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
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 examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.Less
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 examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.
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
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 describes a generalization of their techniques to Hilbert modular forms (towards the proof of the celebrated ‘R=T’ theorem) and applications of the theorem that have been found. Applications include a proof of the torsion of the adjoint Selmer group (over a totally real field F and over the Iwasawa tower of F) and an explicit formula of the L-invariant of the arithmetic p-adic adjoint L-functions. This implies the torsion of the classical anticyclotomic Iwasawa module of a CM field over the Iwasawa algebra. When specialized to an elliptic Tate curve over F by the L-invariant formula, the invariant of the adjoint square of the curve has exactly the same expression as the one in the conjecture of Mazur-Tate-Teitelbaum (which is for the standard L-function of the elliptic curve and is now a theorem of Greenberg-Stevens).Less
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 describes a generalization of their techniques to Hilbert modular forms (towards the proof of the celebrated ‘R=T’ theorem) and applications of the theorem that have been found. Applications include a proof of the torsion of the adjoint Selmer group (over a totally real field F and over the Iwasawa tower of F) and an explicit formula of the L-invariant of the arithmetic p-adic adjoint L-functions. This implies the torsion of the classical anticyclotomic Iwasawa module of a CM field over the Iwasawa algebra. When specialized to an elliptic Tate curve over F by the L-invariant formula, the invariant of the adjoint square of the curve has exactly the same expression as the one in the conjecture of Mazur-Tate-Teitelbaum (which is for the standard L-function of the elliptic curve and is now a theorem of Greenberg-Stevens).
S. N. Afriat
- 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
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: purchasing power; the indirect (utility) ‘integrability’ problem; basic relations and properties (of the structure involved with direct and indirect utility, and other features of demand analysis); adjoint of a relation; adjoint of a function; and limit adjoints.Less
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: purchasing power; the indirect (utility) ‘integrability’ problem; basic relations and properties (of the structure involved with direct and indirect utility, and other features of demand analysis); adjoint of a relation; adjoint of a function; and limit adjoints.
D. Huybrechts
- 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
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 are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.Less
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 are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.
Steve Awodey
- 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
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 groups to limits to exponentials. It also captures an important mathematical phenomenon that is invisible without the use of the lens of category theory. It is argued that adjointness is a concept of fundamental logical and mathematical importance not captured elsewhere in mathematics. Topics discussed include hom-set definition, examples of adjoints, order adjoints, quantifiers as adjoints, RAPL 197, locally cartesian closed categories, and the adjoint functor theorem.Less
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 groups to limits to exponentials. It also captures an important mathematical phenomenon that is invisible without the use of the lens of category theory. It is argued that adjointness is a concept of fundamental logical and mathematical importance not captured elsewhere in mathematics. Topics discussed include hom-set definition, examples of adjoints, order adjoints, quantifiers as adjoints, RAPL 197, locally cartesian closed categories, and the adjoint functor theorem.
Steve Awodey
- 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
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 for a monad, comonads and coalgebras, and algebras for endofunctors. Exercises are given in the last part of the chapter.Less
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 for a monad, comonads and coalgebras, and algebras for endofunctors. Exercises are given in the last part of the chapter.
Haruzo Hida
- 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
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 treatment) in this chapter. In particular, Fujiwara's ‘R=T’ theorem (the identification of the Galois deformation ring and the corresponding Hecke algebra) is proven in the minimal case. In addition to the Taylor-Wiles methods, an explicit formula of the L-invariant (of the adjoint L-functions) as well as an integral solution to Eichler's basis problem are presented for Hilbert modular forms.Less
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 treatment) in this chapter. In particular, Fujiwara's ‘R=T’ theorem (the identification of the Galois deformation ring and the corresponding Hecke algebra) is proven in the minimal case. In addition to the Taylor-Wiles methods, an explicit formula of the L-invariant (of the adjoint L-functions) as well as an integral solution to Eichler's basis problem are presented for Hilbert modular forms.
