On the Stretch Factor of Polygonal Chains

Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain. We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space

On the Stretch Factor of Polygonal Chains

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter $c \geq 1$, we call $P$ a $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 \leq i<j<k \leq n$. The stretch factor is a global property: it measures how close $P$ is to a straight line, and it involves all the vertices of $P$; being a $c$-chain, on the other hand, is a fingerprint-property: it only depends on subsets of $O(1)$ vertices of the chain. We investigate how the $c$-chain property influences the stretch factor in the plane: (i) we show that for every $\varepsilon > 0$, there is a noncrossing $c$-chain that has stretch factor $\Omega(n^{1/2-\varepsilon})$, for sufficiently large constant $c=c(\varepsilon)$; (ii) on the other hand, the stretch factor of a $c$-chain $P$ is $O\left(n^{1/2}\right)$, for every constant $c\geq 1$, regardless of whether $P$ is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain $P$ in $\mathbb{R}^2$ with $n$ vertices, the minimum $c\geq 1$ for which $P$ is a $c$-chain in $O\left(n^{2.5}\ {\rm polylog}\ n\right)$ expected time and $O(n\log n)$ space.Comment: 16 pages, 11 figure

Measuring tactical behaviour using technological metrics : case study of a football game

In football, the tactical behaviour of a team is related to the state of ball possession, i.e., the defensive and offensive phases. The aim of this study was to measure the tactical responses of two opposing teams in the moments with and without ball possession, thus trying to identify differences in results arising from tactical metrics such as weighted centroid position, weighted stretch index, surface area and effective area of play. The herein presented results show statistical differences in both teams, either with or without the ball possession, for the -axis centroid (pvalue≤ 0.001), -axis centroid (p-value ≤ 0.001), stretch index (p-value ≤ 0.001), surface area (p-value ≤ 0.001) and effective area of play (p-value ≤0.001). Such results confirm that teams react depending upon ball’s possession, respecting the tactical principles of width and length, as well as the unit in the offensive phase with ball possession, and also theconcentration and defensive unit in the moments without ball possession.info:eu-repo/semantics/publishedVersio

Fibrations of 3-manifolds and asymptotic translation length in the arc complex

Given a 3-manifold $M$ fibering over the circle, we investigate how the asymptotic translation lengths of pseudo-Anosov monodromies in the arc complex vary as we vary the fibration. We formalize this problem by defining normalized asymptotic translation length functions $\mu_d$ for every integer $d \ge 1$ on the rational points of a fibered face of the unit ball of the Thurston norm on $H^1(M;\mathbb{R})$. We show that even though the functions $\mu_d$ themselves are typically nowhere continuous, the sets of accumulation points of their graphs on $d$-dimensional slices of the fibered face are rather nice and in a way reminiscent of Fried's convex and continuous normalized stretch factor function. We also show that these sets of accumulation points depend only on the shape of the corresponding slice. We obtain a particularly concrete description of these sets when the slice is a simplex. We also compute $\mu_1$ at infinitely many points for the mapping torus of the simplest hyperbolic braid to show that the values of $\mu_1$ are rather arbitrary. This suggests that giving a formula for the functions $\mu_d$ seems very difficult even in the simplest cases.Comment: 47 pages, 13 figure

Measuring collective behaviour in football teams : inspecting the impact of each half of the match on ball possession

The aim of this study was to inspect the influence of each half of matcha and the ball possession status on the players’ spatio-temporal relationships. Three official matches of a professional football team were analysed. From the players' locations were collected the team’s wcentroid, wstretch index, surface area and effective area of play at 9218 play instants. The results suggested that the values of teams’ dispersion and average position on the field decreases during the 2nd half of the match. In sum, this study showed that the half of match and the ball possession status influenced players’ spatio-temporal relationships, in a way that significantly contributes to the collective understanding of football teams.info:eu-repo/semantics/publishedVersio

