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Duality

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/9780691150659.003.0003
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are ... More


Statement B Implies Statement A

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/9780691150659.003.0007
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter recalls the use of the Schützenberger involution on Gelfand-Tsetlin patterns to prove that Statement B implies Statement A. These statements will be discussed two more times in the later ... More


Cartoons

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/9780691150659.003.0008
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces a method of marking up a short Gelfand-Tsetlin pattern based on inequalities between its entries, that encodes the effect of the involution t 7 → t′ and the boxing and ... More


Statement E Implies Statement D

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/9780691150659.003.0014
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the theorem that says Statement E implies Statement D, first by fixing a nodal signature η‎ and presenting a “cut and paste” virtual resotope. It then presents the proof, ... More


173,067,389-VERTEX GRAPH Δ

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/9780198527596.003.0008
Subject:
Mathematics, Pure Mathematics

This chapter studies the graph Δ on the set of elements of type m DG (with stabilizers 211 : M24) in which two vertices are adjacent whenever in D(G) they are incident to a common element of type 2 ... More


POINCARÉ DUALITY

Andrew Ranicki

in Algebraic and Geometric Surgery

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198509240
eISBN:
9780191708725
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509240.003.0004
Subject:
Mathematics, Geometry / Topology

Poincaré duality is an isomorphism between the homology and cohomology of a manifold. The chapter introduces the basic duality and the universal Poincaré duality for non-simply-connected manifolds ... More


Madame White, The Book of Change, and Eileen Chang: On a Poetics of Involution and Derivation

Kam Louie (ed.)

in Eileen Chang: Romancing Languages, Cultures and Genres

Published in print:
2012
Published Online:
September 2012
ISBN:
9789888083794
eISBN:
9789882209060
Item type:
chapter
Publisher:
Hong Kong University Press
DOI:
10.5790/hongkong/9789888083794.003.0012
Subject:
Literature, World Literature

This chapter discusses Chang's aesthetic of revision and bilingualism by examining her two English novels, The Fall of the Pagoda and The Book of Change, These two novels were written in the late ... More


Linked Tori, I

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0005
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates the consequences of the assumption that one Moufang set is weakly isomorphic to another. It first introduces some well-known facts about involutions which are assembled in a ... More


Semi-ramified Quadrangles of Type E6, E7 and E8

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0012
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang semi-ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a ... More


Galois Involutions

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0031
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter focuses on the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions laid out in Hypothesis 30.1. It begins with the fhe ... More


Fugue in E Major: The Well-Tempered Clavier, Book 2

Joseph Kerman

in The Art of Fugue: Bach Fugues for Keyboard, 1715-1750

Published in print:
2005
Published Online:
May 2012
ISBN:
9780520243583
eISBN:
9780520941397
Item type:
chapter
Publisher:
University of California Press
DOI:
10.1525/california/9780520243583.003.0009
Subject:
Music, History, Western

Diagrams intrude on listening by asserting direct linearity and they offer unambiguous ground plans for patterns in time that are existential and often tenuous. This chapter offers a tabular analysis ... More


Harish-Chandra Modules over ℤ

Günter Harder

in Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions: (AMS-203)

Published in print:
2019
Published Online:
September 2020
ISBN:
9780691197890
eISBN:
9780691197937
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691197890.003.0008
Subject:
Mathematics, Number Theory

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted ... More


Involution and Enterprise in Rural Areas: A Twenty-Year Perspective on Rural and Agricultural Change in Morogoro

Stefano Ponte and Dan Brockington

in Prosperity in Rural Africa?: Insights into Wealth, Assets, and Poverty from Longitudinal Studies in Tanzania

Published in print:
2021
Published Online:
September 2021
ISBN:
9780198865872
eISBN:
9780191898174
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198865872.003.0012
Subject:
Economics and Finance, Development, Growth, and Environmental

This chapter analyses the patterns of poverty, prosperity, and rural transformation in Tanzania through longitudinal research examining livelihoods and asset change in a twenty-year period. We argue ... More


Symmetric matrices and congruence

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/9780691161853.003.0004
Subject:
Mathematics, Algebra

This chapter develops diagonal canonical forms and proves inertia theorems for hermitian and skewhermitian matrices with respect to involutions (including the conjugation). These canonical forms ... More


Invariant neutral and semidefinite subspaces

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/9780691161853.003.0006
Subject:
Mathematics, Algebra

This chapter concerns subspaces that are simultaneously invariant for one matrix and semidefinite (or neutral) with respect to another. Such subspaces show up in many applications, some of them ... More


Matrix pencils with symmetries: Nonstandard involution

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/9780691161853.003.0011
Subject:
Mathematics, Algebra

This chapter involves quaternion matrix pencils or, equivalently, pairs of quaternion matrices, with symmetries with respect to a fixed nonstandard involution φ‎. Here, the chapter provides canonical ... More


Mixed matrix pencils: Nonstandard involutions

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/9780691161853.003.0012
Subject:
Mathematics, Algebra

This chapter also studies the canonical forms of mixed quaternion matrix pencils, i.e., such that one of the two matrices is φ‎-hermitian and the other is φ‎-skewhermitian, with respect to ... More


Indefinite inner products: Nonstandard involution

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/9780691161853.003.0013
Subject:
Mathematics, Algebra

This chapter fixes a nonstandard involution φ‎. It introduces indefinite inner products defined on Hn×1 of the symmetric and skewsymmetric types associated with φ‎ and matrices having symmetry ... More


Linked Tori, II

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0006
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = ... More


Unramified Galois Involutions

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691166902
eISBN:
9781400874019
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691166902.003.0032
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter describes the fixed point building of an automorphism of a Bruhat-Tits building Ξ‎ which induces an unramified Galois involution on the building at infinity Ξ‎∞. An element of G (for ... More


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