fractional Brownian field (with Hurst parameter $H<1/2$) is observed in a family of points in the unit square $\mathbf{C}=(-1/2,1/2]^{2}$% . These points are assumed to come from a realization of a homogeneous Poisson point…

We show that if $X$ is a separable locally compact Hausdorff connected space with fewer than $\mathfrak c$ non-cut points, then $X$ embeds into a dendrite $D\subseteq \mathbb R ^2$, and the set of non-cut points of $X$ is a nowhere dense…

Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S of Y has a stratification S^0\subset S^1\subset\...\subset S, where y\in S^k if no tangent…

We calculate the exact asymptotic survival probability, Q, of a one-dimensional Brownian particle, initially located located at the point x in (-L,L), in the presence of two moving absorbing boundaries located at \pm(L+ct). The result is…

The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which gives the Hausdorff dimension for limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let $h>0, \tau\geq…