Currents with corners and counting weighted triangulations

Let $\Sigma$ be a closed orientable hyperbolic surface. We introduce the notion of a \textit{geodesic current with corners} on $\Sigma$, which behaves like a geodesic current away from certain singularities (the "corners"). We topologize the space of all currents with corners and study its properties. We prove that the space of currents with corners shares many properties with the space of geodesic currents, although crucially, there is no canonical action of the mapping class group nor is there a continuous intersection form. To circumvent these difficulties, we focus on those currents with corners arising from harmonic maps of graphs into $\Sigma$. This leads to the space of \textit{marked harmonic currents with corners}, which admits a natural Borel action by the mapping class group, and an analog of Bonahon's\cite{Bonahon} compactness criterion for sub-level sets of the intersection form against a filling current. As an application, we consider an analog of a curve counting problem on $\Sigma$ for triangulations. Fixing an embedding $\phi$ of a weighted graph $\Gamma$ into $\Sigma$ whose image $\phi(\Gamma)$ is a triangulation of $\Sigma$, let $N_{\phi}(L)$ denote the number of mapping classes $f$ so that a weighted-length minimizing representative in the homotopy class determined by $f \circ \phi$ has length at most $L$. In analogy with theorems of Mirzakhani\cite{Mirzakhani}, Erlandsson-Souto\cite{ErlandssonSouto}, and Rafi-Souto\cite{RafiSouto}, we prove that $N_{\phi}(L)$ grows polynomially of degree $6g-6$ and the limit $$\lim_{L \rightarrow \infty} \frac{N_{\phi}(L)}{L^{6g-6}}$$ exists and has an explicit interpretation depending on the geometry of $\Sigma$, the vector of weights, and the combinatorics of $\phi$ and $\Gamma$.Comment: 40 pages, 3 figure

The taut polynomial and the Alexander polynomial

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmuller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Then we give formulas relating the taut polynomial and the untwisted Alexander polynomial. There are two formulas; one holds when the maximal free abelian cover of a veering triangulation is edge-orientable, another holds when it is not edge-orientable. Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case we give a formula that relates the specialisation of the taut polynomial under the Dehn filling and the Alexander polynomial of the Dehn-filled manifold. This extends a theorem of McMullen connecting the Teichmuller polynomial and the Alexander polynomial to the nonfibred setting, and improves it in the fibred case. We also prove a sufficient and necessary condition for the existence of an orientable fibred class in the cone over a fibred face of the Thurston norm ball.Comment: v3: Major changes: 1) Proof that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. (Proposition 5.7). 2) Added a result that gives a formula relating the taut polynomial and the Alexander polynomial in the case when the maximal free abelian cover is not edge-orientable (Proposition 5.17). 28 pages, 9 figure

Shift coordinates, stretch lines and polyhedral structures for Teichmüller space

This paper has two parts. In the first part, we study shift coordinates on a sphere $S$ equipped with three distinguished points and a triangulation whose vertices are the distinguished points.These coordinates parametrize a space $\widetilde{\mathcal {T}}(S)$ that we call an {\it unfolded Teichmüller space}. This space contains Teichmüller spaces of the sphere with $\mathfrak{b}$ boundary components and $\mathfrak{p}$ cusps (which we call generalized pairs of pants), for all possible values of $\mathfrak{b}$ and $\mathfrak{p}$ satisfying $\mathfrak{b}+\mathfrak{p}=3$. The parametrization of $\widetilde{\mathcal {T}}(S)$ by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in $\mathbb{R}^3$. Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with $\mathfrak{b}$ boundary components and $\mathfrak{p}$ cusps, for fixed $\mathfrak{b}$ and $\mathfrak{p}$, the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on $\widetilde{\mathcal {T}}(S)$ whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space $\widetilde{\mathcal {T}}(S)$. Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2

Combinatorics in N = 1 Heterotic Vacua

We briefly review an algorithmic strategy to explore the landscape of heterotic E8 \times E8 vacua, in the context of compactifying smooth Calabi-Yau three-folds with vector bundles. The Calabi-Yau three-folds are algebraically realised as hypersurfaces in toric varieties and a large class of vector bundles are constructed thereon as monads. In the spirit of searching for Standard-like heterotic vacua, emphasis is placed on the integer combinatorics of the model-building programme.Comment: 14 pages. An introductory review prepared for the special issue "Computational Algebraic Geometry in String and Gauge Theory" of Advances in High Energy Physic