Foundations of Incidence Geometry

Author: Johannes Ueberberg
Publisher: Springer Science & Business Media
ISBN: 3642209726
Format: PDF, ePub
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Incidence geometry is a central part of modern mathematics that has an impressive tradition. The main topics of incidence geometry are projective and affine geometry and, in more recent times, the theory of buildings and polar spaces. Embedded into the modern view of diagram geometry, projective and affine geometry including the fundamental theorems, polar geometry including the Theorem of Buekenhout-Shult and the classification of quadratic sets are presented in this volume. Incidence geometry is developed along the lines of the fascinating work of Jacques Tits and Francis Buekenhout. The book is a clear and comprehensible introduction into a wonderful piece of mathematics. More than 200 figures make even complicated proofs accessible to the reader.

An Introduction to Incidence Geometry

Author: Bart De Bruyn
Publisher: Birkhäuser
ISBN: 3319438115
Format: PDF, Docs
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This book gives an introduction to the field of Incidence Geometry by discussing the basic families of point-line geometries and introducing some of the mathematical techniques that are essential for their study. The families of geometries covered in this book include among others the generalized polygons, near polygons, polar spaces, dual polar spaces and designs. Also the various relationships between these geometries are investigated. Ovals and ovoids of projective spaces are studied and some applications to particular geometries will be given. A separate chapter introduces the necessary mathematical tools and techniques from graph theory. This chapter itself can be regarded as a self-contained introduction to strongly regular and distance-regular graphs. This book is essentially self-contained, only assuming the knowledge of basic notions from (linear) algebra and projective and affine geometry. Almost all theorems are accompanied with proofs and a list of exercises with full solutions is given at the end of the book. This book is aimed at graduate students and researchers in the fields of combinatorics and incidence geometry.

Diagram Geometry

Author: Francis Buekenhout
Publisher: Springer Science & Business Media
ISBN: 3642344534
Format: PDF
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This book provides a self-contained introduction to diagram geometry. Tight connections with group theory are shown. It treats thin geometries (related to Coxeter groups) and thick buildings from a diagrammatic perspective. Projective and affine geometry are main examples. Polar geometry is motivated by polarities on diagram geometries and the complete classification of those polar geometries whose projective planes are Desarguesian is given. It differs from Tits' comprehensive treatment in that it uses Veldkamp's embeddings. The book intends to be a basic reference for those who study diagram geometry. Group theorists will find examples of the use of diagram geometry. Light on matroid theory is shed from the point of view of geometry with linear diagrams. Those interested in Coxeter groups and those interested in buildings will find brief but self-contained introductions into these topics from the diagrammatic perspective. Graph theorists will find many highly regular graphs. The text is written so graduate students will be able to follow the arguments without needing recourse to further literature. A strong point of the book is the density of examples.

Groups of Exceptional Type Coxeter Groups and Related Geometries

Author: N.S. Narasimha Sastry
Publisher: Springer Science & Business Media
ISBN: 8132218140
Format: PDF, Kindle
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The book deals with fundamental structural aspects of algebraic and simple groups, Coxeter groups and the related geometries and buildings. All contributing authors are very active researchers in the topics related to the theme of the book. Some of the articles provide the latest developments in the subject; some provide an overview of the current status of some important problems in this area; some survey an area highlighting the current developments; and some provide an exposition of an area to collect problems and conjectures. It is hoped that these articles would be helpful to a beginner to start independent research on any of these topics, as well as to an expert to know some of the latest developments or to consider some problems for investigation.

Geometry of Semilinear Embeddings

Author: Mark Pankov
Publisher: World Scientific
ISBN: 9814651095
Format: PDF
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This volume covers semilinear embeddings of vector spaces over division rings and the associated mappings of Grassmannians. In contrast to classical books, we consider a more general class of semilinear mappings and show that this class is important. A large portion of the material will be formulated in terms of graph theory, that is, Grassmann graphs, graph embeddings, and isometric embeddings. In addition, some relations to linear codes will be described. Graduate students and researchers will find this volume to be self-contained with many examples. Contents:Semilinear Mappings:Division Rings and Their HomomorphismsVector Spaces Over Division RingsSemilinear MappingsSemilinear EmbeddingsMappings of Grassmannians Induced by Semilinear EmbeddingsKreuzer's ExampleDualityCharacterization of Strong Semilinear EmbeddingsProjective Geometry and Linear Codes:Projective SpacesFundamental Theorem of Projective GeometryProof of Theorem 1.2m-independent Subsets in Projective SpacesPGL-subsetsGeneralized MacWilliams TheoremLinear CodesIsometric Embeddings of Grassmann Graphs:Graph TheoryElementary Properties of Grassmann GraphsEmbeddingsIsometric EmbeddingsProof of Theorem 3.1Equivalence of Isometric EmbeddingsLinearly Rigid Isometric EmbeddingsRemarks on Non-isometric EmbeddingsSome Results Related to Chow's TheoremHuang's TheoremJohnson Graph in Grassmann Graph:Johnson GraphIsometric Embeddings of Johnson Graphs in Grassmann GraphsProof of Theorem 4.2Classification Problem and Relations to Linear CodesCharacterizations of Apartments in Building GrassmanniansCharacterization of Isometric Embeddings:Main Result, Corollaries and RemarksCharacterization of DistanceConnectedness of the Apartment GraphIntersections of J(n, k)-subsets of Different TypesProof of Theorem 5.1Semilinear Mappings of Exterior Powers:Exterior PowersGrassmanniansGrassmann Codes Readership: Graduate students and researchers interested in the field of semilinear embeddings. Keywords:Semilinear Embedding;Grassmannian;Grassmann Graph;Linear Code

Lineare Algebra

Author: Gerd Fischer
Publisher: Springer-Verlag
ISBN: 383489365X
Format: PDF, ePub
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Das seit über 35 Jahren bewährte, einführende Lehrbuch im kompakten Taschenbuchformat mit einer umfassenden Stoffauswahl eignet sich als Grundlage für eine zweisemestrige Vorlesung für Studierende der Mathematik, Physik und Informatik. Der Text enthält zahlreiche Übungsaufgaben. Lösungen findet man in dem von H. Stoppel und B. Griese verfassten Übungsbuch. Zur Motivation der Studierenden enthält das Buch eine Einführung, in der die Bedeutung der Linearen Algebra als Grundlage innerhalb der Mathematik und ihren Anwendungen beschrieben wird.

Das BUCH der Beweise

Author: Martin Aigner
Publisher: Springer-Verlag
ISBN: 3662064545
Format: PDF
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Die elegantesten mathematischen Beweise, spannend und für jeden Interessierten verständlich. "Der Beweis selbst, seine Ästhetik, seine Pointe geht ins Geschichtsbuch der Königin der Wissenschaften ein. Ihre Anmut offenbart sich in dem gelungenen und geschickt illustrierten Buch." Die Zeit

Anschauliche Geometrie

Author: David Hilbert
Publisher: Springer-Verlag
ISBN: 3662366851
Format: PDF, Mobi
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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.