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Linear symplectic geometry

Dusa McDuff and Dietmar Salamon

in Introduction to Symplectic Topology

Published in print:
2017
Published Online:
June 2017
ISBN:
9780198794899
eISBN:
9780191836411
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198794899.003.0003
Subject:
Mathematics, Analysis, Geometry / Topology

The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the ... More


Valued Abelian Groups

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

in Asymptotic Differential Algebra and Model Theory of Transseries

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.003.0003
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more detail. Ordered ... More


Generalities

Adam M. Bincer

in Lie Groups and Lie Algebras: A Physicist's Perspective

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199662920
eISBN:
9780191745492
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199662920.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Generalities refers to the definition of basic concepts. The goal here is to define group, field, vector space and algebra. The chapter starts with the definitions of associativity, identity, inverse ... More


Algebra of vector bundles

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0004
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

Any linear operation that can be performed to generate a new vector space from a given set of initial vector spaces can be done fibre-wise with an analogous set of vector bundles to generate a new ... More


Distributional Semantic Models

Stephen Pulman

in Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199646296
eISBN:
9780191747847
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646296.003.0012
Subject:
Mathematics, Applied Mathematics

A recent preoccupation in computational linguistics has been the attempt to combine compositional models of formal semantics of the Montagovian type with corpus-based distributional models of word ... More


Algebras over a Field and Semantics for Context-Based Reasoning

DAOUD CLARKE

in Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199646296
eISBN:
9780191747847
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646296.003.0011
Subject:
Mathematics, Applied Mathematics

This chapter presents the thesis that real associative algebras are appropriate structures for representing vector-based natural language semantics.


Mathematical foundations

Max A. Little

in Machine Learning for Signal Processing: Data Science, Algorithms, and Computational Statistics

Published in print:
2019
Published Online:
October 2019
ISBN:
9780198714934
eISBN:
9780191879180
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198714934.003.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy, Mathematical Physics

Statistical machine learning and signal processing are topics in applied mathematics, which are based upon many abstract mathematical concepts. Defining these concepts clearly is the most important ... More


Machine Modeling

Sadegh Vaez-Zadeh

in Control of Permanent Magnet Synchronous Motors

Published in print:
2018
Published Online:
April 2018
ISBN:
9780198742968
eISBN:
9780191802942
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198742968.003.0002
Subject:
Physics, Condensed Matter Physics / Materials

This chapter presents dynamic and steady-state modeling of permanent magnet synchronous (PMS) machines with the help of reference frames. The modeling starts with a machine model in terms of phase ... More


Vector bundles with ℂn as fiber

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0006
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

Just as there are vector spaces over ℂ, there are vector bundles whose fibres can be consistently viewed as ℂn for some n. This chapter first defines these objects and then provides number of ... More


Vector-Valued Forms

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/9780691191751.003.0014
Subject:
Mathematics, Educational Mathematics

This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the ... More


Vector spaces and matrices: Basic theory

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0003
Subject:
Mathematics, Algebra

This chapter covers the basics on the vector space of columns with quaternion components, matrix algebra, and various matrix decompositions. The real and complex representations of quaternions are ... More


Clifford and Heisenberg algebras

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691151298
eISBN:
9781400840571
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151298.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter recalls various results on Clifford algebras and Heisenberg algebras. It first introduces the Clifford algebra of a vector space V equipped with a symmetric bilinear form B and then ... More


Quadratic Forms of Type F4

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0009
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and ... More


Special Algebraic Notions

Daniel Canarutto

in Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics

Published in print:
2020
Published Online:
December 2020
ISBN:
9780198861492
eISBN:
9780191894374
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198861492.003.0002
Subject:
Mathematics, Applied Mathematics, Mathematical Physics

A presentation of notions that can be found in the literature, though not always in the precise form that we need: unit spaces and physical scales; complex vector spaces together with conjugate and ... More


Hamiltonian Systems Based on a Lie Algebra

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0010
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not ... More


A closer look at the stable completion

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0005
Subject:
Mathematics, Geometry / Topology

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, ... More


The Maurer–Cartan Form

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/9780691191751.003.0015
Subject:
Mathematics, Educational Mathematics

This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The ... More


The action functional and the harmonic oscillator

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691151298
eISBN:
9781400840571
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151298.003.0011
Subject:
Mathematics, Geometry / Topology

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. ... More


Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse

Chris Heunen, Mehrnoosh Sadrzadeh, and Edward Grefenstette (eds)

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199646296
eISBN:
9780191747847
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646296.001.0001
Subject:
Mathematics, Applied Mathematics

Quantum mechanics and linguistics appear to be quite unrelated at first sight. Yet significant parts of both concern compositional reasoning about the way information flows among subsystems and the ... More


Equations of motion

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0005
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter turns to the essential aspects of Newtonian dynamics. It argues that this chapter’s representation of an interaction by a vector means that it is limiting itself to phenomena that do not ... More


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