Self-organization of Vectors

  • Position vectors, difference vectors and displacement vectors are linked by the dot product in a time discrete dynamical system in the n-dimensional Euclidean space. The algorithm presented here produces a variety of structures which are sometimes unstable and they may jump to other more stable attractors after a longer lifetime. First, consider two points (position vectors) in the space. The one point is displaced iteratively by a unit vector, and the second point by the same unit vector, however with reversed sign. Thus, the “center of mass” remains constant. The antiparallel displacement vectors lie on parallels, whose distance will be also kept unchanged. Hence the "torque" of the two points is also constant. Now, more points can be added. The points can be chosen freely combined to pairs of points. It results in a graph, the nodes of the graph correspond to the points, the edges correspond to the selected pairs of points (see SelfOrganization.pdf). The executable program Attractor.jar demonstrates the variety of ways in whichPosition vectors, difference vectors and displacement vectors are linked by the dot product in a time discrete dynamical system in the n-dimensional Euclidean space. The algorithm presented here produces a variety of structures which are sometimes unstable and they may jump to other more stable attractors after a longer lifetime. First, consider two points (position vectors) in the space. The one point is displaced iteratively by a unit vector, and the second point by the same unit vector, however with reversed sign. Thus, the “center of mass” remains constant. The antiparallel displacement vectors lie on parallels, whose distance will be also kept unchanged. Hence the "torque" of the two points is also constant. Now, more points can be added. The points can be chosen freely combined to pairs of points. It results in a graph, the nodes of the graph correspond to the points, the edges correspond to the selected pairs of points (see SelfOrganization.pdf). The executable program Attractor.jar demonstrates the variety of ways in which the vectors organize themselves spontaneously as a function of dimension and selected graph. The algorithm uses bivectors which have to be constant (Bivector.pdf). The transformation of this calculation method is described in the file Map.pdfshow moreshow less

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Metadaten
Author:Wolfhard Hövel
URN:urn:nbn:de:bvb:92-opus4-1265
Document Type:Report
Language:English
Publishing Institution:Technische Hochschule Nürnberg Georg Simon Ohm
Release Date:2015/06/02
Tag:Attractor; Bivector; Center of mass; Dynamical system; Euclidean space; Self-organization; Torque; Vector
GND Keyword:Attraktor; Drehmoment; dynamisches System; euklidischer Raum; Dimension n; Selbstorganisation; Skalarprodukt; Vektor; Schwerpunkt
Note:
Deutsche Version: https://opus4.kobv.de/opus4-ohm/frontdoor/index/index/docId/96
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik
Licence (German):Keine Lizenz - Deutsches Urheberrecht gilt
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