## 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