A Mathematical Gift I: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This ebook will convey the sweetness and enjoyable of arithmetic to the study room. It bargains severe arithmetic in a full of life, reader-friendly kind. incorporated are routines and lots of figures illustrating the most thoughts.
The first bankruptcy offers the geometry and topology of surfaces. between different issues, the authors speak about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a variety of elements of the idea that of measurement, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of third-dimensional manifolds. particularly, it really is proved that the three-d sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a sequence of lectures given by means of the authors at Kyoto collage (Japan).

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Additional resources for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)

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Mehrfach von medizinischer Knotenmagie. So solI das Heilkraut "Heliotrop" nicht gepfliickt, sondem in drei oder vier Knoten gebunden werden, mit einem Gebet, daB der Kranke gesunden und die Knoten wieder 16sen moge, usw. 20 Auch das Binden einer magischen Zahl von Knoten auf eine Schnur sowie die Anwendung dieser Technik bei Schwangerschaften war Plinius und dreihundert Jahre spater Marcellus "dem Empiriker", einem weiteren spatantiken medizinischen Anthologiker, bekannt. 22 Neben der magisch-medizinischen Symbolik lag eine religiose Funktion komplexerer Knoten und Verkettungen (insbesondere aus in sich geschlossenen Faden) in der Symbolisierung des unaufloslichen Zusammenhangs aller Dinge, des ewigen Lebens usw.

Ais schlieBlich, wie in der vierten Episode angedeutet, sachlich tiberraschende Querverbindungen zu anderen Gebieten der Mathematik auftauchten, iinderte sich die Gestalt der Knotentheorie noch einmal. Obwohl die theoretische Struktur der Mathematik als eines hierarchischen Geflechts von Teiltheorien auf soIche Verbindungen vorbereitet war, iinderten die durch Jones' Entdeckung angeregten Forschungen die epistemischen Konfigurationen der Knotentheorie so stark, daB auch die darin erzielten Resultate auf andere Weise zusammengefaBt wurden; eine Serie neu erschienener Monographien bestiitigt dies.

2). 1 Vor der Mathematisierung Knoten und verschlungene Faden gehoren zu jenen raumlichen Gebilden, mit denen Menschen in allen Kulturen schon seit sehr frUber Zeit umgegangen sind. Ganz ahnlich wie Gruppen von Objekten durch das Zahlen auf einer elementaren, aber ftir viele kulturelle Techniken fundamentalen Komplexitatsskala geordnet werden konnen, bilden auch die Formen von Knoten eine Hierarchie einer gewissen Art diumlicher Komplexitat, deren einfachere Stufen Bestandteil einer Reihe von kulturellen Techniken sind.

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