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BUNDLES

Andrew Ranicki

in Algebraic and Geometric Surgery

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198509240
eISBN:
9780191708725
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509240.003.0005
Subject:
Mathematics, Geometry / Topology

This chapter brings together the basic properties of fibre bundles, fibrations, and vector bundles required in surgery theory. The tangent and normal bundles of a manifold are introduced. The ... More


 Gauge theories and fibre-bundle spaces

Frank Arntzenius

in Space, Time, and Stuff

Published in print:
2012
Published Online:
May 2012
ISBN:
9780199696604
eISBN:
9780191738333
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199696604.003.0006
Subject:
Philosophy, Metaphysics/Epistemology, Philosophy of Science

This chapter discusses the implications of gauge theories for our conception of properties, and for the reality of fibre-bundle spaces. One way of understanding such theories is to claim that the ... More


What is a gauge theory?

Richard Healey

in Gauging What's Real: The Conceptual Foundations of Gauge Theories

Published in print:
2007
Published Online:
January 2008
ISBN:
9780199287963
eISBN:
9780191713453
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199287963.003.0001
Subject:
Philosophy, Philosophy of Science

This chapter explains what is meant by a gauge theory. It states the origins of the term ‘gauge’, and describes the prominent role of gauge theories in contemporary physics. Using the theory of ... More


Composite Summary of Cerebral White Matter Fiber Pathways in the Rhesus Monkey

Jeremy D. Schmahmann and Deepak N. Pandya

in Fiber Pathways of the Brain

Published in print:
2006
Published Online:
May 2009
ISBN:
9780195104233
eISBN:
9780199864294
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195104233.003.0027
Subject:
Neuroscience, Techniques, Disorders of the Nervous System

This chapter presents a series of diagrams of rostral to caudal coronal sections of the brain of a rhesus monkey that illustrate in a composite manner the association, commissural, projection, and ... More


 Calculus as geometry

Frank Arntzenius

in Space, Time, and Stuff

Published in print:
2012
Published Online:
May 2012
ISBN:
9780199696604
eISBN:
9780191738333
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199696604.003.0008
Subject:
Philosophy, Metaphysics/Epistemology, Philosophy of Science

This chapter argues that calculus, the theory of differentiation and integration that permeates modern science, can be formulated as a purely geometric theory, which does not entail the existence of ... More


Classical gauge theories

Richard Healey

in Gauging What's Real: The Conceptual Foundations of Gauge Theories

Published in print:
2007
Published Online:
January 2008
ISBN:
9780199287963
eISBN:
9780191713453
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199287963.003.0003
Subject:
Philosophy, Philosophy of Science

This chapter extends the analysis of classical gauge theories to encompass both non-Abelian Yang-Mills theories and the general theory of relativity. It explains what is meant by a non-Abelian ... More


Differential geometry, topology and fibre bundles

Reinhold A. Bertlmann

in Anomalies in Quantum Field Theory

Published in print:
2000
Published Online:
February 2010
ISBN:
9780198507628
eISBN:
9780191706400
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507628.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces all necessary mathematical concepts. Section 2.1 briefly summarizes some topological definitions. Section 2.2 explains the homotopy of maps and the homotopy of groups. Section ... More


Superior Longitudinal Fasciculus and Arcuate Fasciculus

Jeremy D. Schmahmann and Deepak N. Pandya

in Fiber Pathways of the Brain

Published in print:
2006
Published Online:
May 2009
ISBN:
9780195104233
eISBN:
9780199864294
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195104233.003.0013
Subject:
Neuroscience, Techniques, Disorders of the Nervous System

The superior longitudinal fasciculus (SLF) and the arcuate fasciculus (AF) have long been considered synonymous in the human, and these names have been used interchangeably. According to observations ... More


Weyl's “Purely Infinitesimal” Constitution of Field Physics

Thomas Ryckman

in The Reign of Relativity: Philosophy in Physics 1915-1925

Published in print:
2005
Published Online:
April 2005
ISBN:
9780195177176
eISBN:
9780199835324
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195177177.003.0006
Subject:
Philosophy, Philosophy of Science

It is shown how Weyl intertwined phenomenological analysis and mathematical construction in building the foundation of his “pure infinitesimal geometry” underlying gravitation and electromagnetism. ... More


Muscle mechanics I: intact single muscle fibres

Haruo Sugi and Teizo Tsuchiya

in Current Methods in Muscle Physiology: Advantages, Problems and Limitations

Published in print:
1998
Published Online:
March 2012
ISBN:
9780198523970
eISBN:
9780191724480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198523970.003.0001
Subject:
Neuroscience, Techniques

In the field of muscle mechanics, muscle physiologists attempt to obtain information about molecular events responsible for muscle contraction, i.e. the myofilament sliding, mainly by applying ... More


Electrophysiological approaches to the study of axons

PETER K. STYS and JEFFERY D. KOCSIS

in The Axon: Structure, Function and Pathophysiology

Published in print:
1995
Published Online:
May 2009
ISBN:
9780195082937
eISBN:
9780199865802
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195082937.003.0017
Subject:
Neuroscience, Disorders of the Nervous System

This chapter discusses the theory and practice of some of the more common techniques devised for recording electrical activity in single axons or fiber bundles. It focuses on electrophysiological ... More


Product Spaces and Fiber Bundles

Tim Maudlin

in New Foundations for Physical Geometry: The Theory of Linear Structures

Published in print:
2014
Published Online:
April 2014
ISBN:
9780198701309
eISBN:
9780191771613
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198701309.003.0009
Subject:
Philosophy, Logic/Philosophy of Mathematics

Standard topology provides a method of defining a topology on the Cartesian product of two topological spaces. Defining a linear structure on the Cartesian product of two spaces is more involved. The ... More


Principal Bundles

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0003
Subject:
Mathematics, Educational Mathematics

This chapter examines principal bundles. Throughout the chapter, G will be a topological group. It then defines a principal G-bundle and provides a criterion for a map to be a principal G-bundle. ... More


Homotopy Quotients and Equivariant Cohomology

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0004
Subject:
Mathematics, Educational Mathematics

This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M ... More


Topology, Manifolds, Dynamical Systems, Measure, and Bundles

Xin Wei Sha

in Poiesis and Enchantment in Topological Matter

Published in print:
2014
Published Online:
September 2014
ISBN:
9780262019514
eISBN:
9780262318914
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262019514.003.0006
Subject:
Philosophy, Aesthetics

This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological ... More


Spectral Sequences

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0006
Subject:
Mathematics, Educational Mathematics

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined ... More


Equivariant Cohomology of S2 Under Rotation

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0007
Subject:
Mathematics, Educational Mathematics

This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by ... More


Homotopy Groups and CW Complexes

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0002
Subject:
Mathematics, Educational Mathematics

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built ... More


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