Haruzo Hida
- 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
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 of the formula by Ferrero-Greenberg and Gross-Koblitz from the 1970s for imaginary quadratic fields. These results are proven through the comparison theorem (the ‘R=T’ theorem) between the Iwasawa-theoretic version of the universal deformation ring and the universal nearly p-ordinary Hecke algebra over the (infinite) cyclotomic Iwasawa tower. These combined results enable us to compute the adjoint L-invariant of a CM theta family in terms of the U(p)-eigenvalue of the theta family.Less
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 of the formula by Ferrero-Greenberg and Gross-Koblitz from the 1970s for imaginary quadratic fields. These results are proven through the comparison theorem (the ‘R=T’ theorem) between the Iwasawa-theoretic version of the universal deformation ring and the universal nearly p-ordinary Hecke algebra over the (infinite) cyclotomic Iwasawa tower. These combined results enable us to compute the adjoint L-invariant of a CM theta family in terms of the U(p)-eigenvalue of the theta family.
Christopher D. Hacon and James McKernan
- 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
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 is based on the key notion of an ‘adjoint algebra’. The chapter contains an introduction to multiplier ideals, and the celebrated lifting lemma is developed from first principles. Key ideas of the minimal model program for real pairs are also developed from the ground up.Less
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 is based on the key notion of an ‘adjoint algebra’. The chapter contains an introduction to multiplier ideals, and the celebrated lifting lemma is developed from first principles. Key ideas of the minimal model program for real pairs are also developed from the ground up.
Moody T. Chu and Gene H. Golub
- 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
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. This chapter illustrates this point by concentrating on the Toeplitz structure and the self-adjoint quadratic pencils. The possibility of model updating or tuning applications is discussed.Less
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. This chapter illustrates this point by concentrating on the Toeplitz structure and the self-adjoint quadratic pencils. The possibility of model updating or tuning applications is discussed.
Sylvie Benzoni-Gavage and Denis Serre
- 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
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 in Kreiss' sense) yields the so-called uniform Kreiss-Lopatinskii condition. The chapter investigates the case of a characteristic boundary. It gives practical devices to check the K.-L. condition, including the construction of a Lopatinskii determinant. It shows that the adjoint BVP shares with the direct one the K.-L. condition, a fact exploited in the duality method employed in the existence theory. The latter is carried out for the BVP in weighted (in time) spaces under the assumption that a Kreiss' dissipative boundary symmetrizer exists. Its existence is stated for a constantly hyperbolic operator, but the proof will be seen in the next chapter. The evolutionary property is shown with the use of the Paley-Wiener Theorem. Rauch's Theorem tells what the solution at a given time T can be estimated to; this extends the well-posedness to the full IBVP.Less
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 in Kreiss' sense) yields the so-called uniform Kreiss-Lopatinskii condition. The chapter investigates the case of a characteristic boundary. It gives practical devices to check the K.-L. condition, including the construction of a Lopatinskii determinant. It shows that the adjoint BVP shares with the direct one the K.-L. condition, a fact exploited in the duality method employed in the existence theory. The latter is carried out for the BVP in weighted (in time) spaces under the assumption that a Kreiss' dissipative boundary symmetrizer exists. Its existence is stated for a constantly hyperbolic operator, but the proof will be seen in the next chapter. The evolutionary property is shown with the use of the Paley-Wiener Theorem. Rauch's Theorem tells what the solution at a given time T can be estimated to; this extends the well-posedness to the full IBVP.
Bijan Mohammadi and Olivier Pironneau
- 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
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' experience with AD systems using code generation operating in both direct and reverse modes. The chapter describes the different possibilities and through simple programs, gives a comprehensive survey of direct AD by operator overloading and for the reverse mode, the adjoint code method. Several elementary and more advanced examples help the understanding of this central concept.Less
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' experience with AD systems using code generation operating in both direct and reverse modes. The chapter describes the different possibilities and through simple programs, gives a comprehensive survey of direct AD by operator overloading and for the reverse mode, the adjoint code method. Several elementary and more advanced examples help the understanding of this central concept.
D. Estep
- 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
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, we discuss error estimation for a widely used numerical approach for multiphysics, multiscale problems called multiscale operator decomposition. In this approach, a multiphysics model is decomposed into components involving simpler physics over a relatively limited range of scales, and the solution is sought through an iterative procedure involving numerical solutions of the individual components. After describing the ingredients of adjoint-based a posteriori analysis, we describe the extension to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, there are several key ideas underlying a general approach to treat operator decomposition multiscale methods.Less
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, we discuss error estimation for a widely used numerical approach for multiphysics, multiscale problems called multiscale operator decomposition. In this approach, a multiphysics model is decomposed into components involving simpler physics over a relatively limited range of scales, and the solution is sought through an iterative procedure involving numerical solutions of the individual components. After describing the ingredients of adjoint-based a posteriori analysis, we describe the extension to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, there are several key ideas underlying a general approach to treat operator decomposition multiscale methods.
