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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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