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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


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