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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


Hilbert Modular Forms and Iwasawa Theory

Haruzo Hida

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198571025
eISBN:
9780191718946
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571025.001.0001
Subject:
Mathematics, Algebra

The 1995 work by Wiles and Taylor-Wiles opened up a whole new technique in algebraic number theory and, a decade on, the waves caused by this incredibly important work are still being felt. This book ... More


Direct and Indirect Utility

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:
10.1093/0198284616.003.0011
Subject:
Economics and Finance, Microeconomics

This is the fifth of six chapters in Part II about demand and utility cost, a typical area for what is understood as choice theory. It discusses direct and indirect utility. Its six sections are: ... More


Triangulated Categories

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199296866
eISBN:
9780191711329
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296866.003.0001
Subject:
Mathematics, Geometry / Topology

Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria ... More


ADJOINTS

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/9780198568612.003.0009
Subject:
Mathematics, Algebra

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered, from free ... More


MONADS AND ALGEBRAS

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/9780198568612.003.0010
Subject:
Mathematics, Algebra

This chapter presents a third characterization of adjunctions. This one has the virtue of being entirely equational. Topics discussed include the triangle identities, monads and adjoints, algebras ... More


HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS

Haruzo Hida

in Hilbert Modular Forms and Iwasawa Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198571025
eISBN:
9780191718946
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571025.003.0003
Subject:
Mathematics, Algebra

The deformation theoretic techniques of Wiles-Taylor were introduced for elliptic modular forms in the introductory Chapter 1, and are generalized to Hilbert modular forms (following Fujiwara's ... More


MODULAR IWASAWA THEORY

Haruzo Hida

in Hilbert Modular Forms and Iwasawa Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198571025
eISBN:
9780191718946
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571025.003.0005
Subject:
Mathematics, Algebra

This chapter proves the torsion of the anticyclotomic Iwasawa module of a (p-ordinary) CM field, and presents an explicit formula of the L-invariant of the CM field, which is a natural generalization ... More


INTIAL BOUNDARY VALUE PROBLEM IN A HALF-SPACE WITH CONSTANT COEEFICIENTS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0004
Subject:
Mathematics, Applied Mathematics

This chapter drops the assumption of symmetry, or at least it does not assume the dissipativity in a classical sense. The search for a necessary condition for maximal estimates (strong well-posedness ... 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


PARTIALLY DESCRIBED INVERSE EIGENVALUE PROBLEMS

Moody T. Chu and Gene H. Golub

in Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566649
eISBN:
9780191718021
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566649.003.0005
Subject:
Mathematics, Applied Mathematics

In practice, it is often the case that only partial information on eigenvalues and eigenvectors is available. In many cases, just a few eigenpairs can determine much of the desirable reconstruction. ... More


Sensitivity evaluation and automatic differentiation

Bijan Mohammadi and Olivier Pironneau

in Applied Shape Optimization for Fluids

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199546909
eISBN:
9780191720482
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199546909.003.0005
Subject:
Mathematics, Mathematical Physics

This chapter describes sensitivity analysis and automatic differentiation (AD). These include the theory, then an object oriented library for AD by operator overloading, and finally the authors' ... More


Error Estimates for Multiscale Operator Decomposition for Multiphysics Models

D. Estep

in Multiscale Methods: Bridging the Scales in Science and Engineering

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199233854
eISBN:
9780191715532
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233854.003.0011
Subject:
Mathematics, Applied Mathematics

Multiphysics, multiscale models present significant challenges in terms of computing accurate solutions and for estimating the error in information computed from numerical solutions. In this chapter, ... More


Operators

Gary E. Bowman

in Essential Quantum Mechanics

Published in print:
2007
Published Online:
January 2008
ISBN:
9780199228928
eISBN:
9780191711206
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199228928.003.0005
Subject:
Physics, Condensed Matter Physics / Materials

This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity ... More


Basic Tools For Quantum Mechanics

Robert Alicki and Mark Fannes

in Quantum Dynamical Systems

Published in print:
2001
Published Online:
February 2010
ISBN:
9780198504009
eISBN:
9780191708503
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504009.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave ... More


Data assimilation in glaciology

B. Bonan, M. Nodet, O. Ozenda, and C. Ritz

in Advanced Data Assimilation for Geosciences: Lecture Notes of the Les Houches School of Physics: Special Issue, June 2012

Published in print:
2014
Published Online:
March 2015
ISBN:
9780198723844
eISBN:
9780191791185
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198723844.003.0025
Subject:
Physics, Geophysics, Atmospheric and Environmental Physics

This chapter addresses an inverse problem in glaciology, namely how to infer a climatic scenario (i.e. how to reconstruct past polar temperature) from ice volume records. Ice volume observations are ... More


The (Not So?) Basics

Peter Mann

in Lagrangian and Hamiltonian Dynamics

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198822370
eISBN:
9780191861253
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198822370.003.0030
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses matrices. Matrices appear in many instances across physics, and it is in this chapter that the background necessary for understanding how to use them in calculations is ... More


Lie groups and Lie algebras

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/9780199662920.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Lie groups and Lie algebras are defined. SU(2), the group whose elements are 2x2 unitary unimodular matrices is described providing an example of a 3-dimensional Lie group. Infinitesimal generators ... More


Diagrams of Difference: Adjunctions and Topoi

Rocco Gangle

in Diagrammatic Immanence: Category Theory and Philosophy

Published in print:
2015
Published Online:
September 2016
ISBN:
9781474404174
eISBN:
9781474418645
Item type:
chapter
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474404174.003.0007
Subject:
Philosophy, Metaphysics/Epistemology

This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several ... More


The Semi-group on

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.003.0012
Subject:
Mathematics, Probability / Statistics

This chapter deals with the semi-group on the space Β‎⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components ... More


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