Related papers: Directed and irreversible path in Euclidean spaces

We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does…

A path-following control algorithm enables a system's trajectories under its guidance to converge to and evolve along a given geometric desired path. There exist various such algorithms, but many of them can only guarantee local convergence…

For sufficiently tame paths in $\mathbb{R}^n$, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an…

We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1].…

The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…

In 1960, Nash-Williams proved his strong orientation theorem that every finite graph has an orientation in which the number of directed paths between any two vertices is at least half the number of undirected paths between them (rounded…

Let $\Phi$ be a correspondence from a normed vector space $X$ into itself, let $u: X\to \mathbf{R}$ be a function, and $\mathcal{I}$ be an ideal on $\mathbf{N}$. Also, assume that the restriction of $u$ on the fixed points of $\Phi$ has a…

We prove that in every direction in the Euclidean plane, there exists a line containing no double exponential time random (ee-random) points. This means each point on these lines has an algorithmically predictable location, to the extent…

We say a directed graph $G$ on $n$ vertices is irredundant if the removal of any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such that there exists a directed path from $u$ to $v$. We determine the maximum possible…

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex $(i_1,i_2)$ to $(j_1,j_2)$, whenever $i_1 \le j_1$, $i_2 \le j_2$, with probability $p$, independently for each such pair of vertices.…

We study the correlations of directions $P_0 P$ in the Euclidean plane, where $P_0$ is a point in a fixed disc, $P$ is an integer lattice point in the square $[-Q,Q]^2$, and $Q\to \infty$.

A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more…

We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by…

Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce…

The path spaces of a directed graph play an important role in the study of graph $\css$. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple,…

A useful result is that if a bounded complex-valued path is Riemann-integrable, then its modulus is also Riemann-integrable. The extension of this last result to bounded paths taking values in a normed space is affirmed, as being true, in…

Given a directed graph $D$ of order $n\geq 4$ and a nonempty subset $Y$ of vertices of $D$ such that in $D$ every vertex of $Y$ reachable from every other vertex of $Y$. Assume that for every triple $x,y,z\in Y$ such that $x$ and $y$ are…

It was independently conjectured by H\"aggkvist in 1989 and Kriesell in 2011 that given a positive integer $\ell$, every simple eulerian graph with high minimum degree (depending on $\ell$) admits an eulerian tour such that every segment of…

The direct effect of one eventon another can be defined and measured byholding constant all intermediate variables between the two.Indirect effects present conceptual andpractical difficulties (in nonlinear models), because they cannot be…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…