## 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

## BASICS

*Roman Kossak and James H. Schmerl*

### in The Structure of Models of Peano Arithmetic

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568278
- eISBN:
- 9780191718199
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568278.003.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This introductory chapter covers a wide range of topics, from basic notational conventions and coding in arithmetic to important classical results. The well-known theorems, such as Gaifman's ... More

## The Ubiquity of Catalan Numbers I

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0006
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a number of occurrences of Catalan numbers in a number of quite unexpected places and their relationships among them. They include mountain ranges (or Dyck paths), noncrossing ... More

## Knowability

*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/9780691150659.003.0012
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the ... More

## Tractatus in Iohannis Euangelium

*H. A. G. Houghton*

### in Augustine's Text of John: Patristic Citations and Latin Gospel Manuscripts

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199545926
- eISBN:
- 9780191719974
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199545926.003.0007
- Subject:
- Religion, Early Christian Studies

Four types of gospel citation are identified in Augustine's commentary on John. The continuous treatment of the text is made up of the initial citation, most likely to have been taken from a ... More

## INTRODUCTION

*Pavol Hell and Jaroslav Nešetřil*

### in Graphs and Homomorphisms

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198528173
- eISBN:
- 9780191713644
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528173.003.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This introductory chapter is a sampler of the material covered in the book. It introduces the notation and terminology in the book, and provides motivational examples and applications, many taken up ... More

## CATEGORIES OF DIAGRAMS

*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/9780198568612.003.0008
- Subject:
- Mathematics, Algebra

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it ... More

## Intersections and Fixed Points

*S. N. Afriat*

### in Logic of Choice and Economic Theory

- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0031
- Subject:
- Economics and Finance, Microeconomics

This is the last of three chapters on general mathematics as applied to economics, and presents a treatment of intersections and fixed points. The eight sections of the chapter are: simplices and ... More

## APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS

*Ulrich Berger and Monika Seisenberger*

### in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0008
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter is concerned with the problem of finding constructive content in nonconstructive mathematical theorems. By so doing, one gains insight into possible constructive aspects hidden in ... More

## Extension theorems and the existence of flips

*Christopher D. Hacon and James McKernan*

### in Flips for 3-folds and 4-folds

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter provides a detailed and self-contained exposition of Hacon and McKernan's construction of pl flips in dimension n assuming minimal models with scaling in dimension n-1. The construction ... More

## AUTOMORPHISMS OF RECURSIVELY SATURATED MODELS

*Roman Kossak and James H. Schmerl*

### in The Structure of Models of Peano Arithmetic

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568278
- eISBN:
- 9780191718199
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568278.003.0008
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter develops the basic theory of automorphisms of countable recursively saturated models of PA. The key results are: Smoryìnski's characterization of exponentially closed cuts, the Moving ... More

## Classical Scattering Theory

*Peter Monk*

### in Finite Element Methods for Maxwell's Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0009
- Subject:
- Mathematics, Numerical Analysis

This chapter presents basic tools for studying the scattering problem. The Stratton-Chu integral representation of the electromagnetic field is derived as well as the far field pattern of the ... More

## CENTRALITY IN FINITELY GENERATED SOLUBLE GROUPS

*John C. Lennox and Derek J. S. Robinson*

### in The Theory of Infinite Soluble Groups

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198507284
- eISBN:
- 9780191709326
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507284.003.0008
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses the centrality property for finitely generated soluble groups, upper central length, and the Fan Out lemma. It also considers the theorems of Lennox and Roseblade.

## *DIRICHLET PROBLEMS IN PERFORATED DOMAINS

*Andrea Braides*

### in Gamma-Convergence for Beginners

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0014
- Subject:
- Mathematics, Applied Mathematics

This chapter discusses a direct treatment of problems in perforated domains with a Dirichlet condition on the perforation. It relies on a joining lemma by Ansini and Braides, allowing the decoupling ... More

## Snakes

*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/9780691150659.003.0009
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the “Snake Lemma” and uses it to prove the statement that GΓ(t) = GΔ(t′) is “often” true. It first describes an indexing of the Γ preaccordion and of the Δ′ preaccordion; ... More

## Lowness properties and K-triviality

*André Nies*

### in Computability and Randomness

- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199230761
- eISBN:
- 9780191710988
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230761.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter shows the equivalence of K-triviality and lowness for Martin–Löf randomness. This coincidence extends to other lowness properties, such as being a base for Martin–Löf randomness, and ... More

## Connection Games and Sperner’s Lemma

*David Molnar*

### 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/9780691164038.003.0015
- Subject:
- Mathematics, History of Mathematics

This chapter provides an introduction to connection games in general. It also recounts how Sperner's Lemma, a result about labeling a triangulation of a simplex, can be used to prove that someone ... More

## Multiresolution Galerkin Methods

*Karsten Urban*

### in Wavelet Methods for Elliptic Partial Differential Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780198526056
- eISBN:
- 9780191712340
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526056.003.0004
- Subject:
- Mathematics, Applied Mathematics, Mathematical Finance

This chapter uses a Multiresoultion Analysis for the discretization of elliptic boundary value problems. The error analysis is provided and the setup of the linear system of equations is described. ... More

## Modelling Issues

*David F. Hendry*

### in Dynamic Econometrics

- Published in print:
- 1995
- Published Online:
- November 2003
- ISBN:
- 9780198283164
- eISBN:
- 9780191596384
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198283164.003.0015
- Subject:
- Economics and Finance, Econometrics

Data mining has three distinct meanings, but is detectable when it denotes ignoring, or camouflaging, conflicting evidence in a prejudiced search for an acceptable model. The twin dangers of ... More

## The Market

*S. N. Afriat*

### in Logic of Choice and Economic Theory

- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0020
- Subject:
- Economics and Finance, Microeconomics

This is the third of six chapters on the logic of price. It deals with the phenomenon of the market, and starts with a discourse on the ‘logic of price’ and a critique of the supply and demand ... More