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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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' ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
T. T. C. Ting
- Published in print:
- 1996
- Published Online:
- November 2020
- ISBN:
- 9780195074475
- eISBN:
- 9780197560280
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195074475.003.0004
- Subject:
- Chemistry, Materials Chemistry
We will present in this chapter some aspects of matrix algebra that are needed in this book. Most results presented here can be found in standard books on matrix ...
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We will present in this chapter some aspects of matrix algebra that are needed in this book. Most results presented here can be found in standard books on matrix algebra. Proofs are provided for those results that are either easily derived or not readily available elsewhere. For readers who have no knowledge of matrix algebra, this chapter is essential for the rest of the book. They may find the one-chapter treatment of matrix algebra in the book by Hildebrand (1954) helpful and informative. For readers who have some knowledge of matrix algebra this chapter can be skimmed or skipped altogether, depending on how familiar they are with the subject. If they want to find the proofs omitted in this chapter or want to devote more time on the subject, the books by Hohn (1965) and Pease (1965) are recommended. The notations employed in this chapter have no relations, in most cases, with the notations adopted in the rest of the book. This point should be kept in mind in referring back to this chapter.
Less
We will present in this chapter some aspects of matrix algebra that are needed in this book. Most results presented here can be found in standard books on matrix algebra. Proofs are provided for those results that are either easily derived or not readily available elsewhere. For readers who have no knowledge of matrix algebra, this chapter is essential for the rest of the book. They may find the one-chapter treatment of matrix algebra in the book by Hildebrand (1954) helpful and informative. For readers who have some knowledge of matrix algebra this chapter can be skimmed or skipped altogether, depending on how familiar they are with the subject. If they want to find the proofs omitted in this chapter or want to devote more time on the subject, the books by Hohn (1965) and Pease (1965) are recommended. The notations employed in this chapter have no relations, in most cases, with the notations adopted in the rest of the book. This point should be kept in mind in referring back to this chapter.
T. T. C. Ting
- Published in print:
- 1996
- Published Online:
- November 2020
- ISBN:
- 9780195074475
- eISBN:
- 9780197560280
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195074475.003.0009
- Subject:
- Chemistry, Materials Chemistry
The matrices Q, R, T, A, B, N1, N2, N3, S, H, L, and M introduced in the previous chapter are the elasticity matrices. They depend on elastic constants only, and ...
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The matrices Q, R, T, A, B, N1, N2, N3, S, H, L, and M introduced in the previous chapter are the elasticity matrices. They depend on elastic constants only, and appear frequently in the solutions to two-dimensional problems. The matrices A, B, and M are complex while the others are real. We present their structures and identities relating them in this chapter. In Chapter 7 we will show that A and B are tensors of rank one and S, H, L, and M are tensors of rank two when the transformation is a rotation about the x3-axis. Readers who are not interested in how the structures of these matrices and the identities relating them are derived may skip this chapter. They may return to this chapter when they read later chapters on applications where the results presented here are employed.
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The matrices Q, R, T, A, B, N1, N2, N3, S, H, L, and M introduced in the previous chapter are the elasticity matrices. They depend on elastic constants only, and appear frequently in the solutions to two-dimensional problems. The matrices A, B, and M are complex while the others are real. We present their structures and identities relating them in this chapter. In Chapter 7 we will show that A and B are tensors of rank one and S, H, L, and M are tensors of rank two when the transformation is a rotation about the x3-axis. Readers who are not interested in how the structures of these matrices and the identities relating them are derived may skip this chapter. They may return to this chapter when they read later chapters on applications where the results presented here are employed.
Eldred H. Chimowitz
- Published in print:
- 2005
- Published Online:
- November 2020
- ISBN:
- 9780195119305
- eISBN:
- 9780197561249
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195119305.003.0003
- Subject:
- Chemistry, Physical Chemistry
The second law of thermodynamics states that the entropy change in any spontaneous adiabatic process is greater than or equal to zero. It is a disarmingly simple ...
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The second law of thermodynamics states that the entropy change in any spontaneous adiabatic process is greater than or equal to zero. It is a disarmingly simple statement but one that is a cornerstone of scientific theories. It is instrumental in describing the extent and direction of all physical and chemical transformations and contains within it the essential ideas for developing thermodynamic stability theory. Stability theory concerns itself with answering questions such as (1) What is a stable thermodynamic state? (2) Which conditions define the limit to this state beyond which the system becomes unstable? (3) How does the instability manifest itself ? In a real sense, stability theory provides the underlying framework for a macroscopic understanding of phase transitions and critical phenomena, the subject of this text. Many of the results of stability theory related to phase equilibria are well known; an example is the condition that, for a pure fluid in a stable state, the quantity −(∂P/∂V )T,N must be greater than or equal to zero, with the equality condition holding at the limit of stability. Many other facets of thermodynamic stability theory, however, are relatively unfamiliar. For example, any of the well-known thermodynamic potentials E, H, A, and G can be used to develop stability criteria for a given system. Are these criteria always equivalent, or do some take precedence over others? If so, what are the implications of this for understanding phase transformations in physicochemical systems? It is questions of this sort that we take up in this chapter, where we lay the macroscopic foundations for the material developed throughout the rest of the text. In this analysis, we rely heavily upon results taken from linear algebra, a branch of mathematics that provides an ideal tool for developing a comprehensive description of thermodynamic stability concepts. The combination of the first and second law of thermodynamics for a closed system leads to the well-known equation: . . . dE = T dS − PdV . . . . . . (1.1) . . . where E(S, V ) represents the system energy as a function of the independent variables S and V.
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The second law of thermodynamics states that the entropy change in any spontaneous adiabatic process is greater than or equal to zero. It is a disarmingly simple statement but one that is a cornerstone of scientific theories. It is instrumental in describing the extent and direction of all physical and chemical transformations and contains within it the essential ideas for developing thermodynamic stability theory. Stability theory concerns itself with answering questions such as (1) What is a stable thermodynamic state? (2) Which conditions define the limit to this state beyond which the system becomes unstable? (3) How does the instability manifest itself ? In a real sense, stability theory provides the underlying framework for a macroscopic understanding of phase transitions and critical phenomena, the subject of this text. Many of the results of stability theory related to phase equilibria are well known; an example is the condition that, for a pure fluid in a stable state, the quantity −(∂P/∂V )T,N must be greater than or equal to zero, with the equality condition holding at the limit of stability. Many other facets of thermodynamic stability theory, however, are relatively unfamiliar. For example, any of the well-known thermodynamic potentials E, H, A, and G can be used to develop stability criteria for a given system. Are these criteria always equivalent, or do some take precedence over others? If so, what are the implications of this for understanding phase transformations in physicochemical systems? It is questions of this sort that we take up in this chapter, where we lay the macroscopic foundations for the material developed throughout the rest of the text. In this analysis, we rely heavily upon results taken from linear algebra, a branch of mathematics that provides an ideal tool for developing a comprehensive description of thermodynamic stability concepts. The combination of the first and second law of thermodynamics for a closed system leads to the well-known equation: . . . dE = T dS − PdV . . . . . . (1.1) . . . where E(S, V ) represents the system energy as a function of the independent variables S and V.