## Category Theory

*Steve Awodey*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.001.0001
- Subject:
- Mathematics, Algebra

This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible ... More

## Procrustes Problems

*John C Gower and Garmt B Dijksterhuis*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198510581
- eISBN:
- 9780191708961
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198510581.001.0001
- Subject:
- Mathematics, Probability / Statistics

Procrustean methods are used to transform one set of data to represent another set of data as closely as possible. This book unifies several strands in the literature and contains new algorithms. It ... More

## An Introduction to Model-Based Survey Sampling with Applications

*Ray Chambers and Robert Clark*

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780198566625
- eISBN:
- 9780191738449
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566625.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book is an introduction to the model-based approach to survey sampling. It consists of three parts, with Part I focusing on estimation of population totals. Chapters 1 and 2 introduce survey ... 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

## FUNCTORS AND NATURALITY

*Steve Awodey*

### in Category Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0007
- Subject:
- Mathematics, Algebra

This chapter develops a general theory for functors. Topics discussed include category of categories, representable structure, stone duality, naturality, examples of natural transformations, ... More

## CONCRETE GROUP THEORY

*A. A. Ivanov*

### in The Fourth Janko Group

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198527596
- eISBN:
- 9780191713163
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527596.003.0001
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses concrete group theory. Topics covered include symplectic and orthogonal GF(2)-forms, transvections and Siegel transformations, Witt's theorem, and the space of forms.

## Weighting, scaling, and missing values

*J. C. Gower and G. B. Dijksterhuis*

### in Procrustes Problems

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198510581
- eISBN:
- 9780191708961
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198510581.003.0008
- Subject:
- Mathematics, Probability / Statistics

This chapter introduces the different forms of weighting. One form of weighting is when the rows of X1 are weighted. It is also possible to weight the columns. In these cases the ... More

## Mirror Symmetry of Fourier—Mukai Transformation for Elliptic Calabi—Yau Manifolds

*Naichung Conan Leung and Shing‐Tung Yau*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0015
- Subject:
- Mathematics, Geometry / Topology

Mirror symmetry conjecture says that for any Calabi–Yau (CY) manifold M near the large complex/symplectic structure limit, there is another CY manifold X, called the mirror manifold, such that the ... More

##
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

## 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

## Solving the Tower of Hanoi with Random Moves

*Max A. Alekseyev and Toby Berger*

### in The Mathematics of Various Entertaining Subjects: Research in Recreational Math

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0005
- Subject:
- Mathematics, History of Mathematics

This chapter studies solutions of the Tower of Hanoi puzzle and some of its variants with random moves, where each move is chosen uniformly from the set of the valid moves in the current state. The ... More

## Global Existence Theorems: Asymptotically Euclidean Data

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0015
- Subject:
- Mathematics, Applied Mathematics

This chapter shows how the Penrose transform can be used to prove global existence of solutions of various semilinear field equations. It outlines the foundation points of Friedrich's conformal ... More

## Using Transformations in Sample Survey Inference

*Raymond L. Chambers and Robert G. Clark*

### in An Introduction to Model-Based Survey Sampling with Applications

- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780198566625
- eISBN:
- 9780191738449
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566625.003.0017
- Subject:
- Mathematics, Probability / Statistics

Using transformations in sample survey inference is the final chapter of this book and describes the extension of the empirical best prediction approach to the situation where the population values ... More

## Gluing Solutions

*Philip Isett*

### in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0012
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if ... 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

## The Ping-Pong Lemma

*Johanna Mangahas*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, ... More

## Introduction to the WKB(J) Approximation: All Things Airy

*John A. Adam*

### in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0007
- Subject:
- Mathematics, Applied Mathematics

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order ... More

## The WKB(J) Approximation Revisited

*John A. Adam*

### in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0022
- Subject:
- Mathematics, Applied Mathematics

This chapter reexamines the WKB(J) approximation and applies it to some simple one-dimensional potentials, with a focus on the case of a triangular barrier. It first considers the connection formulas ... 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