We prove the following generalised empty pentagon theorem: for every integer ` ≥ 2, every sufficiently large set of points in the plane contains ` collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].

[1] Since the terrorist attacks on the World Trade Center and the Pentagon, nearly every angle of these events has been explored in print, pertaining to everything from the individuals and nations involved to the political and historical precursors and ongoing social ramifications of the attacks. Yet few writers have seriously analyzed the actual arguments that Osama bin Laden and al-Qaeda have...

Topological defects are singular points in vector fields, important applications ranging from fingerprint detection to liquid crystals biomedical imaging. In discretized topological and their charge identified by finite differences or finite-step paths around the tentative defect. As is (half) integer, it cannot depend continuously on each input a discrete domain. Instead, switches discontinuou...

where (35) follows from the independence of X? and X,", and (36) and (37) follow because the channel is memoryless. Similarly, we have (38) 1 1 " In order to show that C, 1 C Cllm, it suffices to show that for any Pdyl and PA^, the corners of the pentagon,

A fullerene graph is a 3 regular planar simple finite graph with pentagon or hexagon faces. In these graphs the number of pentagon faces is 12. Therefore, any fullerene graph can be characterized by number of its hexagon faces. In this note, for any h > 1, we will construct a fullerene graph with h hexagon faces. Then, using the leapfrogging process we will construct stable fullerenes with 20 +...

We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Kl, KM, HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structu...

Sustainable design and constructability are two keys for efficient design and construction that are often addressed separately from each other in projects. This paper examines the integration of these two initiatives on the Pentagon renovation. The examination exposes a synergistic relationship between sustainable design and constructability. This relationship strives to efficiently utilize res...