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## The Structure of Models of Peano Arithmetic

*Roman Kossak and James Schmerl*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568278.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This book gives an account of the present state of research on lattices of elementary substructures and automorphisms of nonstandard models of arithmetic. Major representation theorems are proved, ... More

## TWENTY QUESTIONS

*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/9780198568278.003.0012
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter gives a list of open problems the book's authors have considered most interesting. Two of the problems (7 and 9) have already been solved by Enayat. The chapter includes a website which ... More

## Symmetries and Sasakian Structures

*Charles P. Boyer and Krzysztof Galicki*

### in Sasakian Geometry

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0009
- Subject:
- Mathematics, Geometry / Topology

This chapter presents known results concerning symmetries of Sasakian structures, including a detailed study of the Sasakian automorphisms and various deformations of Sasakian structures. Homogeneous ... More

## MODIFYING THE RANK 2 AMALGAM

*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/9780198527596.003.0003
- Subject:
- Mathematics, Pure Mathematics

This section modifies the classical amalgam H to obtain the amalgam G which eventually leads to J4. The core of the modification is a triple isomorphism which explains the non-vanishing cohomology of ... More

## RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE

*Stefanie Gerke, Colin McDiarmid, Angelika Steger, and Andreas Weißl*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0006
- Subject:
- Mathematics, Probability / Statistics

This chapter investigates the behaviour of the function γ(q). In particular, it considers γ(q) as q approaches 1 from below and above and 3 from below, and explains the discontinuity as it approach 1 ... More

## The exceptional group G 2

*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/9780199662920.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The exceptional Lie algebra G 2 is obtained as a subgroup of SO(7) by making use of the octonionic habc . It is shown that G 2 is the group of ... More

## Unitary groups

*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/9780199662920.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Unitary groups are defined. An explicit choice of generators in the defining representation is displayed and used to abstract commutation relations for u(n). It is shown that the center of U(n) has ... More

## Bases and Automorphisms of the Free Group on Two Generators

*Christophe Reutenauer*

### in From Christoffel Words to Markoff Numbers

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.003.0017
- Subject:
- Mathematics, Pure Mathematics

The chapter begins with a self-contained exposition of the theory of Nielsen on the free groupwith two generators: bases of F(a, b),Nielsen’s criterion for automorphisms of F(a, b), It also ... More

## Flat connections and holonomy

*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/9780199605880.003.0013
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines flat connections. A connection on a principal bundle is said to be flat when its curvature 2-form is identically zero. The discussions cover flat connections on bundles over the ... 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

## Dynamics of automorphisms of compact complex surfaces

*Serge Cantat*

*Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

### in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0018
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter describes the dynamics of automorphisms of compact complex surfaces when it is rich. In particular, the chapter carries to this general setting the interplay between ideas from ergodic ... More

## Cube Complexes, Subgroups of Mapping Class Groups and Nilpotent Genus

*Martin R. Bridson*

### in Lectures on Geometry

- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198784913
- eISBN:
- 9780191827150
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198784913.003.0003
- Subject:
- Mathematics, Geometry / Topology

Based on a lecture at PCMI this chapter is structured around two sets of results, one concerning groups of automorphisms of surfaces and the other concerning the nilpotent genus of groups. The first ... More

## Symmetry Semantics

*Andrew Bacon*

### in Vagueness and Thought

- Published in print:
- 2018
- Published Online:
- May 2018
- ISBN:
- 9780198712060
- eISBN:
- 9780191780264
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198712060.003.0013
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

The borderlineness operator can be given a relatively straightforward semantics in terms of the theory of precisifications. Without the assumption that vagueness can be analysed in terms of ... More

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