Gary E. Bowman
- 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
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 forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.Less
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 forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.
Robert Alicki and Mark Fannes
- 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
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 functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.Less
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 functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.
B. Bonan, M. Nodet, O. Ozenda, and C. Ritz
- 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
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 available from oceanic records, giving information about sea level, and therefore about the amount of water stored in ice sheets, ice caps, and glaciers. The link between the climatic scenario and the ice sheet volume is complex. The temperature affects the ice mass budget (accumulating minus melting ice). Then, the ice mass budget is a key factor in ice dynamics, which is quite complex in itself (non-Newtonian fluid and nonlinear rheology). The few previous studies of this subject have used fairly simple inverse methods. The idea of the approach described in this chapter is to explore, with idealized twin experiments, the ability of the adjoint method to solve the inverse problem of reconstructing past temperature given all available observations.Less
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 available from oceanic records, giving information about sea level, and therefore about the amount of water stored in ice sheets, ice caps, and glaciers. The link between the climatic scenario and the ice sheet volume is complex. The temperature affects the ice mass budget (accumulating minus melting ice). Then, the ice mass budget is a key factor in ice dynamics, which is quite complex in itself (non-Newtonian fluid and nonlinear rheology). The few previous studies of this subject have used fairly simple inverse methods. The idea of the approach described in this chapter is to explore, with idealized twin experiments, the ability of the adjoint method to solve the inverse problem of reconstructing past temperature given all available observations.
Peter Mann
- 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
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 provided. Although matrices can be a little daunting upon first exposure, they are very handy for a lot of classical physics. This chapter reviews the basics of matrices and their operations. It discusses square matrices, adjoint matrices, cofactor matrices and skew-symmetric matrices. The concepts of matrix multiplication, transpose, inverse, diagonal, identity, Pfaffian and determinant are examined. The chapter also discusses the terms Hermitian, symmetric and antisymmetric, as well as the Levi-Civita symbol and Laplace expansion.Less
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 provided. Although matrices can be a little daunting upon first exposure, they are very handy for a lot of classical physics. This chapter reviews the basics of matrices and their operations. It discusses square matrices, adjoint matrices, cofactor matrices and skew-symmetric matrices. The concepts of matrix multiplication, transpose, inverse, diagonal, identity, Pfaffian and determinant are examined. The chapter also discusses the terms Hermitian, symmetric and antisymmetric, as well as the Levi-Civita symbol and Laplace expansion.
Adam M. Bincer
- 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
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 are defined and used to provide a basis for a vector space that leads to the Lie algebra. Structure constants are introduced and shown to provide the so-called adjoint representation. The Cartan metric tensor is defined and Cartan’s criterion for semisimplicity is described. After showing that SU(2) is compact SL(2,R) — the group whose elements are 2x2 real unimodular matrices — is introduced to give an example of a non-compact group. Biographical notes on Euler, Lie and Cartan are given.Less
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 are defined and used to provide a basis for a vector space that leads to the Lie algebra. Structure constants are introduced and shown to provide the so-called adjoint representation. The Cartan metric tensor is defined and Cartan’s criterion for semisimplicity is described. After showing that SU(2) is compact SL(2,R) — the group whose elements are 2x2 real unimodular matrices — is introduced to give an example of a non-compact group. Biographical notes on Euler, Lie and Cartan are given.
Rocco Gangle
- 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
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 concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.Less
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 concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.
Charles L. Epstein and Rafe1 Mazzeo
- 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
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 before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, L, meets bP cleanly. It then examines long time asymptotics, along with a lemma in which P is a compact manifold with corners and L is a generalized Kimura diffusion on P. It also considers the existence of irregular solutions to the homogeneous equations Lu = f, for functions that do not belong to the range of the generator of a C⁰-semi-group on Β⁰(P).Less
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 before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, L, meets bP cleanly. It then examines long time asymptotics, along with a lemma in which P is a compact manifold with corners and L is a generalized Kimura diffusion on P. It also considers the existence of irregular solutions to the homogeneous equations Lu = f, for functions that do not belong to the range of the generator of a C⁰-semi-group on Β⁰(P).
