相关论文: How far can you see in a forest?
A dense forest is a set $F \subset \mathbb{R}^n$ with the property that for all $\varepsilon > 0$ there exists a number $V(\varepsilon) > 0$ such that all line segments of length $V(\varepsilon)$ are $\varepsilon$-close to a point in $F$.…
A set of points with finite density is constructed in $\mathbb{R}^d$, with $d\geq2$, by adding points to a Poisson process such that any line segment of length $O\left(\varepsilon^{-(d-1)}\ln\varepsilon^{-1}\right)$ in $\mathbb{R}^d$ will…
A 1965 problem due to Danzer asks whether there exists a set in Euclidean space with finite density intersecting any convex body of volume one. A recent approach to this problem is concerned with the construction of dense forests and is…
This paper studies visibility problems in Euclidean spaces $\mathbb{R}^d$ where the obstacles are the points of infinite discrete sets $Y\subseteq\mathbb{R}^d$. A point $x\in\mathbb{R}^d$ is called $\varepsilon$-visible for $Y$ (notation:…
Empirical observations suggest that in pure even-aged forests, the mean diameter of forest trees (D, diameter at breast height, 1.3 m above ground) tends to remain a constant proportion of stand height (H, average height of the largest…
Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function. We show that cut-and-project…
A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…
In \cite{FKW} Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of $\mathbb{R}^2$ of positive upper (Banach) density. A second proof of this…
The existence of a partition of the common set of the vertices of two forests into two subsets, when difference of their capacities in the neighborhood of each vertex of each forest not greater than 2 is proved, and an example, which shows…
Let $F$ be an $n$-vertex forest. We say that an edge $e\notin F$ is in the shadow of $F$ if $F\cup\{e\}$ contains a cycle. It is easy to see that if $F$ is "almost a tree", that is, it has $n-2$ edges, then at least…
A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space intersecting any convex body of volume one. A suitable weakening of the volume constraint leads to the (much more recent) problem of…
We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with $m>1$ viewpoints (or guards) located on the terrain surface. A point on the terrain is considered…
In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree…
We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive $\varepsilon_1$ and $\varepsilon_2$ such that every tree that is neither a path nor a star has inducibility at…
Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$…
Borodin and Kostochka proved that for $d_2 \geq 2d_1+2$ and a graph $G$ where every subgraph $H$ satisfies $$ e(H) < \left(2 - \frac{d_2+2}{(d_1+2)(d_2+1)}\right)n(H) + \frac{1}{d_2+1} $$ has a vertex partition $V(G) = V_1 \cup V_2$ such…
Let $G$ be a connected graph and $W$ be a set of vertices of $G$. The representation multiset of a vertex $v$ with respect to $W$, $r_m (v|W)$, is defined as a multiset of distances between $v$ and the vertices in $W$. If $r_m (u |W) \neq…
The z > 2 Lyman-alpha (Lya) forest traces the underlying dark-matter distribution on large scales and, given sufficient sightlines, can be used to create 3D maps of large-scale structure. We examine the observational requirements to…
We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose…
Monitoring and managing Earth's forests in an informed manner is an important requirement for addressing challenges like biodiversity loss and climate change. While traditional in situ or aerial campaigns for forest assessments provide…