The principle of duality in Euclidean and in absolute geometry

2016 ◽  
Vol 107 (3) ◽  
pp. 707-717 ◽  
Author(s):  
Rolf Struve
Keyword(s):  
Absolute Geometry ◽  
Principle Of Duality

Duality beyondPersons and Groups

2020 ◽  
pp. 391-413
Author(s):  
Sophie Mützel ◽  
Ronald Breiger
Keyword(s):  
Social Network ◽  
Social Network Analysis ◽  
Network Analysis ◽  
Network Data ◽  
Future Directions ◽  
Affiliation Networks ◽  
The Social ◽  
Recent Developments ◽  
Principle Of Duality ◽  
Seminal Article

This chapter focuses on the general principle of duality, which was originally introduced by Simmel as the intersection of social circles. In a seminal article, Breiger formalized Simmel’s idea, showing how two-mode types of network data can be transformed into one-mode networks. This formal translation proved to be fundamental for social network analysis, which no longer needed data on who interacted with whom but could work with other types of data. In turn, it also proved fundamental for the analysis of how the social is structured in general, as many relations are dual (e.g. persons and groups, authors and articles, organizations and practices), and are thus susceptible to an analysis according to duality principles. The chapter locates the concept of duality within past and present sociology. It also discusses the use of duality in the analysis of culture as well as in affiliation networks. It closes with recent developments and future directions.

Bireflectionality in Absolute Geometry

1989 ◽  
Vol 32 (1) ◽  
pp. 54-63 ◽  
Author(s):  
Dragoslav Ljubić
Keyword(s):  
General Result ◽  
3 Dimensional ◽  
Absolute Geometry

AbstractIf G is any group then g ∊ G is called an involution if g ≠ 1 and g o g = 1. A group G is called bireflectional if every element in G is a product of two involutions. It is known that 2- dimensional, 3- dimensional, and some types of n-dimensional (n > 3) absolute geometries (in the sense of H. Kinder) are bireflectional. In this article the author proves the general result that every n-dimensional absolute geometry is bireflectional.

scholarly journals Inverse Optimization Method of Resource Allocation with Consideration of Advantage Structure

2017 ◽  
Vol 8 (2) ◽  
pp. 86
Author(s):  
Lili Zhang ◽  
Wenwen Yang ◽  
Dejun Teng
Keyword(s):  
Optimization Model ◽  
Optimization Method ◽  
Indicator System ◽  
Inverse Optimization ◽  
Competence Model ◽  
Interview Method ◽  
Expert Interview ◽  
Principle Of Duality ◽  
Conventional Optimization ◽  
Programming Analysis

The research collects competence items of informal questionnaire by expert interview method, analyzes the results qualitatively and quantitatively and forms formal questionnaire. Through statistical analysis and AMOS modeling, the research obtains workers’ competence model and validates competence model. An identification method of individual advantage characters according competency indicator system is built up relying on programming analysis and parameter optimization. We build the optimization model and its inverse optimization model, starting from the original optimization model, adjusting parameter value as small as possible, the conventional optimization model translated into the inverse model by the principle of duality. A calculation example is used to make sure the method is reliability and feasibility.

Existence of special rainbow triangles in weak geometries

2019 ◽  
Vol 26 (4) ◽  
pp. 489-498
Author(s):  
Victor Pambuccian
Keyword(s):  
Orthogonality Relation ◽  
Absolute Geometry ◽  
Acute Triangle ◽  
Definition Of ◽  
Meaningful Definition

Abstract We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann’s Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

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