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Understanding Polynomial Equations in n Unknowns

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691149059
eISBN:
9781400850525
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691149059.003.0012
Subject:
Mathematics, History of Mathematics

This chapter examines the search for simultaneous solutions which eventually led to the creation of whole new algebraic constructs over the course of the eighteenth and nineteenth centuries. The ... More


Linear Transformations, Error Correction, and the Long Run in Dynamic Regression

Anindya Banerjee, Juan J. Dolado, John W. Galbraith, and David F. Hendry

in Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data

Published in print:
1993
Published Online:
November 2003
ISBN:
9780198288107
eISBN:
9780191595899
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198288107.003.0002
Subject:
Economics and Finance, Econometrics

The focus in this chapter is on the properties of linear autoregressive‐distributed lag (ADL) models for stationary data processes, in order to understand later transformations in non‐stationary ... More


GROUP ORBITALLY CONSTRAINED APPROXIMATION

Moody T. Chu and Gene H. Golub

in Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566649
eISBN:
9780191718021
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566649.003.0009
Subject:
Mathematics, Applied Mathematics

Group actions together with properly formulated objective functions can offer a channel to tackle various classical or new and challenging problems rising from applied linear algebra. This chapter ... More


Riemann Surfaces, Coverings, and Hypergeometric Functions

Paula Tretkoff

in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691144771
eISBN:
9781400881253
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691144771.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and ... More


Conclusion

Anindya Banerjee, Juan J. Dolado, John W. Galbraith, and David F. Hendry

in Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data

Published in print:
1993
Published Online:
November 2003
ISBN:
9780198288107
eISBN:
9780191595899
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198288107.003.0009
Subject:
Economics and Finance, Econometrics

The final chapter summarizes the main themes of the book, and considers the invariance of the matrix of co‐integrating vectors in a linear system under both linear transformations and seasonal ... More


Type A Weyl Group Multiple Dirichlet Series

Ben Brubaker, Daniel Bump, and Solomon Friedberg

in Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691150659
eISBN:
9781400838998
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691150659.003.0001
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are ... More


Probability

Stephen J. Blundell and Katherine M. Blundell

in Concepts in Thermal Physics

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199562091
eISBN:
9780191718236
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562091.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter defines some basic concepts in probability theory. It begins by stating that the probability of occurrence of a particular event, taken from a finite set of possible events, is zero if ... More


Canonical Transformations

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0006
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter focuses on linear canonical unitary transformations of mode annihilation and creation operators, both for bosons and for fermions. The generators of such transformations (which form a ... More


Cardinal Utility: How It Entered Economic Analysis from Pareto to Samuelson, 1915–1945

Ivan Moscati

in Measuring Utility: From the Marginal Revolution to Behavioral Economics

Published in print:
2018
Published Online:
December 2018
ISBN:
9780199372768
eISBN:
9780199372805
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199372768.003.0007
Subject:
Economics and Finance, Microeconomics

Chapter 6 reconstructs the progressive definition and stabilization of the current notion of cardinal utility as utility unique up to positive linear transformations. This notion was the eventual ... More


The algebra of quaternions

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/9780691161853.003.0002
Subject:
Mathematics, Algebra

This chapter concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as axb = c and ax − xb = c. It studies the Sylvester equation ax ... More


Sign Variation of Vectors and TN Linear Transformations

Shaun M. Fallat and Charles R. Johnson

in Totally Nonnegative Matrices

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691121574
eISBN:
9781400839018
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691121574.003.0005
Subject:
Mathematics, Applied Mathematics

This chapter develops an array of results about TP/TN matrices and sign variation diminution, along with appropriate converses. If the entries of a vector represent a sampling of consecutive function ... More


Small-Volume Expansions of the Displacement Fields

Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, and Abdul Wahab

in Mathematical Methods in Elasticity Imaging

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691165318
eISBN:
9781400866625
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691165318.003.0004
Subject:
Mathematics, Applied Mathematics

This chapter deals with small-volume expansions of the displacement fields. It first introduces the notion of elastic moment tensor, a geometric quantity associated with a small-volume inclusion, ... More


Complex Numbers and Grassmann Numbers

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0003
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter sets out the basic algebraic properties of both complex numbers (c-numbers) and Grassmann numbers (g-numbers). The formal rules of addition and multiplication together with the ... More


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