Flips in edge-labelled pseudo-triangulations

We show that O(n2) exchanging flips suffice to transform any edge-labelled pointed pseudo-triangulation into any other with the same set of labels. By using insertion, deletion and exchanging flips, we can transform any edge-labelled pseudo-triangulation into any other with O(n log c + h log h) flips, where c is the number of convex layers and h is the number of points on the convex hull

On the spanning ratio of theta-graphs

We present improved upper bounds on the spanning ratio of a large family of θ-graphs. A θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2 π/m. We show that for any integer k ≥ 1, θ-graphs with 4k + 4 cones have spanning ratio at most 1 + 2 sin(θ/2) / (cos(θ/2) - sin(θ/2)). We also show that θ-graphs with 4k + 3 and 4k + 5 cones have spanning ratio at most cos(θ/4) / (cos(θ/2) - sin(3θ/4)). This is a significant improvement on all families of θ-graphs for which exact bounds are not known. For example, the spanning ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. We also improve the upper bounds on the competitiveness of the θ-routing algorithm for these graphs to 1 + 2 sin(θ/2) / (cos(θ/2) - sin(θ/2)) on θ-graphs with 4k + 4 cones and to 1 + 2 sin(θ/2)·cos(θ/4) / (cos(θ/2) - sin(3θ/4)) on θ-graphs with 4k + 3 and 4k + 5 cones. For example, the routing ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 4.0490

Improved Bounds for Guarding Plane Graphs with Edges

An edge guard set of a plane graph G is a subset Γ of edges of G such that each face of G is incident to an endpoint of an edge in Γ. Such a set is said to guardG. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G: (1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (Comput Geom Theory Appl 7:201–203, 1997) that G can be guarded with at most 2n5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n8 edges for any plane graph. (2) We prove that there exists an edge guard set of G with at most n3+α9 edges, where α is the number of quadrilateral faces in G. This improves the previous bound of n3+α by Bose et al. (Comput Geom Theory Appl 26(3):209–219, 2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that n3 edges suffice, removing the dependence on α

Flipping edge-labelled triangulations

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge e, which is the set of all edges reachable from e via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight Θ(nlog⁡n) bounds for edge-labelled triangulations of n-vertex convex polygons and combinatorial triangulations, contrasting with the Θ(n) bounds in their respective unlabelled settings. The Ω(nlog⁡n) lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to O(log2⁡n), although we no longer have a matching lower bound. Moving beyond conve

Inventário masculino dos esquemas de gênero do autoconceito (IMEGA)

O propósito deste artigo foi elaborar e validar o Inventário Masculino dos Esquemas de Gênero do Autoconceito (IMEGA). Baseado nas estruturas fatoriais das escalas masculina e feminina do Inventário dos Esquemas de Gênero do Autoconceito (IEGA), este instrumento avalia os esquemas masculino e feminino do autoconceito dos homens. A amostra utilizada foi composta por estudantes universitários do sexo masculino. Para a validade de construto do IMEGA, foram realizadas análises fatoriais (Principal Axis Factoring - PAF), com rotações oblíquas e ortogonais, para ambas as escalas e análise da consistência interna dos fatores (alfa de Cronbach). Os resultados demonstram que ambas as escalas são compostas por estruturas multifatoriais que se assemelham às estruturas fatoriais do IEGA. Devidamente validado, o IMEGA pode ser utilizado para avaliar os esquemas masculino e feminino do autoconceito de indivíduos do sexo masculino

On Plane Constrained Bounded-Degree Spanners

Let P be a finite set of points in the plane and S a set of non-crossing line segments with endpoints in P. The visibility graph of P with respect to S, denoted (Formula presented.), has vertex set P and an edge for each pair of vertices u, v in P for which no line segment of S properly intersects uv. We show that the constrained half-(Formula presented.)-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of (Formula presented.). We then show how to construct a plane 6-spanner of (Formula presented.) with maximum degree (Formula presented.), where c is the maximum number of segments of S incident to a vertex

The θ5-graph is a spanner

Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most 50+225≈9.960. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly se

An introduction to Randell Mills' Grand Unified Theory of Classical Physics (GUT-CP) : an Eindhoven University of Technology Honors Program 2007-2008 research project

This document represents the results of a year of research as part of the Honors Program of Eindhoven University of Technology. We heard about Randell L. Mills abandoning Quantum Mechanics and proposing an alternative theory. This sounded interesting enough to contact Gerrit M.W. Kroesen whose group was looking into Mills’ experiments. After some first brainstorms we decided that the best way for us to contribute to the research would be by looking at the theory Mills is proposing. Kroesen arranged a visit to Mills in February. We would be able to ask questions about the theory and get things explained. This resulted in a shift in our focus. We went on to collect a list of questions regarding the theory, it’s assumptions, propositions and derivations. During the visit Mills explained his theory to us giving special attention to the questions we send him on forehand. This helped greatly our understanding of his theory and the relation to the more accepted Quantum Mechanics. After the visit we started to write an introduction to his theory, explaining in short the key concepts. We wanted to help people going to work in this field to get to know important concepts and critics without having to look trough the whole derivation. This introduction became the paper you are reading right now. We will give a short history of atomic theory including a short summary of the atom-model of Quantum Mechanics. Mills criticizes Quantum Mechanics, so we extracted his main arguments against it. Later we explain the key concept of Mills’ theory: his electron model, and the implications it may have for our energy economy. We will conclude with a short mention of the experiments taking place to falsify or verify Mills’ claims

Making triangulations 4-connected using flips

We show that any triangulation on n vertices can be transformed into a 4-connected one using at most ⌊(3n - 6)/5⌋ edge flips. We also give an example of a triangulation that requires ⌈(3n-10)/5⌉ flips to be made 4-connected, showing that our bound is tight. Our re- sult implies a new upper bound on the diameter of the flip graph of 5.2n - 24.4, improving on the bound of 6n - 30 by Mori et al. [4]