A convex polygon is defined as a sequence (V0, . . . , Vn−1) of points on a plane such that the union of the edges [V0, V1], . . . , [Vn−2, Vn−1], [Vn−1, V0] coincides with the boundary of the convex hull of the set of vertices {V0, . . . , Vn−1}. It is proved that all sub-polygons of any convex polygon with distinct vertices are convex. It is also proved that, if all sub-(n − 1)-gons of an n-g...

We prove the following main theorem of the theory of (r, q)-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions: (1) each internal face is an r-gon, where r ≥ 3 ; (2) the degree of each internal vertex is q , where q ≥ 3 , and the degree of each boundary vertex is at most q and at least 2 . Then it also possesses the following third property: (3) the vertices, ...

Given a strictly convex plane curve γ and a point x in its exterior, there are two tangent segments to γ from x. The equitangent locus of γ is the set of points x for which the two tangent segments have equal lengths. An equitangent n-gon of γ is a circumscribed n-gon such that the tangent segments to γ from the vertices have equal lengths, see Figure 1. For example, if γ is a circle, the equit...

Seven-particle scattering amplitudes in planar super-Yang-Mills theory are believed to belong to a special class of generalised polylogarithm functions called heptagon functions. These are functions with physical branch cuts whose symbols may be written in terms of the 42 cluster A-coordinates on Gr(4, 7). Motivated by the success of the hexagon bootstrap programme for constructing six-particle...

This paper raises the empirical point that modal subordination is not always obligatory, and that moreover, this is a point of lexical variation. Some modals, like will, which I call definite modals, undergo modal subordination obligatorily, and some, like gonna, which I call nondefinite modals, do so optionally. I propose a dynamic framework in which, following from Frank (1997), information s...

Given a simple n-gon P in the plane, normalized to contain the unit disk, we define a map from P to the unit circle so that (1) the map extends to be 8-quasiconformal on the interior, (2) it contracts arclength on the boundary and (3) the images of all n vertices can be computed in time O(n). Thus we obtain a fast K-QC-approximation to the Riemann map with constant K independent of the domain. ...

We verify an old conjecture of G. Pólya and G. Szegő saying that the regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area.

In this paper, we consider the problem of placing a small number of angle guards inside a simple polygon P such that each point in the plane can determine if it is in or out of P from a monotone Boolean formula (and (.) and or (+) operations only) composed from the angle guards. Each angle guard views an infinite wedge of the plane. This class of new art gallery type problems is called sculptur...

The Hamiltonian potentials of the bending deformations of n-gons in E3 studied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra Pn of the pure braid group Pn on the moduli space Mr of n-gon linkages with the side-lengths r = (r1, . . . , rn) in E3. If e ∈ Mr is a singular point we may linearize the vector fields inPn at e. This linearization yields a flat connection...

In this work we apply the techniques that were developed in [M.N. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), in press] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulato...